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THE NEARNESS OF GRACE
A PERSONAL SCIENCE OF SPIRITUAL TRANSFORMATION
Arnold J. Mandell
Table of Contents
Acknowledgements ………………………………………………………… 3
Chapter 1: In Search of the Miraculous .…………………………………. 4
Chapter 2: Doesn’t Everybody ……………………………………………. 22
Chapter 3: Transmogrifications Of Energies ……………………………. 42
Chapter 4: Sensual In-Between Entropies ………………………………. 64
Chapter 5: Some Entheogenic Entropies …………..……………………. 87
Chapter 6: Pentecostal Phase Transitions ………………………………. 122
Chapter 7: Amphetamine Roll-Up And Splitting .………….……………. 144
Chapter 8: Faith And Rationality …………………………………………. 168
Appendix: An Intuitive Guide to the Ideas and
Methods of Dynamical Systems for the Life Sciences …………… 186
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ACKNOWLEDGEMENTS
Appreciation is expressed to the Fetzer Institute for their support of this work.
Particular thanks are due their imaginative Vice President, Dr. Paul Gailey, who shared my
vision and hope that these somewhat disparate themes could be blended into a meaningful
whole. Time and the reading by others will tell whether this idea was realized. The Fetzer
Foundation and Dr. Gailey have facilitated exploration into blends of science and
spirituality, particularly in the context of personal meaning. They also have a history of
supporting serious work in this era’s most powerful and rigorous exercise in holism as
represented by the mathematical and applied mathematical fields of modern dynamical
systems theory. Fetzer’s very special environment and years of dedication have
encouraged the variety of personal meanings within science to emerge and be recognized
as legitimate and important parts of the research enterprise. It would be difficult to imagine
a more propitious context for this effort.
The book is dedicated to my daughter Buna, and to my intellectual and creative
companion, Dr. Karen Selz, whose deep and lovely mind wrote much more of this book
than is formally acknowledged.
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CHAPTER 1:
IN SEARCH OF THE MIRACULOUS
More than a half-century of naïve persistence and driven search for unity in
the biophysics of mind and personal spirituality as the basis for healing
transformation has led me into many laboratories. The motivation may have been
genetic. My father said that we were descended from several generations of Jewish
mystics, none of them able to attain the salaried status of rabbi or cantor. These
ecstatic men lived lives of peripatetic eccentricity, stirring congregations with
provocative insights and uncomfortably personal inquiry. But only for a little while.
Soon they were asked to leave the synagogue and often their Eastern European
Jewish townships called shtetels as well.
My father, in the first generation of our family without rabbis in over a
Century, was a businessman-musician, who in the early mornings studied Talmudic
commentaries. He taught me about why it was that most interpretations of the book
by the rational, physician, lawyer, philosopher, Moses Maimonides, called Guide for
the Perplexed, were in error in their assumption that man cannot understand God’s
nature with his mind. He took issue with the opinion that the union of a person’s
intellect and Spirit with Him was not possible as long as a person was living. Ibn
Tibbon, Maimonides’ best-known early translator and interpreter, relegated the
cognitive, analytical, physical and alchemical transformational sciences to the
earthly, not spiritual realm. My father disagreed. He espoused the work of the 13 th
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Century proponent of a school of Jewish ecstatic mysticism, Abraham Abulafia,
whose interpretation of the Guide and his own Commentary on the Secrets taught
that the human mind, if transformed into a “state of active intellect,” could become
one with Spirit, realizing the Kingdom of God in rational mystical experience in a
state of excitement with new ideas. The new consciousness achieves deep
knowledge of both the “upper” and “lower” realms of what he called “reality” both
spontaneously and directly. He said that without personal transformation, this
knowing is not possible.
Abulafia’s lesson was that the mundane intellect of man has the potential for
transformation into another kind of mind in a spiritualization of thought. This occurs
via developmental stages that begin with intellect and imagination and culminate in
what he called prophetic emanations. The exercises leading to this transformation
are to be strongly willed and practiced with regularity. This work results in ascension
to an ecstatic state accompanied by great intuitive powers, which Abulafia called
“prophesy.” Ibn Adret, the Chief Rabbi of Spain at the end of the Thirteenth Century,
banished Abulafia from the Country, a Century before the Spanish Inquisition
ousted all the Jews.
Following what my father said was required in the practice of Kabbalah, a
13 th Century tradition of esoteric and mystical interpretations of the Scriptures, I
learned the secret meanings of each of the twenty-two letter Hebrew alphabet.
Much like the Platonic view of mathematics, that it existed before the physical
universe, these symbolic equivalences were believed to be eternal in the
transcendental realm. One of the rare written accounts of this oral tradition is in the
thirteenth-century Hebrew Book of Splendor called the Zohar which describes the
Hebrew alphabet as the heavenly code of the cosmos.
I learned that the Tegragrammaton’s repeated letter Hei, being fifth in the
Hebrew alphabet, represents the number five. In the Kabbalistic tradition, Hei
implicates the functional five-partition of the human inner self or soul. The five parts
are: nefesh, instinctual drives; ruach, mood, affect and emotions; neshamah,
cognitive activities of the mind; chayah, efforts to understand and attain
transcendence; yechidah, experiencing the world as a cosmic unity. Later in life as
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a psychoanalytical neuroscientist with a computational bent, the partitions divided
thoughtful, forewarning forebrain from automatic and stereotyped hind brain, the
signal analyzing thalamocortical system from the emotional and impulsive brain
stem-limbic, the symbolically logical left from intuitively geometric right
hemispheres. We divide the neurotransmitter moods of dopamine aggression from
the transcendentally erotic serotonin and the organized dynamical states of
periodicity and quasi(multi)periodicity from the real world complexity of chaos. I
learned that it is comforting to divide an unknown whole into two or more
unknowable parts.
The Jewish guru and Hebraic tutor of my childhood, Rabbi Isadore Kliegfeld,
smiled when I told him about my sudden loss of panic during nighttime Hebrew
letter meditations. He said that I had had received personal evidence that these
powerful symbols could call forth the transformational powers of God. He said that I
had been given a blessing, in Yiddish, a nachas. Maybe panic is not that far from
the transcendence of an activated mind.
In my tenth summer, behind closed door in a hot back bedroom, first by
accidental touch and then by more systematic chaffing, I evoked a pleasurably
urgent and yawning feeling that began in the lower part of my abdomen and back. It
filled me with thought emptying fullness that a sudden involuntary burst of pelvic
contractions found resolution in an hour or two of an unexplainable sadness. I had
been struggling to understand my father’s well warn copy of William James’s
Varieties of Religious Experience and I wondered if I had been visited by one of the
altered states he described. Was this what he meant by a transformative
experience? A few months later, a late night meditation produced physical
evidence, a thick, sticky, salty sweet stuff that by morning stuck my sheets together.
Later that year, in my father’s library, I found a translation of the 1500 BCE Egyptian
Book of the Dead. It contained a creation myth of two Gods in which “rubbing with
my fist, my heart came into my mouth and I spat forth Shu and Tefnut.” Psalm 23,
read rather regularly in Sunday school, began to make me wonder about the
meanings of“…rod and staff that comforts…” and what was meant by “…my cup
runneth over.” Among the ten regions of the Zohar, connecting the inner world of
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man to the upper world, is the tree of ten sefirot in which Yesod , the phallus,
occupies a central place. Now we’re allowed to know that G-spot stimulation of the
para-urethral glands in the female can result in spurt as well as a cup that runneth
over.
Other occasions of the temporal disappearance of the self-conscious I
occurred while doing the theorem and proof work of high school geometry. Axioms
and the rule bound processes of deduction created difficult journeys from that which
was given to what should be found. Rocking back and forth in a desk chair for
hours, chewing on fingernails, cuticles and pencil ends, time disappeared in a none
self aware state of work-a-day well-being. Sri Aurobindo’s Bhagadvad Gita
described this state as one of the rewards of karma yoga. Abulafia’s Kabalistic
School emphasized the importance of hitbodedut, detachment and seclusion in
concentrated thought, as a technique for the attainment of spiritual “intensification.”
Stacks of lined yellow paper piled up full of blind alleys as I lived in humbling
dumbness. One of my teachers of mathematics described it as the working
mathematician’s dark night of the soul. A breakthrough to a route from premises to
proof brought an expansive rush.
Engagement in a struggle to fuse two differing contextual worlds may be
transporting. Geometric visions can be used to do imageless algebra in a brain
state that feels like intuition. The brain does something like this: Let the number of a
sequence of unit squares, each side of measuring 0 to 1, be the denominator of a
series of fractions, say fifths. Now put five of these boxes in a row. Then the
sequence of all possible fifths, 0/5,1/5, 2/5,..5/5, is inscribed by cutting the vertical
sides of the five sequential squares with a diagonal from the lower left of the first
one to the upper right corner of the last. This line cuts each sequential square’s
front boundary with vertical lengths, 0.0, 0.2, 0.4…1.0 in a series of decimal
fractions equivalent to the sequence of all possible fifths, the proper fractions 1/5,
2/5…5/5. It was Abulafia’s kabalistic belief that symbolic, (algebraic), operations in
(geometric) spaces can unify the “upper” and “lower” worlds in the eternal tensions
between the body and soul, the inner world and the cosmos, the conflict making the
global system both sensitive and stable. The geometric-topological approach to
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modern dynamical system’s theory describes a convolution of the expansive
motions (as in the upper world) and contractive motions (as in the lower world)
embedded naturally in the curved time and space geometries of what are called
hyperbolic spaces. Each point in this space can be visualized as a little saddle in
which orbital flows from pommel and back flow down to the seat, bringing points
together in contracting motion, and flows away from seat down along the sides are
expanding the distance between nearby points.. In the middle of the saddle,
simultaneously expansive and contracting orbits demonstrate hyperbolic stability
composed of intersecting destabilizing and stabilizing influences. Loss of this
countervailing hyperbolic dynamical stability results in global system
transformations called bifurcations and/or phase transitions.
Transformation as a loss of stability is a theme of a recent poetic translation
of portions of the Zohar called Dreams of Being Eaten Alive by David Rosenberg.
He writes that at some time in the difficult journey through the oftenincomprehensible
Zohar, in order to gain entrance to the kabalistic cosmos, there
arose what he called “heartbreak.” “No matter how much intellectual study is
involved, the reader cannot understand the text unless he or she has offered his
heart to be broken on the altar of poetry…and prayer.” Surrender may be the source
of the strange, uplifting feeling of worked through dumbness.
My mother, once a conservatory teaching assistant in piano, sat beside me
while I practiced almost daily, weekends included, from the age of two until the
midteens. Her quiet analytic counter-point sounded mathematical, “You can hear
that that this harmonic progression goes through intervals of fourths of dominant
seventh chords.” I felt the persistent lack of harmonic resolution as growing tension
in my groin. “If you transform each of the 12 notes in a chromatic scale, multiplying
it by five (in what mathematicians call) mod 12 (the numbering system goes from
one to twelve, not ten, before it repeats), one can recover the circle of fourths, the
commonest harmonic chord progression in music.” Though her computational talk
supported rational thought, in my adolescent heat, the addition of Charley Parker’s
flatted fifth and ninth to the dominant seventh chord led suddenly somewhere else
and she knew it. Hearing my arrangement of a Beethoven piano piece become a
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mix of classical and modern jazz themes that I called “How High the Moonlight
Sonata,” she laughed lasciviously as though tickled by this sensual violation of
musical canon. A boogie-woogie Bach two and three-part invention brought more
excited disapproval.
Mysterious are the conditions of attentive (preoccupied) and none attentive,
(fugued out) disappearing time. I found a musical way for it to happen when
improvising: continue to shuffle a small set of notes that stay within the melodic field
of the tonal center of an unchanging tonic chord. In contrast, most melodies and
their chords leave the tonal center to which they return in harmonic and melodic
progression. We can call these conventional tonal centers unstable fixed points.
They are attractive repellers of melodic and harmonic expectation. It has been
mathematically proven that these hyperbolic systems are globally stable. In
contrast, a melody that remains stuck in the tonic chord, a purely contracting stable
fixed point, is technically a chant. Paradoxically, it can be shown that this kind of
fixed point is globally unstable. Rigid things can more easily fracture. The rich,
altered states of consciousness that emerge while hearing the beat of Tibetan
monks meditating, the Sufi chant-dances of Rumi and the John Coltrane and
McCoy Tyner’s endless, single chord, tenor/piano dialogues exemplify the
bifurcation to hallucinatory new stuff arising spontaneously from the experience of
unchanging repetition. Constant repetition of the conditioned (expected) stimulus
drove Pavlov’s dogs, especially those with “nervous temperaments,” into frozen,
catatonic states. Abulafia’s 1280 book on ecstatic techniques, Hayyei Ha’Olam
HaBa, recommended the recitative rearranging of a finite set of Hebrew letters,
frontward and backward, many times, using prayer melodies, until “…the heart will
suddenly become aware of the intellectual, divine and prophetic…” and hitbodedut
will rest upon him. The instructions were “…combine letters (and associated musical
notes)… reversing and rolling them around rapidly until one’s heart begins to feel
warm.”
It was in my freshman year at Stanford University when I met Michael
Murphy, later to co-found Esalon, the California center for mystical pursuits and
naked mud bathing. He is the author of Golf in the Magic Kingdom and with George
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Leonard, Integral Transformative Practice. I watched him go through a dramatic
personal transformation after participating in Professor of Asian Studies, Frederick
Spiegelberg’s seminar (with meditation lab) about Sri Arubindo’s interpretation of
the Hindu Bible, the Bhagavad-Gita. Shortly after the semester, he climbed into an
abandoned tower on campus to continue his meditation. He remained there for
several months, refusing to come down even after the Stanford Student Health
Service sent a medical school psychiatrist to investigate. I was more than curious
about how it was that this hard drinking, and like his brother Dennis, all night poker
playing, Phi Gamma Delta party boy, had suddenly become a transcendent ascetic.
My girl friend Mary and I signed up for Spiegelberg’s seminar in Indian
Religions. We were made breathless by his accounts of administering a Rorschach
Test to the Indian Saint, Swami Sivananda. He recounted discussions about God
with the artists Paul Klee and Max Ernst and the philosophers Rudolph Otto, Paul
Tillich, Martin Heidegger and Martin Buber. As homework, Mary and I practiced
breathing awareness mediation twice a day. During the year, Spiegelberg
sponsored a visit by the aging but still very lively Aldous Huxley to our seminar. He
also brought us Alan Watts and several lecturers from the Jung Institute of San
Francisco. Shortly after hearing Huxley talk about the spiritual power of a particular
exercise of will and loving thoughts, Mary and I began the daily practice of karessa,
some call it coitus reservatus. I was eighteen and she was nineteen. We found that
withholding an orgasm in order to achieve nirvanic extinction of all desires and
passions was difficult. We spent hours in karessa meditation, trying to experience
the detachment described in the Bhagadvad Gita. This biblical explication of karma
yoga told how it was that the warrior, Ardjuna, instructed by God Krishna in the form
of his charioteer, was able to detach sufficiently to do his assigned job of killing
without emotional involvement. Ken Wilbur, a modern, self proclaimed pandit, an
academically oriented articulator and intellectual justifier of the dharma, the spiritual
work of Hindu and Buddhist practice, contrasts the nirvana (literally “end”)
composed of emptiness in time and space, dharma Kaya in which “…no objects are
arising…” with the lesson of the Bhagavad-Gita. Its message involved realizing
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ones spiritual unfolding within the stream of real time and space, finding emptiness
in the world of form and inaction in the world of action.
We worked at karessa so ardently that there was barely enough time left to
do our assignments in biology and chemistry. In a darkened room, Mary and I lay
legs locked, lying on our sides, moving slowly and rhythmically, humming Om and
waiting for our ascension. We worked at making the journey through Sri Aurobindo’s
soul planes of higher mind, illumined mind, infinitive mind, over mind and finally, the
supermind of infinitely empty no mind. This somewhat unusual way to study for a
three credit course in Asian Studies at Stanford grew naturally out of the central
message of Spiegelberg’s seminar that whereas “…deriving a universal theology is
not possible, having the universal experience is required for an understanding of
any of the world’s theologies.” The controversial Bishop of the Episcopal Diocese of
Newark who teaches that Christian forms continue to evolve, John Shelby Spong,
D.D. says, “…every biblical word represents an attempt on the part of our ancestors
in faith to make sense out of a God experience in their time and place. The
experience …is eternal and real. The explanations will never be eternal and real.
They will last only as long as the (cultural) mind-set that created them.”
Mary got an A+ grade, topping Spiegelberg’s class with a final examination
essay, which, in literary detail, described her episodes of samadhi, yoga’s state of
unity with the creator. Her 25 page blue book contained accounts of walking fugues,
spontaneously strong genital sensations, changes in tastes and smells, sudden
feelings of rising spinal-abdominal kundalini, middle of the night dreams of oceanic
orgasmic fusion with God. She failed to mention that she was describing her usual
pre-menstrual state.
During these college years, I learned about two Isaac Newtons The first I met
at elementary physics lectures; the unit was about how things worked called
mechanics. Logically and computationally consistent but taken on faith, I learned
about an invisible field force between masses called gravity that decayed in strength
like the inverse of the square of their distances apart and operated in my intuitive
world like an electromagnetic spirit. Less occult were the expressions of
gravitational fields as contact forces, computed for the tension in the string of a
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pendulum or the pressure of the floor on a weight resting upon it. Faith in this realm
came from exercises in physical object visualization followed by manipulation of
self-consistent algebraic symbols. I learned about experiments attesting to the
“reality” of these ghostly fields (that now include electric, magnetic and strong and
weak nuclear forces), and yet it was the physicists that already believed them who
designed the machines to demonstrate them. It was Gregory Bateson, Margaret
Mead’s houseboy, lover, photographer and social anthropologist who said, “Newton
didn’t discover gravity, he invented it.”
One college summer I found a second Isaac Newton, perhaps not so
estranged from the first. He appeared in the form of a marble bust in the chapel of
Trinity College at Cambridge University, holding the prism he had used to explore
the polychromatic properties of light like a talisman. In his essay called Newton, the
Man, the early 20 th Century Cambridge Don and economic theorist, John Maynard
Keynes, said that the Newton of the chapel followed “…certain mystic clues which
God had laid about the world to allow a sort of philosopher’s treasure hunt to the
esoteric brotherhood.” Michael White’s biography, called Newton the Last Sorcerer,
described his work as an attempt to integrate the magic of the Old World with the
science of the New Age. Newton’s awe over what he saw as the wonders of the
universe maintained him in private theological study throughout his life. Arthur
Waite’s Alchemists Through the Ages describes how Newton’s alchemical
orientation toward the earth’s fundamental substances such as fire, air, wind and
water, their powers and potential for transformation, was joined imperceptibly with
his metaphysics and physics. In his hands, experimental observations involving
gravitation, celestial mechanics and optics, though motivated by esoteric alchemical
theories, generated experimentally accessible phenomena and testable ideas.
The French mathematician, Jacque Hadamard, in his The Psychology of
Invention in the Mathematical Field, said that mystical preoccupations were never
far from the minds of most of the English and European mathematicians and
physicists of the 18 th and 19 th Centuries. This orientation served as an impetus for
them to pay attention to the almost imperceptible whispers of their emergent
thoughts. E.T. Bell, the historian of mathematics and mathematicians said even
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Descartes, the essential Enlightenment rationalist, was responsive to his “…call of
the Spirit…” Napier the inventor of logarithms wrote an exegetical commentary on
the Book of Revelations. The mathematician and physicist, Pascal, believing that
contact with a religious relic had cured his terminally ill sister, wrote long tracks
about whether or not the Devil could work miracles. The great mathematician,
Cauchy, was known for his persistent efforts to convert fellow mathematicians to
Roman Catholicism. Gauss, who was not particularly religious, said that a difficult to
prove theorem did not result from hard work but “…the grace of God.” In letters
between Liebniz, who along with Newton was the inventor of calculus, and a
member of the family of great mathematicians, John Bernoulli, used scriptural
quotations and biblical diagrams as part of their theoretical correspondence.
Perhaps the greatest mathematician of the 18 th Century (or ever), Euler, in his
Letters to a German Princess, discussed the functional characteristics of spirits and
the connections between body and soul. Bell said Euler “…never discarded a
particle of his Calvinist faith.”
It was to the working out of a law of mechanics called “the principle of least
action” that Ernst Mach attributed the beginning of the separation of physical
mechanics from formal theology. The flavor of this change is captured in his 1893
The Science of Mechanics that stimulated Bridgeman’s 1936 more formal
philosophical analyses of physical theory, from a position that came to be called
operationalism: the restriction of physical concepts to those definable in terms of the
experimental operations required to demonstrate or prove them. Mach said that
these events marked the move of formal metaphysical thinking about mechanics
and the physical sciences more generally into the personal and private realm of
belief and meaning.
Maupertuis, an eccentric friend of Frederick the Great and president of the
Berlin Academy, proposed the principle of least action as evidence of the infinite
wisdom of the Creator. As an early psychopharmacologist, Maupertuis
recommended the use of opium to facilitate creative thought and was famously
parodied for doing so by Voltaire in his 1752 story in which he is portrayed as the
naïvely foolish Dr. Akakia. The physical law of least action belongs to a set of ideas
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that are called variational analysis. They involve the natural (or miraculous)
selection of maxima or minima in quantifiable physical processes. Of all possible
two-dimensional shapes with the same perimeter, the circle contains the greatest
area; in three dimensions, it’s the sphere. In his Principia, Newton reports his work
determining the optimal shape of round solids, with circles of revolution having the
same effective cross section, in order to minimize frictional resistance to gravity in a
medium.
The principle of least action says that imparting energy; say by a kick, to a
physical body on a rigid two-dimensional surface like the earth, results in it taking
the shortest route possible from its initial to final position. The related 1650 Fermat’s
“principle of least time” is about light. As Feynman explains in his Lectures in
Physics, “…out of all possible paths that light might take from one point another,
light takes the path that requires the shortest time.” Feynman, using elementary
relations from high school geometry, proved that the least time principle could lead
directly to Snell’s law of the refraction of light at the interface of two different
conducting media such as air and water. His analogy was the optimal choice of the
path to take in order to rescue a pretty girl drowning in the ocean. Whereas the
shortest distance to the girl leads directly into the water, faster running along the
beach to the point that minimizes the distance required for the intrinsically slower
rate of swimming increases the distance traveled but reduces the time required to
reach her.
Euler attributed the optimization principle to an expression of the meaning
and purpose of a loving God. Infused with this spirit, he developed mathematical
methods describing smooth variations in position of an object in motion, the Euler
differential equation, in which differential coefficients are varied to prove the
principle of least action for mechanical motion. He gave the law Maupertuis’s name.
Mach quoted Euler’s conclusion, “As the construction of the universe is the most
perfect possible, being the handiwork of an all-wise Maker, nothing can be met with
in the world in which some maximal or minimal property is not displayed.” Such faith
based mathematical formalisms were rejected by Joseph Lagrange, an early 19 th
Century mathematician, who, among many other things, proved that every natural
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number could be expressed as the sum of at most four squared numbers. It was his
strongly held opinion that metaphysical speculation was both foreign and inimical to
the conduct of mathematics and science. His work in the calculus of variations led
to the development of a system of algebraic manipulations seeking the value of
constants, Lagrange multipliers, in place of solving Euler’s differential equations. It
makes it possible to immediately write down a computable expression for the
maximum of a mathematical equation. The technique is now routinely taught to high
school students and with no mention of the role of belief in the perfection of God in
its discovery.
* * *
I was a fortunate freshman medical student. After a visit to his office and a
stimulating discussion about some of the correspondences between the ideas of
psychoanalysis and neurobiology, Robert Heath, Tulane Medical School’s Gary
Cooper-like charismatic chairman of the psychiatry department, offered me a place
in his animal and human neurophysiological laboratory. Between classes, evenings
and weekends, I used a Horsely-Clarke apparatus, one of the world’s first
stereotaxic devices. It allowed the precise placement of electrodes into functionally
specific regions of a cat’s brain. The electrodes were cemented to the skull in place
and their wires connected to a device by which the frequency, amplitude and wave
shape of the electrical stimulation could be oscilloscopically monitored and
electronically controlled as the conscious cat walked around the room. I spent hours
observing and recording changes in spontaneous behavior that followed activation
of various nuclei in the cat’s brain with small electrical currents.
Deep in the part of brain that resides in the upper neck, called the lower brain
stem, the region thought to regulate functions such as breathing, heart rate, blood
pressure, gastrointestinal motility and global states of consciousness such as
wakefulness and sleep, I found stimulus sites that, after 15 seconds of electrical
activation, led to several minutes of hissing and objectless rage. One cat attacked
an empty chair. These regions when activated also inhibited spinal reflexes such as
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the knee jerk of the standard neurological examination. Such phenomena were
already well known in the late 1930’s in what W.R. Hess and later John Flynn,
following electrical stimulation of cats in the lateral hypothalamus, called
“hypothalamic rage.” In the late 1940’s and 1950’s, work by National Institutes of
Mental Health’s Paul MacLean attributed it to the actions of parts of the emotional
“limbic” brain, particularly the fear-rage-attack coloring of experience by the
temporal lobe’s amygdaloid nucleus. Modern imaging studies in man have shown
that this source of emotional coloring is activated by new information, even before
the more rational parts of the neocortical brain processes it. How we feel about
something new arises before what we think about it. These survival-oriented states
of fight or flight are known to be biologically universal and demonstrable in even
single cell organisms.
A greater contribution to my brain metaphysics followed observations that
after several seconds of stimulation of other brain stem sites, the cats became alert
but quiet, staring into space for several minutes. Then, they circled slowly and
curled up on the ground. This was followed by several minutes of grooming and
loud purring. Difficult to handle cats became transiently tame, some coming close
for petting. I found that these same sites also increased the amplitude and reduced
the threshold for the cat’s knee jerk reflex. Responsiveness increased with
calmness. Particularly interesting was the finding that electrical induction of this
purring state could immediately stop on-going stimulation-induced episodes of
hissing rage. I referred to these experiments with my friends as my
neurophysiological studies of Old Testament vengeance and New Testament
forgiveness. It seemed that the hissing rage would produce eye for an eye and a
tooth for a tooth hypertension, the talon principle of the Old Testament and Koran.
New Testament forgiveness would yield low blood pressure health and Jesus was a
healer. It was about this time in the early 1950’s that Northwestern University social
psychologist, Jim Olds, found that rats could be trained to push levers to obtain
current delivery via electrodes in various parts of their brains. Shortly after, Joseph
Brady, then of the Walter Reed Army Institute of Research, showed that squirrel
monkeys would do the same. With depth electrodes attached to wires running to a
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miniaturized electronics box strapped to their belts, some of Robert Heath’s
schizophrenic patients spent hours pressing their switches with beatifically
expectant smiles.
It was after several months of cat experiments that Professor Heath
suggested that we spend some time interviewing a hospitalized, chronically ill
female patient, Donna, before and during the time she was being studied with
recording and stimulating depth electrodes in the human neurophysiology
laboratory. Donna, bony thin in a lose fitting green hospital gown and sandals, had
dark red toenails, blonde hair and eyes shadowed darkly. In her mid-thirties, she
had never married and, when she could, worked as a beautician. She told us that
since her menarche at 13, she increasingly often had episodes of spontaneous
ecstatic rushes along with sudden visions of strong white light. She attributed these
experiences to visitations of “…an unseen Christ.” She showed me a stack of
notebooks filled with hand written accounts of her religious experiences
interspersed with biblical quotations and difficult to follow discussions of what she
called the Christian ideals underlying the Civil War. She read parts of it to us. One
of her memorable stories was about being invited to a Children’s Crusade that had
begun in Georgia, led by a great grandson of Stonewall Jackson. “We were trying to
find the Lord to see if He would part the waters and open up an escape route from
General Sherman’s march to the sea.”
From a relatively poor family of Southern Baptists in rural Louisiana, she had
lived in a state psychiatric hospital for almost three years. Her diagnoses ranged
from borderline schizophrenia to temporal lobe epilepsy. The collateral interviews
with her mother from several years before had been placed in the hospital chart.
They recounted that in the patient’s middle to late teens she had become suddenly
promiscuous, frequently approaching strange men in city parks. Obsessed with
fellatio and swallowing sperm, she told her mother that she was receiving a holy
sacrament. More recently, the increasing incidence of ecstatic episodes and
compulsive note taking coincided with the complete loss of interest in sexuality in
any form. Her talk was now full of moralizing detail about the shoulds and should
nots of daily living. She referred to herself as a non-Catholic nun who was married
17
to Christ. The brain waves recorded from electrodes deep in her brain demonstrated
transient episodes of spiking in a midline limbic structure called the septum and in
the right hippocampus, deep in the temporal lobe. Paul MacLean and others since
have shown that electrical stimulation of these and related brain regions could
produce pleasure and grooming reactions in cats and prolonged penile erections in
squirrel monkeys.
Many years later, I spoke about Donna with the Harvard professor of
neurology, Norman Geschwind. He took me to his twice a week epilepsy clinic. In
an effort to demonstrate what is now known as the Geschwind Syndromes of
between seizure, inter-ictal personality changes in patients with temporal lobe
epilepsy, he stood in front of the patients’ waiting room. In a loud voice, he asked
that all people keeping diaries and personal notebooks please stand up. Several did
so, some displaying their notebooks in outstretched hands. The pages that I saw
were filled mostly with religious writing, biblical quotations and exclamation points.
Gathering the positive responders together, he asked them in turn what religion they
were. Several answered the question with the question, “When?” It turned out that
many reported having several experiences of religious conversion. Geschwind
called them “Jamesian Episodes” after William James’ Varieties of Religious
Experience. He then asked when was the last time they engaged in sexual activity.
For most of them, including those that were married, it had been years. Thought the
men said they were not impotent, experiencing early morning spontaneous
erections, they claimed a complete loss of interest in sex though feeling warmly
affectionate toward people generally. As he anticipated, the patients were
emotionally intense and unstoppably loquacious, needing to speak at length about
their moral philosophies. They persisted in following us around the clinic waiting
room, several speaking at once. In his lectures and papers, Geschwind called this
last feature, difficulty in separation, interpersonal “stickiness.” First reported by the
French electroencephalographer, Henri Gastaut, a history of multiple ecstatic
religious experiences, increasing emotional intensity and lability, hyposexualilty (not
impotence), moralizing religiosity, compulsive and frequently poetic writing and
tendency to cling to people is now called the Geschwind Syndrome of temporal lobe
18
epilepsy. Some say it is relevant to the likes of Apostle Paul, Sister Teresa and
Joan of Arc.
One evening in the human neurophysiology laboratory, I was invited by Dr.
Heath to join him and several other brain scientists behind a two-way mirror to
watch an interview with Donna while electrical current was being put through her
recording electrodes. We watched and listened as a psychiatrist interviewed her
about her past. The patient was speaking about her childhood. Unseen by the
patient, the neurophysiologist, with us behind the mirror, was intermittently pushing
the button evoking brain stimulation with very low current applied to the septum. Dr.
Heath told me to listen for subtle changes or discontinuities in the flow of the ongoing
conversation that he said might reflect alterations in her thoughts and
feelings. .
“The first time we were allowed to take a break from Sunday school for the
church service and I got to hear the choir and the pipe organ, I suddenly got a
feeling of happiness that I hoped would last forever. My Sunday school teacher told
us how much Jesus loved us and that’s what the music made me feel like. For the
first time in my life I felt completely safe.” Though the two way mirror I saw the
psychiatrist nod silently. “When I learned about the real meaning of Christmas and
Easter, it was frightening and beautiful.”
Within a few seconds after the neurophysiologist, behind the mirror and
unseen by the patient or her interviewing physician, pushed the switch on the
stimulus generator, the patient stopped talking. After a little more silence, her
interviewer encouraged her to continue,
“You were talking about how beautiful the holidays were. Tell me in what
ways?”
“I don’t want to talk about that anymore.” She blushed and looked very
uncomfortable. The neurophysiologist’s hand remained on the switch. She
continued to speak with her psychiatrist.
“I have to ask you a favor and I don’t know why. I hope you don’t get upset.
The thought won’t leave me alone.” She seemed embarrassed even as her body
relaxed against the back of the chair languorously.
19
“Of course not, Donna. You know that with me you can say anything.”
Her face reddening further, she stuttered something unintelligibly and then
was silent.
“Pardon me, Donna, I didn’t hear what you said.”
“Would you mind if I rested my legs on your shoulders?”
Further Readings for In Search Of The Miraculous
The Hebrew Alphabet, A Mystical Journey, Edward Hoffman, Chronical Books, San
Francisco, 1998
The Book of Letters, A Mystical Alef-bait, Lawrence Kushner, Jewish Lights
Publishing. Woodstock, Vt., 1990
Studies in Ecstatic Kabbalah, Moishe Idel, State University of New York Press,
Albany, N.Y. 1988
Beyond the Human Species, The Life and Work of Sri Arubindo and The Mother,
Georges van Vrekhem, Paragon House, St. Paul, MN, 1997
Bhagadvad Gita, Sri Aurobindo, Lotus Press, Twin Lakes, WI, 1995
Play of Consciousness, Swami Muktananda, Syda Foundation, South Fallsburg,
NY, 1978
Alchemical Psychology, Old Recipes for Living in a New World, Thom F. Cavalli,
J.P. Tarcher/Putnam, NY 2002
Studies in Schizophrenia, A Multidisciplinary Application to Mind Brain
Relationships, Robert G. Heath, Harvard University Press, Cambridge, MA 1954
20
Role of Pleasure in the Brain, Robert G. Heath, Harper-Row, N.Y. 1964
Psychiatric Aspects of Neurological Disease, D. Frank Benson and Dietrich Blumer,
Grune and Straton, N.Y. 1975.
Mathematics –The Music of Reason, Jean Dieudonne, Springer-Verlag, N.Y. 1991
Mathematics for the Liberal Arts, F. Richman, C.L. Walker, R.J. Wisner and J.W
Brewer, Simon and Schuster, N.Y. 1998
The Feynman Lectures on Physics, R.P. Feynman, R.B. Leighton and M. Sands,
Addison Wesley, Reading, MA, 1963
21
CHAPTER 2:
DOESN’T EVERYBODY
Varieties of religious experience and the potential they bring for personal
change are embedded in and perturbative of our unique and common personalities.
The obsessive compulsive may have an easier time with the rigid restrictions of
Fundamentalism or be more resistant to the flagrancy of none rational mystical
experience. The hysteric may find subjective evidence for the Holy Ghost more
accessible and rules of behavior beside the point. The potential for double-jointed
multiplicity in personal styles and quick transitions between them characterize what
is called the borderline personality. It is in these ways that temporary and
permanent brain styles in us and important others supply much of the ground for the
possibility of spiritual transformation and the often attendant alterations in
personality. How can we think about this facilitator and source of resistance to new
spiritual practice?
A skinny, knobby kneed, small breasted, mousy haired, bright-eyed
psychotherapy patient of mine at UCLA’s Neuropsychiatric Institute Outpatient
Clinic was among the highest priced Santa Monica call girls serving Beverly Hills.
Answering my unaskable question about her thousand-dollar fee, she explained that
she was living proof that, in her profession, what was more important than physical
beauty was “griv sense.” She explained that by her middle twenties, she had
22
developed the ability to anticipate the most highly prized but often embarrassing-tosay
longing for a particular sexual act without being asked. She told me that she
had to “empty out my personal sex manual” to feel the cravings of her clients. What
the john most wanted appeared suddenly in her mind in the form of a cartoon. A
university criminologist later explained that the word “griv” was probably derived
from what pick pockets call grift sense, the ability to intuit who was likely to have
enough money in their billfold to justify the risk, even if they appeared in the worn
clothes and dated cars of old money.
In his 1913 Dernieres Penses, Henri Poincare′, France’s seminal theorist in
nonlinear dynamical systems theory, described intuition as a mental faculty which
allows us to “…immediately see the end from afar…” In the context of mathematical
epistemology, the instantaneous images of a geometer contrast with the labored
sequential logic of the mathematical analyst. Poincare′ claimed that inclinations
toward one or the other of these two cognitive styles and their associated
mathematical tools arise from different kinds of minds. He contrasted the 19 th
Century German mathematicians, Weierstrass, who he said reduced his general
theory of functions to “…a prolongation of arithmetic…without a single (pictorial)
figure in any of his books…” with Riemann who called geometry to his aid in
describing functions. He created “…an image that no one can forget… once he
understood it.”
Experiencing the behavior of others, we create a set of anticipations about
whom and how they are that align with parts of ourselves. Aware of one aspect of a
person, we imagine the others. With a small amount of initial information, we
connect the dots, fitting features we have seen and heard to personality
configurations stored by informal category in our brain files. Our conclusions about
them “being one of those” can both facilitate and impair our perceptions. Eastern
metaphysicians, Western mystical religionists, socially liberal secular humanists,
Shannon information theorists and today’s students of dynamical systems in brain
and behavior can, in different ways, make the case that the content of these
stereotypes reflect a pattern of constraints, our personal limitations resulting from
the rutted roads of worldly experiences. Baba Muktananda, the Hindi Saint from the
23
Indian village of Ganeshpuri, called them our samsara. These limit the formlessness
of anticipation that underlies sensibility. Our samsara reduces the uncertainty that
could serve as grounds for new perceptions and understanding of others. Preemptive
distortions reduce the bandwidth available for new information. They impair
the range of empathic relations with others as well as ourselves. These restrictions
in possibilities and choices are expressed in enduring patterns of behavior, thinking
and feeling that mental health practitioners call personality and character. When
confronted with these constrictions, the self justifying and diagnostically revealing
thought about a feature of one’s personality is, “…doesn’t everybody? “ This pride in
our shape contrasts with the teachings about emptiness of one of Baba’s favorite
Indian holy men, Zipruanna, who sat all day, loin clothed naked in a garbage dump,
instructing his students and followers about knowing and being nothing.
We quantitate deficiencies in formlessness using statistical measures of
entropy. They characterize the system’s behavior as a distance from the state of
highest entropy also known as maximal randomness. Professor Karen Selz of
Emory University did a study in which her human subjects, after taking a battery of
personality inventories, were asked to remove as many dots as possible from a
computer screen full of them in three minutes. They were to do so by left clicking on
each of them with the mouse key. Two seconds after a dot was removed, it
reappeared and became subject to removal again. As they went about the dot
removal task and unbeknown to the subjects, the orbit inscribed by their dot
removing mouse travels was recorded for later graphic representation and
quantification. Most subjects with the usual broad mixture of personality traits
inscribed a wide variety of orbital line styles: little wiggles, big wiggles, large and
small loops, little smooth slides and big and little jumps. The counter-intuitive
coupling of stylistic rigidity and whole system instability (as in non-hyperbolic fixed
points described in the previous essay and below) is in evidence at the personality
and graphical extremes of her subject group.
A fastidious, rigidly organized, severely obsessive-compulsive subject
repeatedly removes the same dot, only occasionally moving to a neighboring one to
do more repetitious left key mouse clicking. Very little of the large computer screen
24
of possible mouse travels is occupied. All the action is centered on a small set of
points. When such a minimal entropy person is injured and feeling helpless, their
stuckness can grow bizarre. Ruminative fixation in self-critical and persecutory
ideas extend into poisoned food anorexia, circular pacing, weight loss and middleof-the-night,
worried insomnia. Suffused with sin, they ask forgiveness for soiling the
chair by their sitting in it or smelling up the room with their body odor.
At the high entropic extreme, the mouse orbits of the seductively dramatic,
new reality-creating hysteric includes big jumps, disorganized whorls and large and
small restless and short attention span scribbles that tend to fill up the entire screen.
The fragility of fixation at this end manifests itself in breakdown into impulsively outof-control
and floridly dramatic displays. Their decrease in contact with reality
precipitates social chaos around them. The Montreal behavioral neurologist, Pierre
Flor-Henry, using electroencephalographic and psychological test data, described
the difference between these two extreme forms of personality expressions as the
overly dominant expressions of one or another of the left obsessional or right
hysteric hemispheric emotional styles. As examples, Flor-Henry said that a left half
brain depression feels like hopeless and agitated indecision and the depression of
the right brain is an experience of emptiness like homesickness. Left-brain
happiness is being exactly correct and right brain joy rushes like being especially
chosen.
The church going obsessional resonates with the sermon of the punitive
priest who invokes the tension and relief of sin and salvation. The practice can
result in a life long addiction to the transient high of this temporary forgiveness. In
other churches, the hysterical character gets spiritual respite in disassociative
visitations of the Holy Ghost and attendant signs and wonders. At Wednesday night
healing services, new hope arises from personal surrender in a floor hitting,
backward collapse called dying in the Lord. Both of these antipodal personalities
contrast with the more receptive state of in-between entropy (with enough entropy
available to form messages) which predicts more flexibility and higher potential for
undistorted information processing. Relatively style-less and ego-less people are
more open to hearing a variety of Gods in themselves and others. High alertness
25
without presupposition, ecstatically aware and selfless, it is God’s gift realized, a
joyfully awake and nonjudgmental empty state of transcendence. As we sit, we work
at feeling this in the brain of the enigmatically smiling stone Buddha.
The externally inactive state of high internal activity, the Bhagavad-Gita’s
formlessness in the world of form, inaction in the world of action, has a natural
mathematical representation in the simultaneously expanding and contracting
motions of hyperbolic dynamics and its associated entropic descriptors. How can
this kind of formlessness equip us for almost instantaneous knowing? In a resting
state of uniform hyperbolicity that only looks like randomness, accurate impressions
of others can arise quickly and from only a few data points of observation. In the
late 1960’s, University of California mathematician, Rufus Bowen, proved the now
famous shadow theorem. This says that in dynamical states of hyperbolicity, directly
observable on the screen in computer simulations, the first few points of the ongoing
wild dynamical dance that appears to jump randomly from here to there on
the computer screen, counter-intuitively will quickly outline the entire skeleton of its
future global shape, its geometry, though more time of observation is required to
realize this structure in full detail. The contracting motions on the stable surface of
action, called a manifold, “iron down” all the points onto the unstable manifold that
serves to outline the shape of the attractor of all starting points. In such a system,
observation of just the first few points outline the whole. Intuition, anticipatory
knowing and that which some call prophesy, may be expressions of the hyperbolic
brain’s mind doing dynamical shadowing.
To review briefly, hyperbolic brain flow is made up of three decomposable
components: (1) The apparently predictable one along the main road of the action,
going straight ahead and round and round on a throughway called the center
manifold—analogous perhaps to what might be a sequentially logical development;
(2) Intersecting the center manifold transversally is a field of influence moving the
action away from the center manifold with out-of-the-box motion, exploring side
paths of unpredictably new, creative possibility called the unstable manifold, we
might think about inspired risk-taking, impulsive associations in thought; (3) Another
transversally intersecting field of influence, which conservatively, rationally, “irons
26
down” the expansive flow back onto the road, the entire constrictive field called the
stable manifold. This influence herds points into shadowing the main road of the
dynamics, like the hair of the dog that stay close to the real body of the animal in
motion. It is in this way that just a few often slightly off the mark points nonetheless
shadow the real (called fiduciary) orbits of the attractor, outlining its global geometry
with just a little information.
The intuitive reason shadowing works is built into these natural countervailing
tendencies of hyperbolic dynamics, which on one hand tends to spread out nearby
initial points and brings disparate others together. The latter inclination is the one
that smoothes down the escaping points onto surfaces of actions that
mathematicians call manifolds. However, the details of the orbital paths don’t look
that orderly due to the mixing of the sequence of points in hyperbolic motion. The
mixing process on manifolds has been analogized to that of the bundled pink loops
of the stretching (expanding) and folding (contracting) taffy puller at the carnival
candy stand. The process gets sequences of small particles of candy out of
sequential order while maintaining the taffy’s overall geometrically ovoid shape.
Disorder is local with the entropy being generated by the repeatedly shuffling of the
line up of the original orbital sequence. This results in the impossibility of any pointto-point
prediction for more than a few points even though the over all shape is
maintained. Exactly what minute a habitually late sleeper awakes can’t be
predicted. On the other hand, the skeletal manifold of the global structure is entirely
in evidence from almost the beginning. Late risers remain late risers even without a
precise, minute-to-minute, predictable schedule.
It is also interesting that a uniformly hyperbolic dynamical system, unlike the
fixed-point attractors of stylistic fixation, resist perturbation-induced changes in
global dynamical form. In an apparent paradox worthy of metaphysical allusion, the
dynamically hyperbolic kind of formlessness has structural stability. The global
geometric predictability of this point-to-point, completely unpredictable system can
be both the subject and object of Zen frustration and thoughtful meditation.
During weekly professorial rounds at Los Angeles’s Neuropsychiatric
Institute, I assigned a standard exercise for psychiatric residents on clinical rounds,
27
which involved limiting their contact with a patient to five minutes. This was followed
by detailed discussion of everything we’d seen and heard. I’d ask them to predict
what we’d find in the many pages of personal interviews and nurses observations in
the clinic chart. The student psychiatrists with the most street smarts, called
emotional intelligence by Daniel Goleman, were particularly quick at shadowing and
thus predicting the patient’s global dynamical pattern.
Do personality patterns exist? Evidence from biometric studies of the
hereditary aspects of personality style in animals and humans suggest that
relatively few global component properties underlie a variety of complicated-looking
manifestations of behavioral style. Primary colors are the source of all hues.
Harvard psychologist, Jerome Hagen, has reviewed the history of this idea in his
book, Galen’s Prophecy. While there are differences among personality research
programs, almost all rating scale and questionnaire-based studies result in clusters
of traits that reflect statistically associated properties which when taken together are
called temperament. This idea is close to what we mean by personality. These
relatively few response clusters are given descriptive names such as introversion,
extroversion, neuroticism, impulsivity, sociability, task persistence and tolerance of
ambiguity. As defined by psychological inventories, studies of families show that
these styles are heritable in the range of 60%.
Hans Eysenck, in over four decades of work and more than 5000 published
papers from London’s Maudsley Hospital, derived common global factors of
personality using questionnaires. The best known was called the Eysenck
Personality Inventory. His studies resulted in evidence for only a few fundamental
behavioral axes, behavioral manifolds, which describe extremal properties of
personality types analogous to stable and unstable manifolds: introversionextroversion,
shyness-sociability, low and high activity level and emotional
constriction versus impulsivity.
To make the issue of personality as dynamical system more realistically
complex, we can call on some examples of the rich history of behavioral genetic
studies using animals such as the mouse. They can be selectively bred for
underlying personality factors, such as dominance, fear, aggression or exploratory
28
courage. Not surprisingly, social interactions, as configured by the mouse’s own
personality style, contributed significantly to their behavioral patterns. As an
example, the C57BL strain of laboratory mouse has strong tendencies toward
impulsively wild behavior. To be anthropocentric and using Hagen and Eysenck-like
behavioral dimensions, we could describe the C57BL mouse as exhibiting high
psychotocism, P, energetic sociability, high energy, E, and low emotionality, low
neuroticism, N. The C57BL also loves alcohol and will dominate the low E, shy, low
P, retiring, alcohol avoidant, high N, emotional, anxious, frequently defecating albino
BALB strain of mouse when they are placed together for a limited time in a novel
situation during the daylight hours. Over a more extended time, however, the BALB
mouse comes to dominate the C57BL, beginning with attacks in the dark and finally
as the persistent and patient survivor over days of aggressive fighting. BALB’s low
E, social fear eventually turns into rage and aggression. The C57BL is quick to mate
and ejaculate but very slow to recover sexually, so that the less post-orgasmically
refractory BALB also wins in long term sexual competition in a cage full of fecund
females. Modern social psychological approaches to human personality are
beginning to approach the interactions of genetic brain proclivities and collective
social dynamics in this way.
Employing Eysenck categories of personality characteristics, similar results
about style as influenced by genetic selection can be seen in humans. The
correlations between factor scores based on B. Loehlen’s studies using the
California Personality Inventory in twins demonstrated as much as threefold higher
correlations among identical twins for extroversion (E) and neuroticism (N) factors
compared with matched fraternal twins. The primacy of some of the in-born
biological roots of these personality styles is suggested by G. Methany’s finding of
higher correlations between identical as compared to fraternal twins when studied at
the age of two months. The similarities in personality and temperament measures
included activity level, regularity, approach-withdrawal, intensity, persistence,
distractibility and adaptability.
More recent familial studies of the heritability of personality characteristics
included childhood shyness, neuroticism, depressive symptoms, aggressiveness,
29
behavioral inhibition and anxiety, behavioral flexibility, narcissism, deviant motor
activity levels, novelty seeking, harm avoidance and reward dependence. These
studies were conducted by R.R. Crowe, J.F. Rosenbaum, A. Methany, and J.L
Robinson and indicated familial congruity of these characteristics among first and
second degree relatives in the range of 40-50%. This level of heritability in
genetically unrelated family members was found to be less than 20%.
Low entropy fixations of personality can also evolve developmentally.
Experiments in young animals have shown that stress-induced high levels of
adrenal hormones exaggerate the normal developmental process of trimming back
unused neural connections, called pruning, the normally complexly over-grown
sprouting pathways. The pruning actions of the pituitary-adrenal stress hormones
come to dominate sprouting actions of neural growth factors and their protection of
neuronal axonal branching and connections during development. The research
program of Bruce McEwan of Rockefeller University and others document nerve cell
loss resulting from the neurohormonal concomitants of stress. This reduction in
neuronal connectivity and neuronal cell content has been conjectured to contribute
to the pathological simplification of neuronal projections and neural network
complexity, reducing information processing capabilities. The still intact machinery
underlying the global patterns of neurological activity, such as those that underlie
personality styles, is arranged around these pruned, unoccupiable holes of lost
brain possibility. If this range of potential behavior is extremely reduced, the
behavioral syndrome is often called a personality disorder. Those that have one are
the predictable Johnny one notes of response to perturbation: thrash out, lie without
reason, get drunk, binge on promiscuity, steal unneeded things from department
stores, or withdraw into interpersonal isolation.
A more abstract and quantifiable way of representing the pathological
simplification-induced emergence of low entropy, stereotypical personality style is
inscribed on the head stone of the post-suicidal grave of Ludwig Boltzmann. This
father of modern statistical physics expressed the idea in the form of a
transformation: the (maximal) entropy, S, of a system is the logarithm of the
number,Ω, of its available ways of being, (i.e., S = log Ω). That is, one way a
30
reduction in the dynamical entropy of a system can occur is by reducing the number
of its available states. As the repertoire of ways of personal responding, log Ω, is
reduced, so is the brain system’s entropy, S.
Reality constrained patterns of behavior, as in successfully adaptive
personalities, lie in some optimal in-between place between the maximal and
minimum measures of entropy. The dynamical state that is postulated to yield inbetween-valued
entropies is called nonuniform hyperbolicity. This is best seen when
the values of the experimental observations are plotted in a two dimensional phase
space with each point represented by two values: along the x-axis is plotted the
value observed, along the y-axis is graphed the change in the value from the last
observation. The signatory motions of these observations plotted in phase space
are irregularly varying in rate of expansion (near by initial values are separating in
time) and contraction (greatly differing initial values are coming together in time).
Values are not fixed, rhythmically varying nor in random motion. These
nonuniformly hyperbolic motions are seen in speeded up, talking head videos
showing bursts of hand gestures and in normal neuronal activity. Silences have
widely varying lengths and bursts of hand movements and neuronal discharges are
irregular in duration and character. The statistical pattern of neuronal inter-burst
intervals is not the convergent Gaussian distribution of I.Q. or heights but the
nonconvergent, long tailed, Levy distribution of flood incidences and, according to
Mandelbrot, stock market crashes.
The labored logic and inscrutably compact mathematical formalisms of the
Nobel Prize winning physicist, Ilya Prigogine, and his Belgian school, explain the
thermodynamics of these long lasting niches of restricted variation in our personal
style as energy requiring dissipative structures. Compulsive nail biting, driven
promiscuity, readiness to be suspicious are seen as a persistence of deviations
from the maximum entropy of formless, flexible, receptive end states. The system is
trapped in possibility reduced, energy requiring, samsaric niches of what Prigogine
called minimal entropy generation. We unique and oddly shaped and entropy
leaking balloons maintain our characteristic distortions through energy-requiring,
31
persistent efforts at insufflation. The maintenance of neurotic defenses and
eccentric habits can be fatiguing.
* * *
The children at Kids in Distress Residential and Day Care Center in
Southeast Florida, called KIDS, tended to be small for their ages. As a psychiatric
consultant to the Center, I often summarized an evaluation of both their physical
and intellectual development as “delayed.” Looking like almost completely formed
adult-like personalities, however, they were developmentally “advanced.” I heard in
a child analytic seminar at the Psychoanalytic Institute of Southern California that
traumatized children often hurry through the dangerous developmental ambiguity of
openness and flexibility to the predictable, fixed attitudes and behavior of adults. It
was common to find prematurely wise young children serving as parents in
chaotically dysfunctional families. In residence at the Center, set free from their
pathogenic homes by social workers and family law judges, these premature
caregivers lost sleep worrying about who was taking up their obligations to the
sisters and brothers left behind.
Trauma-induced possibility pruning was often obvious in the young refugees
at Kids in Distress. Having been soaked in alcohol containing, nutritionally deficient,
crack-laced amniotic fluid, young babies were then left in dirty cribs behind locked
doors to cry themselves into exhausted despair. Their mothers were working the
streets for drugs. The children that survived often demonstrate personality styles
that are reduced in variety. They came to use a few, individualized, and stereotyped
techniques for survival. Some children’s insulated detachment was hollowly
disguised as interpersonal caring. Others used driven and rigid compulsion to
maintain the appearance of conscientious good citizenship. For some children,
paranoid thoughts were realistic expectations. .
Arriving at the Center I heard “Dr. Arnold! Dr. Arnold!” in high-pitched
screams. Several children ran up to me at once, demanding to be held. Some
leaped into my arms for a hug. Trying to get and hold their visual gaze was another
32
matter. Their eyes darted back and forth across my face, not stopping at my eyes,
as though checking for danger. It felt like a strange mix of physical clinging and
interpersonal distantiation. Many articles in the International University Press’s
Psychoanalytic Studies of the Child book series, described these prematurely
formed child personality types: the paranoid scouts, the detached as if children
pretending to feel, the desperate to please obsessionals, the charismatically
seductive hysterics and the unconscionable psychopaths.
Experiments simulating trauma and neglect in young animals also
demonstrate acceleration in biobehavioral development. Possibilities, the number of
available states, Ω, brain entropies as S = log Ω, become casualties of traumatic
and neglected early life. Like one trick ponies, these abused and abandoned
children take up singular patterns of behavior that seem to work and stick to them.
One doesn’t anticipate seeing such narrowly fixated personality patterns until late
adolescence or adulthood. They appear at ages too young to qualify for the
character pathology coding of the Diagnostic and Statistical Manual IV. Yet the
labels of adult personality disorder seem inescapable when one sees a four-yearold
child trapped in a compulsive hand washing ritual or a panty flashing five-yearold
girl with a seductive gait.
Four-year-old Alicia rubbed the lumps in my right hip pocket containing
caramel candies. Her blue eyes twinkled. Her long blonde hair was in bangs and
her lips in a pout. She kept a hand on her hip and tilted her pelvis as she spoke.
Listening to children’s stories, she straddled the reader's thigh and rocked. Alicia
had a history of sexual abuse in a home that was a hang out for drug dealers. There
were rumors that she talked to strange men late at night on the phone. On
admission to the Center, she was found to have genital herpes. Both of her parents
had been in and out of prison for drug-related crimes. The Center’s staff spoke of
Alicia’s seductive smiles, incessant demands, irritable complaints and tantrums.
With the back of her hand held against her forehead, she said that it was too hot to
pick up the toys she had scattered around the fenced yard. Ordered to comply,
Alicia took three steps into Florida’s summer heat and fainted. Each morning, she
spent the better part of an hour in front of the mirror, trying on all four of her dresses
33
and their scarf and belt accessories before choosing one for her appearance at the
breakfast table.
Five-year-old Grace was a suspicious and dictatorial presence in the
Center’s kindergarten class. Articulate and righteous, she confronted children and
staff alike with evidence for the unfairness she found everywhere. In legalistic
defense of her rights and sometimes those of her peers, she used her strong wide
face, penetrating look and quick and observant mind aggressively. Her somewhat
intimidated childcare worker maintained Grace's cornrowed hair with care. Sensitive
to criticism and quick to anger, she competed with her teacher for control of the
class. Her drug abusing young mother had escaped from her own mother’s
authoritarian house, leaving six-month-old Grace in the care of her commanding
grandmother, a matronly church elder. Recent studies by David Reiss and
associates at George Washington University assessed psychosocial dynamics in
genetically varied families. They found that genetic similarities amplified the
expression of individual characteristics of interpersonal relating through what might
be called personality resonance. Relatives often commented that Grace and her
grandmother, being alike, deserved one another. Shortly after her fourth birthday
Grace was removed from her grandmother's home while the circumstances
surrounding the accidental scalding of the bottom half of her body in an overheated
bath were being investigated. She began her first conversation with me, “Hey doctor
baldy, why are your bottom teeth so crooked?”
Damon was darkly handsome, with teasing eyes and a gleaming smile.
Talking to his legal guardian on the pay phone in the afternoon of his second day at
KIDS, he was heard to be making charges of mistreatment by the staff. He asked
his guardian, loud enough to be heard throughout the day room, “What does it take
to get someone fired around here?” Six years old and abandoned by his mother at
the age of three, Damon came to KIDS with a history of provoking administrative
conflicts at several children’s shelters. His record showed that once he successfully
used accusations of beatings to get a staff member fired employing charges that
were later shown to have been fabricated. He argued persuasively, manufacturing
events and quoting imaginary conversations with smooth confidence. He could
34
change stories midstream without apparent loss of continuity or confidence. He
learned the power of a claim of abuse, and used the threat of it to control his
environment. Damon talked other children out of their candy allotments, cheated at
games and stole clothes from other children’s lockers.
Debbie, age eight, was the eldest of four children. Her mother was a street
prostitute with an expensive drug habit. Debbie was thin, restless and worried. A
self-appointed mother from the age of four, Debbie felt responsible for the care and
feeding of her brother and two sisters. With a history of physical and sexual abuse
by a series of her mother’s boyfriend-pimps, Debbie spent most of her time cleaning
and recleaning their small apartment and worrying about obtaining enough food for
her brothers and sisters. Her mother was often gone for one or two days at a time,
and food supplies were not dependable. On several occasions, Debbie was caught
stealing food from all night grocers. The investigative social worker reported that
Debbie had learned to sell oral sex to the men who loitered behind a neighborhood
bar. She used the money to buy food. For several days after admission to the crisis
home, Debbie was anxious and sleepless. She worried endlessly about the welfare
of her sisters and brother despite reassurances that they were in caring foster
homes. She checked on them as frequently as allowed by phone. In a playroom
therapy session, wielding a rubber knife, she pointed to a scar on her left forearm
and told a story about the time that she cut herself with a kitchen knife and fed her
blood to her infant sister when there wasn’t any food in the house. Debbie kept her
room very tidy, did all her chores and sometimes those of other children. Even after
several months in residence, always-busy Debbie didn’t have even one close
relationship with any of the other children or members of the staff.
Despite the superficial differences, there are subtle and pervasive similarities
among the personality styles of Alicia, Grace, Damon and Debbie. Like overgrown
and tasteless cabbages, pale and four feet across, growing from seeds over-treated
with gibberellin or auxin plant hormones, the inner lives of these prematurely big
little people are relatively empty of stable interpersonal objects. The pantheon of
indwelling companions are either malignant, absent or both. There is a deficiency of
internalized significant others with qualities we more healthy neurotics paste onto
35
new faces which we then love and hate. Instead, every interpersonal arrangement
is new, suspect and run on a cash-and-carry basis. We are made to feel like there
are no seats for us inside of them. Even Debbie, with her history of selfless motherly
devotion to her “children,” felt like an empty husk, encased in the exoskeletal armor
of compulsive correctness. With their inner life unpeopled, the best we on the
outside can hope for is to be valuable to them as tools, like forks and chairs.
In new and potentially therapeutic settings, for example a genuinely loving
foster family, these children manipulate, testing for the feared loss and abuse that
first generated their detachment. They provoke the very mistrust they fear. The
sexually exploited child is seductive. The physically abused child provokes attack.
Personality constellations which can be adaptive, when narrowed and fixated,
become impediments to new and reparative experience. It is in this way that
personality disorders are self-maintaining.
An irony is that these interpersonally empty and rigid patterns in personality
tend to occur in the most constitutionally robust of the abused and neglected
children. They are those who have escaped early death from failure to thrive,
severe neuropsychological impairment, chronic depression, severe social
withdrawal or the pediatric psychotic disorders. The children with sufficient flexibility
to adapt quickly and survive often settle into empty-centered rigid caricatures of
adult personality styles.
Of course, well-defined and characteristic personality patterns do not require
abandonment and abuse or the pathological simplification of traumatic deforestation
of neuronal connectivities in order to emerge. Demanding social selection of
particular personality proclivities that are competitively advantageous for highly
sought positions also results in the appearance of well-defined personality styles.
Common examples are the technical types, “techies,” “nerds,” whose work require
long hours alone to master and execute, as in doing mathematical proofs, solving
problems in theoretical physics, unraveling computer programming problems or
writing highly technical tracks. These activities can be aided by the personality
inclinations of shyness and distantiation, the experience of discomfort in social
occasions along with a rich private fantasy life. Diagnostically oriented mental health
36
professionals (and lonely mates) may label these interpersonally distant,
engineering rocket science people, “high functioning” sufferers of Asperger’s autistic
spectrum disorder. Things going on inside get most of the attention, having more
impelling importance than those on the outside involving other people. A recent
study by Cambridge University’s Autism Research Center compares the
empathizing (E) versus systemizing (S) ability of normal controls and adults with
Asperger Syndrome and find the quasi-autistic adults are deficient in E and superior
in S. They call it the E-S theory of autistic spectrum diseases. Psychotherapists of
these autistic spectrum personality types, patients who characteristically do not
seek therapy but are forced into the office by marital or family conflict, speak of their
long, patient and mighty struggles to make intimate contact with these clients. A
more philosophical question involves issues of what are acceptable individual
differences and why it is that these high functioning, highly paid and successful
professionals have any diagnosis at all.
It is not surprising that the highest paid members of corporations producing
technical products and services such as IBM and Oracle are those rare individuals
in technical sales that are able to combine the skills and insights of introverted
scientists and technicians with those of the gregariously successful salespersons. In
business schools such a blend is seen in people who combine talents in both
marketing and finance. In architecture this combination might take the form of a
graphic-design artist with computational mechanical engineering skills. Recruiters
know that it is difficult to find people for what is called engineering sales.
From all over the United States, professional instrumental musicians that
began to experience severe technical difficulties that defied their teachers as well as
more extended practice time came to see Chicago’s music guru, Carl Boardstadt.
He was a nationally known consultant to classical and jazz professionals in the
1920’s and 30’s. His particular specialty involved those who had “hit the wall,” those
whose progress toward advanced musical mastery and accession into the higher
echelons of the business had been truncated. His recommendations were often
eccentric indeed. For the wind musician with breadth control problems, it might be
blowing uniform bubbles through a long tube held at increasing depths of a filled
37
bathtub or feeling the seductively diaphragmatically oscillating belly of a taxi dancer.
Pianists with speed problems worked at specially constructed up-side-down
keyboards with the rationale being that finger lifting was more rate limiting than
finger placing. He said that his most hopeless cases were those whose
personalities didn’t fit their choices of instrument, too often made by what position
remained open in the high school band rather than following a personal interview.
He claimed that trombonists should be sensually languorous; clarinetists, nervously
impatient; double reed instrument players, obsessional and withdrawn; brass
players, athletic and exhibitionistic.
* * *
As one of the team physicians of the San Diego Chargers in the years 1971-
1975, I spent several days a week in their summer training camps, on the team
plane to and from games, in the locker room and on the sidelines during games. I
was involved particularly in player drafts. Unbeknown to candidate players and
other teams, we used a system of what social scientists call unobtrusive measures
of their personalities as part of their evaluations. College football players are sent
questionnaires each year by professional teams asking about a variety of life events
and attitudes including their goals for the future. Filled out by hand, they served as
repeated measure, handwriting samples. Twenty years of them were available in
the Charger’s record room. Using 30 standard signs from the French graphology
literature and three trained raters, we evaluated the hand writing characteristics of
players, National Football League wide, who obtained and retained playing, not
reserve, positions in the League for at least three years.
After studying handwriting profiles from close to a thousand established NFL
players, and hundreds of hours of individual interviews of members of many teams,
it became clear that, athletic abilities being equal, success was more likely when the
player’s personality type fit his football position. What amounts to a series of
selective filters are operated by coaches, scouts and managers throughout the
playing careers of these players in grammar schools, high schools, universities and,
38
ultimately, the NFL draft. Choices obviously involved more than height, weight, time
in the 40-yard dash and performance in motor coordination tasks. The players
behavior, carefully studied on the field, in multiple camera angle game films, direct
and collateral interviews and observations under game conditions constituted a high
level of selective pressure that brought with it the emergence of characteristic
personality types. Tens to hundreds of thousands of candidates are winnowed down
to several hundred highly paid players in this selective process.
Distinctive personality patterns accompany success at a particular position.
Structure loving, politically more conservative, choreographed in detail and
repeatedly rehearsed, offensive players keep their lockers more organized and tidy.
More rebellious, resentful of structure, politically more libertarian, thematically
instructed but principally opportunistic, defensive players, particularly linemen and
linebacker’s lockers had messy lockers. Defensive team players were most often in
trouble with the law. Offensive lineman including centers, guards, tackles and some
tight ends tend to be patiently enduring and tenacious, their aggression taking the
form of stubbornness. This contrasts with the temperamental explosiveness of the
defensive line and linebackers. We could speak of the volubility of centers, the loyal
and caring kindness of offensive tackles, the narcissistic exhibitionism of wide
receivers, the murderous rage of the defensive end, the sullen and paranoid
depressiveness of the defensive back, the joyfully impulsive unpredictability of
broken field running backs and the good citizenship egolessness of the blocking
fullback. Some quarterbacks lead and play fearlessly in a religious state of grace,
some are members of the Fellowship of Christian Athletes. Others lead as
fearlessly, but in the style of an unconscionably calm psychopathic bank robbing
professional.
Influenced by our findings, the San Diego Chargers drafted the Hall of Fame
quarterback and one time ABC Monday Night Football commentator, Dan Fouts.
Skinny and hurt several times during his college years as a quarterback in Oregon,
he was passed over in the NFL draft until the third round. The scouts “knock” on him
was that they thought that he lacked psychological and physical toughness; the
ability to get up after a hit and to ignore the on coming tons of defensive linemen
39
while calmly and quickly surveying the routes of several potential receivers. The
pattern found in his handwriting features, however, resembled those Johnnie
Unitas, the Hall of Fame quarterback of the Baltimore (then) Colts who, in spite of
his small size, famously played with great courage and physical toughness. In
chronic and severe back pain, he played regularly until retirement in his early 40’s.
Fouts drafted in the third round with a small five-figure bonus, proved to be a great
bargain for the Charger franchise.
Given the theoretically infinite number of ways that a personality can be, it is
remarkable that the latest Diagnostic and Statistical Manual of the American
Psychiatric Association, DMS-IV, describes only eight types, which form three
subsets of exaggerated expressions of stable personality styles called personality
disorders. All eight personality disorders can be grouped into: (1) Cluster A - Odd
and eccentric types, whose anxiety is related to the felt threat of disintegration and
annihilation of the self and whose style is dominated by mistrustful paranoia, a
schizoid, detached and emotionally flat pattern or the isolated strange eccentricism
of schizotypal characters; (2) Cluster B - Unstable and impulsive types whose
anxiety is related to loss of the stable self and whose style is dominated by
irresponsible antisocial behavior, chronic instability with high amplitude fluctuations
in behavior called borderline, or patterns of excessive emotionality and dramatic
display associated with histrionic characters; and (3) Cluster C - Fearful types
whose anxiety is related to hypersensitivity to criticism, guilt and feelings of
inadequacy or loss of control, and whose style is dominated by interpersonal
avoidance, clinging dependency, or rigid lock up into obsessive-compulsive efforts
to do the right thing and avoid disapproval. This remarkably small array of
stylistically consistent global behaviors selected from a practically infinite number of
imaginable possibilities establishes a small set of invariants of some, perhaps
abstract, property. These characteristic patterns inspire our search for the implied
brain and behavioral conservation laws that may underlie them.
40
Further Readings for Doesn’t Everybody
The Evangelicals, David F. Wells and John D. Woodbridge, Abingdon Press,
Nashville, 1975.
Godtalk, Travels in Spiritual America, Brad Gooch, Knopf, N.Y. 2002
The Value of Science, Essential Writings of Henri Poincare’ Stephen Jay Gould,
Modern Library, Random House, N.Y. 2001
From Being to Becoming, Ilya Prigogine, Freeman, San Francisco, 1980
The Development of Mathematics, E.T. Bell, McGraw Hill, N.Y. 1945
Deterministic Chaos, An Introduction, Heinz, George Schuster, VCH, Weinheim,
1989
Lectures on Dynamical Systems, Structural Stability and their Applications, Kotic K.
Lee, World Scientific, Hongkong, 1992
The Psychobiology of Behavioral Development, Ronald Gandelman, Oxford, N.Y.
1992
Handbook of Character Studies, Psychoanalytic Explorations, Manfred Kets de
Vries and Sidney Perzow, International Universities Press, Madison, 1991
Cognitive Style, Five Approaches and Relevant Research, Kenneth M. Goldstein
and Sheldon Blackman, Wiley, N.Y. 1975
41
CHAPTER 3:
TRANSMOGRIFICATIONS OF ENERGIES
After several of months of running, 12 miles most days, I felt an energetically
calm, self-containment and a growing loss of interest in things sexual. My
increasingly impoverished fantasy life led my training psychoanalyst to suggest that
I was running away from the critical, females issues of my psychoanalysis. He said I
was becoming more out of reach as I became more socially pleasant. This was
decades before Prozac, Paxil and other serotonin reuptake inhibitors were inducing
similar hyposexual, withdrawn states of cordiality in millions of Americans. Recall
that Norman Geschwind, the Harvard Professor of Neurology, reported similar
conditions of high energy sexual disinterest and abstract metaphysical
preoccupation in patients with right temporal lobe epilepsy. For reasons other than
the loss of church property rights and the spread of syphilis to the clergy, it felt like I
was being readied for Pope Gregory VII’s Eleventh Century celibacy reforms for
abbots and clerics of the Catholic Church.
It was true that my feelings of dependence on my analyst for understanding
and approval were being reduced as I ran into less emotional involvement. I was
becoming a more rationally objective observer of others and myself. It wasn’t the
first time that my over-ardent practice led to this warning. Baba Muktanada, my
Hindi guru, told me to reduce my daily sitting time of meditation. He said my spacey
42
social smile belied a growing disinterest in the welfare of others. I was getting
hooked on the hard training high of not really being there for other people.
Several articles in Runner’s World said that many runners become addicted
after even a few months of running over six miles per day. It’s true that over fifteen
years I missed less than 10 days of running per year. I ran in driving rain, sweltering
heat and dangerous places. In New York’s Central Park after dark, I followed a
freshly strewn trail of torn woman’s garments that ended in shredded panties and a
bra on the Park’s bridle path. In Oklahoma City at 104,° I was chased and bitten by
a terrier. In Munich at 4:30 AM, before delivering a morning lecture, the black
uniformed police stopped me for a shakedown. In Ann Arbor, I shuffled along in two
feet of snow. By the Seine, at 14°, paranoid barge hounds barked in big dog
baritones. I ran on the Hebrew University track a block away from a loud Palestinian
bomb left in a refrigerator near a busy street corner. Breathless at nine thousand
feet in Aspen, gagging on the strong manure smell of Sacramento Valley farms, in
the hot wetness of Houston and dry heat of Palm Springs. I wore out three to four
pairs of Nike running shoes per year. What I did not tell my training analyst was that
this felt like a chase after God. As in most spiritual transformations, His messages
and music could emerge quite suddenly.
Even after stretching, it was painful to begin and that was my daily sacrifice. I
was readying myself to follow the God of the Hebrews and make the “three days
journey into the desert” as in Exodus and Paul’s recommended presentation of my
body “as a living sacrifice, holy and well pleasing to God.” After three miles of
running, the hip pain, back stiffness and leg heaviness lifted, difficult breathing
became easier. A burst of new energy appeared suddenly. The first pop usually
took the form of assertive feelings fueled by new personal power, an undoing of the
lethargy and depression of a helpless sinner. New and big, I felt like I could fix
almost anything. Up bubbled an aggressive speech to the Dean about his refusal of
our recent request for an increase in departmental research space. As for the
National Institute of Health’s recent return of one of our grant proposals, it was now
clear that the reviewers were wrong. I would resubmit but this time ask for twice the
amount of money. I rehearsed a new list of necessary and routine laboratory chores
43
for my most rebellious post-doctoral student. I would tell my teen-age son that he
must wait another year for his own car. I felt generally intolerant.
In an article in Runners World, I labeled my run’s first global brain state
transition, the first second wind. It energized me with the cool firmness but ready-tobe
angry righteousness of modern religious orthodoxy: Orthodox Jews gunning
down Hamas terrorists as retribution for bus bombing children which was itself a
retribution; Muslim suicide bombing as vengeance for cultural contamination;
Catholic Bishops refusing the Eucharist to pro-choice politicians; Charismatic
Christians gay bashing defense of the sanctity of marriage; Mohammed’s early Sufilike
poetry of love turning into territorial aggression and Jew killing in his later years.
Once in while, unpredictably, past the first hour of running and after the first
second wind, a fatigue easing second burst of energy followed the second stage of
exhaustion. I called this running-induced, second global brain state transition to a
softer loving energy, the second second wind. Colors became intense, clouds
breathed and my body lightened. Running once again became easy. I was flooded
with empathic and generous thoughts. I understood that the Dean was faced with
too many space demands to satisfy; the grant reviewers’ criticisms of the budget
were meant to be constructive. I recalled that strong minded, rebellious postdoctoral
students often made the most creative contributions to science. I realized
that my son’s urgent desire for his own car was a proposal in the direction of the
independence that would be required of him the following year when he was going
to be hundreds of miles away at a university. Filled with benign optimism, I felt the
compassionate perspective afforded those with energy but without envy, anger or
fear. William James, in Varieties of Religious Experience, A.C. Underwood’s book,
Conversion, Christian and Non-Christian and Gobi Krishna’s The Awakening of the
Kundalini, among many others before and since, describe the sudden appearance
of long lasting states of optimistic energy and loving empathy that can emerge after
long episodes of suffering, especially following periods of privation of spiritual
meaning and the loss of a previously strong faith. These episodes are painfully
chronicled by St. John of the Cross in his Dark Night of the Soul.
44
In the long distance running model of spiritual transformation, the first energy
appears suddenly in the middle of painful fatigue and feels like a vigorous
implementation of Halachic commands or Canon Law. The second burst of energy
emerges from readiness for resignation and ends in humane comprehension and
empathy. In some Christian monastic practice, a similar transition is represented in
the ritual of Tenebrae (or Darkness). Fifteen lit, unbleached candles are
extinguished, one by one over the night, while reading the Psalms. The practice is
said to represent the desertion of Christ by his disciples, as the church grows darker
over the night. After the singing of the Benedictus, the one remaining light is
quenched, plunging the church into total darkness. In Myth and Ritual in
Christianity, Alan Watts suggests that the loss of the last light of Tenebrae induces
the realization that “I am nothing.” This reduction in egocentrism, along with a darkpiercing
alertness is said to facilitate an invasion by a loving God that precipitates
the fasting, sleep deprived and praying petitioners into long lasting ecstatic states.
These uses of energy and its attendant characteristics are not physically
specifiable but rather hermeneutic of a force. It is both a potential and a realization,
observed and inferred. It is the “energy stuff” of Freud’s libido, Wilhelm Reich’s
orgone energy, Pavlov’s drive, Rudolph Steiner’s etheric formative force, the
arousal and attention of brain wave and consciousness research, the Ch’I of
Chinese medicine, the Hindu divine energy of Shakti, the Hebraic ruach, the
Cabalist’s Yesod, the Sufi’s Baraka, the Christian Holy Spirit, the Yogic breath
energy, prana, Mesmer’s animal magnetism, Galvani’s life force, Goethe’s
Gestaltung, Madam Blavatsky’s astral light, Georg Groddeck’s it, Henri Bergson’s
elan vitale, Schroedinger’s entropy, Abraham Maslow, Ruth Benedict and
Buckminister Fuller’s synergy, Bertalanffy’s anamorphosis, Colin Wilson’s x factor
and George De la Warr’s biomagnetism. Of course, by nationality, culture and field
of study, there are many more examples, each locally defined by its particular
context and haunting with its promise of universality.
Energy in the context of mathematical physics is intuitional, abstract and
relational. It is not created or destroyed, but rather transformed. Consistent with his
deceptively simple style of physical intuition training of the young, Feynman’s
45
discussion of thermodynamic energy and its conservation in Lectures in Physics
begins with the premise that it is a numerical quantity that does not change when
one or many alterations in the system occurs. His heurism for energy and its
conservation involves the premise that Dennis the Menace has 28 indivisible blocks,
a number which his parents find constant at the end of every day of play. If one day
a count yielded 27, an investigation would reveal that a block could be found
elsewhere, say under the rug. If at the end of the day, the count was 29, the extra
one had to come from somewhere else, perhaps Dennis’s playmate Bruce. If
Dennis locked some of his blocks in the toy box and threw some into a bathtub of
dirty water and (1) A block weighed three ounces; (2) The box alone weighed 16
ounces; and (3) Each block raised the water level of 6 inches by one fourth of an
inch, then this metaphoric energy relation can be expressed:
(blocks seen) + (weight of box)-16 ounces + (height of water)-six inches = constant (28)
3 ounces 1/4 inch
Feynman notes that this representation of an energy relation, computed as a
number of blocks, will always remain the same. If there were no blocks in sight, and
one used this energy conservation relation with blocks as units of energy, we find
no blocks as such in the expression at all.
The abstract and formal idea of energy in physics first arose in
mechanics and was generalized to electrostatics and electrodynamics. If one
idealizes these systems, eliminating real world factors such as friction, temperature
gradients, temperature dependence of the properties of materials, viscosity,
hysteresis and other nonlinear behavior, then the energy conservation law says that
in an isolated and interacting set of systems, the sum of the energies of the several
systems remains constant. If, on the other hand, a system interacts with its
surroundings, not isolated and interacting, then the increase in the energy of the
index system is equal to the work done on the system by its surrounds. Like pre-
Enron bookkeeping of corporate cash flow and balancing ones personal checking
account, energy, like money, does not disappear; it is only changed in expression.
As in the context of currency equivalent value, energy can represent a very general
quantity applicable to a wide array of specific objects and activities. The results of
46
the early studies by Professor Seymore Kety of Harvard and Dr. Harold Himwich of
the Thudicum Laboratory in Galesberg, Illinois, using measures of whole brain
oxygen and glucose utilization as indices of energy generation and utilization by the
brain, surprised many of us. They indicated that energy use by the whole brain was
relatively constant when states of relaxed awakeness, mathematical cognition and
deep sleep were compared. Of course, modern studies have indicated that relative
regional brain energy utilization is state dependent and may vary quite widely.
More spiritual aspects of energies and their transformations were made
clearer during several month visits to Baba Muktananda’s, now Gurumayi
Chidvilasananda’s, Sidha Yoga Ashrams. Baba Muktananda loved and worshipped
his Hindu Guru, Bhagawan Nityananda. Baba had been a restlessly wandering,
guru-hunting, young man. Nityananda said he had “wheels for feet.” After many
years of devoted meditation, chanting and service, sadhanna, all the while being
prohibited from eating mangos, his favorite food, his passive, taciturn, ecstatic guru,
Nityananda, presented voluble, energetic, joyful Baba with the guru’s rather
aromatic and worn sandals. This symbolically acknowledged Baba’s successful
absorption of the guru’s transforming spiritual energy, shaktipat, the power of his
enlightenment.
At Nityananda death, Baba, using world tours, spiritual fellowship meetings,
satsangs (public conversations) and spiritual training sessions called “intensives”,
organized Ashrams in West Coast sites such as Oakland and Venice, and on the
East Coast, in South Fallsburg, New York, buying several old residence hotels in
the Borscht Belt. Baba was introduced to America by one of his first advance men,
Be Here Now Baba Ram Das, Timothy Leary’s co-investigator in the Harvard
Student LSD project when his name was Richard Alpert. EST’s Werner Erhard was
another of Baba’s advance men.
Baba discipled and disciplined a sister and brother who, when 18 and 11
respectively, were sent to live in his Ashram in Ganeshpuri India by their parents.
The girl was known as Malti when she served as a translator for Baba and
Gurumayi Chidvilasananda after receiving the energy of her enlightenment. The
younger brother was given the name of Baba’s guru, Nityananda. When Baba took
47
a guru’s ecstatic death, Samadhi, both Gurumayi and young Nityananda became
co-gurus. Following three years of the usual covert power struggles of succession in
organizations, Gurumayi took over the guru lineage of Siddha Yoga. Her lively
brother’s worldly preoccupations with jazz drumming and confessions of promiscuity
led to his giving up of the orange robe of the denunciate, sanyasi, for the blue robe
of worldliness, exchanging one kind of energy for another.
Brad Gooch who visited Gurumayi’s Ashram in Ganeshpuri, India, wrote in
his recent book, Godtalk, that she looks like a “synthesis of Indira Gandhi and
Bianca Jagger.” In what reads like a Hunter Thompson episode in an unwritten book
called Fear and Loathing Along the Guru Trail, Godtalk’s explication of Siddha Yoga
was dominated by yellow journalistic rumors such as the one about Baba’s use of a
gynecologist’s table with stirrups for non-ejaculatory Tantric practice with some
female followers. This unconfirmed claim remains, as Gooch says, in the realm of
“…he said, she said.” Gooch’s exploration almost ignores the deeper meanings of
Kashmir Shavism, Buddhism and Kundalini Yoga that compose the philosophical
foundations of Siddha Yoga. The importance of knowing, loving and becoming one
with the God within trivializes all but ungenerous or hurtful interpersonal behavior.
Even the tougher version of the Ten Commandments in Leviticus 19 would not
necessarily disagree.
When a Los Angeles Times reporter tried to chide Baba about being driven
about in his “worldly” Mercedes sedan, he explained that a very wealthy Indian
merchant had given it to him and “…I have to put my behind somewhere.” Similarly,
why would Gooch’s account of Baba’s Tantric practice, even if true, ruin the imago
of him in my mind unless I had already surrendered to the pantheon of good and
evil absolutes of Judeo-Christian taboo? My knowledge of these non-materialistic
meanings of apparent materialism began with one of the favorite finds of Baba’s
youthful days of guru hunting: Zipruanna, who, wearing only a loincloth spent all
day, every day, on a stool in the middle of a garbage dump. Remarkable changes
occurred in people who spent time there in his presence. Baba said the identity of
guru was established by the results experienced by those that spent time in his
presence. It could not be defined by the physical features or ritual conduct of the
48
interaction. People become spiritually energized and change in Zipruanna’s smelly,
garbage-filled presence. I keep a picture of him on my desk.
Gooch, in his implicitly and superficially righteous preoccupation with what he
considered disenfranchising human vulnerability, recalls how the medieval church
used the difficult to impossible vow of chastity for political control of their priesthood.
He seemed to have missed Baba’s lessons about the remarkably simple sounding
practices for mobilizing the energy of the God-receptive state. Once in this new
state, the rest of the metaphysical work almost takes care of itself. I, like many
others, adopted Baba’s mantra, Om Namah Shivaya, “I worship the God within me
(and you)” that he was given by his guru. The inner chant of this mantra brings me
to an internal quiet in which things become clearer. Meditation, chanting and service
to the guru was motivated by his promise that my egoistic concerns ranging from
the number of publications on my curriculum vitae, to the size and adroitness of my
penis, would disappear autonomously in the Baba state of bliss. This sounds very
much like the role of the transition to an “active intellect.” in Abraham Abulafia’s 13 th
Century Commentary on the Secrets. Arduous study of the spiritually dense writings
of Sri Aurobindo during the days with Professor Spiegelberg at Stanford gave me a
peak into the simple but difficult to execute idea of “simply” becoming the
transcendently comprehending state of existence-consciousness-bliss.
Whereas Baba would occasionally lapse into terse Sanskrit verse and its
multiplicity of potential meanings, Gurumayi keeps things simple. Sitting silently and
immobile at satsang for hours, she radiates transformational energy, shakti, that
makes ruminations about human affairs seem unimportant. The work is about
getting the self concerned head noise of ones preoccupations sufficiently out of the
way to allow the discovery of the God who has been waiting patiently within. A
fellow ashramite gave me a photograph of my first audience with Gurymayi. It
showed me on my knees in front of her. She appears to be dismissing me with a
baleful, almost disdainful look as my introducer, gesturing broadly, was, unasked,
reciting a list of my professional bona fides. The picture caught her waving me off
with a long, peacock-feathered stick. Obviously unimpressed, she is sending me
back to my all night, every night, tent cleaning labors at the Ashram. Rich Indian
49
businessmen, whose large donations were a major source of support of the
Ashrams, faired little better. They seldom received a personal audience or favorable
seating at Darshan, the evening public time of question and answers with the guru.
In contrast with the relatively easy public availability, mischievous play, provocative
humor and worldly sophistication of Baba, the ambience of Gurumayi is more
private, simple, serious and subtle. It is as powerful, but in another way.
In response to Gurumayi’s ascension to Siddha Yoga’s singular guru, I
imagined hearing Baba saying that God energy was at least androgynous, if the
dimension of sexual identity was relevant at all. Baba taught that divine energy, by
necessity, is expressed through a wide variety of particular personalities and
cultures and should not be confused with the details of its manifestations. This
included the sexual identity of the chosen Vehicle. Guramayi’s central theme, as I
understand it, concerns the simple, quiet and pervasive powers of love and faith.
Some say Baba took the path, marga, of selfless action, karma-marga, whereas
Gurumayi took the bhakti-marga, the road of loving devotion and faith. The third
marga is jnana-marga, my inclination, is the road of intellectual study and
knowledge. Aldous Huxley related the choice among these three categories of yoga
practice, to the physical and personality types of William Sheldon’s 1954 Atlas of
Man. Karma yoga corresponded to the mesomorphic body type and the assertive
boldness, high energy, and interpersonal callousness of the somatotonic
personality. Bhakti Yoga was the characteristic choice of endomorphic body types
with the viscerotonic personality traits of sociability, good will, tolerance and love.
Huxley associated Jnana Yoga with ectomorphic body type and the cerebrotonic
characteristic of shyness, sensitivity and intellectuality.
My summers with Baba at his temporary Ashram in Venice, California and
the permanent American Ashram in South Fallsburg, New York, were spent in daily,
very early morning, chanting of the gurugita after most of the night spent taking
down, cleaning and putting up large tarpaulin meeting tents. I was assigned this
simple, arduously manual, all night work after being interviewed and found out to be
a professor and chairperson of a medical school department. Baba instructed his
assignment committee that many if not all professorial egos would benefit from what
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Andrew Carnegie famously called the dignity of real work. Spicy one dish vegetarian
meals, twice a day meditation and brief stolen naps consumed the rest of the day. I
found myself meditating for longer and longer times, chasing the promised Blue
Pearl that Baba said appeared behind the eyes near the supreme meditative end
point.
Beside care with the titration of meditation-induced interpersonal
disconnection, detachment with love is the desired end point of most Hindu and
Buddhist meditative practice, another set of “side effects” of the energy arising early
in the course of too much meditation is called kriyas, spontaneous episodes of
involuntary behaviors and postures of the body such as unprovoked chanting and
writhing and stereotyped hand positions called mudras. Baba told us one of his
kriyas took the form of spontaneous erections that occurred during his first
experiences with deep meditative states. I recall a woman physician and fellow
ashramite in Los Angeles telling me that her panties often got so soaked during
meditation that she worried about being stuck to her cushion. Beyond these initial
somatic overflows of Divine Energy, shakti, emerges a vision of the Blue Pearl,
bindu, Baba’s “gift from the Goddess Kundalini.” As he entered this stage, he said
that his mind filled with “joyous contentment.” Jewish mysticism of the 1300’s
acknowledged the neighborhood relations of Eros and the Sacred.
More formal and scientific uses of the word, energy, like all objects of thought
embeddable in a mathematical context, are abstract and relational. In his book,
Mathematics-The Music of Reason, Jean Dieudonne′ treats mathematical objects
as objects of thought. Dieudonne′’s book documents the 19 th Century transition
from concrete, visualizable, classical mathematics to abstract, nonvisualizable
relational ideas. This conceptual transition to abstract, relational thought objects that
are no longer representable by pictures or accessible to our senses of mathematics
and physics is yet to reach the concrete DNA-causal religionists of modern
molecular biology. In 20 th Century mathematics, Dieudonne′ observes that “…the
primary role in theory is played by the relations between mathematical objects
concerned rather than the nature of the objects themselves…these relations are
often the same for objects which appear to be very different and therefore they must
51
be expressed in ways which do not take these appearances into account…and can
be specialized at will…” DNA sequences are, as MIT molecular biologist, Eric
Lander observed, nothing more than an elementary “…list of parts…” In fact, since
about 1% of the nucleotides are relevant to functional genes, one might say that the
important members of this list of parts are distributed very thinly among many more
apparently unimportant ones. The next frontier will certainly involve an
understanding of the dynamics of the interactions among elemental parts and in
more abstract laws about molecular biological relations; a focus on the dynamics,
not the structural parts, that regulate and control their expression.
* * *
I made a pilgrimage to spend eighteen months within Rene’ Thom’s
penumbra, living among mathematicians in his “ashram” in Bures sur Y’vette,
France. Thom was one of the founders of the Institute des Hautes D’Etudes, IHES,
Institute for Advanced Scientific Studies, created to stanch the flow of high-level
scientific talent away from France after the Second World War. It is in Bures sur
Y’vette, deep in a green forested valley, 50 or so miles South of Paris, in a building
packed with small, thin walled, big windows-on-the-woods offices. Each office
contained a single hard chair, an old office desk, two walls of blackboards and a box
of white only chalk. The use of colored chalk was felt to be without mathematical
rigor because its use substitutes colors as dimensional descriptors for more
demanding abstract and formal representations. Color was cheating. Meditation in
this ashram was practiced by staring, pacing, scribbling, and humming, mumbling,
belching and farting through the Institute’s thin office walls. The building, though
almost completely occupied, was otherwise silent. The Institute was populated by
such world-class mathematicians and theoretical physicists that once inside that
building, I felt so intimidated that I almost never spoke above a whisper. Listening to
excellent William Thurston’s casual use of a tiled bathroom floor to motivate a
unique partition of a topological space, I was attacked by the awe of an early
morning visit to an almost empty Notre Dame Cathedral in Paris or standing in front
of Michelangelo’s radiant marble statue of Mary and Jesus the Infant in the Vatican.
52
Though the environment was one of tranquil academic scholarship, I lived charged
with anticipated performance anxiety about the seminars on the brain as a
dynamical system I was scheduled to present to these (I feared) ready-to-bedisdainful,
prize-winning, pure mathematicians and theoretical physicists.
My dorm-style sleeping room at IHES was, in winter, painfully cold and
drafty; the narrow iron bed’s thin mattress contained lumps of persistently disturbing
dreams, the small scratched table for work shim-irreparably wobbled. A faded
poster of Van Gogh’s garden was tacked crookedly on the door facing the toilet in
the dank, dimly lit small bathroom. A dwelling for distracted young mathematicians.
A retired but still famous Parisian chef cooked many course, elegant meals every
afternoon. The food was accompanied by so many liters of unlabeled red wine and
peer pressure to be French and socially drink it that it became a choice between
dulled, blunted,. sleepy post-prandial afternoons or living on bread, many cheeses,
apples and Perrier water, alone in my room. I chose the latter.
Thom’s gifts to us theoretically oriented non-mathematicians were
diagrammatic, easy-to-visualize pictures that allow the intuitive capture of counterintuitive
discontinuities in functions. How we might imagine that a smooth and
continuous change in a cause of something can lead to a big, discontinuous change
in the results. His system of topological (shape not size) diagrams was useful when
considering up to four causal variables and one to two dependent variables that
described how things behaved.
For an important real life example, in modern clinical pharmacology, the
smooth dose-response curve consistent with the physician’s intuition that if a little
drug didn’t work, a little more may do so, should become an up and down search for
the dose-region for the desired effect which may involve a lower amount than a
previously ineffective drug dose. The therapeutic effect may occur in the middle of a
narrow dose range with too much or no effect occurring out of this span. In many
physical systems, sudden and global transitions in state, from incoherent light rays
to coherent lasing and from laminar flow of fluids to turbulence, emerge
unexpectedly when causal parameter are moved into what some call the critical
region of the values of control parameters. Outside this region, cause and result
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were behaving linearly and smoothly whereas within this region we observe global
and dramatic changes via a forced discontinuity in what Thom called a catastrophe
and others use related words such as bifurcation or phase transition. The transitions
from painful fatigue to running rage and then to ecstatic transcendence feels like the
gifts from two kinds of Gods, the first, bearing the righteous lawfulness of the Old
Testament, the second bringing the empathic forgiveness of the New Testament
Jesus. Catastrophe and bifurcation theories predict and keep track of these
transitions using mathematically describable changes in global characteristics of the
“motion” using technical descriptors such as eigenvalues, germs and jets.
Thom taught me my first catastrophe, called the cusp, in words during our
late afternoon walks along a shadowed green wooded path on the grounds of the
Institute des Hautes Etudes, outside of Paris. My homework consisted of trying to
visualize his verbal descriptions. It was not until weeks later that he drew the
geometric object being discussed on the blackboard. With eyes twinkling and in his
provocatively playful style, he said,
“Imagine an empty rectangular box with the front edge of its roof buckled
into an `S’ and the back edge, an unfolded, left-to-right gradually rising simple
smooth curve. If one moves the causal force from low to high, from left to right along
the back of the box, the changing effect (represented by height) would be smooth;
moving from left to right in the front encounters a sudden drop off at the S shaped
buckling, a discontinuity in roof height indicating a discontinuity in effect. The energy
equivalent height of the roof graphically indicates the amount of result. The roof is
the manifold upon which the result of causal change is portrayed. The two
dimensional floor of the box represents a graph of the two causal parameters, the
increasing amount of normal factor going left to right along the `x’ dimension, the
increasing amount of splitting factor (taking one from the back to the front to the
region of the buckling) going back to front along the `y’ dimension.”
He gave me some examples of systems that showed cataclysmic changes in
effect from smooth changes of normal and splitting factors. About the onset of a
war: “At the back of the top surface of the box, the manifold, the normal factor
increasing from left to right is the amount of the perceived threat. The splitting factor
54
decreasing from front to back is the cost (and ability to pay) for war. Without the
financial capacity to make war, threat goes from left to right smoothly at the back of
the box as tension gradually increases without the onset of armed conflict. When
effective fighting capacity is cheap and/or already well funded, the country well
armed, the increases in threat go from left to right at the front edge of the box and
encounter the cliff of catastrophe and war is declared. Cost of, or ability to wage war
varies from the front to back, and serves as the splitting factor. Considering prison
riots, social tension is the normal factor and alienation (degree of identification with
prison authority) is the splitting factor.” Using factial expressions of dogs sketched
by the Konrad Lorenz, Christopher Zeeman then of Warwick Mathematics Institute
in England, considered countenances reflecting increasing rage as the normal
factor, the amount of fear was the splitting factor. Increasing rage at high fear
increased smoothly at the back of the box; at low fear, increasing rage falls off the
cliff to an animal attack at the front of the box.” He paced as he talked, occasionally
looking up to see if I was following him. He continued,
“A light above the box casts a shadow from the roof to the floor, outlining the
gradually widening fold created by the transition from the smoothly rising back of the
roof to its `S-shaped’ front. This triangle on the x-y causal floor is the region in which
the discontinuity in the result surface roof results and is called the bifurcation set. An
increasing amount of the causal `normal factor’ is represented from left to right
along the `x’ dimension, the results of which change smoothly at the back of the roof
but encounter a discontinuous jump up or fall down crossing the inaccessible
crevice in the `S’ fold at the front of the roof. Again, the triangular shadow on the
floor made by the fold indicates the parameter region in which discontinuous
changes in the result surface occur. The reason the parameter that determines the
front to back location of the left to right movement of the `normal factor’ is called the
`splitting factor’ becomes obvious. Its value determines whether the results induced
by increasing amounts of `normal factor’ will be smoothly changing or generate a
discontinuous jump. The entire visualizable object is called a cusp catastrophe and
it along with higher dimensional parameter region-inspired shapes such as the
55
swallowtail and butterfly buy back the smooth DE deterministic intuition lost with
discontinuous changes in results.”
He grinned mischievously as he asked, “Can you see it?”
Thom’s catastrophes serve as accessible and powerful theoretical settings
for the use of energy as a generalizable, one dimensional, dependent, resulting
effect, influenced by one or several, sometimes conflicting, independent, causal,
variables. For more examples: the weight of a ship (smaller to greater, left to right,
along the x, normal dimension) and the position of center of gravity (smaller to
greater, front to back, along the y splitting dimension) are causal with a jump in roofheight
energy from stability to capsizing, a discontinuity emerging from initially
smooth changes in stability. As above, gradually increasing tension (the left to right
normal factor) and alienation (the back to front (splitting factor) in inmates generate
a sudden increment in energy, from subtlety increasing tension in relative quiet to
the sudden outbreak in a riot in the prison population. Embryological notochord
somitogenesis, (that which become the vertebrate of the spinal column) has a
smooth (left to right) causal influence that Chris Zeeman named a normal factor. It
is the smooth growth of the material wave of mesodermal (to become muscle,
connective tissue and bone) tissue. Zeeman called the front to back dimensional
gradient of influence, the secondary wave of adhesiveness, the splitting factor. The
value of this secondary wave co-determined a critical-valued interaction between
these causal parameters leading to a discontinuous change in the “energy”
equivalent continuity of developmental growth and vertebral column segmentation.
A little more technically: Thom’s basic mathematical contributions were in
differential topology and analysis with particular emphasis on what is called
structural stability of surfaces representing and supporting actions called manifolds.
For example, in a graph of a function, say F(x), such that a change in cause x
determines what happens to the result y= F(x), the stability question involves what
happens when one perturbs F(x) with a littleδ, i.e. δ + F(x). Do the topological
properties of the surface representing the potential range of actions of the system
(such as nearness of an originally close point set, continuity and connectedness of
the surface, its dimensionality, its compactness as a generalization of finiteness)
56
remain the same after perturbation? Note that the inter-data point metric distances
are not considered. If they do, the two dynamical objects being compared are
topologically equivalent. The test of this equivalence requires the mapping one set
onto the other with, at most, smooth distortions of either or both surfaces.
In the context of catastrophe-related bifurcation theory, if a δ converts a
steady valued fixed point to an oscillating cycle on a manifold of potential actions,
also called a state space, then the fixed point system was not structurally stable. In
phase space, this is seen as a change-in-causal-parameter induced transformation
of a dot to a circle. If the one frequency circle is perturbed to a manifold of the
system’s actions consisting of two independent frequencies, the circle takes the
topological form of the crust of a doughnut, one frequency graphed spiral winding
around the doughnut, the other winding along the doughnut around its orifice, the
circle is not structurally stable. If δ distorts the frequency-amplitude relations on a
surface such that the manifold of possible actions is distorted from a doughnut to a
tea cup, both topological manifolds being one holed surfaces and therefore
topologically equivalent, the system is structurally stable. Perturbed systems that
maintain the sequence of points in time in sequential order (though the distances
between the points may be different), are generally structurally stable.
The seductive possibility, one which Thom realized so successfully, was
that in the language of distance-independent differential topological forms, there
would exist a small, finite set of shapes categorically describing the causes and
result parameter spaces from which, even without specific quantities, universal
qualitative (including discontinuous) behavior could be described and sometimes
predicted. A formal yet general categorical system within which a small set of
universal discontinuous changes in global qualities could be rationalized seemed
seductively applicable to the enlightenment transitions, spiritual transformations,
appearing suddenly after months and years of disciplined spiritual practice. The
Platonic view is that the universal forms of discontinuous change existed before
they could be about anything specific, before the universe was born.
In this era of nonlinear dynamics and dynamical system, common dynamical
scenarios give accounts of smooth changes in causes leading to discontinuous
57
changes in results. The Nobel Prize winning solid-state physicist, Phillip Anderson,
in a short but memorable piece in Science in the 1970’s said it tersely, “More is
different.” This general, qualitative mathematical theory of discontinuous change
models nicely the sudden delivery of the first and second second winds from
gradually and continuously increasing running distances as well as the abrupt
transmission of the guru’s “energy”, shaktipat, from smoothly increasing amounts of
chanting, meditation, guru service and Baba love. Gradually changing forces
leading to sudden changes in an energy-equivalent result are found in most
rigorous form in Rene′ Thom’s singularity-bifurcation-catastrophe theory applied to
rational mechanics and geometric optics. Here the existence of already solvable
computational formalisms makes this more qualitative approach superfluous. On the
other hand, the power of this both basic and applied mathematical orientation and
method lies in its approach to the qualitative understanding of variously induced
global and sudden changes in an energy-equivalent observable in biological,
psychological, spiritual and social systems, fields of study in which little abstract and
formal lawfulness presently exists. Oxford’s Chris Zeeman’s more accessible
applications of Thom’s deeper, more generally ramifying, almost mystical (due to
their apparent wide generality) results, include approaches to real world problems
such those above as well as the sudden change in excitable membrane potential
accompanying the generation of the heart beat and neuronal discharge;
mechanisms of opinion change, stock market crashes and, as noted above, the
social science of riots. Whereas Thom’s On Structural Stability and Morphogenesis
can be said to be scriptural, Zeeman’s Selected Papers, 1972-1977 constitute the
Book of Common Prayer of this church.
To review and place catastrophe and bifurcation theories in the context of the
differential equations of mathematical physics and biology, causal determinism
implied by differential equations conventionally requires continuity and smoothness
in behavior to be credible. Our intuitions as well as the formal conditions for the
generic differential equations of mathematics and physics imply that smoothly
increasing amounts of cause lead to smoothly increasing results and yield at least
local predictability: a little more leads to a little more, a little less leads to a little less.
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This smoothness-dependent intuition of determinism breaks down in nonlinear
equations as well as in a wide variety of the machines of experimental physics, from
the sudden coherent lasing of previously incoherent light to the vortices and
turbulence in suitably bounded rotating or flowing fluid. It took me a while for these
topological still shots and movies of the head to become real. Nevertheless, the
enrichment of intuition was well worth it. Of course one could smoothly increase the
normal factor weight of a ship until it gradually sank, but if one moved the center of
gravity splitting factor to an eccentric position in the ship in the parameter region of
the bifurcation set, a sudden global capsize before weight-induced gradual sinking
made sense. I could see it. Indeed, increasing normal factor tension in a prison
population that was identified, not alienated, from the officials and mores of the
penal institution, would increase social symptomotology gradually. However,
increasing the splitting factor of social and institutional alienation results in the
cataclysmic change of a riot with increasing tension. I could see it.
Do we need to know the causal equations to anticipate instability and
discontinuity in our lives? Zeeman making Thom’s thoughts accessible to us plain
mortals said no. He suggested that we could use several diagnostic
phenomenological signs to make a good guess about whether we are near or within
the bifurcation set. Depending upon the route that the causal variables take through
the shadow of the bifurcation set, we may see very large fluctuations in our
observable. The Dow or S&P stock indices in the neighborhood of a sudden large
change is often presaged, sometimes for weeks, by a marked increase in volatility,
fluctuations between extreme values. Theorists call the statistical properties of a
time series of values behaving this way anomalous variance. For several months, I
did psychotherapy with a genuinely spiritual Catholic priest who only some Sundays
served the Eucharist, the corporal presence of our Lord at Communion, wearing no
trousers or underpants beneath his robes. A sudden change in a stock index in
response to the “shock” of a terrorist attack takes much longer to settle down if a
cataclysmically bigger change is in the neighborhood. This extension of the
system’s usual relaxation time is sometimes called critical slowing. In the bifurcation
59
regime of a schizophrenic break down, critical slowing can be both global and literal
as the patient freezes in catatonic postures.
In the neighborhood of the bifurcation set, big jumps in the stock index, up or
down, are possible under almost the same surrounding conditions. This stock
analyst-humbling phenomenon is called bimodality. Jimmy Swaggert’s Saturdays
were often spent watching the show at naked dance parlors and buying videos at
the pornography shops of Metairie Highway near Schwegmann’s Grocery outside
New Orleans. Sundays found him on national television engaged with infectiously
real, transcendent experiences in the public arena of the pulpit. The ecstatic
congregation was deeply moved by his eloquent and tearful sermons about sin and
salvation. Counter to most suspicions, this is less conscious fakery than the
genuinely felt alternating states intrinsic to the bimodality in neighborhoods of
spiritually unstable, born again transitions.
Similarly, beginning with nearly the same initial values near the boundary of
the bifurcation set, very similar motions lead to dramatically different results. This
counter-intuitive behavior has been called divergence. At UCLA’s Neuropsychiatric
Institute, I interviewed a pair of lively teenage, genetically identical male twins raised
by a loving family in Los Angeles’s Valley. One was president of his high school
class, a Sunday school nursery school volunteer and a Saturday soup server to the
poor. The other twin sold pot and cocaine to support his habit. Deep and potentially
dark mysteries live in these spiritual bifurcation sets. They leave us pondering child
sexual abuse by deeply religious clergy and the massacre by mass suicide of a
New Christian congregation by James Jones. We wonder why it is that
fundamentalists (Jewish, Christian and Muslim) have the most ecstatic and direct
validating experiences of God and do the most shooting and bombing of other
people. In Burt Lancaster’s portrayal of bifurcation set dweller, Elmer Gantry,
charismatic believer and exploitative psychopath, were simultaneous and both
credibly real.
Another feature of the occupancy of this bifurcation region in control space is
that the values producing a sudden jump that occur passing through going one way
along the “normal” dimension usually jump back much further along when moving
60
the other way. Theorists call this characteristic sign of bifurcation land, hysteresis. It
is generally known that sudden healing changes of the first born again experience
can arrive magically fast whereas a run at it a second time, another born again state
after the loss of the first one, comes, if at all, with much more effort and difficulty.
Members of Alcoholic’s Anonymous know that getting on the AA wagon the first
time may be quick, joyful and easy. Getting back on this wagon after a fall is much
more painfully slow and demanding, analogous to the Carmalite monk; St. John’s
lost faith engendered suffering of the Dark Night of the Soul.
Viewing the instabilities and extremes near the boundary of a bifurcation
brings inquiries and advice about why a rational compromise, some form of
disciplined moderation, would not be more desirable. It turns out that in this
parameter regime, the in-between state is intrinsically inaccessible. The pocket in
the S shaped fold of the upper manifold cannot be attained, at least for very long, by
varying the values of the two parameters. However, if one increases the number of
controls, it might be possible to stabilize a small island in a parametric sea of
instabilities. In an application of this strategy, Smith College and Harvard Professors
James Callahan and Jerome Sashin used a geometric representation of the difficult
to stabilize region of normal weight on a double cusp manifold representing the
behaviors of patients with eating disorders with both anorexia nervosa and bulimia.
They varied five controls to stabilize a very small result area representing normal
eating by varying the control values for ability to verbalize feelings, to imagine
solutions, to defend against anxiety with unconscious forgetting called repression, to
make contact with realistic rationality and to modulate feelings with say exercise,
meditative practice or psychopharmaceuticals.
My experiences with the so-called borderline personality, with the tendency
toward sudden and global personality change, from Sunday school teacher to Harlot
in the space of a breath, has been both sexually exciting and personally ruinous for
me in my life. I could feel the instabilities in these dwellers of the bifurcation pockets
and my heart raced at the promise of mutually unconsidered impulses, the blurring
of orificial identities, the experiments with sexual roles and modes and the
incipiency of collapse into regressive mud play. Most of all, I anticipated that their
61
screaming orgasms, potentiated by a natural inclination to bifurcate, would be so
messianic as to carry me along to a transcendentally erotic new place.
Unfortunately, paranoid rages, bursts of promiscuity and hopeless inconsistency of
goals and efforts dominated the remainder of our living days.
Further Readings for TRANSMOGRIFICATIONS OF ENERGIES
Religions in Four Dimensions; Existential, Aesthetic, Historical, Comparative, Walter
Kaufman, Reader’s Digest Press, 1976
Religious and Spiritual Groups in Modern America, Robert S. Ellwood, Prentice-
Hall, Englewood Cliffs, 1973.
The Evangelicals, What They Believe, Who They Are, Where They are Changing,
David F. Wells and John D. Woodbridge, Abington Press, Nashville, 1975
A Nation of Believers, Martin Marty, Univ. Chicago Press, Chicago, 1976
Conversion: Christian and Non-Christian, Alfred C. Underwood, George Allen,
Unwin Ltd., London, 1925
Eros and the Sacred, Paul Avis, SPCK, London, 1989
Mukteshwari, The Way of Muktananda, SYDA Foundation, Ganeshpuri, India, 1972
Godtalk, Travels in Spiritual America, Brad Gooch, Knopf, N.Y. 2002
The Beat of a Different Drum; The Life and Science of Richard Feynman, Jadish
Mehra, Clarendon Press, Oxford, 1994
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The Shape of Space, Jeffrey Weeks, Dekker, NY, 1985
The Topological Picture Book, George K. Francis, Springer-Verlag, NY 1988
Mathematical Models of Morphogenesis, Rene Thom, Wiley, NY 1983
Catastrophe Theory, Selected Papers, 1972-1977, Christopher Zeeman, Addison-
Wesley Reading, MA 1977
63
CHAPTER 4:
SENSUAL IN-BETWEEN ENTROPIES
Since the early teens, I’ve been beguiled by girls and women that have what
might be regarded as exquisite sensibility, perhaps more precisely, exquisite self
sensibility. These inhabitants of the near transformational neighborhoods of
bifurcation sets, are grandly responsive receivers of emotionally significant
information arising from their insides and the world. They are the canaries in the
deep mines of human experience. Not the usual one lively-eye, one sober-eye,
binocular difference of most of us, both their eyes sparkle, their feeling antennae
await a happening and each is regarded as new. I spot these brains in a crowd
within minutes and am compulsively drawn to know them better, to become part of
them, to vicariously experience and serve them. They seem to have little inhibitory
control of even weak sensory information on its way to their strong, global feelings.
Near ecstasy and excruciating pain await. They feel their anticipations with their
body, down to their painted toes. Their receptivity brings me lower abdominal
warmth in remembrance.
At sixteen in my Dad-purchased second hand Ford convertible, I was parked
with my new girl friend on Sarasota’s Lido Beach, hearing and seeing dark shadows
of the Gulf of Mexico’s waves hit white sand against the night sky. I took her flat
party shoes off to message her feet. When I kissed her left foot and sucked gently
on her toes, she gasped and became faint. She told me that a strong electric shock
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had run up her back. The passionate licking and sucking of her musky, moist, pink
labial lips brought what she said were explosions of pink and blue lights. She had
several ecstatic multicolored crises in a row, sometimes without pause. She begged
me to stop. I was as pleased as a sexually inexperienced young man in love could
have possibly been.
Bowled over by what seemed to be the uniquely sensual properties of her
brain, I began to wonder if her sensitivity was more general when she asked me to
keep the windows open or top down, even in the cool of a Florida January, because
the exhaust smell in my car was suffocating, though I couldn’t smell it. The car had
been checked and registered negative for abnormal fumes and leaks by Anderson
Ford. She asked me never to wear any kind of after-shave lotion because it choked
her. Jazz music on the car radio had to be played quietly. On-coming headlights
gave her headaches. Her mother, sometimes desperate, called me for help during
her daughter’s episodes of premenstrual emotionality and early menstrual
discomfort. During these times, we would drive together for hours as she explained
the many different colors of lower abdominal pain and how this particular kind
yawned darkly before it cramped. It was more purple then any of the others. I tried
to explain what I intuited but didn’t understand to her mother about the her gift of
unfiltered information coming through her nerve endings, her ever readiness for
surprise and her brain’s unwillingness or inability dampen or ignore what it didn’t
like. She saw things in art, heard things in music that I only saw, and heard after her
telling. She had tearful smiles listening to Debussy’s Afternoon of a Fawn. The
flatted fifths of Charley Parker and the laconic riffs of Miles Davis made her anxious.
Since then and for all these many years, the same sensually susceptible
brains showed up in my life carrying a variety of woman’s names and I never lost
my fascination for them. I learned that their heightened awareness extended to the
spiritual realm with unusually strong metaphysical inclinations and readiness for
transcendent experience. They seemed to live closer to the direct experience of
God. Attending Assembly of God and other Pentecostal midweek service, I found
that praying in tongues and dying in the Lord came as easily and dramatically to
them as their orgasmic experiences. At the same time, distant bad news could
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suddenly become immediate and loud in a litany of threatening thoughts that
hooked and persisted through sleepless nights. They taught me to see genuinely
the delicate beauty of flowers and to know in my stomach that some forms of
sadness felt hollow like homesickness. In medical school I found that that many of
them were the clinic patients, women and men, with unusual sensitivity to chemical
odors, think Gulf War Syndrome, and fibromyalgia, which I heard as unusually
sensitive awareness of normal sensory information about posture and position
coming in from the bones and muscles of the body but experienced as pain. This
background of odorific and somatic information is usually repressed from
consciousness by the rest of us. Their medical histories contained detailed accounts
about how each of their organs was feeling at the time, sensations that the
textbooks say we are incapable of consciously knowing. Internists and psychiatrists
often dismissed their accounts as signs of somatoform disorder, psychological
conflicts expressed in the language of body feelings.
In the psychophysiological laboratory, I learned these brains tended not to
habituate. Each of a series of noises continued to elicit startle responses that could
be picked up in brain wave recordings or in the running record of a
psychophysiological, lie detector, machine. In psychoanalytic training, I learned that
these brains remembered their dreams more richly than the rest of us and that
treatment with over twice a week analytic sessions was potentially dangerous. The
psychoanalytical situation-engendered fantasies and feelings could get too strong
and exaggerated, too real.
Professor Iris Bell of University of Arizona’s Alternative Medicine Research
Program has, studying these brains, found slower reaction times, defects in divided
attention psychological tasks, longer latencies to the first dream, and unusual
patterns of odor reception called cacosmia or dysosmia. Using brain wave and
cardiac interbeat interval data as markers, Bell reports the increase in the amount of
alpha awake brain waves and decreases in cardiac interbeat interval variation
associated with increasing sensitivity, rather than habituation, with repeated
exposure to a variety of smells over time.
In spite of these brains usually requiring what is known as high maintenance
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in relationships, I continue to be erotically spellbound, in love with them in all their
forms. Questions about how to think about these exquisitely sensitive women, Bell’s
Syndrome exists but is rarer in men, continue to drive aspects of my scientific
research. It has been variegated quest, which began with trying to find a general
conceptual framework that would help my understanding of this unique capacity to
be aware and process large amounts of internal and external information that
escape the awareness of most of us. As one might guess, this search led to
fundamental ideas about information and its inverse, the entropy indicating the
amount of information transport capacity, with respect to their characterization,
quantification and measurement.
To get to the end from close to the beginning, we recall that it was Claude
Shannon and his followers who both mathematically proved and experimentally
verified that a receiver must have more entropy, less already fixed knowledge and
more wondering, than the sending source, in order for the message to be sensitively
and reliably received and encoded. Sensibility seems to have something to do with
the readiness for information transmission afforded by the brain’s high entropy,
minimal fixed information states, in its resting dynamics. Their remarkable
receptivity derives from a baseline brain state like the formless emptiness of the
bodhisattva’s “…no form, no sound, no feelings, no perceptions, no
consciousness…” of transcendent Tibetan Buddhism as described in the Heart
Sutra of The Dalai Lama.
In Chinese Medicine, xu, meaning emptiness, contrasts with shi, the word for
fullness, both of these complementary opposites having multiple specific meanings.
Most metaphysically relevant is the characterization of xu as the emptiness of the
deepest reality of being and the highest state of human spirituality. Like that aspect
of Lao-Tsu’s ineffable Dao, The Way that is empty, xu indicates a mind devoid of
desire, being lucid and serene. In the context of dynamical form, xu shares the
structureless, non-imagery of maximal entropy systems and shi the lower dynamical
entropy of fixations on form, desires and beliefs. Shigehisa Kuriyama’s The
Expressiveness of the Body, elucidating historical and conceptual divergences of
Greek and Chinese Medicine, notes that xu was the supreme end of self-cultivation
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and the secret to vigor and longevity. “…to achieve fullness of life one had to abide
in empty nothingness, xuwu.” In Lao-Tsu’s Tao-Te-Ching, “…the Way is gained by
daily loss, loss upon loss until…by letting go, it all gets done…”
William James, in The Principles of Psychology, tried to capture the subjective
dynamics of the brain as an on-going preconscious stream of statistical wave
processes. He envisioned autonomously increasing and decreasing coherence
emerging spontaneously and from sensorial evoked thoughts via the confluence
and disaggregation of statistical wave processes, “…wave crests and hollows…”
that achieved temporary statistical stability by “…feelings of relation, consubstantial
with our feelings or thoughts of the terms between which they (only temporarily)
obtain.” In the more receptive, higher entropy brain systems, fleeting forms change
without continuity, jumping from one to another with “magical rapidity,” but being not
already engaged, are available for use for self-organized structure evoked by new
information. Without ordered, low entropy, preconceived ideational defects in the
resting random brain field, the full attentional statistical machine is available to
sensitively respond in self-organized, quasi-stable states of cognitive, conative and
affective integration. They then disappear; this brain relaxes quickly, ready for new
experience. This contrasts with those brains that are dominated by islands of order
composed of personality fixations and rigid belief systems, low entropy defects,
which interfere with sensorially responsive self-organization.
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As in most systems of authoritarian premises, precise definitions and what
appears to be strict logical continuity, as in discussions of Torah among Orthodox
Jews and Canon Law by Catholic bishops, classical equilibrium thermodynamic
ideas that are borrowed for use out of the context of their origins, risk the calumny
of their physicist practitioners. We have probably already earned more than a little
distain from those quarters with our use of none-minimal or none-maximal but inbetween
entropies. This phrase cannot be found in the literature of physics or, as
such, in the writings of communication and information theory. In the modern theory
of nonlinear motion called dynamical systems, in-between entropies can be
generated by chaotic systems that are non-uniform in their rates of separation of
near by points and convergence of far-away points in dynamics that have been
previously described as nonuniformly hyperbolic.
The energies and their transformations that fuel and support karmic escape
from the personality fixations of samsara and accession to unmanifest Divine Life
can occur without the loss of the richness and multiplicity of apparent reality. Big
internal changes without external sign can occur in the arrangements of the
ineffable and mysterious formless silence within which we have associated with
states of high, but not maximal, in-between entropy. For examples, the Indian Saint,
Sri Aurobindo, in the early 20 th Century, the Catholic metaphysical anthropologist,
Teilhard de Chardin and currently American pandits (spiritual seekers with
intellectual and academic inclinations) such as Ken Wilber, among many others
over the millennia, direct us toward the goal of Nirvanically changeless emptiness
without the properties of space or time. At the same time, we maintain an astute
and effective yet distantiated appreciation for existential realities. The non-dual
enlightenment of Integral Being or Yoga involves realizing emptiness through the
world of form. There is a way of thinking about and even computing that “nothing
within” and its changes.
As John R. Pierce suggested in the 1981 revision of his book that made the
theorems of the father of communication theory, Claude Shannon, so accessible,
“…if we want to understand information-related entropies, it is perhaps best to clear
our minds of any (physical) ideas associated with the entropy of physics.”
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Nonetheless, historical comments about what the classical thermodynamic term,
entropy, is and is not about are in order.
We recall that Richard Feynmann, in his well-known 1962 class notes,
Lectures on Physics, said that the subject of thermodynamics is the study of
relationships among the heat, energetic and organizational properties of materials,
without knowing their internal structure. Historically, the relational formalisms of
equilibrium thermodynamics emerged before our knowledge of the internal structure
of matter. For examples, the pressure in an insulated container of gas is due to
molecular bombardment of the container walls, which increases with heat or
compression of its volume. Compression of its volume increases its temperature
and expansion of its volume leads to cooling. Note that these relationships hold
without specifying the constituents and the specifics of a particular gas or solid.
In his lectures, Feynman’s intuitively accessible examples of reversible
thermodynamic properties are reminiscent of his on camera performance at the
Senatorial hearings about the Challenger disaster. Recall that he dropped an O-ring
in a glass of iced water demonstrating cold-induced rigidification of the rubber ring,
which he postulated to be the cause of the fuel leak and resulting explosion. In his
Lectures, he said that if one holds a rubber band between ones lips as a crude
thermometer, stretching a rubber band heats up the lips and relaxing it cools them.
Working the same system in reverse, and equilibrium thermodynamic systems are
classically reversible, we find that heating a rubber band makes it contract. These
changes involve complicated alterations in the internal arrangements of the
polymeric strands of rubber, their structural properties, the details of which, for the
purpose of global thermodynamic characterization, need not be known. The
relationships between physical state, energy and temperature in this material were
predictable from thermodynamic laws even without specific knowledge of the
complex internal structure and physical dynamics of rubber.
Thermodynamic theory, which makes deep conceptual connections between
quantitatively measurable primitives such as heat, hotness and work and the
invisible in the form of derived ideas such as energy and entropy, yielded an
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enormously rich and logically consistent intellectual framework from within which to
characterize macroscopic behavior composed of unknown molecular mechanisms.
Ideas about entropy grew out of William Thomson's (a.k.a Lord Kelvin)
thermodynamic laws about energy conservation and its allowable transformations.
Later Clausius decomposed the energy into that which was available for mechanical
work, called work-content, and that which was not, called transformation content.
He referred to the transformation content, a reflection of what changes in the
internal order properties of the system that occurred as a concomitant of changes in
energy and heat, as the entropy.
Rudolph Clausius added the word entropy as a thermodynamic property to
the conceptual armamentarium of theoretical physics in about 1865. This followed
the earlier work of the French engineer, Nicolas Leonard Sadi Carnot, who was
trying to develop a theoretical framework within which efficiencies in heatgenerating
engines might be understood. It implicated positive, > 0, changes, d, in
entropy, S, with changes in time, t, i.e. dS
dt
> 0, entropy is increasing in time, as a
concomitant of the inevitable mechanical inefficiencies in an energy driven system.
The resulting losses in the form of wasted energy show up as increases in
molecular motion, which could be estimated from the increases in heat. Wasted
energy dissipated as heat increases the amount of random motion and volume
occupied by the surrounding molecules in physical processes involving heat,
pressure, vaporization, condensation and work; all elements of that era’s dominant
physical metaphor, the steam engine.
The highly developed, multifaceted, often quite abstract formal
characteristics of the inferred property, entropy, prevent glib definitions and
generalizations. In the context of Kelvin-Clausius theory, the entropy of a closed
system will remain the same if it is isolated from any matter or energy exchanges
with the environment. If heating a system such that the change, d, in heat, Q, is
positive, i.e. dQ > 0, it experiences a rearrangement in its microstructural motions,
but the temperature is left unchanged. The (inferred) entropy, S, increases (i.e., dS
> 0) as the ratio of change in added heat, dQ, over the unchanging, absolute
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temperature, T. Thus, one definition of entropy change is dS = dQ/T. In classical
contexts, dS is expressed in units of heat called Joules per degree of absolute
temperature in units Kelvin, the temperature in Centigrade plus 273.16 o . The bestknown
physical image involves the heat-energy transfer to and from heat baths
called reservoirs as intermediate actions of the work of the heat driven engine
executing what has come to be known as the Carnot Cycle. The same formulation
emerges in this more concrete context: the heat, Q, transfer, dQ, at a particular
absolute temperature, T, dQ/T, has been used to define an entropy change, dS =
dQ/T related to some not-need-to-know-about specific alteration(s) in a system’s
internal physical properties.
If one allows some loose thinking about heat-induced increases in the
statistical randomness of molecular motion in the above reservoir that is associated
with the loss of useable energy, the positive entropy change, dS > 0, is vaguely
relatable to the kinds of information entropies to be discussed below. If a gas
trapped in an insulated, physically isolated, closed cylinder is allowed to expand
infinitely slowly, reversibly, called adiabatically, pushing up the piston that closed off
its end, the gas will become cooler, energy having been expended doing the work of
lifting the piston. Defined as an isolated system (of course no where in the real, nonlaboratory,
world can this condition of absent exchanges of energy or matter with
the environment be found), it is a reversible process, because returning the energy
of the work by, again, infinitely slowly pushing down on the piston and compressing
the gas to its original volume, returns it to its former temperature-defined energy
state. In this historically prominent thought-toy of physics, there has been a
reversible change in energy but no changes in the entropy, dS = 0. The gas’s heat,
temperature (and energy and volume) can be completely restored in this
metaphysically mythic classical thermodynmical tale of an entropy-conserving,
reversible process.
While fixed entropy and independence of the specific path is the case for the
above noted abstract reversible cycle, in the real, irreversible orbits of most physical
and all biological systems, entropy increases, dS > 0. Walter Nernst’s 1907 heat
theorem yields a zero point from which to determine a difference measure in the
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postulated, real physical world of ever-increasing entropy. He showed that at an
absolute temperature of zero, entropy is zero. We can illustrate an approach to this
singular state by placing a heated metal rod in ice water which would result in a
decrease in the entropy of the rod’s molecular motions by dQ/ T 1 < 0, the cooling
reducing the complexity of molecular motion in the metal bar and an increase in the
entropy of the water by dQ/T 2 > 0 indicating an increase in the amount and
complexity of the surrounding water’s molecular motions. Of course the heat moves
from metal rod to the water as T 1 →T 2 making dQ > 0 positive and the entropy
change, dS = dQ/ T 2 - dQ/ T 1 , also positive. In another simple example, producing
friction by rubbing a surface generates heat, dQ > 0, at a temperature T. This
induces a positive change in entropy, dQ/ T > 0, in the form of increasing amount
and complexity of the patterns of molecular motion in the air surrounding the rubbed
surface.
Using another related and well-known thermodynamic thought toy, the
original isolated, insulated body of gas in the cylinder is partitioned by a membrane
into two chambers, one containing all the gas with its temperature, pressure and
ability to do mechanical work and the other a vacuum without these properties. This
equilibrium state is changed into another equilibrium state by suddenly removing the
membrane, filling both chambers with gas and, while increasing its entropy
irreversibly, dS > 0, removes at least some of the gas’s ability to do piston raising
work. In the context of classical thermodynamics, it is in this way that irreversibility
can be defined by its associated increase in entropy. Though there has been no
change in total energy in this insulated closed system, an increase in entropy
means a decrease of the energy available for work. The increased disorder in the
gas is associated with the loss of ability to convert heat, thermal energy, into
mechanical energy. Historically important and still available elementary texts by
Enrico Fermi (1936), Mark Zemansky (1957) and Herbert Callen (1985), among
many others, explicate clearly the formal, but far from biologically relevant, classical
theory of the physical entropy of closed equilibrium thermodynamic systems.
Growing in part out of the formal thermodynamics of physics, statistical
mechanics offers yet another set of intuitions about the not-necessarily-known
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molecular details associated with changes in entropy. These ideas are closer to
applicability in problems of making measures on the behavior of biological systems.
Very generally, in the statistical mechanical context, an increase in entropy means a
decrease in the order, which can be a quantitative observable reflecting a decrease
in predictability and/or knowledge about the system. For example, we can locate the
molecules of the gas more accurately when they are all on one side of the
membrane-partitioned cylinder compared with the situation when the membrane is
suddenly removed. This accompanying increase in ambiguity and decrease in
knowledge in locating a set of gas particles reflects a statistical mechanical view of
increases in entropy. Can anything general be said about the bounds on an
increase in entropy? The statistical developments of the Yale mathematical
physicist, Josiah Willard Gibbs (about 1875), consonant with the logical arguments
of the Greek mathematician, Constantin Caratheodory (about 1910), conclude that
the entropy increase goes to the maximum allowed by the constraints imposed by
or upon the system. A change in likelihood as a probability is a characteristic way to
quantify the entropy change, reflecting an alteration in knowledge or its reciprocal
complement, uncertainty. The system’s entropic uncertainty said more colloquially,
and relevant to the Bell Syndrome’s women of my life, is its capacity for surprise.
A statistical mechanical approach to the total entropy of a bounded set of
molecules in motion involves summing this property across all the participating
molecules. We let N be the number of particles involved. As a problem in Newtonian
mechanics, each of the N particles is represented in 6N dimensional phase space.
That means that each point represents one of the N molecules in the three
dimensions of location space plus three dimensions of motion space as its velocity,
more specifically, the product of mass times velocity called momentum. This adds
up to 6 dimensions of measurement. This so called phase space reconstruction of
the molecules of a gas as individual particles are a daunting task, though fast
computers and new algorithms are making computations from first principles more
generally attainable. Those based on the first principles of short-range repulsion
and long-range weak attraction among particles and the bumper-car collision
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dynamics between them can now be implemented if the system of particles being
simulated is sufficiently small and the computer simulation is for very short times.
To transform the entropy into something more statistical and global, we
return to the theoretical work of Ludwig Boltzmann whose formalism was used
previously to quantitate pathological developmental simplification. He assumed that
given a set of constraints, say the closed volume, V, of a box, B, of a fixed size, V
(B), the orbit of each particle would eventually explore all the space in the box that
was available to it. Boltzmann’s entropy became a constraint dependent, n-
dimensional volume measure, with the assumption that the entropy, S, equals the
logarithm of this volume measure, S = ln V (B). To calculate a value for the entropy,
compute the volume of the molecular motion as determined by the invariant
constraints of the system, such as the volume, temperature, pressure and/or its total
number of molecules. We may partition, discretize, the volume up to some limit of
resolution such that it is divided into Ω small boxes, each containing the
representation of a particular state.
Making the same assumptions of closed system, equilibrium
thermodynamics, such a system is completely isolated from outside sources of
matter and energy, it spends equal time in each of its Ω available states. In such a
case, the characteristic occupancy time of any state is inverse to the number of
states available, e.g. 1/Ω, and the system’s entropy is maximal for that set of states.
Under these conditions, S = k ln(Ω), where the k term is the Boltzmann constant
that contributes to the numeric units of entropy, as above, in Joules of heat /degrees
Kelvin of the temperature. If the system is in contact with a heat bath, but cannot
exchange matter with its environment, it is called diathermally isolated. The
distribution of times spent in the available states of a classical diathermally isolated
system of gas molecules can be represented by what is called a Boltzmann
distribution of probabilities of state occupancies, ρ (as a function of their energy
level, more measurably, their responsiveness, susceptibility, to heat). Here the
characteristic time of the system spent in each state varies as the particular state’s
probability.
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Leaving the framework of physical thermodynamic entropies entirely, the
entropy of information was introduced in the context of communication engineering
in electrical and electronic devices. The metaphorical machine for the current age of
entropy, analogous to the role of heat and steam engines in classical
thermodynamics, is the computer. Energy in this context is a relatively trivial
property. Ammeters and other monitors of load are unable to discriminate between
a computer actively engaged in encoding and computation or one simply
maintaining its dynamic memory while resting in computational readiness. This
situation is very analogous to the results of early work discussed previously on the
metabolic rates and sources of the whole brain’s energy, oxygen and glucose
metabolism, by National Institutes of Mental Heath’s Seymore Kety and Louis
Sokoloff and the State of Illinois Thudicum Laboratory’s Harold Himwich. Using
whole head arterial-venous, energy-in, energy-out, differences, they could not
demonstrate differences in rates of whole brain metabolism between states in which
the human subjects were engaged in solving mathematical problems or deeply
sleep. In today’s brain imaging research, using a variety of physical reflections of
the brain’s metabolic activity, it is the differences in regional distributions of
metabolic activity that are relatable to subjective and behavioral states, not
differences in total amount of energy expended. In graphically coded
representations of the regional metabolism of the brain in action, one or another or
many areas “light up” and others “grow dark” in correlation with changes in thinking,
feeling and action.
The entropy first developed by Claude Shannon was formalized for use in
1948 in what was then called communication theory and now information theory. It
represented a measure of the ambiguity and uncertainty that had the potential for
being resolved by new knowledge. In this context, entropy and information were
obviously complementary descriptors. A message that informs us about which of
ten possibilities should be chosen contains less information than one that informs us
about the proper choice to be made from among a thousand possibilities. The
entropy of communication theory is a measure that is computed on uncertainty. The
information reception capacity of a system is dependent upon the amount of
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uncertainty in the receiver that pre-existed the receipt of the message. In the binary
coding scheme of digital electronic operations, the unit of information is the bit, a
choice made between 0 or 1 in the resolution of a two state ambiguity at each place
of some power of two number of places. Our relatively common computers these
days have 32 or 64 bit processors. If these 0,1 choices are made in a random
sequence in which each step is independent of the previous one, the sequential
probabilities,  � ��  � are multiplicative: e.g. the probability of getting two 1’s (heads
in a fair coin) in a row are the product of each 0.5 probability: ρ 1 = 0.5 × ρ 2 =�0.5 =
ρ 1 ρ 2 = 0.25. Using the common base ten system of logarithms to demonstrate the
algebraic fact that multiplicative probabilities are logarithmically additive (and
ignoring the minus sign that comes with making logarithms of the decimal fractions
of probability), we notice that log 10 (0.5) = 0.693147 and log 10 (0.25) = 1.386294 and
that 0.693147 + 0.693147 = 1.386294.
The dot-dash choices of Morse code machines, the go, no-go gates of
transistors, the open versus closed ion channel-mediated neuronal membrane
discharge and the left, right spins of the single electrons of today’s quantum
computers lead naturally to an information encoding of multiplicative sequences as
the sum of logarithms in base (equal to the number of available states) two, each ρ�=
0.5 choice called, log 2 (0.5) = 1, a bit. Shannon’s 1938 master’s thesis mapped
George Boole’s algebraic scheme for doing yes-no, either-or computation onto
current switching devices such that circuit closed was “true” and circuit open was
“false.” Using Boole’s laws such as “Not(A and B)” always equals “(Not A) or (Not
B)” led to schemes for circuit routing through electronic gates which also serve for
information storage in gadgets ranging from cell phone directories to computer hard
disks.
Following Claude Shannon, each logarithmically additive entropy term is
expressed as the sums, Σ ι � of its probability, ρ ι , times the probability’s logarithm,
Σ ι (ρ ι �× log 2 ) (ρ ι ��in base two. A logarithm is an exponent of its relevant base such that,
for example, the logarithm, base two, of 2 × 2 × 2, 2 3 , = 3 and 3 bits can encode
eight binary (0,1) numbers: (000, 001, 010,011,100,101,110, and 111). Shannon
used a hill-like, called convex, entropy function S (ρ)= -Σ(ρ ln (ρ)). The amount of
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information required to gain knowledge of an event is dependent upon the
probability of its occurrence. log 2 (0.5) = 1 is the maximal entropy when modeling the
equilibrium entropy of an independent random 0,1, (heads or tails) series of
informational states as might result from flipping a fair coin a large number of times.
This value would be maximal when the coin was fair, ρ(heads, tails) = 0.5, and the
entropy would be 2(number of allowed states)×0.5(probability of occupying each
state)×log 10 (0.5) = 0.693147...or in bits, log 2 (0.5) = 1.
More generally, if system’s behavior is distributed equally among its possible
states, the Shannon entropy is maximal and equal to the logarithm of the number of
defined states, for example, log 2 (2) = 1. Shannon’s classical equation about
information content says the amount of information, I = -ρ log 2 ρ, measured in bits.
The minus sign in this reciprocal relation indicates that the information content of
data, I, goes up as the probability of occurrence of the observed data, ρ, goes
down. Since soon we will be talking about brains and their various styles of
information encoded content as well as its transmission, we note the other famous
Shannon theorem dealing with limits on the channel capacity, C, for information
transport is C = Wlog 2 (1+S/N) where W is bandwidth, the range of frequencies
available for information transport, S is the strength of the signal and N is the
strength of the noise. Recall that the log 2 (1) = 0 so only the signal-to-noise ratio,
S/N contributes to the value of the product of the multiplication by bandwidth, W.
Transparent clinical examples come from studies of the perceptual and cognitive
decline in normal geriatric patients in which the range of aural frequencies (W)
heard without augmentation decreases with age as does the frequency range (W)
observed in their resting brain waves. The inattentiveness of the obsessively
worried ruminator can be used as an example of brain channel capacity being
reduced by the amount of on going head noise, an increase N, which, of course,
reduces the value of S/N and therefore C.
Measures of the informational complexity of systems in motion, in contrast
with the information content of a static equilibrium state, are of dynamical entropy.
Dynamical entropy is often called H, in contrast with thermodynamic and/or
informational entropy, S. One can begin with a representational image of the
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location, velocity and directional tendency of every point generated by a dynamical
system by an arrow on the surface of action, the manifold, of a dynamical system.
This field of arrows indicating directional and strength of motional tendencies is
called a vector field. A vector represents its location at the base of the arrow, its
velocity by the length of the arrow (called the modulus) and the direction of the
motion by the direction of the arrow. If we regard all moduli as equal to one, every
vector on the surface has the same length. The resulting graphs are called direction
fields. Looking at a stop-action photograph of any point on this surface, its
associated vector informs about where the system would take it over the next unit of
time. The whole surface can be marked by initial points, which the dynamical
systems move as they generate patterns of orbits of moving arrows in time. The
following two brain and behavioral experimental circumstances make this depiction
and its relevance to dynamical entropy more concrete.
We review in more detail the concrete and visualizable findings from
experiments requiring the quantification of characteristic patterns of motion in
animals and man. They can be embedded into a similar surface-like setting, which
might be called a behavioral manifold. For examples, my students from the past,
Martin Paulus and Mark Geyer, now Professors at the Medical School of the La
Jolla branch of the University of California studied the effects of psychotropic drugs
on the patterns made on the floor by rats of various genetic strains while they
wandered about, in exploratory behavior in a bounded space. Monitored by a video
camera placed above the ceiling less cages, the patterns made by the paths taken
by the rats over time were reconstructed as vectorial orbits on a behavioral
manifold. This manifold was then repeatedly partitioned, covered with, from just a
few large, in graded progression, to many smaller boxes, each partition composed
of rectangular lattices of a particular size. Units of time were also partitioned into
range of units from larger to smaller durations of observation. Differences in the
rat’s genetic strain as well as injections of stimulants, antidepressants or
antipsychotic drugs resulted in characteristic and discriminable path geometries
mapped onto the behavioral manifold as orbital patterns. Each path was encoded
as a sequence of size-dependent numbered boxes that were entered and occupied
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or left. The new information being generated by the pattern of spatial orbits took the
form of sequences of numbers or symbols representing the sequence of labeled
boxes. The complexity of these numeric or symbol sequences was then quantified
in a variety of ways including the use of two fundamental measures of dynamical
entropy.
One measure reflects how many new, previously unexplored boxes were
entered by the rat per unit of time. This rate represents a percent of the possible.
The second measure reflects how much of the time did the rat in each box visited
as a distribution of the probable. The rate of expansion of the possible and the
relative time in occupancy of these possibles, the probables, form the bases for the
computation of these two kinds of entropies. For example, the work of Paulus and
Geyer showed that the administration of a very small amount of stimulant drug,
compared with a salt water control, led to an increase in the first measure of the
number of new, previously unexplored, boxes entered per unit time. With respect to
the second measure, the stimulant drug augmented exploratory activity was also
more uniformly distributed over the possible boxes, making for more uniform
probability. Administration of higher doses of stimulant drugs, at a critical dose, led
suddenly to more spatially and temporally restricted and stereotyped patterns of
motion of the rats, compulsive circling alternating with frozen sniffing. Both
contributed to a decrease in the possible and nonuniformity in the distribution of the
probabilities. In man, low doses of amphetamine tend to increase the rate and
creativity of thought streams and high doses generate fixed ideas and paranoid
delusions. In the statistical approach to nonlinear dynamical systems, timedependent
generation of new possibilities is called topological entropy, H T and the
entropy associated with the distribution of probabilities is called the metric entropy,
H M. These kinds of entropies have also been used to quantitate characteristic
patterns of in human behavior as well.
We have previously mentioned these measures as used in human
experiments by Karen Selz, a Research Professor of Psychiatry at Emory University
in Atlanta. Recall that she devised a set of experiments leading to unobtrusive
measures made on human subjects by asking them to remove, as many as they
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could, the dots in a lattice, one by one, from the computer screen, by clicking on
each point with a mouse. In some experiments, after removal, the dot reappeared in
fifty milliseconds, in the “fast return condition”, or after one-second delay in the
“slow return condition.” Unbeknown to the subject, the path made by the motions of
their mouse on the computer screen over time while removing dots were
reconstructed as a path on a fine to coarse grained box-partitioned behavioral
manifold. Entropic indices of the rate of expansion of the possible, number of new
boxes entered, reflecting H T , and the relative occupancy of the partition of the
possible, reflecting H M , the distribution of probabilities with respect to the boxes,
could then be computed. For examples, Selz found that the spatial and temporal
patterns of computer mouse motions made in this dot search and destroy task
correlated highly with the subjects’ age, sex and personality types as defined by
profiles from the Minnesota Multiphasic Personality Inventory, MMPI, and the
Structured Clinical Interview, SCI, associated with the standard Diagnostic and
Statistical Manual, DSM IV. She found that subjects whose personalities were like
my high self-sensibility girlfriends demonstrated high indices of both H T and H M .
The actions of nonintegrable nonlinear differential equations, not solvable by
the usual techniques of integration, can be transformed into graphical images by
plotting their orbits in abstract phase spaces with the three physically measurable
coordinates of location x (or some other temporarily fixed value), velocity y (the rate
of change in the location or measured value) and z acceleration (the rate of change
of the rate of change in location or value) in x, y, z space. Graphical representations
of the system in action in phase space can serve in place of analytic solutions to the
equations. This idea was one of Henri Poincare’s major contributions to
mathematics and physics, and has come to be the centerpiece of the qualitative
theory of differential equations. The often point-to-point unpredictable but globally
and qualitatively characteristic geometric shapes of the orbital patterns in abstract
phase space are the objects of interest. There are visualizable representations such
as cycles as circles and statistical measures made on these objects such as the H T
and H M entropies and the in-betweenness (neither maximal nor minimal) of their
difference.
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A global statistical context for these qualitative differential systems was
inspired by the Russian mathematician, Andrei Nikolaevic Kolmogorov. In his now
famous foundational talk about the stability of classical mechanical systems in the
final session of the 1954 International Congress of Mathematics, he gave public
birth to, among other ideas, what has come to be called the ergodic or statistical,
measure theory of dynamical systems. Here, ergodic means the existence of an
invariant statistical measure on the phase space attractor of the system that can be
obtained using a variety of equivalent methods and beginning the count at any of its
points. Two phase space objects generated by a dynamical system may look
different in phase space but their statistical measures may all be the same, i.e.
invariant. These qualitative orbits in a box-partitioned space can be visualized as
Paulus and Geyer’s rats exploring a space and Selz’s path sequences of computer
screen dot quenches produced by clicking on them with a computer mouse.
A precursor of Kolmogorov’s ergodicity was the earlier ergodicity of Ludwig
Boltzmann. This describes a suitably partitioned system such that equivalent values
come from quantitating the behavior of one single orbit exploring the space of the
lattice of boxes over very long times time as those obtained from a single aggregate
photograph of all orbits run from all possible starting places simultaneously. The
ergodicity of gas-like molecular randomness implicates systems being in one of only
two possible equilibrium statistical states: measure zero (at most occupying a single
point, zero, minimal entropy) or its “complement,” full measure one (occupying all
available space in a state of maximal entropy). Joseph Goldstein, a well known
teacher of meditation, giving advice recorded in Daniel Goleman’s 1977 book on the
subject said that all methods of nirvana directed meditation amounted to “…simple
mathematics …all systems aiming for One or Zero—union with God or emptiness.”
In place of the maximal or minimal values for the H T and H M entropies of these
states of transcendence, we in the world of samsara are stuck in states of inbetween
entropy which invariant statistical measures of on phase space shapes
help quantify.
To generalize measures made on rat and computer mouse paths to more
general and idealized systems, after plotting an orbital path in a phase space, we
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may partition the space of values taken by the journey of the orbital action
generated by the equation over time with rectangular grids of increasing fineness.
The result is an equipartition of phase space such that there is at most one orbital
point in each rectangle of the grid, with, of course, many rectangles in the finer grids
being empty. This final grid partition is called a generating partition. The proportion
of the available boxes of the partition occupied by points is called its area or volume
measure. This measure has been given a variety of names including Liouville, Haar
and Lesbegue measures. If every box is occupied, it has measure one. If at most
one box, it has measure zero. If we allow partitions to be non-uniform and/or not
fine enough to be generating and apply probability weightings for how many points
fall into each particular box of the grid, the method is called the Sinai-Ruelle-Bowen
or SRB measure after Kolmogorov’s students and followers, the Russian, Ya Sinai,
the Belgian Frenchmen, David Ruelle and the American, Rufus Bowen.
Similar to the SRB measure, the distribution of box occupancy probabilities
multiplied by their logarithms and summed over all cells of the partition yields a
statistical measure that is close to the informational entropy of Claude Shannon as
described above. It is called the metric entropy ( H M = -Σ(ρ i ln(ρ i )), where H means
entropy and ρ i is the proportion of the total observations that occupy cell i of the
phase space or state space partition. It was the above noted Russian father of
modern dynamical systems, Kolmogorov, who in 1956 proved that the Shannon
metric entropy is a quantifiable invariant of systems even in very complicated
motion. Stanford University's Donald Ornstein won a Field’s Medal (the under forty
year old mathematician’s Nobel Prize) for his late 1960’s work proving that the
Shannon metric entropy, H M , was the only invariant for a large class of appropriately
defined, expansive (near by points separating in time) dynamical systems. Recall
that we refer to metric entropy reflecting the relative occupancy as probability
among the possible boxes (or states) as H M . H M is maximal when the percentage
occupancy of all occupied boxes is uniform.
IBM’s Roy Adler in New York and Brian Marcus in California, Hebrew
University’s Benjamin Weiss, Warwick University’s English mathematicians, William
Parry, Peter Walters, Mark Pollicott and others developed and proved the relevance
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of a related measure of the rapidity of dynamical expansion, the generation of new
information seen as the rate of entering new boxes of the partition, a logarithmic
rate of expansion of the possible. Counting the number of previously unoccupied
squares entered by the dynamical systems orbit per unit time over the generating
partition, for instance, yields an estimate of entropy that, as in the rat and computer
mouse examples above, is called the topological entropy, H T . H T , is about how
much new information is being generated by the system per unit time. Theorems
have been proven that H T is a maximal estimate of the global dynamical entropy
with H M proven to be a minimum estimate. Monitoring single or aggregate molecular
motion in a system with the maximum randomness of a space filling gas, we find
that, on the average, every box is entered and occupied uniformly such that H T = H M
or said another way, H T – H M = 0.
As evidenced by the above described experiments in rats and people, the
same entropic relations (but usually not with maximal or minimal measure) can be
found in biological systems. We have previously described the manifold geometry of
a generic (typical, idealized) nonlinear dynamical systems as hyperbolic defined by
the presence of simultaneous but decomposable components of the motion
including the straight ahead and round and round actions on the center manifold,
the new possibility generating, expansive, away from the center manifold motions
along unstable manifolds and the back to the center manifold, contracting motions,
along the stable manifolds. Uniform expansive and contractive influences in the flow
leads to mixing of the order of the initial sequence of the values inscribed by the
orbits. This results in maximization of the entropies and satisfaction of a
concomitant of the uniformly hyperbolic condition, H T – H M = 0.
These clean and mathematically proven findings do not hold for the quasimess
that is human neuropsychobiology. Enmeshed as most of us are in only
intermittently random or nonuniformly hyperbolic systems with the in-between
entropies of the only apparently real world of maya, H T – H M ≠ 0. How the H T – H M
= 0 of uniform hyperbolicity fails, H T – H M ≠ 0, and along with it the dispassionate
detachment of entropic emptiness and fullness, becomes a problem not unrelated to
the existence and quantitative qualities of personality styles and their dissolution
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with return toward but not reaching the maximally entropic openness, flexibility and
naïve credulousness of the in Jesus and Holy Ghost occupying transcendent
dynamical states. We are all stuck somewhere in the range of measures indicating
in-between entropies.
Further Readings for Sensual In-Between Entropies
Ecstasy in Secular and Religious Experience, Marghanita Laski, Tarcher, Los
Angeles, 1961.
The Role of Neural Plasticity in Chemical Intolerance, Barbara A. Sorg and Iris R.
Bell, Ann. N.Y. Acad. Sci. Vol. 933, 2001
The neuropsychiatric and somatic characteristics of young adults with and without
self-reported chemical odor intolerance and chemical sensitivity, I.D. Bell, C.S.
Miller, G.E. Schwartz, Arch. Environ. Health 51:9-21, 1996.
Application of entropy measures derived from the ergodic theory of dynamical
systems to rat locomotor behavior, M. Paulus, M. Geyer, L. Gold, A. Mandell, Proc.
Natl. Acad. Sci. (USA) 87:723-727, 1990.
Long-range interactions in sequences of human behavior, Martin Paulus, Phys.
Rev. E. 55:3249-3256,1997.
Mixing properties in human behavioral style and time dependencies in behavior
identification: The modeling and application of a universal dynamical law. Karen A.
Selz, UMI, Ann Arbor, 1992.
A family of autocorrelation graph equivalence classes on symbolic dynamics as
models of individual differences in human behavioral style, Karen A. Selz and
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Arnold J. Mandell, In (ed. R.R. Vallacher and A.J. Nowak), Dynamical Systems in
Social Psychology, Academic Press, San Diego, 1994.
Toward a neuropsychopharmacologicy of habituation: a vertical integration. Arnold
J. Mandell, Math. Modeling 7:809-888,1986.
Thermodynamics, Enrico Fermi, Dover, N.Y. 1956.
Thermodynamics and Statistical Mechanics, Peter T. Landsberg, Dover, N.Y. 1978.
Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane
and C. Series, Oxford, Oxford, 1991.
The Mathematical Theory of Communication, Claude E. Shannon and Warren
Weaver, U. of Illinois Press, Urbana, 1963.
Science and Information Theory, Leon Brillouin, Academic Press, N.Y. 1962.
Brain Metabolism and Cerebral Disorders, Harold E. Himwich, Williams and Wilkins,
Baltimore, 1951.
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CHAPTER 5:
SOME ENTHEOGENIC ENTROPIES
In the spring of 1968, members of my laboratory team were looking for new
brain metabolic pathways of the essential amino acid tryptophan, the dietary
precursor of the human mood, sleep and libidinal neurotransmitter, serotonin. After
struggling for several months to identify an apparently new compound, which turned
out not to be new but only new in the brain, we collected evidence for a human
brain enzyme that could catalyze the production of an LSD-like hallucinogen,
dimethyltryptamine, DMT. Tracing its metabolic origins, we found that DMT was
derived from tryptamine, a common metabolite of the essential and omnipresent
amino acid, tryptophan. This enzyme and its metabolic product were located in
highest concentrations in brain stem systems that influence the neural regulation of
the heart, blood pressure, temperature, breathing, vomiting and primitive approachavoidance
behavior. It was also found in limbic brain nuclei thought to modulate the
emotional coloring of perception and thought. Richard Wyatt, working at the
National Institutes of Mental Health found DMT in the urine of schizophrenic
humans. He also showed that DMT increased significantly if tryptamine’s normal
pathway for degradation was blocked by monoamine oxidase inhibitors, such as
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Nardil, Marplan, Eutony, Parnate and others of a then common family of
antidepressant drugs.
The presence of a DMT-generating enzyme in human brain was particularly
exciting because we knew from the work of Harvard botanist, Richard Shultes and
others, that DMT and the monoamine oxidase inhibitor, beta carboline, are
combined in a mixture of the leaves of a shrub and the bark of a vine, both
Amazonian plants, used together by the shaman of Peru, Colombia and Ecuador for
thousands of years to evoke mystical experiences in themselves. In their state of
chemically-facilitated, spiritual transformation, they were better able to engage in
healing and divination of others. More recently this and other similarly acting
biochemicals have been called entheogenic, “connecting to the sacred within.”
Consistent with our neurochemical findings in human brain, the shamanic
concoction, called by many names including ayahuasca and yage, combined the
DMT containing plant, Psychotria viridis, with an extract of a vine with the powerful
monoamine oxidase inhibitor properties of the beta carbolines found in
Banisteriospsis caapi. In 1975, working with a graduate student, Louise Hsu, we
found that the mammalian brain could also synthesize beta carbolines. This family
of compounds from the vine protects the tryptamine substrate as well as DMT from
metabolic degradation such that it could circulate in the blood long enough after oral
ingestion for enough to cross the blood brain barrier to induced prolonged and
dramatic alterations in perceptions, feelings and thoughts. In addition, the
carbolines of the Benisteriospsis component extended the time of action of DMT
beyond the 15-30 minutes of effect of DMT when injected alone in human subjects.
We found it fascinating that the human brain made combinations of DMT and beta
carbolines similar to the blend that indigenous shamamic chemists discovered as an
entheogenic from plant sources.
Ralph Metzner, in the introduction to his 1999 collection of papers called
Ayahuasca concluded that “…it is widely recognized by anthropologists as
being…the most powerful and most widespread of the shamanic hallucinogens.”
William Burrough in a 1953 City Lights published book written with Allen Ginsberg,
The Yage Letters, said that yage “…gave entrance to a city where all human
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potential is spread out in a silent market…” It was generally believed that with
adequate spiritual preparation, ayahuasca could generate transcendent states that
allowed access to ones inner being and the beings of other worlds that could serve
as sources of mystical knowledge and healing. The Shams dervish of the 13 th
Century, wandering the Turkish portion of the Silk Road, used the word sohbet to
describe the inner land of mystical conversations about mystical subjects that their
turning meditation, whirling, and the shaman’s entheogenic compounds such as
DMT give entrance.
The question was whether our finding of DMT and its human brain enzyme
had been an artifact, an accidental laboratory fluke. Members of my neurochemical
research teams at the University of California Medical Schools in Irvine and La
Jolla, notably Dr. Lee Poth, now a professor of pediatric endocrinology at the
Uniform Services Medical School in Washington D.C., demonstrated that the DMT
synthesizing enzyme existed in the brains of recent accident victims that as far as
we were able to learn from their family and social histories, had been completely
psychologically normal. More than a little bit startled by this finding and worried
about making a sensational scientific mistake, we repeated the experiments with a
variety of controls with the same findings. Though our original estimates of the
human brain enzyme concentration were on the high side, we confirmed the general
finding and published them in Science in 1969 and Nature in 1970. Our carboline
work was published in the Journal of Neurochemistry in 1975. A year or so after our
Nature paper was published, the Nobel Prize winning neuropharmacologist at the
National Institutes of Mental Health, Julius Axelrod, confirmed the presence of the
DMT biosynthetic enzyme that converted the tryptophan product, tryptamine, to
DMT in mammalian brain tissue. We were both delighted and relieved.
We speculate, perhaps too grandly, that this finding, along with the beta
carboline human brain synthesizing capacity, supplies one of many possible
neurobiological and neurochemical mechanisms for the claims of the cross-cultural
universality of mystical experience. We all had human brains with these enzymes.
The idea that the phenomena accompanying primary religious experience were
common to all cultures was a major theme of the life’s work of the philosopher-
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psychologist, William James, and was studied using fieldwork by anthropologists
such as Bronislaw Malinolowki as described in his classic book, Magic, Science and
Religion. Was this neurochemical-behavioral organization an evolutionarily adaptive
mechanism selected so that some spiritually gifted individuals self-selected from a
severely stressed population could escape and then lead the rest of us out of a
sense world that had become intolerable? Could this be an antidote for the
hopeless, without materialistic solutions and trapped in a belief system of spiritual
nihilism? Was this a brain chemical transcendence escape and spiritual delivery
system for the suprapsychological survival of those in dire need? As the 13 th
Century Islamic mystic, Jelaluddin Rumi, has written, “If a tree could fly off, it
wouldn’t suffer the saw…” and more concretely, “…if you can’t go somewhere,
move into the passageways of the self…,” a spiritual escape via a neurobiological
road to the God-space within.
What followed were a few years of occasional exploration of an “inside out”
understanding of the mystical states evoked by the entheogenic family of chemicals.
There were varieties of settings for these personal experiments. I found myself
LSD-lost, circling endlessly in the tall silence of a Northern California redwood
forest. I tried on Hunter Thompson’s mescaline lenses for the experience of Las
Vegas unfiltered. I was expertly mentored in these quests by a distinguished
collection of guides: Cultural anthropologist Michael Harner who taught me about
the yage and datura use among the shaman of the Jivaro; Social anthropologist,
Barbara Meyerhoff introduced me to the personal renewal rituals of the peyote
cactus-using Huichol Indians of the Southwestern Sonora Desert; Neurochemically
sophisticated Sidney Cohen, founding director of the National Institutes of Health’s
Institute on Drug Abuse, told me stories of his involvement with Aldous Huxley and
Barbara Brown in the Los Angeles covey of early American LSD explorers; organic
chemist Albert Hoffman, Sandoz’s designer of a series of ergot alkaloids including
LSD, told me stories of his accidental post-sniff hallucinations while returning home
on a bicycle; An anonymous group of us conducted personal experiments with
Sacha Shulgin, the University of California at Berkeley professor who first
synthesized and tested the mescaline-derived, Ecstasy series of compounds; We
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did some work with the dissociated anesthesias (producing wide awake but not
there states) having consulted with John Lilly, a brain scientist who used these
agents as a courageous self-medicating explorer of sensory isolation tanks; I met
several native shamanic practitioners including the Huichol Indian that was the
model for Don Juan in Carlos Castanada’s five volumes of pseudoethnography
written up in my essay “Is Don Juan Alive and Well?” in The Pushcart Prize of 1977.
Issues of culture and brain chemistry came together in several accounts about
entheogenic, mescaline-containing peyote use among the Huichol Indians in a book
edited by Kathleen Berrin and Thomas Seligman of the San Francisco Art Museum
called Art of the Huichol Indians.
Over these years I collected many nauseating, upper and lower bowel
wrenching and ecstatically transcendent and exhausting day-long episodes of the
angular geometries of visual pattern-generating DMT, the animistic breathing of
bush and flower breathing peyote cactus, the darkly forbidding shadows of the
psylocybin-containing mushrooms, the irreversible rocket launches into the
electrically buzzing, kaleidoscopic circus of LSD-containing vials from Sandoz and
the optimistic, trust engendering, expansively warm rush of six of Sacha Shulgin’s
gregarious, rave dancing, chlorinated, methoxylated and ethoxylated
phenylethylamines which he had, years before, synthesized for “an undisclosed
purpose” for the Dow Chemical Corporation under contract with the U.S. Army
Chemical Corps. The best known of the latter group remains part of the rave culture
as Ecstasy.
These agent’s peaks are flooded with exaggerated, caricaturizing images of
people’s faces and a belief in the mindedness of animals and even the embodiment
of inanimate things. Evoked are simultaneous and diametrically conflicting
interpretations of the same social context, heteromodal sensory fusion called
synesthesia so that sound bespoke color and smells induced music, habitual
thoughts rearranged as new ideas in what is experienced as exciting new insights,
and, most of all, that which Louis Lewin, Berlin’s early 20 th Century Freud of
psychotropic drugs in his book Fantastica, called gladness of the soul. Timothy
Leary wrote of entheogenic escape from the habitual human brain’s mental-
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manipulative and socio-sexual circuits gaining access to the rapture and ecstasy
brain pathways on the way to the new planet within.
What is seldom written about is the aftermath of chemical entheogenic
agents. After the several hours of fireworks, all of these entheogenic agents, some
more than others, gifted me with weeks to months of more self-sufficient, emotional
fullness and ease in the conduct of living that was less contaminated by narcissistic
preoccupation or defensive distantiation. I was left with increased interpersonal
sensitivity and a noticeable repair of my deficiencies in aesthetic sensibility,
particularly for the visual arts and landscapes. What were once two dimensional,
trivial, beside-the-point, scattered copses of trees and apparently casual arrays of
plant life in the Boboli Gardens behind the Palazzo Pitti in Florence, became the
grandly structured, botanical wonder of increased dimension, communicating awe
filled new perceptions of its previously unseen beauty. For the first time, I found
myself walking slowly and stopping for several minutes, wordless, spellbound, in
front of the modern art pieces of New York’s Guggenheim Museum. Lost in the
experience, I found myself exclaiming to no one in particular, “I can see!”
The delicacy and deliciousness of post-entheogenic agent’s new and
beautiful everything made me tiptoe watchfully so as not to injure an ant. Feelings of
omnipersonal kindness and generous compassion were without prideful selfreflection.
This state of grace felt like an invasion of a shimmering presence that
made contact with my other, generally unknown to me, life. It brought new
perceptions, feelings and ideas for which I was moved to give thanks. I began to
think I understood a little bit about what was meant by living in the Spirit and
merging with God. Mircea Eliade, the French, University of Chicago Professor of the
History of Religions, in his classic The Sacred and Profane, calls the revelation of
the sacred in ordinary objects, people and events an hierophany. In the state that
this requires, “…all nature is capable of revealing itself as cosmic sacrality….” The
entire world can become a hierophany with what Abraham Abulafia called an
activated mind, the Jewish soul of emergent properties called the Nefesh.
This entirely new world, Rudolf Otto in his 1917 Das Helige (The Sacred)
called it ganz andere, (wholly other, something else), seemed to emerge
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spontaneously along with an instantaneous knowing-how-it-is-with-you-and-I-andall-of-us
that made even vicious killers appear sympathetic. Is this what the
Charismatic New Testament Book Churches mean by redemption through
forgiveness of others, requiring the genuine sincerity of this thought before
qualifying for Communion? Is this Christ’s undemanding gift of grace as in Romans
4: where Paul observed that all of us fall short of the full glory of God unless justified
freely by His grace. Was this the New Testament’s spiritual technological advance
from the Old Testament’s and Koran’s eye-for-an-eye? Did this chemically triggered
transcendent experience differ significantly from the supernatural transformation of
individuals by the Holy Spirit of Christian revivalist teachings? Martin Marty,
University of Chicago’s Professor of Modern Church History, dates the
institutionalization of this personal transformation in the United States to the post-
Civil War period. Did this mean that the mysteriously selfless love of Christian
agape and the altruism of E.O. Wilson’s sociobiology lay waiting in the brain and
could appear spontaneously, by grace, without lawful directive, repetitive recitation
or the discipline of catechism?
As one might have suspected, the urgency of my inner and outer search for a
new spiritual ecology of mind was driven by more personal needs. My spiritual
hunger was made acute a couple of years before our laboratory’s DMT discovery
when as a 30 year old Assistant Professor of Psychiatry and Neuroscience at UCLA
in West Los Angeles, I was living in a small, heavily mortgaged house in Brentwood
with my graduate student wife and two young sons. A testicular lump was an
accidental discovery made while showering. After surgical biopsy and radical lymph
node dissection, the professor of urology gave me a diagnosis of right testicular
choriocarcinoma. All by itself, my testicle had given birth to a mass containing all the
embryological tissues of a fetus, and had thrown in some maternal placental cells
as lagniappe. Unlike now, when the group of testicular neoplasms are treated
successfully with a high survival rate (think Lance Armstrong), at that time, follow up
research of this young man’s disease by the Army Medical Corps promised a fiveyear
survival rate of only 5% to 10%. The news filled me with fear and the ensuing
hopeless resignation detached me from life with a dread broken up only by
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episodes of rageful envy of everyone else in the world that had been spared. My
wife escaped into an alcoholic flirtation with her major professor; my sons grew
increasingly ensconced in the generous and kind neighborhood homes of their
playmates. I metered as many hours as possible in equity growing, long lonely days
in a small, dark, couch filled, university office, listening to Beverly Hills, Brentwood
and West Los Angles citizens as they psychoanalyzed their mysterious lack of
emotional fulfillment from materialistic fulfillment. Legend has it that Gautama’s
sudden insight about the universality of this sated, bored condition occurred in 528
B.C. after 49 days of sitting in the lotus position under the bodhi tree, now called
ficus religiosa. In contrast with Buddha’s illumination, my psychoanalytic traininginduced,
Freudian-Darwinian instinctual conflict, driven by fears of starvation and
castration, drew me tighter into the world of meaningless, coin flip probabilities.
Our house was a block away from a West Los Angeles synagogue and we
knew the Rabbi and his family well. Our sons played together frequently. The Rabbi
tried to bring comfort to me on my death watch, with hours of discussions about
trans-individual, ethnic belonging and a deeper foray into philosophical humanism.
Both felt completely irrelevant to my condition. As an intern tending to those dying at
night in Ochsner Foundation Hospital in New Orleans, it seemed to me that Jews
tended to die more noisily than Catholics. For my personal escape from low-lying
dread, I needed the metrically linear time of chronos to become the metric-free,
topological, continuous surface of the twisted circular ribbon of a Mobius loop, with
the view from each moment a kairos, a stretchable infinity of each moment’s internal
multiplicity of times.
The ruthlessly reasonable Hebraic historicity, configured by the tooth-for-atooth,
Mosaic and Roman talion law, the reciprocal, economic, exchange-calculating
brains of Barkow, Cosmide and Tooby’s The Adapted Mind (1992) and the terrifying
stories of the Five Books of Moses, made the hopelessness of this sinner’s plight
inevitable. It felt like my dichotomous choice of God-type was between One of
merciless fairness and the He and She of unconditionally forgiving generosity. The
mind set of logical problem solving applied to the question about which of these two
represented the true character of God lead to a momentarily distracting, metaphoric
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ecclesial exercise: what were the minimal number of four magical cards need we
turn over with preconditions or results on the upsides and downsides if what was
showing was: (1) Beatifically good; (2) Cursed with extraordinarily bad luck; (3) Not
dependent upon personal virtue; (4) Inordinately fortunate in all of life’s trials. The
pay-as-you-go God people would need to pick up (1) and find fortunate life and (2)
to find the fate of the non-believer to establish that God was coldheartedly true and
fair with the results of flipping (3) and (4) being none contributory. The grace-to-allsinners
God people need to turn over card (3) to find good life and (4) to find
sometime sinners nonetheless fortunate to confirm their belief in the unconditionally
of the loving generosity of God and making finding out about the underside of cards
(1) and (2) unnecessary. This liturgical discussion and gamble with God’s cards,
perhaps a caricature of the Talmudic, rational discussions with the rabbi, felt
irrelevant to my spiritual needs.
Missing was mysticism’s promise of the disappearance of I into a union with
the divine, the Heart Sutra’s eternal emptiness of form and the eternal form of
emptiness that gifts with spiritual perspective and not-necessarily-logical intuition
about unseen Absolute Reality. Forced either-or, binary, card-turning cognition in
the search for God’s logic is unrewarding. As the Dalai Lama, in his Heart of
Wisdom Teaching, says, “…all phenomena are emptiness, without defining
characteristics, they are not born, they do not cease…" In trying to penetrate the
mystery and promise of this emptiness, it was difficult to surrender my internal
parody of what sounded like that day’s Southern California New Age stuff about
global nonaggression, sexual politics, Beadles music, distressed jeans and pot. In
the synagogue of my neighborhood, experience with a deeply felt, never-you-mindabout-anything
God of detachment with love, was not on the menus of Friday night
or Saturday morning services. All I could feel was a faithless and nonnegotiable
fear.
In the work of many mysticism-positive scholars, a classic being Evelyn
Underhill’s Mysticism, 1961, it has been speculated that this ineffable state as a
union with a powerful unknown, transcending description in language, becomes
more socially prominent during times of cultural efflorescence. She pointed to the
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flowering of mysticism in epochs of the high cultural achievements at the close of
the Classical Period in the Third Century, the Medieval Period in the Fourteenth
Century, the Renaissance in the Seventeenth Century and, now, as we know, in the
Western World toward the end of the Twentieth Century. An increase in general
acceptance of talk, writing and practice focused on mystical experience is said by
many to accompany historical high points in intellectual, literary and political
achievement. One might include as a component of our growing cultural richness,
the new science about chemical dialogues with the brain. Although no central
nervous system agents were ever allowed in the ashrams of Baba Muktananda, it
was common during some evening sessions of questioning, called satsangs, for him
to acknowledge that one or a few experiences with entheogenic agents can open
many recalcitrant folks to the existence of the God within. This, in turn, led them to
the drug free spiritual exercises, sadhana, of love, self-truth, and spontaneity (each
according to their nature) as well as abstinent discipline, meditation, chanting and
yoga to maintain the knowledge. We might speak of participating in the creation and
maintenance of the spiritual ecology of ones inner and outer being. Underhill said
that the cultural richness of an efflorescent epoch is taken inward and accompanies
personal and societal mutations into states and institutions involving higher spiritual
consciousness.
In addition to an increase in the common outward manifestations of having
had a mystical experience, such as an increase in compassion, forgiveness and
more respectful and reverential attitudes toward the Earth and all its creatures
(currently taking the forms of deep ecology, ecofeminism, herbal medicine, organic
farming and the like), these times bring more public consideration of the nature of
reality itself, apart from its material manifestations. The theme of the life’s work of
the Dominican priest, Thomas Aquinas, made master of theology by papal
dispensation in 1259, involved the existential recognition of this dichotomy of
existence, esse, and essence, nature and grace, the material world and God.
William James wrote famously about mystical experience penetrating the thin veil
between these two worlds. Those with a mystical orientation attribute reality to inner
experience in relationship to a transcendental, supernatural world. Whereas
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everyday events are subject to perceptual ambiguity and its attendant variety of
interpretations, mystical union is claimed to bring the existence and meaning of
Absolute Reality into direct experience. This kind of knowing is more akin to the
Platonic view of mathematics, that theorems have been everlastingly existent, from
before our physical world, then it is to the here and now, physically based, finite
computations involving the experimental machines of physics.
The philosopher-mathematician father of phenomenology, Edmund Husserl,
criticized the physics-want-to-be orientation of the 1860 empirical, objective
measure psychologies of Fechner and Wundt. He understood the best of their
findings as simply correlations between subjective and observable events. Using
mathematical discoveries as examples, Husserl spent his life arguing for the
possibility of abstract truths relevant to mind being more reliable and valid if grasped
via direct experience. Knowing by what the popular mid-twentieth century writer of
science fiction, Robert Heinlein, called grocking it. This is antithetical to the attitudes
of today’s human cognitive and brain sciences which disallow such knowing as
deeply suspect unless accompanied by objectively definable observables such as
changes in electrical or imaging indices of brain activity in one neural region or
other. The modern psycholinguistics of brain mechanics can be called
neolocationism. Using modern technology to measure regional blood flow, energy
metabolism and/or electrovoltage or magnetic field activity, stories of function are
spun that closely resemble those imagined more than a century ago by the first
locationists, such as Ramon Cajal. These neuroanatomists spent thousands of
hours looking at cell clusters and their connections in stained slides of human brain
tissue using microscopes and imagined their singular and integrated function.
Today, Lewis Judd, long time chairperson of the Department of Psychiatry at
UCSD in La Jolla, carries a full sized, polymeric, three-dimensional model of the
human brain when teaching his students about human subjective experience and
interpersonal behavior. In his weekly grand rounds, he explains that day’s
psychiatric patient’s problems pointing here and there at regions in this plastic
surrogate for our electrical jellied brain. Few, if any, of the psychiatry students in his
class was inclined to ask the foundational question: how it is that a finger point and
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a name of a brain place can describe, much less explain in the language of physical
or physiological mechanism, a patient’s illogical thoughts, feelings of hopelessness,
irrational rage or prayerful gratitude. There remains a wide gap between ideas
about the mechanisms of human symbolic processing and those involving the
structures and functions of neuronal components and their connectivities in the
brain, particularly when perceived as regionally segmented meat. Yet this report of
Professor Judd’s finger-pointing plastic brain ritual should not elicit surprise since
iconic manipulation is certainly not new to the practices of priesthood.
In contrast with neuropsychiatry’s behavioral attributions to brain parts as an
explanatory pantheon of mysterious doers, absent of mechanical specifics, the
fields of physics turn to more abstract and general mathematical and statistical, socalled
phenomenological laws, such as those of thermodynamics and statistical
mechanics. The accounts of Feynman’s abstract and general thermodynamic
development of conservation of energy as well as equilibrium thermodynamics
discussed previously serve as relevant examples. These abstract models have
been found to capture the behavior common to diverse physical systems involving
(often still unknown) differing physical mechanisms. Consistency of description,
reliability, weighs in before predictive validity, which, with maturation of the research
area, gradually becomes detailed mechanistic understanding with the eventual goal
being derivation from the first principles of physics. The painful truth is that that in
spite of evocative claims made to the contrary in the 1990-2000 Decade of the
Brain, this level of understanding at the interface of neurobiological hardware and
software remains unbreached. Some recent attempts are interesting.
One of the current research themes about real single neurons in real brains
(in contrast with the silicon chip modules used in neural network computer
simulations), involve widely distributed neurons that discharge in temporal
synchrony. These phenomena have been described by Max Planck’s Wolf Singer,
Christoff Koch of California Institute of Technology and Florida Atlantic University’s
Steven Bressler and others with words such as synchronization, phase locking,
coherence and binding. Binding is an intuitively seductive word that premises that
two, even widely spatially separated, brain regions that manifest neuronal signals of
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activation locked together in time are assumed to be functionally integrated. Another
time-dependent neuronal characteristic of current interest involve neurons or
neuronal clusters that beat with almost strict periodicity, the oscillatory pacemakers.
For example, the program of research by Professor Al Selverson at University of
California at San Diego, among others, has elucidated the role of these rhythmic
pattern generators, both autonomous and those emerging from particular patterns
of network connections. A wide variety of functional links involving neuronal
pacemakers has been demonstrated. They range from the oscillatory transport of
calcium through membrane channels in neurons and heart muscle, smooth muscle
oscillations of the pylorus muscle of the stomach, the neuronal ganglion driven
chewing motions of the jaws of invertebrates and the retina-to-brain hypothalamic
cells gating human circadian rhythms coupling our body’s hormonal clocks to light
cycles.
Though regular rhythmicity in neuronal discharges is an intuitively attractive
idea and relatively easy to quantitate using simple sine wave trigonometric
transformations, in the real brain it is statistically rare. The commonest neuronal
discharge pattern observed is that of intermittent bursting, clusters of neuronal
discharges in time in which the inter-discharge intervals irregularly stretch and
contract like the bellow pleats of a syncopated accordion. Bursts of repeated firing
of some unpredictable length followed by silences of equally mysterious durations.
Their behavior can be represented as statistical measures using non-normal, long
tailed distributions and in-between entropies described previously. For a whole
human example, although the rhythm of manic depression is commonly thought to
involve periodic cycles, careful study using motility patterns of the timing through life
of these episodes of extreme mood states by Professor Allan Gottschalk at the
University of Pennsylvania and others have demonstrated an irregularly intermittent
bursting pattern in manic-depressive episodes, getting more frequent with age.
Neuronal inter-discharge intervals seldom demonstrate what is called a regression
to the mean like the normal distribution of heights, as one increases the number of
people measured, the tighter the distribution around the mean. Neurons, much like
our own irregular pattern of doing things (in spite of our plans), the statistical
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distributions of neuronal interspike intervals have increasingly long tails. Contrary to
the behavior of a normally distributed observable, the larger the series of neuronal
spike observed, the more likely that a longer interspike interval than had been seen
before will occur. Counter-intuitively, long intervals tend to be followed by more long
intervals as more shorts follow short intervals. Manic attacks cluster in time as does
a number of other brain and body diseases. Maybe it is intuitively obvious that bad
stuff tends to cause more bad stuff and good stuff is self-propagating. Having
suffered recently does not mean fate owes you one. The brain’s syncopated
segmentations of time can be translated into a creatively arrhythmic dance.
What makes neurologizing conversations like these about subtle human
experience possible are the human subjective scenarios we have agreed to short
hand with names of brain parts and neurochemicals. The how is where conceptual
connection is filled with post 19 th Century Spanish microscopic neuroanatomist,
Santiago Ramon y Cajal-like, intuitions about the functional role of brain structures:
we think motor automaticity and pacing when hearing the brain place names such
as caudate, putamen and cerebellum; we think limbic lobe when musing about
sexuality, rage and depression; we short hand left versus right hemispheric places
for verbal and sequential versus intuitive and geometric shape cognition; we point to
the frontal lobe for the future work of executive control, anticipation and paranoia;
the hypothalamus for primitively expressed appetites and to the brain stem for our
vital functions such as breathing and blood pressure. With respect to the brain
juices, we say dopamine for aggressive activity, norepinephrine for attention and
sensory discrimination and serotonin for hunger, mood and sexual inclination. No
matter how avant guarde our experimental techniques such as monitoring local
functional blood supply by fMRI, regional brain glucose utilization maps, timedependent
changes in skull surface voltage using a cap studded with
electroencephalographic, EEG, leads, monitoring these voltage field via their
transverse magnetic fields by the frozen helmets of magnetoencephalography,
MEG, we conclude our work by calling forth named but still enigmatic brain parts
and their juices as mysteriously powerful little men and women executing
remarkably complex and subtle tasks, sometimes even when called upon.
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Current neurochemical research using molecular biological tools such as
mice knockouts (the ablation of specific proteins though interference with their
nucleotide-mediated protein biosynthesis), for example, the production of animals
missing a subunit of their hippocampal glutamate receptors associated with the loss
of some memory functions, conclude the memorial mechanism to be a specific
cellular region, such as hippocampal CA3 cells. Technology advances but continues
to support a primitive philosophic animism of named brain parts which pop science
icons like the late Francis Crick called “The Amazing Hypothesis.” He and his fellow
brain philosophers implicate brain mechanisms such as the amygdaloidal nucleus
man who can emotionally color even affectually neutral information that is
transported through him. Imaging data showing amygdala man lighting up is used to
tell us that circulating sensory information through the differentially behaving
amygdaloid nucleus is used for fight or flight interpretive significance. Emotionally
expressive human faces light up inferior parietal cortex. The Iowa University
Professors, the husband and wife Damasios, have located even the criminal
psychopath man in specific locations in the brain. As we have argued, perhaps ad
nausem, using multimillion-dollar imaging and molecular biological technology and
no new thoughts that weren’t around during the era of the 19th Century’s
neuroanatomists, specific brain regions continue to gain implicative properties like
the task-specialized gods of the Roman and Greek pantheons. Crick implied that
God is a brain part.
At the same time, those of us that have been in the brain business for a
while, recall skyscraper window washers, standing steady, high up on rope lashed
planks, suffering from congenital absence of the cerebellum, the supposed sine qua
non brain part supporting motor coordination and balance in humans. More
generally, there is much evidence that if young enough and willing to work, many of
the functions of missing parts of the brain can be taken on remarkably well by other
brain parts thought not to be involved in these functions at all. In addition, since
evidence of neuronal responding to loud noise or bright light perturbation can be
found almost everywhere in the hyper-connected human brain, because anticipation
and brain time inversions make before and after indicate little about human
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neuropsychological causality, and inhibitory on or off and activating on or off are a
priori functionally equivalent with respect to the logic gates of information encoding,
transport or storage, the modern study of brain mechanisms in emotion, cognition
and behavior remains almost as mysterious as ever.
* * *
The only human mind-brain observations that are doubted consistently, and
treated as unpublishable by the editors of the journals of science, are those that
result from direct human experience using subjective reports from within. They are
called unscientific. Often ignored are logically consistent mathematical and
computational contexts, which, as abstract and general tools of thinking and
imagining, have the capacity to frame, rigorously define and describe thinking about
both the subjective and objective aspects of brain-generated phenomena. These
mathematically configured metaphors can lead to consistencies in description, this
is behaving like that, in what are called equivalence relations expressed both as
intuitive imagery; for a concrete example, a one holed bagel and one handled tea
cup are topologically equivalent because, sculpting in clay, they can be smoothly
transformed into each other. We have seen that invariant measures in computable
statistical flows can come out of a mess of data. Professor Paul Rapp of the
University of Pennsylvania has been able to mathematically encode the verbal
content of the patient’s free associations and the therapist’s responses, using tape
recordings of hours of psychoanalytical treatment. Examples of quantifiable qualities
found useful in this regard involve a variety of characteristic statistical patterns in
what are called entropies and information as well as various measures of what with
a wide range of definitions is called complexity. These quantifiable properties,
measures, can help in the struggle with the intrinsic tension of Absolute Reality
between the “eternal emptiness of form and the eternal form of emptiness.” We
resort to measures of entropy, information and complexity when confronted with our
ignorance, “emptiness,” great or little, with respect to either cause or result, about
what exactly is going on. Entropy in its forms relevant to information quantifies our
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ignorance, the emptiness and its mystery. Computations of the entropy of systems
in motion convert questions and answers concerning the detailed workings of the
leg’s neuromuscular machinery to global statistical descriptions of more abstract
thematic motifs, forms, expressed in the dance. Patterns of behavior of these
properties can suggest intuitive ideas and imagery about global mechanisms,
approach/avoid, smooth/discrete, wild/tame, as well as correlated and objective
physical observables.
To learn more about this abstract, topology tinged (none numeric) style of
model building, we can go to school on a long studied physical example. It connects
a simple and well understood real world observable with abstract statistical patterns
resulting from motions using the one-to-one correspondence (the equivalence
relation called isomorphism) between their entropies. As we have discussed, the
Stanford mathematician and Field’s Medal Winner, Donald Ornstein, proved that in
statistical studies of even point-to-point unpredictable, chaotic systems, entropy is
the only isomorphism. The hardware of this physical example is what the statistical
physicists call a dilute gas of some fixed number, n, of uniform hard spheres,
moving scatterers, that, absent of dissipative friction, wander continuously around,
changing their directions when bumping into each other. In a two dimensional
bounded arena of randomly rolling balls, this game has been called Sinai’s billiards.
It was named for previously mentioned Ya Sinai, an eminent Russian
mathematician He is now at Princeton and was previously a student of Andrei
Nikolaevic Kolmogorov, the Russian guru of many of the Twentieth Century’s worldclass
Russian mathematicians. Kolmogorov axiomatized the field of probability and,
more relevantly, initiated the theory of statistical descriptions, the ergodic theory, of
nonlinear dynamical systems. In the language of statistical physics, we will see that
the same system produced by high number of elements executing Newton’s
deterministic laws can be generated by a so-called random system such as that
resulting from flipping a suitably biased coin. Our example can also serve as a
metaphor, used extensively in the mathematical biology of the late Professor Art
Winfree, for the temporal features of life on a topological circle: the natural
irregularities of the recurrent beat of the heart, the in and out breathing of lungs, the
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up and down voltage of brain waves, the pendulum swings of our blood hormone
levels, the cyclic procession of our days, months and years and at large scale, our
body’s journey from dust to dust.
The angular deviation theta, θ� �from the initial reference direction of a single
moving sphere, gets rotated to a new angle theta, θ
→ θ
’ �by a collision with another
sphere. It has been shown that the new angle θ ’ is the previous angle, ��times twice
the average distance traveled between collisions called the mean free path, here
symbolized by delta, δ, divided by the diameter, D, of the sphere. Algebraically,
θ ’ �=�
2δ θ ��the new angle is equal to twice the mean free path divided by the diameter
D
of the spheres times the original directional angle of the sphere’s motion. �If we
symbolize the time between collisions with tau, τ, after an elapsed time of
experimental observation, t, we can say that the deviations from the initial direction
of the sphere changes like ( 2 δ � t/τ . The exponent, t/τ, represents the time of the
D
experimental observation divided by the average time between collisions of the
spheres, i.e. the time we’ve been watching, t, is expressed as units of inter-collision
interval, τ. Of course, the circular deviation in the angle from the initial direction
rotates repeatedly around a circle as the number of collisions increase. If a point on
a circle marks the angular change resulting from each collision and the system runs
long enough, it has been shown that the circle will eventually be completely covered
by points.
An estimate of the entropy, S, being generated by each sphere labeled with
some index i, Si, is positive because the recurrent motion is deviating continuously
from the initial direction. It can be computed for each sphere as the logarithm of the
intercollision time-averaged deviation from the initial direction, S i = 1 log( 2 δ
) and the
τ D
entropy of the whole n hard sphere system is the sum of the n entropies, which can
be expressed as n× S i . If we keep books by registering the points when each
sphere’s  ’ makes a stop on the top half of the sphere’s circle as 1 and the bottom
half as 0 (and we must arbitrarily decide between 0 and 1 if it falls exactly on the
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division between top and bottom and do so in a consistent way), then we can keep
score with a random looking binary series such as 11001001010…. that describes
the sequence of rotations. The advantage that accrues by doing so is that this coin
flip counting eliminates details in favor of a computable over all measure and
supports several forms of entropy calculations for its use in deciding if this system is
behaving like that system, an equivalence relation. One can imagine a series of coin
flips with 1 being heads and 0 being tails such that the statistics of a characteristic
series is determined by the fairness of the coin. As noted above, Donald Ornstein’s
famous theorem says that the entropy of these kinds of hardware and software
systems is the only general basis for finding correspondence between
characterizations of two such irregularly behaving systems. The important idea here
is that a series of 1’s and 0’s may not be identical but the two systems can be
isomorphically equivalent with respect to their entropy.
Notice again that the physical process of hard spheres bouncing off each
other on a flat surface has been captured by an abstract representation in binary
numbers that, like a series of coin flips, can be quantified as entropies (which would
be maximal for an ideal, fair coin). After describing the process of real number
representation by the binary code, we will show how entropies can be computed for
these binary series. We remind ourselves that we are struggling to obtain some kind
of knowing in a representative system manifesting the tension and mystery between
emptiness and form.
We can translate all finite real numbers into this language, making them
accessible to standard entropy computations. The following discussion of the
process of transforming numbers into binary series is in the spirit of the famous
number theory theorem that every natural number (the positive integers such as 1,
2, 3, 4…) can be expressed as the sum of at most four squared numbers. Encoding
any number by a series of 0’s or 1’s in what is called a binary transformation, begins
with its separation, called partition, into a sum of powers of 2, for example, 100 = 64
(2 6 ) + 32 (2 5 ) + 4 (2 2 ). A short hand description of this sum begins with a form
indicating the presence or absence of each successive power by a 1 or 0 coming
before the relevant power of two; i.e. 100 = 1 × 2 6 + 1 × 2 5 + 0 × 2 4 + 0 × 2 3 + 1 ×
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2 2 + 0 × 2 1 + 0 × 2
0 (in which the last term, arbitrarily, is 2 0 = 1, since anything to
the power 0 = 1). This can be written even more simply as a series of 0’s or 1’s,
their presence indicating whether the power represented by each place in the left to
right descending sequence of powers of two participates in the sum of the partition.
It is in this way that in binary numbers, 100 = 110010. As another example, if we
similarly partition the decimal number 729 = 512 (2 9 ) + 128 (2 7 ) + 64 (2 6 ) + 16 (2 4 ) +
8 (2 3 ) + 1(2 0 ), we find that its binary transformation results in 729 = 1011011001,
the 0’s representing the descending powers of two that are absent in the powers of
two partition. One can compute the binary representations of lower valued numbers
immediately; for example, 4 = 1 × 2 2 + 0 × 2 1 + 0 × 2 0 so that there is a 1 in the
multiply-the-power-of- two column and 0 the power 1 and power 0 columns so in
binary representation, 4 = 100. Similarly, 6 = 1 × 2 2 + 1 × 2 1 + 0 × 2 0 making the
binary transformation of 6 = 110.
It was the co-inventor (with Isaac Newton) of the calculus, Gottfried Wilhelm
Leibniz, in about 1665, who fully developed the binary representation of all decimal
numbers. In a state of wonderment about the simplicity, power and completeness of
this 1 and 0 encoding, he is said to have the beliefs that 0 symbolized the
emptiness of the universe’s beginnings, 1 represented the complete fullness of God
and that this transformation served as metaphoric evidence consistent with God’s
creation of the universe out of nothing.
The simplicity of binary expressions as in the dynamics of hard spheres or
rotations on the circle as well as the transformations such as 729 = 1011011001
make them propitious for exemplifying the methods for computing the entropies of
the growth rate of the possible, called the topological entropy, H T , and the probable,
the metric entropy, H M , which was introduced in a previous chapter called “Sensual
In-Between Entropies.” The following exemplify the computations of measures of
topological and metric entropies, H T and H M , another computable idea called
algorithmic complexity, AC and finally, the well known (to statisticians) standard run
score, src. Their descriptions have as their purpose a demonstration for the reader
that these apparently abstract, perhaps nebulous sounding, words can be
transformed into well-defined, concrete, quantitative and computable form of reality.
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If acceptance of this idea does not constitute a problem for the reader (and you do
not find it fun to follow along with a computer math program and/or a pencil), then
the following several paragraphs can be quickly scanned or skipped entirely.
The computations of H T and H M begins with keeping track of how many 0 →
1 and 1 → 0 transitions are found going from left to right in the binary series. For
example, in the binary expression of 729, 1011011001, one starts counting with a
1→ 0 transition followed by a 0 → 1 transition and then a 1 → 1 transition and so
on. A useful way to record the count is via entries into a 2 × 2 matrix for score
keeping in which the horizontal rows are labeled 0 on top and 1 below and the
vertical columns are labeled 0 on the left and 1 on the right. The number of each
kind of transitions (from the vertical label to the horizontal label) are counted and
summed in the appropriate box of the two box by two box matrix; for examples: for a
0 → 0 transition, a tally mark is entered in the upper left corner of the matrix; for a 0
→1 transition, a tally mark is entered in the upper right corner; a 1→ 0 tally goes in
the left lower corner and a 1→ 1 is tallied in the right lower corner. The resulting
transition incidence counting matrix, M t for the 729 binary transformation series
looks like M t = 1 3
indicating one 0 → 0, three 0 → 1, three 1 → 0, and two 1 → 1
3 2
transitions have been tallied. Although this series alone is too short for computing
reliable statistical measures, if we assume that the pattern of transitions observed in
this short series is stationary, that is its transition behavior will remain the same if
the binary series continued on to be infinite in length, the assumption being that the
dynamics of now will be the same as always, 729 will stay 729, then we can use two
forms of this transition matrix in the computation of the topological entropy reflecting
the growth rate of the possible, HT, and the metric entropy from the statistical
weights of allowed choices among them, the probable, H M .
To obtain the entropy representing the growth rate over time of the new
possibles, the computation of H T , the topological entropy, involves first transforming
M t into an transition incidence matrix, M t,i a 0 or 1 matrix indicating whether each
box has been entered at all (or not). Since in the binary representation of 729, all
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four boxes of M t are occupied, the M t,i =
1 1
1 1
indicates that all four kinds of
transitions are possible. Since we remain in the context of a 0,1, two state system,
the growth rate of the possible equals the logarithm, base two, of the sum of the
entries in the boxes of the left-top-row to right-bottom-row diagonal called the trace
and HT = log 2 (1 + 1) = log 2 (2) = 1. Consistent with intuition, since every transition is
possible, the topological entropy of M t as indicated in its M t,i is maximal (= 1).
Another expression equal to the sum of the trace (the sum of the upper left to lower
right diagonal) in a square matrix, is its leading eigenvalue, most often symbolized
with a lambda, λ 1 . The logarithm of the leading eigenvalue of the transition
incidence matrix is equal to its topological entropy. Symbolically, H T (M t,i ) = log 2 (λ 1 )
= log 2 (2 ) = 1. Standard elementary linear algebra texts describe how to compute
eigenvalues, these relations and related operations as well as their foundational
theorems.
Before computing the entropy of the distribution of probabilities among the
possibles as the metric entropy, H M , let us notice again that the occupancies in the
four entry boxes of the transition matrix M t are not uniform, M t =
1 3
3 2
. This leads
naturally to the intuition that for this series of binary transitions, HM, in contrast with
H T , will not be maximal, i.e. not equal to 1 and the nonuniformity of H T and H M is a
computational expression of what we mean by a state of in-between entropy. These
entropies are identical and their difference = 0 for transitions reflecting maximal
entropy, as might be realized in a very long series of fair coin flips in which the
entropies = 1. Entropy will be minimal when flipping a two headed coin, here the
entropies = 0. More compactly, the non-uniform probabilistic, metric entropy,
differing from the maximal topological entropy indicates that the system is in a
dynamical state of in-between entropy, written as H T - H M ≠ 0.
In the computation of the metric entropy, H M , the M t is transformed into a
transition probability matrix, M t,p , called a Markov matrix named for one of the two
great Russian mathematicians, both students of Pafnuti Lvovich Chebyshev, the
Markov brothers. The entries of each row in the M t are transformed into transition
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probabilities, so that the sum of the decimal fraction parts of all the boxes in each
horizontal row add up to 100%, or as a real number, 1.00. Recall that in the
example we’ve been using, the binary expansion of the natural number 729, the
1 3
transition incidence matrix is M t = and its Markov matrix is top row, 1/4, 3/4
3 2
and bottom row 3/5 , 2/5, i.e. M
t,p =
0.25 0.75
. Matrix multiplication of Mt,p by itself
0.60 0.40
repeatedly is equivalent to tracking the temporal evolution of the transition matrix’s
probabilities until the resulting matrices move toward, converge onto, a steady state;
each self matrix multiplication step represents what results from the passage of one
unit of time. The convergence to equilibrium values is continuous and gradual.
When the steady state is reached, both rows become identical. For this example,
M t,p × M t,p or M t,p 2 =
0.5125 0.4875
; M
0.3900 0.6100
t,p 4 =
0.4527 0.5472
; Mt,p 8 0.4445 0.5554
= ;
0.4377 0.5622 0.4443 0.5556
M
t,p 16 =
0.4444 0.5555
0.4444 0.5555
which for the first four decimal places remain the same for
additional times of self multiplication. Note the convergence of the top and bottom
rows to the same asymptotic values. Books discussing the multiplicative and other
behavior of these nonnegative matrices are numerous and frequently appear in
matrix algebra texts under the rubric of the Frobenius-Perron theorems.
Using the entropy formalism of Claude Shannon as developed previously, H M
is computed as the sum across either of the identical rows of each probability times
its logarithm, �ρ � �× log(ρ � �����ρ 2 × � �log(ρ 2 )) remembering from above that we are working in
base 2 logarithms and to change the minus sign (resulting from taking the
logarithms of decimal fractions) to plus: H M (M t,p ) = .4444 × log(.4444) + .5555 ×
log(.5555) = .9911 The nonuniformity of the box occupancy probabilities is reflected
in the difference between the topological (maximal estimate) and metric (minimal
estimate) entropies and is therefore quantifiable and computable: H T - H M ≠ 0 = 1.00
- 0.9911 = 0. 0089. If the maximal and minimal estimates of the entropy were equal
and all the probabilities boxes in each row asymptotically contained the same
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probabilities as in M t =
0.5 0.5
, it would retain these values across an infinite
0.5 0.5
number of self multiplications such that HM = .5 × log(.5) + .5 × log(.5) = 1 and H T -
H M = 1.00 – 1.00 = 0. 0.
Complexity is a more general and variously defined descriptive expression
than that of the topological and metric entropies and as such brings with it many
kinds of definitions and computational approaches. One choice that’s intuitively
appealing assumes that the relative complexity of an expression representing, say
an outcome of an observation or experiment, is reflected in the minimum length of
the most compressed program (algorithm) from which, given a suitable dictionary of
symbolic equivalencies, one can reconstitute the original expression. Increases in
what some have called algorithmic complexity, AC, are reflected in the growth of
this minimally descriptive symbol series length. Karen Selz’s approach to
compression and AC, similar to one proposed by Paul Rapp, involves the
identification and symbolic representation of repeated blocks of symbols called
words. For example, given an arbitrary, exemplifying binary series:
011011101010001010101001001010011, we first find the longest repeated word
[1010100] and represent it with the symbol, a, yielding a shortening in the original
series, 011011a010a1010011. The next longest repeated word is [011] is replaced
with b, yielding a further compression, bba010a1010b. The next remaining binary
word is of length equal to the previous one, [010], which, when replaced by c results
in the series bbaca1cb. This can be further compressed to the final representation
with four symbols and for the sequentially repeated b, one exponent of degree two,
b 2 aca1cb. From this representation and a dictionary of letter equivalent words, the
original binary expression can be recovered. For a quantitative index of the
algorithmic complexity, AC, of the compression, Selz computes the sum of the
number of distinct symbols plus the sum of the natural logarithms of the exponents:
4 + log (2) = 4.6931. The binary representation of 729, 1011011001, discussed
above, is compressed by making two [101]’s = a and two 0’s = b resulting in a 2 1b 2 1.
Having three distinct symbols, a,b, and 1, and two exponents of two, its algorithmic
complexity is equal to, AC = 3 + 2 × log(2) = 4.38.
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In addition to H T , H M and AC, if computable in a meaningful way, the
deviation of the binary series under study from the idealized random behavior of a
fair coin could serve as another index of complexity. Common descriptions of the
amount of randomness in a series are indices of run length If a run length is defined
by number of elements in a series of the same symbol before it stops, counting the
number of run boundaries by reading along the binary series and counting the
number of switches from 0 →1 or 1 → 0, then the binary expression of 729,
1011011001, has six runs. The great analytic probabilist, William Feller, among
many others, including the distinguished 18 th Century Swiss family of
mathematicians, the Bernoulli’s, proved that computing a standard run score, srs,
involves three terms, the theoretical expectation, E, of the number of runs, r, that is
E(r), the number of runs actually observed, Obs(r) and the variance of the
expectation of the number of runs, Var( E(r). If the srs is less than zero, then the
binary series is more random than that resulting from the flipping of a fair coin.
Interestingly, when a normal group of subjects are instructed to simulate what they
think of as a random coin flip determined series of 0’s and 1’s, their srs tends to be
lower than zero, over-estimating the degree of irregularity that randomness
represents. Long runs occur by chance far more often than intuition would dictate. If
srs is more than zero, than the binary, coin-flip series is more ordered than random.
If srs equal to zero, the binary series is not discriminable from fair coin flipping
randomness.
The expected number of runs, E(r), can be estimated by a fraction formed by
twice the product of the number of heads times tails divided by the sum of the
heads and tails to which is added one. That is, E(r) = 2 × 6 × 4 + 1 = 5.8. The average
6+
4
variation around this expectation called the variance, Var, of the expectation,
Var(E(r)), is estimated by a fraction formed by (take a breath) twice the product of
the number of heads times tails × twice the product of the number of heads times
tails minus the number of heads and minus the number of tails, all over the product
of the sum of the heads and tails squared, times the sum of the number of heads
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(2× 6× 4)(2× 6× 4 −6−4)
and tails minus one. That is, Var(E(r)) = = 2.03 . From these
2
(6 + 4) × (6 + 4 −1)
three terms, we compute srs =
Er () − Obsr () 5.8−
6,0
= =− 0.140 . We conclude that
Var( E( r) 2.03
the standard run score of the binary series is less than zero and therefore more
random than the expected random behavior of a fair coin.
Recall from the last chapter that Karen Selz, Martin Paulus and others have
shown that various personality types and psychiatric diagnoses are associated with
characteristic deviations of srs from zero. When the winners of the 2002 Annual
World Rock, Paper Scissors Championships held in Montreal Canada were
interviewed, they said that sensing their opponent’s characteristic style of deviations
from randomness in what we would call the continuum from maximal to minimal
entropy determined their successes. We characteristically use all of these measures
to estimate quantitate the deviation from randomness standard run score, srs,
algorithmic complexity, AC, as well as H T and H M ,, the topological and metric
entropies.
* * *
The encounter with mystical Absolute Reality, though sought by arduous
contemplative and other practice, emerges spontaneously, most often during times
of apparent mental emptiness, detachment, a state in which rationally instructive
thought and the choral background of brain voiced, emotion-ladened, commentary
have disappeared into the entropic soup of formless silence. It is this indescribable,
ineffable, stillness that we think serves as the psychophysiological anlage of
mystical experience. The mathematical systems yielding quantitative metaphors,
descriptive ideas about dynamical entropic statistical emptiness and form inspire the
use of mathematical structures in place of localized lumps in brain meat as
personalized icons of doing.
Our wedding of well-defined mathematical objects to metaphoric elements of
more general nonverbal intuition has a long tradition. Rene Thom’s 1990 book,
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Semiophysics, discusses mathematical mechanisms and their representations in
mind and the real world, analogizing mathematical objects and the intuitions they
generate to mechanical tools. Similar ideas are found among the four liberal arts of
the ancients: Number, Geometry, Music and Cosmology. The epistemologies of all
four require, then and now, the intuitive use of mathematical objects, conscious or
unconscious. Examples can be found in conceptual issues of Geometry and
Number with implications for relationships between man’s physical and
psychological worlds. One set of articulations were attributed to the shapes of
Platonic solids found among the Neolithic stone circles in Aberdeenshire, Scotland,
2000 years before Plato. Each symbolized particular physical and psychological
themes. All manifested equal edges and every face of each solid was the same
perfect polygon. The solid with four equilateral triangles manifesting four vertices
and faces, the tetrahedron, represented the physical element, Fire, and the
personal psychological climate of a choleric, fiery nature. A Platonic solid composed
of eight equilateral triangular faces, two tetrahedrons annealed, the octahedron,
signified Air in physical composition and optimistic hopefulness in psychological
disposition. Six square faces together making a cube, evoked the elemental
physical component, Earth, and its human expression as a phlegmatic, apathetic
personal style. Twenty faces, all equilateral triangles, constitute an icosahedron
indicating Water and a dominant feeling state of melancholic sadness. Like
onomatopoeic words and pictorial script, the three dimensional geometry of these
Platonic solids feel like what they came to symbolize.
The personality styles symbolized and evoked by the Platonic solids continue
to be used to this day. For example, they compose the basic elements of the
constitutional categories of remedy in homeopathic medicine as introduced over
200 years ago by Dr. Samuel Hahnemann in his classical Organon of the Medical
Art. The assignment of clinical remedy in homeopathic treatment combines
consideration of the presenting physical symptoms and signs, the what, with
intuitive discernment of the patient’s constitutional type, the who. To the
homeopathic physician, tetrahedral fire is suggested by the traits of personal
magnetism, courage and inspiration as well as egotism, strong desire and rage.
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Octahedral Air people intellectualize objectively in confident and insensitive
aloofness. Those symbolized by cubic Earth are realistic and practical, a what-yousee-is-what-there-is
belief along with rigid, materialistic ways. Icosahedral Water
types experience emotions strongly and are sensitive, intuitive, nurturing and can be
overly sensitive and dependent.
What intuitions and observations relevant to self, subjective and objective,
emanate from the stylistic properties of feelings as derived from a time series of
observations of their associated actions suggested by their statistical measures, H T ,
H M , AC and srs? The yield is rich and unexpected. We find an enjoining of values of
these measures, characteristic and invariant for each person, with the brain and
behavioral actions of entheogenic agents and Zen meditation in contrast with
worldlier focused attitude adjusting experiences and drugs. The range of their
potential values helps rationalize a person’s inclinations along the continuum of
attachment and detachment. This quantifiable dimension augers positively and
negatively with respect to the requirements for mystical experience as poetically
described by the ambivalent warrior prince, Ardjuna, in conversations with Lord
Krishna in the Bhagavad-Gita, the most famous and influential component of the
Mahabharata of Hindu scripture, A similar theme relevant to the occupancy of a
propitious range of values for the measures, H T , H M , AC and srs, is found in what is
often called the Second Nobel Truth as explained by the Buddha, Siddhartha
Gautama, in lectures recorded in a deer park near Benares. We begin with the
results of some drug experiments conducted by behavioral neurophysiologists and
end with suggestions about the intuitive relevance of the conceptual content of
these measures to the universals of mystical experience and perhaps to elements
of spiritual transformation.
As described previously, the brain and behavioral process of habituation is
characterized by a decrease in the strength of an observable response to the
repetition of an evocative stimulus. Imagine the decrease in our startle responding
when a once unexpected loud noise continues to occur. Sir Charles Sherrington,
the early Twentieth Century British pioneer in neurophysiology showed that animals
and humans gradually stopped the withdrawal of their limbs with stimulation of its
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skin when it was repeated several times. Columbia University’s Nobelist in the brain
sciences, Eric Kandel studied the neural mechanisms of habituation as a primitive,
accessible and fundamental example of learning, the association of a nonresponse
to a usually evocative stimulus, in Aplysia californica. The sea snails learned not to
respond to a local irritation with a gill-withdrawal response when exposed to it many
times. They learned to stop paying attention to the perturbation. The background
noise appears to disappear after a little time in the Mall. Though his exploration of
its synaptic mechanisms involved the neural circuit of the gill-withdrawal reflex in the
marine snail, its generality and human relevance is well established. Hundreds of
papers can be found reporting the results of studies of habituation in normal
humans under all kinds of circumstances as well as in psychopathological
conditions. That it samples something both fundamental and persistent is suggested
by studies in children by one of Kandel’s students, Michael Lewis. He found that the
rate of habituation of a startle response to a bright light in one-year-old human
infants predicted success in many kinds of learning and other cognitive functions
when the children were tested again at the age of four. Pavlov’s experiments
studied habituation of the classically conditioned salivary response to meat powdercoupled
bell sounds in dogs in which the bell was followed by nothing, not only led
to inhibition of the salivary response with unreinforced trial repetition but
generalization of the inhibitory state such that dogs were observed to freeze in
motionless catatonic states for hours. In the language of our statistical measures,
the fixation of the dog’s behavior would manifest minimal entropy in the form of H T =
H M = 0 and the lowest complexity values for AC and srs. Entheogenic agents like
LSD or mescaline inhibit the process of habituation and fixation, maximizing the
entropy of behavioral measures, H T , H M → 1 and high complexity values for AC and
srs.
Mark Geyer and David Braff, Professors at the University of California in La
Jolla and Michael Davis, a Professor at Yale’s School of Medicine, found that
entheogenic agents, such as mescaline and LSD, as well as naturally occurring
indoleamines, such as DMT, which occurs naturally in human brain, prevented
habituation of startle responding in mammals. Each sound repetition was treated as
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though it were new. The baby is Buddha is an Eastern philosophical aphorism that
captures the fresh spiritual state of each moment’s openness and readiness, the inbetween
entropies for new information surprise. Geyer and Martin Paulus found that
entheogenic agents such as Ecstasy also increased the complexity of the patterns
of spontaneous motor movement made by rats exploring a bounded space. Recall
that they partitioned the floor to document the exploratory motion in the context of a
sequence of location transitions, readying the data for the computation of some of
the measures previously described. Following the administration of entheogenic
agents, the partitioning of the space that the animals were exploring, into a lattice of
discrete boxes and the encoding of each square with a symbol, the computable
entropic and complexity measures such as H T , H M , AC and srs were increased. In
contrast, the administration of amphetamine-like stimulants led to a different kind of
behavioral activation than that induced by entheogenic agents. The measures of H T ,
H M , AC and srs reflected decreases in entropy and complexity. As University of
California’s David Segal and others documented in the 1960’s, high doses of
amphetamine led to animals into in a minimal entropic state, they were frozen in
stereotyped rocking, nodding and circling motions. High dose amphetamine-treated
humans develop rigid fixation of ideas, low H T , H M , AC and srs, in man this is seen
as inescapable obsession and paranoid delusion. There is considerable medical
evidence that Hitler took large doses of amphetamine (Benzedrine) daily for the last
20 years of his life.
The entheogenic drug-induced phenomena of naïve openness and absence
of fixation, states of high entropy and complexity, behavior generating higher than
control measures tending toward maximal values of H T , H M , AC and srs , are
subjectively reflected in the results of personal experiments of University of
Chicago’s Heinrich Kluver as described in his Mescal and Mechanisms of
Hallucinations (1966). Observing himself after the self administration of a crude
preparation of peyote cactus, he said that it led to glad feelings of unfamiliarity and
a marked reduction in his tendency for boredom (habituation), a detachment from
old ways of thinking and a new openness to a rush of seen again for the first time
experiences. Everything in his personal world, no matter how mundane, became a
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source of new interest and fascination. New thoughts replaced old ideas in a
continuing process of new formulation. All of these things feel like they emerge
spontaneously, making ideas about being born again and personal renewal
concrete. We remember that Timothy Leary and his wife in their privately circulated
pamphlet, Neurologic, described their entheogenic drug-induced escape from the
habitual order as supported by the learned and established “…mental-manipulative
and socio-sexual brain circuits…,” an escape to a fresh new planet of possibilities.
Louis Lewin, the early Twentieth Century German pioneering ethnopharmacologist
described his subjective responses to peyote as a flood of lively, numerous, random
fantastic creations of perception and thought, all demanding his fresh attention.
To complement these subjective reports, experimental tasks involving
habituation, such as the disappearance of a brain wave sign of arousal to sound or
light stimulation, called alpha blocking, the eyes-closed resting pattern of 8-14
cycles per second, hz, waves perturbed into the arousal pattern of >20 hz, did not
habituate when the subjects were pretreated with entheogenic drugs. This finding
was also true for the results of years of meditative practice. In his 1974
Psychophysiology of Zen, Hirai reported that Soto Zen monks, after many years of
practice in mindful, one pointed, be here now meditation, unlike normal controls,
continued to show alpha blocking surprise, brain wave arousal patterns, throughout
the course of repeated stimulation with auditory clicks.
James Austin in his monumental book, Zen and the Brain (2000) summarizes
other studies of habituation in TM practitioners and other mediators in which eyes
open versus eyes closed, the set and setting and variations in other experimental
variables blurred these results to some degree. He develops the case that years of
meditation-induced brain states of emptiness, we would say of maximal entropy and
minimal form, set the stage for the ecstatically insightful flood accompanying the
sudden insight into a Zen koan’s solution or the transcendent startle induced by a
roshi’s shout. A meditative struggle concerns how one can think about not thinking.
That is, thinking of nothing. This is generally thought to be the most important part
of Zen meditation, called zazen. Achieving high values for brain and behavioral H T ,
H M , AC and srs supply the formless infrastructure for ecstatic transformation.
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In healthy people, an awareness of self is not lost during this time of invasion
by and fusion with what feels like an independent agency. At full force, the mystical
experience is transfixing, tending to paralyze movement and speech, and at the
same time bringing with it the capacity for clear sensory and sensory-integrative
lucidity. This new seeing brings previously unnoticed things to attention and makes
old things new. Perhaps most striking is the passive (unsought) experience of the
unification of erstwhile disparate, apparently unrelated thoughts and feelings. The
yield can be the sudden emergence of deep relationships between apparently very
different constructs, beliefs and formalisms leading to unanticipated and unsought
integrative connections. In mathematics, this experience can lead to entirely new
kinds of theorems and proofs; in the physical and biological sciences, a previously
unseen organization of the data generating new global relationships and potential
scientific laws. In our spiritual life, the ineffable richness of the direct experience of
God.
Mysticism-negative interpretations of these experiences have always been
attendant. To the extent that the mystic’s inward turn is seen as a detachment and
implicit derogation of the external, consensually real world, it is often seen as
alienating from established institutions of religion and government. Psychoanalytic
practitioners may label it a regression to primary narcissism. Most churches tend to
discourage its practice as counter to the dominant social hierarchy and its
governance. Governments pass laws against its practice and manifestations, a
current example being modern Chinese governmental reactions to the Tibetan
Buddhism of the Dalai Lama and the yogic practices of the Falon Gong. Agencies of
established society such as the institutions of licensed medical practice make the
dominance of the inner world of mysticism subject to diagnoses ranging from the
narcissistic character disorders to interpretations of the reported extraordinary
experiences as manifestations of schizophrenia, manic-depressive disorder or
temporal lobe epilepsy. Rejection and fear of the transcendent states lead to
uninformed and politicized anti-narcotic laws, grouping heroine and cocaine with the
entheogenic (recall: engendering connection with the sacred within) agents such as
the Huichol Indian’s peyote and the Amazonian Indian’s yage, obstruct and socially
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taint the personal use of plants and practices that facilitate access to the mystic
way. Rational, socially responsible and otherwise kind and tolerant Presbyterians,
Unitarians and Reformed Jews can be suspicious and rejecting of what appears to
them as the politically tinged mass hysteria of praying in tongues and other rituals of
Charismatic Christian rebirth and renewal or the ecstatic states of Orthodox Jewish
chant-dancing.
Modern brain and behavioral scientists, remaining under the philosophical
spell of logical positivism and its requirement for operational definitions and
(external) experimental disconfirmability, operate from the position of strong doubt
when mystical experience is addressed. What is striking and strange about how
science plays the game of mysticism research is exemplified by the publishable
increment in credibility concerning a meditation-induced change in state of
consciousness when Boston University’s William Benson reported the
accompanying relaxation response, a sudden decrease in heart rate---much like the
dive reflex of a seal or what the heart rate does when you duck your head suddenly
forward into a sink full of water. Decades are spent getting professorial tenure for
research yielding things we have already experienced and know directly and for
ourselves. Recall that the existence of visual imagery in the human, doubted by an
experimental psychology of the time in which William James self-exploratory
observations were viewed as revolutionary, was made more credible by evidence
for the existence of a subjective spatial metric: verbally reporting subjects, when
timed, took longer in their minds to go from one room to another one that was down
the hall then going to the room that was immediately next door.
We use brain chemical, pharmacological, neurophysiological and
neuroanatomical localization and computation of characteristic statistical patterns in
time dependent brain and behavioral observations to the same end.
Further Readings for SOME ENTHEOGENIC ENTROPIES
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Phantastica: A Classic Survey on the Use and Abuse of Mind-Altering Plants. Louis
Lewin, Park Street Press, Rochester, Vermont, 1998 (First published in 1924)
Indole(ethyl)amine N-methyltransferase in the human brain. M. Morgan (Poth) and
A. J. Mandell, Science 165:492-493, 1969
Enzymatic formation of tetrahydro-beta-barboline from tryptamine and 5-
methltetrahydrofolic acid in rat brain fractions. L.L Hus and A.J. Mandell, J.
Neurochemistry 24:631-636
The Sacred and Profane, The Nature of Relgion. Mircea Eliade, Harvest Books,
Harcourt, San Diego, 1957
Hashish and Mental Illness. J. J. Moreau, Raven Press, N.Y. 1973 (First published
in 1848)
The Neurochemistry of Religious Insight and Ecstacy, A.J. Mandell in Art of the
Huichol Indians, Fine Arts Museum of San Francisco, Abrams, N.Y. 1978
Altered States of Consciousness: A Book of Readings. Charles T. Tart, John Wiley,
N.Y. 1969
Soul; God, Self and the New Cosmology. A. Tilby, Doubleday, N.Y. 1992
Pihkal: A Chemical Love Story. Alexander Shulgin and Ann Shulgin, Transform
Press, Berkeley, CA 1991
Psychochemial Research Strategies in Man, A. J. Mandell and M.P. Mandell,
Academic Press, N.Y. 1969
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The Biology of Transcendence, J. C. Pierce, Park Street Press, Rochester,
Vermont, 2002
Psychiatry and Mysticism, S.R. Dean, Nelson-Hall, Chicago, 1975
Zen and the Brain, James H. Austin, MIT Press, Boston, 2000
Perspectives in Biological Dynamics and Theoretical Medicine. Eds. S.H. Koslow,
A.J. Mandell and M.F. Shlesinger, Ann. N. Y. Acad of Sci. Volume 504, 1987
Consciousness and the binding problem, W. Singer, Ann. N.Y. Acad. Sci. 929:123-
146.
Mixing properties in Human Behavioral Style, Karen A. Selz, U.M.I., Ann Arbor, MI.
1992
Dynamical Systems and Ergodic Theory, M. Pollicott and M. Yuri, London
Mathematical Society, 1998.
Introduction to the Modern Theory of Dynamical Systems, A. Katok and B.
Hasselblatt, Cambridge University Press, Cambridge, 1995
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CHAPTER 6:
PENTECOSTAL PHASE TRANSITIONS
By their late teens, my two offspring, sons of an Alcohol Anonymous, born
again, originally Christian Science mother and a spiritually struggling and mostly
secular Jewish psychiatrist father, had been unfulfilled in their hungry search for the
experience of a personally meaningful God. After years of perhaps too academic
conversations with their parents, visits to a variety of houses of worship, talks with
University of California religion professors and evenings with a Ph.D. psychologistrabbi
and friends at the neighborhood synagogue, they turned somewhere else.
Some of their high school friends who were Evangelical Christians took them to
their Assembly of God, Pentecostal and other Christian, direct experience of God,
churches. They came to love what they sometimes called their Wednesday night
and Sunday morning “rock and roll,” services.
Struggling with the post-Vietnam cynical mistrust of authority and the
marijuana apathetic nihilism of the 60’s and 70’s, and clearly not enticed by what
they regarded as their father’s vacuous mélange of New Age Eastern Religions and
secular brain science, they spoke about their sudden and life-changing experiences.
They studied, memorized and quoted the Scriptures as part of their commitment to
their word churches. As erstwhile cynical teenagers, now positive and brimming with
faith, I secretly called it denial, they described what was happening to them as New
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Birth. They told me that, paraphrasing Paul in Romans, they had been saved and
were living New Life, not earned by good works as in Hebraic Law, but by faith
through God’s Grace. Jesus had “paid their bills” through His sacrifice at
Gethsemane. They both tried to explain inexplicable feelings of new energy, the
unseen hand of spiritual guidance and peace. One told me that the wind of the Holy
Ghost had taken him to the front of the pulpit, tearfully, thankfully, on his knees, to
accept Jesus as his personal Savior. They described how they had opened their
lives to the spiritual strength of living in Jesus.
Many things about them changed: their tastes in food, from hamburgers to
vegetables and fruit; from the jazz of John Coltrane and McCoy Tyner and the
cynicism of Frank Zappa’s “…only fourteen and knows how to nasty…,” to playing
strum guitar and singing the hymns of Wednesday night healing services; from t-
shirts hanging out of raggedy, Southern California, boutique store purchased,
stressed jeans, to polished dark shoes, starched white shirts and gray or tan khaki
slacks, sometimes with ties. They became cool, respectful, rational and more
distant with me. They repeated often the scriptural story about young Jesus,
accidentally separated from his parents on a visit to Jerusalem. When by standers
asked Him about where His parents were, He answered, “I have no mother and
father.” They told me that they, like God’s son Jesus, were filled to completeness
with the Father and the Holy Ghost.
On one hand, their experiences sounded like those of the activated mind
state of Abraham Abulafia, a suddenly emergent Nevesh and my father’s
metaphysical talks about personal transformation. My personal secularcomputational
brain God spoke to me of the mechanisms of sudden personality
change, a phase transition in complex systems, in the context of the nonlinear
dynamics of brain and behavior. On the other hand, their global changes in mind felt
both alien and threatening. When I came to learn their churches’ full list of
expectations, rules, requirements and sociopolitical policies, I found that I could not
identify with this system of spiritual knowing at all. It felt rigid, righteous, unforgiving,
even angry, and it frightened me. I never anticipated that my culturally enriched,
intellectually sophisticated sons would be quoting Pat Robertson and Jerry Falwell.
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The Freudian psychoanalyst of my younger days tried to write off these (to me)
cataclysmic changes as manifestations of male sons’ unconscious oedipal strivings
to father kill and thus become. After some mulling, my theory did not wash.
They spent time accompanying themselves on guitars, singing hymns and
shouted Corinthian Paulisms to small curious crowds gathered in beach parking
lots, city parks and inner city street corners of Southern California. They passed out
pamphlets containing New Testament tracts and formulaic aphorisms promising the
post-repentance blessings of Jesus. The eldest, articulate, bright and prematurely
worldly, had been an ardent memorizer and appreciator of Shakespeare, especially
the mystical Tempest, the music of Aaron Copeland and Igor Stravinsky, the
improvisations of Charlie Parker and Cannon Ball Adderley and the provocative
literature of the time including Jack Kerouc’s On the Road and Hunter Thompson’s
Fear and Loathing in Las Vegas. They loved riffing with the Voltairean pungency of
Frank Zappa’s lyrics. Now, nihilistic humor had become an anathema.
Several weeks after my eldest son’s transformation, I found him in the
garage using a hammer and an empty barrel for disposal as he destroyed his
modern jazz and early rock record collection. He ridded himself of all of his fiction
and most of the nonfiction books in his young but relatively large personal library.
His new energy and high purpose emerged as a clearly defined set of rules of
behavior, a strong stand against abortion, frequent talk about the need to escape
from the contaminating influence of MTV culture, as well as our years of talk about
the biological and physical sciences. Both boys were particularly critical of my
Darwinian flavored attempts at scientific explanation of man’s inner life using the
selective and adaptive neurobiology of brain mechanisms and behavior. They spent
increasing amounts of time with Church friends, seldom seeing their old ones. The
eldest’s college goals turned from plans for a U.C. Berkeley equipped career in
literature and creative writing to a none spiritually challenging, objective and
practical, Christian free market finance and accounting degree from U.C.’s Business
School.
Gone were shared magical hours of intellectually stimulating, humorous,
even scholarly discussions. In place of evidential talk in areas of philosophy,
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literature and science, their opinions and claims derived exclusively from biblical
quotation. Their particularly favorites were Paul’s letters and some of the later
prophets, particularly Jesus-auguring Isaiah. “In the beginning was the word…”
became the real reality. The meaning of life was Scripture as explicated by their
book church pastors. They scribbled notes in the margins of their Bible pages
during sermons They were displeased when I interpreted the wild imagery and 666
symbolism of Revelations from the point of view of the historicity of encoded political
messages, meanings hidden for the safety of the early Jews in their world of Greco-
Roman governance. Twenty-five years, before the glut of books by Tim LaHaye, my
well-educated sons claimed that Revelations was literal and foretold the coming
tribulation that augured the end of the world and ascension to heaven of the
believers. My youngest, since childhood a well-read history buff, now viewed New
Testament scripture as sui generisly, divinely and literally true. They said the
conduct of their lives their meaning had been clarified by the biblical truths revealed
to them by The Book. What I did not say was that much of the talk seemed to me to
be an intellectually and spiritually impoverished miasma of cant and righteousness.
At the same time, their remarkable transformation appeared to be the expression of
a powerful and mystical force, the scientific understanding of which has been the
ostensible focus my life’s work. Why did their alterations appear so alien, strange
and forbidding?
Born to a home of psychoanalytically and scientifically oriented political
liberals, these precociously bright and worldly sophisticated young men were
suddenly transformed into, unrecognizable to me, radical Christian
Fundamentalists. They are now in their late thirties and remain just as ardent,
Christian patriotic, Right Wing voters to this day. The eldest is now an executive in
Morris Cerullo’s San Diego based, worldwide missionary movement, raising money
for revival and media ministries. He travels to and is involved with hundreds of
Fundamentalist Christian churches in countries ranging from Argentina and Africa to
the Middle East and Russia. He hasn’t allow me to contact his children, my
grandson and granddaughter, because, in vague talk and mostly silent implication, I
and people like me are seen as sources of potentially satanic, worldly
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contamination. He feels wronged by the way I am. He once chided me about what
he saw as my futile spiritual search in what he called the “health food” Eastern and
brain religions. My youngest, only a little less ardent and critical, visits occasionally,
and, hands in the air and speaking in tongues, prays to the Lord for my salvation.
Of course, this sudden and long lasting personal transformation in the
direction of Fundamentalism is well known and almost commonplace in modern
American and European Jewish, Christian and Moslem college educated middle
class families. The Saudi Arabian World Trade Center bombers were, mostly, well
supported children of the educated middle class We recall the famously tragic
American radical Moslem, Richard Reid, the would be airplane shoe bomber. My
stomach clenched as I heard Richard’s sophisticated and obviously caring father
share his confusion and struggle to rationalize what had happened to his son. The
commonality of this kind of spiritual and life transformations in the educated young
makes each event no less painful. On the other hand, we know that healing
transformations in the name and spirit of the Christian God can lead to quite positive
realities. They are effective in even quasi-secular disguise as in Alcoholics and
Narcotics Anonymous, Synanon and in the rehabilitation of the Charismatic
Christian, ex-alcoholic, Southern Methodist politician, George W. Bush.
Paul Holmer, Professor of Theology at Yale Divinity School gives thanks to
the evangelicals who “…keep alive the radical breach that the gospel is from the
nous of this world…they (Fundamentalists, Evangelicals) look marginal if you are
churchy…intolerant if you are ecumenical…anti-intellectual if you are trying to
systematize… in their roughness and …abrasiveness.” I bring personal and painful
witness to these claims. To get to the personal meaning and mechanisms of these
transformations, I had to start from somewhere. I am wedded to the belief of the
Jewish ecstatic, Abraham Abulafia, and not those of Moses Maimonides, that the
human mind in an altered state of activated intellect, man’s Nevesh, can understand
such mystical happenings. I would continue to work at it.
One of the early personal church experiences with my sons’ religious path
came after accepting an invitation to go with them to a Sunday service at their
current charismatic church. By then, the eldest was married with children, the
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youngest, unmarried, was teaching bilingual mathematics in high school. I had
waited several years for this occasion.. The meeting took place in a large, gray,
unmarked warehouse building that was crowded in back with high stacks of storage
cartons. The large, cement floored, open space in front of the storage boxes was
occupied by rows of metal folding chairs. They faced an unadorned, elevated
wooden platform upon which was a lectern and microphone. Behind the lectern
stood a casual array of a dozen or so young people, singing hymns and playing a
variety of instruments. These included piano, two or three guitars and upright bass,
tenor saxophone, trombone, trumpet, mouth organ and two snare drums. Sounding
a bit like a Salivation Army Band, they played and sang, “They cast their nets in
Galilee just off the hills of brown; such happy simple fisher folk, before the Lord
came down…the peace of God, it is no peace, but strife closed in the sod. Yet let us
pray for but one thing, the marvelous peace of God.”
The building, used for commercial storage, packaging and mass mailings
during the week and a Charismatic Christian word church on Sunday, was located
at the rear of an unfinished strip mall. A new and well-polished yellow Cadillac
Deville was the only vehicle parked in the no parking zone immediately in front of
the entrance to the warehouse. My youngest explained that the car belonged to Carl
Austin, the self-discovered and declared pastor, who spontaneously rose up to lead
without academic religious training or a conventional ordination. The bright yellow
car was explained as evidence of the power of God. Paraphrasing Mark, my son
told me “…he who does not doubt in his heart and believes that those things he
says will come to pass, he will have whatever he says…whatever things you ask for
when you pray, believe that you receive them, and you will have them.” The car
served as a glorious instantiation of the church’s major promise of the rewards of
faith.
Pastor Carl Austin, a tall, blonde, portly man in his early thirties with a
resonant tenor voice, was the youngest of several children of a poor Midwest farm
family. He had been a state college drop out and without a career or a job. His
sermons contained stories about how he had caught spiritual fire at a revival
meeting conducted by Kenneth Hagin of Kenneth Hagin Ministries, aka Rhema
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Bible Church, Tulsa, Oklahoma. The pastor’s witness of the Holy Ghost acting
through his life was his personal cure, by transformative Grace, of a triad of selfdestructively
sinful addictions: alcohol, gambling and promiscuity. Self-chosen and
self-declared, he now served this two and a half year old growing congregation of
over 200, mostly young, working families. The young men in attendance at the
warehouse church were in shirts and ties, very unlike the more casual garments of
even dressy occasions in Southern California at that time. Women were dressed
simply and modestly. Most of the children were in Sunday school in a small
neighboring store in the strip mall during the adult service. The few that
accompanied their parents were remarkably well behaved I was told that most
families tithed 10% of their income. They quoted Hebrews, “…king of the
righteous…to whom also Abraham gave a tenth part of all….” They believed that
their tithe would be returned manifold and the yellow Cadillac Deville served as
Pastor Carl Austin’s personal evidence. From these funds, the congregation
supported the pastor, his car, the rental expenses of the Sunday warehouse church
and an orphanage in a small Mexican border town. Some of these children, several
neurologically disabled, were bussed to the Sunday service for healing. They sat
together in a section in the front of the congregation and were the beneficiaries of
the second Sunday collection plate, passed around after the first one that was
designated for the church and its pastor.
The first Sunday sermon I heard in the warehouse followed several awkward
minutes of Pastor-directed warm up hugs of neighboring strangers while the choir
sang hymns. The songs were accompanied by instruments playing the melody in
unison sans harmony, and accented by the beats of two loud drums. As the volume
and pace of singing increased, I saw several episodes of ecstatic looks and fainting,
dying in the Lord and shouts of praise with upraised hands. The intermittent
elevation of the hands during prayer and song appeared to be spontaneous. I was
told that the arms were up as antennae, feeling the energy of Lord all around us.
The pastor’s topic was forgiveness. From Ephesians, “…let all bitterness, wrath,
anger, clamor, and evil speaking be put away from you, along with all malice… be
kind to one another, tender hearted, forgiving one another, just as God in Christ
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also forgave you.” In the middle of his sermon, which built slowly in tension and
volume, the pastor introduced a forty-ish, sparkly eyed, somewhat overweight, dark
haired, slightly made up woman who the Pastor said was a witness for the ultimate
in Christian forgiveness. She was someone from whom all of us could learn. She
was the mother of the 7-year-old boy that he, the Pastor, had, four years before,
accidentally killed during a drunken driving episode in his “other life.” That was the
one he was living before he was saved. I was told that he presented her in a service
at least once a year. The woman said that her successful struggle for forgiveness
led to her being saved. She quoted Ephesians, ”And you who were dead in
trespasses and sins hath he quickened.” She looked radiant and hugged the pastor.
When my sons introduced me to him as we filed out at the end of the service, the
pastor told me that my visit was important to the congregation. He told me that Jews
were special in Charismatic Christianity since we would play an important role in the
return. He said he hoped he would see more of me. My boys seemed pleased to
have invited me.
I accompanied them to their church most Sundays, and often for what they
called the “rock and role healing services” on Wednesday night, for over two years.
Within three or four months I found myself, the first time while awakening out of a
deep sleep, mumbling sounds that I was told sounded like some unknown
language, I was praying in tongues. At some services it happened spontaneously
accompanied by an almost ecstatic feeling accompanying the surrender of willful
control. This was usually accompanied by the release of new energy. I recall
thinking that the spontaneous, nonsensical linguistics shorted out my verbal and
obsessionally logical left brain allowing the unbridled expression of my hysterical
right brain. Sometimes in agreement with an insight offered in a sermon or when
particularly moved by a hymn, I found my hands lifting skyward, right hand and arm
higher than left, with a high feeling of trust and delicious surrender of conscious
cognitive control.
Reading the New Testament’s Acts, I learned that we were re-enacting the
scene of the Apostles in the upper room. Those gathered there were the ones
chosen by the risen Jesus to be able to see Him, the list including Peter, James,
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John, Andrew, Philip, Thomas, Bartholomew, Matthew, James, Simon and Judas.
“…they were all filled with the Holy Spirit and began to speak with other tongues, as
the Spirit gave them utterance…” The secular psychoanalyst in me tried to make an
analogy with the joyful jazz lyrics of Ella Fitzgerald’s scat singing, I’d done a little of
that during my small jazz group pianistics as an a adolescent. I thought about how
verbally paralyzed stutterers could be articulate when singing what they mean when
they could not talk it. I wondered about the relevance of the spontaneous poetry of
slams and Hip Hop rapping. We attended what my sons called charismatic black
Baptist churches in South Los Angeles and Long Beach. These often four hour
services usually featured two wonderfully harmonic echoing choirs with organ and
drum punctuation of the speech-singing, sermonizing Reverend. Large and
beautifully dressed black women sang operatically and danced gracefully down the
aisles. I joined my sons in this joyful noise for these long services and, exhausted, I
was forced to go home for a Sunday afternoon nap.
In spite of what could be regarded as validating experiences with the real life
Holy Spirit, I continued to be generally confused and even more deeply estranged.
An inner voice kept recalling my spiritual failure as a parent and being traitorous to
my Jewish ethnic identity by Christian church attendance. I tried to understand how
my sons had traveled from where I thought we were living together to this entirely
new world. How did it happen? Could the path going there and back be
meaningfully reconstructed and then reversed? This idea is consistent with the
medical dictum that knowing the cause, the treatment logical follows. My education
had shown me such assumptions of reversibility need not be true.
Contrary to the beliefs of early physical mechanics, medical psychiatric
history takers and psychoanalysts reconstructing childhood events, the modern
physics and mathematics of complex systems says phase transitions in complex
systems are probably not reversible, at least not simply so. One of the features of
global changes in complex systems, often called bifurcations or phase transitions
(think heated water going suddenly to a boil), is their dramatic discontinuities in
behavior. Knowing only the initial and end state, phase transitions in complex
systems do not allow for point-to-point backtracking or specific linear-causal
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understanding. These discontinuous and global transformations are the stuff of
miracles, especially for physicists. Even with respect to initial and end-states, rather
than using straight forward phenomenological observation, the mathematical and
physical theories of phase transitions are usually dependent on not necessarily
intuitive, derivative physical quantities. Their verbal representations are often not
concrete but metaphoric. This retreat to derived and abstract, far from the primary
data computables, may be more evidence of man’s many insufficiencies in
understanding of the mysteries that are often placed in the spiritual realm.
* * *
Driven by an effect that contributes to cause, like the faith-driven
abandonment to God that generates more faith, a drop of water hanging from a
faucet is pulled down by its own gravitational field as the thinning neck of the drop
facilitates its own further thinning. A gobbet connected by a thick neck to the main
drop begins to separate. The neck between them thins and breaks, and one
becomes suddenly and irreversibly two. A continuous structure has suddenly
become discontinuous in finite time at what is called a singularity. Since the single
measurable feature that dominates the water’s behavior around this singularity is
the diameter of the thinning neck, a derivative physical, one-dimensional
observable, neither the details about where it all began (called the initial conditions)
nor the path it followed to get to the moment of fracture, are predictively relevant
with respect to the sudden transition. Considering this kind of phenomenon going on
in our brains, choosing between theories of behavior that involve changes in brain
cell groups and/or brain chemicals versus those that involve behavioral quantities,
may be neither possible nor necessary. The challenge is to place the problems of
cataclysmic change in brain and behavior in sufficiently abstract and universal terms
that can be represented in some low dimensional, computationally accessible space
of variables.
The simplification and stereotypy of behavior around singularities reduce the
number of features that are required to discuss the dynamics of change in what
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would otherwise be a complicated beyond reach situation. One of the properties
found around singularities, is the loss of absoluteness in contextual characteristics
such as the scale of the observation. We no longer can say that what we are
studying happens in inches or miles, in seconds or days, now or in the past. In the
place of a single unit of relevant measurement, we have a distribution of spatial and
temporal feature sizes that stretch toward both the infinitely small and the infinitely
large.
We can illustrate a dynamical transition involving the passage of the system
through a singularity by using the metaphor of another kind of water experiment. If
we pour a small amount of water through a filter full of coffee grounds, or watch our
coffee maker do it, the first spurt of water makes an incomplete path of wet grounds
in the bed of dry ones. The next bit of water soaks this path more thoroughly and
may form additional and multiple, new and branching, incompletely penetrating
paths. Eventually, on just one more of these pourings, a connection in the paths
occur, such that the water snakes all the way through the coffee grounds and the
first brown drop of coffee falls into the pot. At this flow singularity and opposite to
the dynamic of a faucet water drop, a discontinuous system of pathways becomes
continuous in finite time in a process called percolation,
Trying to set up a predictive model, we can count the number of water
deliveries that occur before the first drop finds its way through. Repeating the
experiment many times yields a span of the number of pours required to reach the
singular point of percolation. If we do the experiment enough times, the distribution
of the number of pours required to reach percolation will range from one toward
infinite. In the neighborhood of the transition, time as recorded as the number of
small pouring events may stretch.
In aa comparable system, as elegantly described by Detrich Stauffer in his
Springer-Verlag book on percolation, multiple hot spots in the woods can suddenly
fuse into a forest fire. Isaiah said, “…glorify the Lord in the growing fires of dawn…”
Faith fires spreading through a faithless dense forest, its hot irregular front damped
by the disbelief of water-filled leaves, or disillusionment gaps of already burned out
trees, can, under the right motivating conditions of dryness, wind velocity, tree
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density, kindling temperature and desperation-induced willing of faith, sweep
through the entire woods in a sudden blaze. This is the spirit of percolation.
Computer simulations of percolating blazes generate a multiplicity of life times of
forest fires near the singularity that represents the transition to a global
conflagration.
Mentioned previously is Rudolf Otto’s 1917 book about the characteristics of
religious experience, Das Heilige, The Sacred, which described phases in the
discontinuous transition from everyday life to the wholly other (ganz andere) reality
of the world of the sacred. They include intense, numinous experiences of fearsome
ambiguity, dawning awareness of awesome mystery, revelation and appreciation of
the majestic power and finally, entrance into a reality of an entirely other place and
time than the natural and secular which Mircea Eliade called profane. In his 1958
book, Patterns in Comparative Religions, this well-known historian of religion called
the revelatory occurrence of sacred reality an hierophany. Eliade’s classic work,
The Sacred and the Profane, contrasts the homogenous, spiritually formless and
relative world of the profane with the results of passage through spatial and
temporal singularities to a place and time that are not of this world.
Poincaré said that the brain did not know of absolute space, but rather
established a model of it through internal reconstructions of sequential sensory
experiences that accompanied our exploratory movements. Activity generates the
internalized, partial differential, equations (describing changes in the observable
with motions in space) required for representing the dynamical cartography of the
world. It was Poincaré’s habit to topologize the dynamics of motion in mathematical
problems that lacked analytic solutions. In this way, simple algebraic operations
replace some of the insoluble problems of the calculus. Eliade’s sacred space
defining singularity in the plane that breaks profane homogeneousness, a center
point that is no longer a circle, can be viewed also as Poincaré’s topological center.
His topological brain theory found expression in the formal representation of internal
space as the invariant product of an organism’s displacement groups of imagined or
real physical movements around such singular fixed points. The operational object
called groups defines this kind of algebraic, mathematical structure and motion.
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associated with the loss of habitual temporal-spatial contextual moorings. A mind at
time one and the same mind at time two are unconnected. They are wholly other.
In much the same sense, for Eliade, sacred time, like space, is neither
homogenous nor linearly continuous. Sacred time is circular, recoverable and
reversible. Past, primordial, mythical time can exist in the present. Religious
festivals are recurrently ontological, allowing the recovery of the sacred time such
that their past and present expressions are the same. Rebirth is new birth. In the
language of the North American Indian Tribe, the Yokuts, the term for world
(cosmos) and year are the same. A year and the world has gone by, only to start
again. The Dakota Tribe says that the Year goes around the World. As Elaide has
said, “…at each New Year…the world (is) recreated and to do this is also to create
time…the sick man becomes well because he begins life again with its sum of the
energy intact.” Healing by becoming another or renewed self may become a frontier
science in the yet unexplored field of phase transition medicine.
The quality of separateness, discontinuity in states, as occurs in the samedifferent
inside world, is much like that found in the stages of anesthesia. Each
stage of anesthesia is ganz andere from the others. In Stage I anesthesia, fast
frequency, low voltage brain waves are observed and accompanied by a two
Martini-like, mildly activated, sedated but exhilarated high. Stage II, the next deeper
stage of anesthesia, is marked by the sudden emergence of intermittent bursts of
high amplitude brain waves, and animals and man demonstrate bizarre postures,
hallucinatory phenomena, fixed staring, and sometimes movements that look like
acting out some symbolic drama. This stage marks the beginnings of the loss of
responsiveness to painful stimuli. In the sudden drop into Stage III, a low voltage
mix of mostly slow and some fast brain waves can be seen associated with
depressed consciousness, complete insensitivity to pain, slow regular respiration
and an unexcitable cardiovascular system. Stage IV is the deepest stage of
anesthesia. This state is characterized by very low voltage, almost flat brain waves,
a loss of spontaneous breathing, the collapse of blood pressure and, finally, cardiac
irregularities and death in cardiac arrest. These are both discontinuous and global
brain state phase transitions.
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primary process by Freud and his followers. This forgotten language of the
unconscious, an archaic needs and fear-driven tongue lurking beneath our
supposedly objective discourse, comes to dominate themes of communication in
the middle of these unfinished spiritual transitions. The Rorschach Test of master
meditaters and LSD users overflow with conflictual primary process images, as
does the talk of patients on the verge of schizophrenic decompensation. The
primitive symbolism of primary process provides the major current in the overwritten
prose of the hyper-religious temporal lobe limbic epileptics described previously and
called the Geschwind Syndrome and in the regressed and iconic transference
concerns of patients with tendencies for global and sudden phase transitions,
prostitute to saint, righteous obsessional to conscienceless psychopath, called
borderline personality disorder.
Primary process represents a dynamical brain state, one unburdened by
linearly predictive connections with reality. It is a state without even a transient
single defining physical time or other fixed measure of order. It is without the causal
logic or knowledge of an outside reality that a brain implies in supposing to know. Its
primitively instinctual style and goals contrast with more physically time-locked,
reality oriented thinking which Freud called secondary process and Penn-Lewis
referred to as ordinary and religiously lawful “reasoning faculties.”
An absence of absolute time and space scales with which the executive ego
orders internal and external time and events, and therefore their relations, results in
primary process thinking characterized by condensations of several, often
incompatible, representations into one. Dueling, conflictual and simultaneous
feelings and thoughts float from their relevant objects to others. In the transitional
transcendent state, there may be confusion of self with others, of objects with their
labels, of parts with the whole and of symbols with the things that they symbolize.
This facilitates living in the spirits of the Father, the Son and the Holy Ghost at the
same time. Mixed inextricably with saintly awareness and charisma, there are
signatures of instinctually driven and configured primary process. Freud’s classical
work on slips of the tongue concerned the intrusion of these instinctual thought
stream condensations from the world of the ganz andere and displacements into
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everyday life. In this intense and quasi-fluid state, saintly priests slip seamlessly into
sexual predation; an ecstatic Jewish Orthodox fundamentalist shoots 29 praying
Moslems in a cave near Abraham’s burial plot for Sarah in Hebron; what were
lovingly mystical, Jelaluddin Rumi’s Afghanistan (Balkh) descendents become
people bashing and women stoning morality police; committed and mesmerizing
Christian televangelists attend peep shows and seek child pornography; devoted
Islamists crash airplanes into tall New York buildings.
In the physics of condensed matter, two common forms of multi-molecular or
polyatomic cooperative arrangements are the crystalline condition and in some
ways its opposite, the amorphous glassy state that results from rapid cooling
through a melting temperature. The microscopic atomic arrangement in glasses, in
contrast with the crystalline state, exhibits no spatial periodicity or long-range order.
In contrast with fluids, the friction of passage of molecular elements of glasses past
each other, their shear viscosity, is large enough such that their macroscopic
shapes are maintained in the very slow flow for very long times. In-between the
crystalline and glassy states their exists a multiplicity of possible unstable
arrangements which result from what physicists call frustration, the inability of a
system to find a unique, lowest energy, ground state. The generic example of a
ferromagnetic crystal has two types of ordering principles: (1) The mutual alignment
of the atomic magnetic moments, visualizable as the lining up of dipole, positive to
negative, magnetic arrows; (2) The geometric crystalline low energy ground state
described above.
When the symmetry of these two ordering principles are incompatible,
imagine an arrangement of neighboring atoms that prefer anti-alignment of the
magnetic moments which are placed on a geometrically triangular rather than a
square lattice, there is no single arrangement that can satisfy both magnetic and
geometric principles. What emerges in this state of frustration is the potential for a
multiplicity of nearly equal energy states. Water has the potential for both geometric
ice crystal symmetry as well as arrangements of hydrogen proton (+) to oxygen
electron (-) magnetic moments (with well-ordered oxygen lattices but disorder
among the hydrogen positions). It is therefore not surprising that a multiplicity of
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indirectly by my sons and church elders about joining a study group for personal
conversion.
I was surprised to learn that discussions of current political topics were a
regular part of these discussions as well as the Sunday and Wednesday night
services. We received a weekly political action committee report. Their issues
involved abortion, school vouchers, sex education in schools, family planning,
school prayer and carefully chosen Christian elected officials for school boards and
the Congress. As a congregation, we frequently held hands in small circles and
prayed for the electoral success of our issues and candidates. Twenty years later,
this movement has evolved into the public political morality play of the Republican
base of George W. Bush.
Laying on of hands, dying in the Lord, speaking in tongues, dancing in the
aisles and praying with up stretched arms were routine in the hymn dense services.
The goal for all was the spiritual transformation of mind as in Romans, “…be not
fashioned according to this world, but be ye transformed by the renewing of your
mind that ye may prove what is the good and well-pleasing and perfect will of
God…” The pastor told us that the world ruled mind could not grasp spiritual things
as in Corinthians “…they are foolishness unto him and he cannot know them,
because they are spiritually understood.”
My research took me to a collaborative project at a European mathematics
institute for three months. I returned to our town very late on a Saturday night. I
planned to surprise my sons by appearing at their usual choice of the middle service
the next day. I drove up to the warehouse church fifteen minutes before the service
was scheduled and found that the parking lot of the strip mall was nearly empty.
There was no Cadillac parked at the front door. I banged on the double door when I
found it locked. More then a little surprised, I called my eldest. He told me that four
weeks before, the pastor disappeared, I later found that his disappearance
accompanied that of the congregation’s bank account, and no one knew where he
had gone. He had not warned or informed anyone in the congregation about his
plans. Calmly and without apparent awareness of my surprise and distress, my
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eldest asked me if I would like to attend the late Sunday morning service at their
newly chosen Charismatic Christian church. He gave me its address and told me
that the service started at 11:00 AM. There still was enough time for us to meet
there. I wondered how the Pastor Carl Austin would use this incident in sermons
about sin and redemption to his next congregation.
Further Readings for Pentecostal Phase Transitions
Religious and Spiritual Groups in Modern America. Robert S. Elliwood, Prentice-
Hall, Englewood, N.J. 1973.
The Name of Jesus. Kenneth E. Hagin, Rhema Bible Church. Tulsa, Oklahoma.
1979.
War on the Saints. Jessie Penn-Lewis, Robert Lowe, N.Y. 1973.
Discipleship, David Watson, Hodder and Stoughton, London, 1981.
Mysticism. Evelyn Underwood, Dutton, N.Y. 1911.
A Nation of Believers Martin Marty, Univ. Chicago Press, Chicago. 1976.
Introduction to Percolation Theory. Dietrich Stauffer. Taylor and Francis. London.
1985.
Modern Theory of Critical Phenomena. Shang-Keng Ma, Benjamin/Cummings.
Reading, MA. 1976.
A Modern Course in Statistical Physics. Linda E. Reichl, Univ. Texas Press, Austin,
1980.
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Manic-Depressive Illness. Fred K. Goodwin and Kay R. Jamison, Oxford Univ.
Press, N.Y. 1990.
The Pharmacological Basis of Therapeutics. Louis S. Goodman and Alfred Gilman,
MacMillan, N.Y. 1975.
Statistical Mechanics of Phase Transitions. J.M. Yeomans, Clarendon Press,
Oxford. 1992.
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CHAPTER 7:
AMPHETAMINE ROLL-UP AND SPLITTING
We try to understand the metaphysics and inner dynamical life of the
committed, judgmental, fundamentalist believer. In these sacerdotaly rigid and
faithful, disenfranchisement and righteous intolerance toward other denominations
are simultaneous with spiritual compassion, mercy and forgiveness for the members
of their own. This splitting between the good people and latent evil doers is seen by
psychoanalysts and dynamically oriented brain scientists as an all too common,
sometimes psychopathological, solution to the inevitable ambiguities of living. I am
certainly not alone in being fearful of Fundamentalists: Jewish, Christian, Moslem
and Hindu. From the overpass above the freeway, bearded Jewish Orthodox men
rained rocks onto the roof of my rented car because I was driving on Sabbath. A
research project had taken me to Jerusalem Mental Health Center’s neurochemical
laboratories for collaborative work with mostly secular Jewish scientists. Halachic
considerations, those of Jewish lawfulness, comparable to the constraints of Muslim
shirah, forbids working, even driving, on the Sabbath. Orthodox Jews live walking
distance from synagogues or benefit from a rabbinicaly blessed, network of
symbolically covered walkways for going longer distances on the Sabbath. This
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Sabbarian grid of permission obviously did not cover driving on the free way to the
mental health center.
It is the splitting of us from them that leads to the breakdown in empathy and
compassionate identification with others. Studies of the dominance of direction of
rotation within a closed space in small mammals have shown that amphetamineinduced
intensification makes the choice of right versus left (or left versus right)
rotation, broken symmetry, more statistically significant. In contrast, the Hefner
Foundation of Switzerland has shown that entheogenic drugs such as psilocybin in
man facilitate seeing both of the conflicting, simultaneously presented, right eye and
left eye images in place of the usual dominance of just one of the two
representations. A precondition of compassion might be that a person’s brain be
able to see and comprehend both or several sides of apparently conflicting points of
view at the same time. The Fundamentalists do not see things that way. In the
Koran, Mohammed says, “…give sustenance to the poor man, the orphan, the
captive…and for the unbelievers We have prepared fetters and chains and a
blazing fire….” In the New Testament’s Mark we find the final words of the risen
Jesus, “…whoever believes and is baptized will be saved but whoever does not
believe will be damned.” The Crusaders’ claimed scriptural support for their
murderous marches to reclaim Jerusalem.
Carl Jung wrote about the New Testament’s Revelations in his Answer to
Job: “…a terrifying picture that blatantly contradicts all ideas of Christian humility,
tolerance, love of your neighbor and your enemies and makes nonsense of a loving
father in heaven and rescuer of mankind. A veritable orgy of hatred, wrath,
vindictiveness and blind destructive fury that revels in fantastic images of terror
breaks out…overwhelming a world which Christ endeavored to restore to the
original state of innocence and loving communion with God…” As Princeton
University philosopher, Walter Kaufman, has noted in his Religion in Four
Dimensions “…compassion for unbelievers is implicitly condemned and
proscribed…Augustine argued expressly against compassion for the damned and
Luther used invectives against his (religious) enemies…” How can this be God’s
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setting for the spiritual work toward that promised in John “…that you love one
another; even as I have loved you, that you also love one another.”
In contrast with what has been described in previous chapters as the
entheogenic drug-induced transitions to a spiritual mind, one is tempted to describe
these Fundamentalists’ states as the amphetamine religions. The Los Angeles
Ram’s Hall of Fame defensive end, on very high doses of amphetamine (125
milligrams compared with the diet dose of 5 milligrams) taken four hours before the
Sunday games, the Baptist minister, Deacon Jones, used his famous and
consciousness annihilating head slap to daze the opposing offensive tackle in order
to gain access to and injure the other team’s quarterback. Before taking the handful
of Dexedrine spansuls, he would tell me, “See you on Tuesday.” Along with the
Deacon’s destructive aggression was the other invariant feature of the actions of
high doses of amphetamine, compulsive stereotypy, the fixity and driven repetition
of over simplified actions and thoughts along with the loss of breadth of vision and
adaptive flexibility. Deacon consistently rushed inside, took the inside lane, in spite
of offensive linemen, who having studied previous game films, being set up to
expect his route. They used this knowledge to take him out of the play. In modern
theological parlance, judgmental rigidity and thinly veiled disapproval take the place
of the more flexibly curious and lovingly humane feelings of the participants in the
evolution in spiritual understanding of today’s liberal Protestant process religions.
These are the ones that believe that the properties of God evolve along with our
biology, our brains and our growing scientific understanding of ourselves and the
world.
Angry splitting is not just a stimulant drug effect. Recall my experience of the
sudden emergence of a first second wind after a mile or so of my daily ten miles of
running. It was frequently accompanied by inner bursts of obsessive, paranoid
thoughts. Taking five milligrams of amphetamine felt much like the first second
wind. I am full of energy with arrogant feelings of power, mind fixated in grand and
simple ideas that I believe to be absolute and correct. I feel irritably intolerant about
anyone or anything different. It is my virtuous duty to set everyone straight.
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In the 1980’s, Moishe Zar, a desert castle dwelling, settlement organizing,
ardent Orthodox Jewish Zionist, now 65 years old, was the leading vigilante of the
West Bank He planted bombs in the cars of Arab mayors and plotted to blow up the
Dome of the Rock. Buying up farmland from the Palestinians beginning in 1979,
many of whom were then killed by their own because they were seen as
collaborators, Zar and his group of young volunteer settlers took over harvesting the
Palestinian’s olive trees and shooting rifles over the heads of those that would take
them back. Fundamentalist Christians share his vision that the coming of the
Messiah, the second for Christians, the first for the Jews, is dependent upon the
complete return of all of the land of Israel to the Jews.
I recall that in the middle 1940’s, my father took me to a fund raising dinner
for the local chapter of the Jewish Antidefamation League. The whispered talk was
about blowing up a warehouse in which anti-Semitic pamphlets were stored,
planned for the middle of the night when it was unoccupied. Even at the age of 10, I
could tell that their quiet anger and firm commitment made these threatened men
feel less vulnerable. I understood a little more about the motivation for this proposed
nighttime property destruction when, the following year, my father explained the
reason for our being refused overnight rooms at several motels as we drove along I-
95 in Southeast Florida. It took us until late night to find a place to sleep. This was
America’s muted version of what Hitler and his legions were doing to Jews that, at
that time, was not generally known, except for Walter Winchell, in America.
Resonant with our chemical-cultural theme are the many reports that Hitler was
taking an amphetamine drug, Benzedrine, daily and in high doses for the last 20 or
more years of his life. One can hear the characteristic, amphetamine-induced,
higher pitched, rants in his recorded radio tirades. Compare the pitch and strained
voice quality of the singing of Bob Dylan in his early records made while he was on
speed with the gravely, much lower pitched voice, now that he is not. In our
behavioral neuropharmacology laboratory at the Brain Research Institute at UCLA,
Professor Charles Spooner and I used an audiographic oscilloscope to monitor the
sounds of baby chicks whose peeps became higher in pitch and rate following
injections of amphetamine. The earliest members of the methadrine-amphetamine
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chemical family were synthesized by the great organic chemists of the German
pharmaceutical industry in the early 1930’s.
The sequence of parallel streets in the neighborhood of my home and first
grammar school in Kansas City, Missouri were my street, Virginia, then Tracy,
Forrest and Troost. My school, Bancroft Elementary, was on Tracy and one block
down that street was the Lutheran Day School established by German immigrants
under the aegis of the Missouri Synod. Starting in the third grade in 1943, I was
intermittently and unpredictably chased by rock throwing, “damn Jew” and “Christ
killer” shouting boys from the Lutheran Day School. I had my choice of running for
safety directly from Tracy to my family’s half duplex at 4232 Virginia Street, or
moving away from school via Troost and then down several blocks and around to
sneak back to my home on Virginia without being spotted. One run-for-it afternoon,
my parents took me to the emergency room of the Menorah Hospital to have my
scalp sewed up where a sharp rock had landed.
When I asked my synagogue’s young people’s spiritual counselor, Rabbi
Kleigfeld, to explain the feelings and actions of these children of Martin Luther’s
Post-Reformation Christian Church, he answered that I already knew about similarly
difficult places and times of our Twelve Tribes’ like Rome, Medieval Europe, the
Spanish Inquisition, Persia (Iran) and, it was rumored, in Germany as we spoke.
“Conversion or death” was its most benign form, in places like Spain and Iran, many
Jews faked it, staying alive and practicing Judaism secretly. Kleigfeld told me that
the causes of this historical theme of persecution of Jews were complex.
Among the frequently unmentioned events recorded in the later part of the
worldly life of Mohammed, who lived from 570 to 632 AD was, ”…in the name of
Allah, the Compassionate, the Merciful…” his participation in the crushing of the
Jewish tribe of al-Nadhir in 626 A.D., the beheading of 800 Jewish men of the tribe
of Qurayza who refused to accept Allah as their God in 627 A.D. and putting to the
sword the Jews of Khaybar in 629 A.D. As in the section of the Koran called The
Cow, Mohammed proposed to “…fight against them (the infidels) until idolatry is no
more and Allah’s religions reigns supreme…” In contrast, the more entheogenic
spiritual orientation of the ecstatic followers of Mohammed in his earlier years
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speaks of the multiplicity of valid Ways to Deep Truth. The acceptability of many
ways is supported in the tales from the millennial oral tradition of the Sufi Masters in
their Teaching Stories. One of them, What Befell the Three, is attributed to the early
18 th Century Sufi teacher, the Dervish Murad Shami. In it, an apparition is mobilized
by the concentrated Truth seeking efforts of three Sufi Dervishes named Yak, one,
Do, two and Se, three. When this “…white smoke head of the very old man…” was
asked what he was, he answered “…I am what you think me to be…have you never
heard the saying ‘There are as many ways to the Deep Truth as there hearts of
man.’” In the narratives about the lives of the Mevlevi Islam dervishes called
Munaquib el-Arafin (1353), Jalaludin Rumi, the Sufi saint, instructs his ill and
troubled petitioner to ask forgiveness from the Christian he recently spat on saying
“…whether a ruby or a pebble, there is a place on His hill, there is a place for all…”
Cole Barks and Michael Green’s The Illuminated Prayer (2000) notes that the Rumi
follower, Bawa Muhaiyaddeen, a modern Sufi guru, was said to be keenly aware
how quickly spiritual entheogenic systems can become amphetamine-like and
“…develop rigid marching orders …which turn into a dumb obsession with other
people’s behavior…”
It appears that entheogenic and amphetamine spiritualities can coexist
contemporaneously, in Islam as well as in all the other of the world’s great religions.
One day, sneaking home from school, taking the long way around via Troost,
I was spotted and chased up some stairs into an apartment building’s dark hall.
Terrified, I swung hard and hit the leading angry and noisy head with a propitiously
found snow shovel that had been left near the apartment’s entrance. An ambulance
was called to tend to the twelve-year-old, transiently unconscious, Lutheran boy. He
recovered completely within a day and the chases after school and my desperate
escapes stopped suddenly, never to reappear. After several months, our family
crossed the socioeconomic divide in Kansas City to a more tolerant, upper middle
class, Southside neighborhood near Rockhill Road, to a suburban home, one block
from Missouri’s border with Kansas. There, persecution for my Jewishness took
more subtle forms such as not being permitted to play teen-age golf with my friends,
though invited, on their Blue Hills and Kansas City Country Club’s golf courses. It
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was decades later that the first Jewish member of the KCCC was the founder of H
and R Block. Unable to afford membership in the single all Jewish country club of
the region, I practiced for my high school golf team on Armour Hills Public Golf
Course, where, at the time, mostly white working class golfers played.
How can it be that spiritual states include both personal humbleness and
loving mercy toward some of mankind and judgmentalness, nonacceptance and
commitment to seduction, threat and even violence in the service of invoking
changes in the beliefs of others. How can the high energy calm of being home at
last in the born again condition with its new freedom from self assaults about sin,
most importantly that of disbelief, but also peccadilloes such as drunkenness,
promiscuity and familial abuse, be associated with readiness to judge, harass even
persecute others. Psychoanalysts would say that it is a riddance mechanism, the
projection of unwanted personal traits onto others. From the standpoint of rational
thought, this seems more like non-Aristotelian cognition, two, not either-or,
countervailing orientations toward mankind held simultaneously. The newborn
parishioners of these charismatic amphetamine churches express their fealty to
God with strongly held beliefs that diagram logically as contradictions. The
perception of the world’s peoples into believers and infidels, good and evil, our
people and your people, ourselves and the others. It is generally believed among
social psychologists that it is the perceived nonpersonness of others, which allows
the cruelty that empathic identification with them would never permit. Splitting feels
like resolution, its stereotypy reducing the complexity of spiritual thought as well as
true to life perception.
A concrete laboratory example of amphetamine conversion, the sudden
transition to a high energy, fixated, and delusional state called amphetamine
psychosis, is supplied by experiments in humans conducted by Professor John
Griffith at Vanderbilt University in the 1960’s. These experiments would not be
allowed by today’s human research committees or medical ethicists. Each one of a
group of psychologically screened-as-normal graduate student volunteers, at an
individually unique amphetamine dose, developed suddenly a personally unique
and peculiar system of new beliefs, obsessionally held as rational thoughts. Ten
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milligrams of amphetamine were administered to volunteer subjects every hour until
every subject crossed their particular threshold for personality change. The
graduate students underwent a global mind-brain-person transition at differing total
doses of amphetamine. The subject’s world was suddenly transformed into one of
enemies and friends. The syndrome dissipated over several hours when the drug
was stopped and the plasma levels of amphetamine and its metabolites declined.
As amphetamine makes memory formation and recall stronger, the subjects were
embarrassed when remembering what strange and forbidding yet uneatable things
they so strongly believed. These included such things as: they as good people were
caught in a network of bad person Russian spies; some threatening others arranged
for poison gas to be seeping out of the water faucet; the white coated scientists
were CIA undercover intelligence officers hoping to get information about their small
pornography collection. The subject’s world had become divided in, for each
person, a stereotyped way.
After a couple of weeks of return to normal living, the experiment was
repeated. Each subject again developed his or her individually unique set of goodguy,
bad-guy delusional beliefs and at the same dose of amphetamine as before.
Like those of strong faith, their ideas once again resisted the logical arguments
made by the professional staff: that the new realities were neuropsychological and
had an obvious pharmacological origin. While on the drug, all stuck to their story,
even while being shown the movie record of their first drug-induced episode. There
is reliable scientific literature describing kamikaze pilots on high doses of
amphetamine in an ecstatic state of Shinto nationalism. With their planes loaded
with explosives, they deliberately crashed their planes onto American aircraft
carriers in the Pacific Theater of World War II. One wonders if these drug-induced
states occur in the drug-free condition in today’s abstemious Muslim suicide
bombers.
A more abstract and general way of thinking about the sudden emergence of
fixation, repetitiousness and splitting in feelings and thoughts involves the
emergence of regular limit cycle oscillations in a complex system that was behaving
previously in a stable but flexible way. Locking up into a fixed, closed loop, is a
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common way for electrical circuits, computer programs, brain mechanisms and
other complicated systems, even cultural or spiritual movements, to behave when
one or more important control parameters crosses a threshold. Doyne Farmer of the
Los Alamos’s Prediction Company once said about this vulnerability in complex
system, “Those things can hardly wait to roll up.” The limit cycle lock-up occurs
most often as a sudden, discontinuous change, called a bifurcation, into
autonomous self-oscillations from an equilibrium state around which there was
some random variation. A bifurcation, a discontinuous change in outcome from a
smooth changes cause, characteristically occurs when the amount of an important
influence, a metabolic state, a drug, a psychodynamic conflict or level of emotional
stimulation crosses some critical value. The switch from one type of dynamical
behavior to another looks like the system has suddenly changed into something
else with an entirely new kind of life of its own. In the new life of rolled up, locked-up
repetitious motion, almost all new starting conditions follow pathways that lead into
the same limit cycle pattern. Evangelical Christians talk about all born again life
being in Jesus, fixed in a complete set of moral, social and political beliefs, ideas
and judgments. The limit cycle gets its name because the end state of the orbits of
almost all starting points of the dynamics winds up being drawn into the same fixed,
repetitious pattern of a stable cycle. Visualizing the simulation of one kind of
bifurcation to a limit cycle on a computer screen, we see a slightly jiggling point
explode suddenly into an orbit of ceaseless rotations around a circle.
Ralph Abraham, the University of California at Santa Cruz pioneer in
graphical approaches to nonlinear systems, describes, cinemagraphically, the
emergence of limit cycles from a single point. He starts with a picture of an attractor
of water flow in the shape of a basin. All water that enters the basin, rolls down its
sides to the bottom, to what physicists say represents a potential energy minimum.
A little more technically, this attractor basin is composed of the set of all points such
that the orbits that flow from them tend to end up inside the basin as time goes
toward infinity, no matter where they start. Changing the value of a control
parameter of the system changes the shape of this basin-like landscape, of the
surface of the systems dynamical actions called a manifold, which can intuitively
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predict how the fluid will flow upon it. If we start with a simple bowl, a parabolic
basin, then the attractor itself is a point at the bowl’s very bottom. Changing the
value of some influential parameter may induce the sudden formation of a small hill,
growing at the center of the basin’s bottom. Now fluid flow in the attractor bowl runs
down to a path around the hill at its bottom. The autonomous motion of the fluid
flows takes place now in a circular orbit. The basin of the new attractor is the
original bowl minus the point at the top of the central hill. The fluid flow around the
hill at the bottom of the basin is circular and is called a limit cycle. Note that the
direction of the rotation of the limit cycle can circle in one direction or the other. In
some computational simulations, motion alternates between directions. This
suggests the aspect of the born again amphetamine religions, splitting. There is an
unstable and intermixed probability of right versus left turning directions and their
alternation. This vulnerability to directional splitting and often unpredictable
alterations in action themes can represent what seem to be paradoxical
combinations of both good and evil in the same strongly faithful, for example, the
apparent bidirectional morality of generous and loving, pederast priests.
These mathematically flavored images of the sudden emergence of a limit
cycle in complex systems was made biologically concrete to me by research
conducted by one of my first graduate students, David Segal. He is now a professor
of psychiatry at the University of California in San Diego. His program of work
involved the administration of very gradually increasing doses of amphetamine to
rats while their behavior was being monitored and recorded by a continuously
running video camera. He documented the behavior of rats in a walled rectangular
space within which, without drugs, they first wandered about randomly and then
settled down to rest in an individually selected, favorite home corner. Segal called
all of these phenomena, patterns of exploratory behavior. At doses of amphetamine
below 2.5 milligrams (mg) per kilogram weight (kg), the exploration of the entire
bounded space proceeded faster than was the case with their salt-water treated
controls, their paths being more uniformly distributed throughout the box. They
spent less time resting in their home corner. At almost precisely 2.5 mg/kg, the rat’s
behavior changed dramatically into an entirely new pattern of continuous circling. As
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was the case in the abstract manifold picture of bifurcations to limit cycles, some
rats tended to circle their chamber to the left and some to the right and switching
between them was often seen.
The influence of amphetamine and other brain dopamine neurotransmittermediated
drug manipulations on directional turning tendencies in rats, mice and
cats were the focus of brain and behavioral research of Professor Stanley Glick of
the University of Massachusetts. The asymmetry of dopamine concentrations in the
two sides of the brain, particularly in the medial prefrontal cortex and the brain
stem’s nucleus accumbens, predicted both the paw preference for pellet reaching
and direction of turning in several studies in rats. These findings were statistically
true over a population of rats, but not necessarily predictive for any single one.
Reminiscent of the conflict between good and evil in our human spiritual analogy,
naturally right turning male rats and left turning female rats, when compared with
the opposite paired group, were greater voluntary ingesters of alcohol placed in
their water bottles.
Splitting as a part of the phenomenology of limit cycle bifurcations, with
directional implications for good and evil, has neurological support in humans as
well. In the context of contrasting right versus left hemispheric temporal lobe
syndromes, recall that temporal lobe seizures with a right side excitatory focus
leads to the development of the Geshwind Syndrome, a high, softly energetic and
saintly state of spiritual preoccupation and voluminous writings, loving and generous
kindness toward all and the complete disappearance of sexual interest but not
sexual potency. A left temporal lobe excitatory focus leads to the development of
the Kluver-Bucy Syndrome of indiscriminate aggressiveness and hypersexuality.
Experimental simulations of this syndrome in cats lead to them mounting and
attacking living and nonliving things, even chairs. A variety of manipulations of the
symmetry of brain dopamine concentration and dynamics by its characteristic drug,
amphetamine, interact with lateral brain lesions such that we conclude that the
stimulant-induced limit cycle lockup remains a phenomena influenced by drugs, sex,
genetic predisposition and several other experimental conditions. This situation is
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perhaps not so different in variety and complexity from the range of representations
in art and literature of the left hand of evil and the right hand of grace.
Oscillations that appear spontaneously in nonlinear systems without external
periodic input were known to Henri Poincaré in 1882, and were systematically
studied and made accessible to non-mathematicians by early 20 th Century Russian
mathematicians and physicists, well represented by a 1949 book, Theory of
Oscillations by the Russian engineer-mathematicians, A. A. Andropov and C.E.
Chaikin. Another relatively early classic is Nonlinear Oscillations by Nicholas
Minorsky. The most common form of transition from a fixed point to a limit cycle was
pictured as changes in the surface of the action, the bowl-hillock manifold in the
paragraphs above, and is called a Hopf bifurcation. Recall that bifurcation means a
discontinuous change in an observable over a continuous change in what is known
as a control parameter, such as dose of amphetamine or intensity of an experience.
The mathematical mechanism resulting in circular directional motion represented by
the (eigen)vectorial states, was named for the German mathematician, Eduard
Hopf. His 1942 paper was a mathematical proof of its existence and was discussed
in the context of fluid flows that role up such that circling vortices arise from smooth,
called laminar, water flow, at a critical value of the flow rate. Hurricanes are another
example of these kinds of dynamics.
The Hopf bifurcation to limit cycles has been found in several, many
dimensional, physical, chemical and biological systems. The latter include calcium
conductance oscillations in the excitable membranes of muscle, heart and the brain,
cardiac arrhythmias such as ventricular flutter as well as oscillations in population
numbers in foxes and rabbits, predator-prey systems. California Institute of
Technology’s Professor, James Old and Johns Hopkins Professor, Joseph Brady
made experimentally obvious the potential for the rigid irrationality implied by the
brain’s inclination to be locked up into limit cycle behavior. They demonstrated that
animals, from rats to monkeys, could get locked up in apparent self torture,
repeatedly and endlessly pushing a bar to deliver current to pain systems in the
their brains. These pushes induced almost unremitting screams in monkeys and
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what appeared to be rageful biting and then immobilized resignation in behaviorally
depressed rats.
Freud’s last paper, Analysis, Terminable and Interminable (1939), featured
examples of what he perceived to be the unsolvable mystery of helpless
psychological entrapment in repetitious patterns of self-destructive behavior. He
blamed the Iliad’s and Odyssey’s villainous immortal, Thanatos, the everthreatening
spirit of death and destruction to contrast with the good, life giving Eros.
The Yiddish word for a personified Thanatos is Moloch ha-Moves. A range of
fixations in self-excitatory, repetitious, self-mutilating behaviors is documented in
domesticated animals. Dogs, particularly German Shepherds and Labrador
Retrievers, can lock up in compulsive grooming cycles of what is called acral lick in
which endless licking of paws or flanks lead to the break down of skin into seepingsore
dermatitis, which, in turn, stimulates more licking.
Mark Twain wrote a story about his getting stuck in ceaseless mental
repetitions of a catchy, clangy poem. He could not stop reciting it to himself even
after days of sleep loss and anorexia. He was finally cured by relating his problem
and the poem to his pastor who he then unwittingly heard creating a community
epidemic by including the rhyme in his following Sunday’s sermon. Psychologists,
who study this form of human mental limit cycle attacks, call this state of internal,
repetitiously recited, poetic stuckness, earworms.
There are additional invariants of sudden transformations into spiritual-mindbrain
bifurcations into a limit cycle lockups and, as discussed, one of them is
psychological splitting. In psychoanalytic theory, as first suggested by Freud in his
1937 written and posthumously published paper, Splitting of the Ego in the Process
of Defense (1940), splitting implies two simultaneous and contrary psychological
reactions, one can be conscious and the other unconscious. They can both emerge
in conflictual situations involving adaptive efforts of the personality to deal with the
opposition between some form of powerful instinctual pressure and attendant
perceived or imagined danger. Otto Fenichel’s Psychoanalytic Theory of Neurosis
(1950) elucidates multiple manifestations of splitting of the I (more technically, the
ego) into a conscious part that knows reality versus an unconscious part that denies
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it. In some situations, a logical view contends with a more irrational, magical one.
Today, the morning group praying, evening hymn singing, Christian Republican
Right Wing feed their feelings of being on the side of God by dividing people into
those that are like them and good and those that President Bush and Attorney
General John Ashcroft calls the evil doing “bad guys.” As noted previously,
psychoanalytic theory posits that the evil doing others may represent the projected
repository of our own unacceptable impulses and inclinations. It became quite clear
in my own psychoanalysis and psychoanalytic training that it is in healing our split
and knowledge of our own unacceptable things that will lead to our understanding
and forgiveness of others.
As we dig deeper into global brain-mind dynamics of emergent high-energy
fixation, stuck repetitiousness and splitting, we encounter their universality in the
structures of mathematical thought. Did we just make them fit? Do these thought
forms map onto internal and external physical reality? Are these abstract concepts
and operations simply products of our biological brains manifested as psychological
mechanics and used to explain to ourselves what we perceive and think? Does a
square have external reality or is it a universally imagined something, and, as such,
represented only in our minds and the pictures of it we draw? Is mathematical
understanding simply inborn perceptual skills combined with developed and
practiced logical cognition? Or, do we take the Platonic view of mathematical
relations: these abstractions are the ultimate realities, antedating and persisting
through the past, present and future of the universe and omnipresent.
Where can the conceptual boundary be drawn between the physical reality of
the Babylonian surveyors use of the Pythagorean theorem to calculate distances,
that the sum of the squares of the lengths of the two legs of a right triangle is equal
to the square of the length of its hypotenuse, and its abstract, pencil-marks-onpaper,
algebraic development as in the definition of Pythagorean numbers, a,b, and
c such that a 2 + b 2 = c 2 . The dichotomy between the abstract and concrete,
consistently blurred in our work, is between a natural science with ideas that can be
disconfirmed, directly or indirectly, by experimental observation and the thinking of
mathematics as an a priori field in the sense of Kant. The modern Platonic view
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such as that held by Rene Thom is that once accepting a set of natural givens,
called the axioms, the rest of the knowledge of this reality grows in the form of
theorems that relate to the axioms and each other through their logical consistency.
Knowledge of reality is moved by the ever-forward mathematical refinement of a
priori conditions to do away with the theorems’ exceptions, called counter examples.
The Hebraic Bible’s view of signifiers such as words and symbols is close to,
but not identical with, the Platonic view of mathematical formalism. According to the
Torah, God made the word with words. God spoke and the world became real. The
Aramaic for “I create in speaking” is avara k’davara , or as the magician says, as he
waves his wand over an apparently empty black high hat, abracadabra. The
Hebrew word for word, davar, also signifies thing. This view contrasts with the
mathematical formalists, among them Hilbert, who considered the signifiers of
abstract mathematics simply symbols used in a game, the rules of which being
arbitrary, must include proofs of consistencies among them. Consistency from the
point of view of physics was addressed by Hertz, in Die Prinzipien der
Mechanik,(1894), where he expressed the formalist theoretical physicist’s work as
“…within our own minds we create images or symbols of the external objects, and
we construct them in such a way that the logically necessary consequences of the
images are again the images of the physically necessary consequences of the
objects.”
In another set of related contrasts, the constructionist mathematician will
argue that mathematical assertions are only true if they can be demonstrated, found
or constructed. In contrast, the classical school of mathematics can develop the
case for the truth of mathematical statements if they are consistent with field’s
network of theorems and proofs, even if, up to the current time, no specific example
of this truth can be demonstrated. The former can be thought of as a builder, the
latter as a discoverer. For example, suppose we try to make a proposition about
perfect numbers where a perfect number is defined as being equal to half the sum
of its divisors. Using the perfect number 6, we find that its non-identity divisors are
1, 2, and 3 and half of their sum = 6. Our proposition: either there exists an odd
perfect number, or else there exists no odd perfect number. An expression of this
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forced decision between yes and no is called the excluded middle. The
constructionist mathematician, an orientation without the excluded middle, asserts
that “an odd perfect number exists” would only be meaningful if one could show that
such a number had been found or constructed. The classical mathematician would
find the phrase “no odd perfect number exists” meaningful without a concrete
example, if the assumption of its existence would lead to a no (versus yes)
contradiction encountered in the proof-relevant network of established theorems
and their relations. The symbolic operations of these formal schools of mathematics
and their relationship to the objective and ideational realities of brain-mind-spiritual
life have been viewed by some as Western cultural products rather than
expressions of secular or spiritual Absolutes. Still others have argued that cultural
relativism is not relevant here because mathematicians worldwide constitute a
monoculture.
With respect to the real world existence of abstract mathematical structure,
our Platonic bias must be obvious. The thrilling experience of a new reality I get to
know from finally understanding how a theorem works and the rush of peering into
the grandeur of the Grand Canyon feel like the same kind of full-of-wonder high to
me. I blend them here without reservation. Perhaps this world of spiritual abstraction
is closer to the orientation of the school of intuitionist mathematics. Its founder,
L.E.J. Brouwer, required that every mathematical construction be so immediately
apparent to the human mind that no formal proof was necessary. This became my
form of spiritual transcendence, which led naturally to a mathematical, mystical
faith.
We carry the explication of this kind of reality further. Reflections of the good
and evil, right and left, moral directional biases and their relative weightings in born
again bifurcations to invariant circles called limit cycles, can be symbolically
represented in what are called the complex eigenvalues of matrices describing the
system’s set of orthogonal motions with changes in their control parameters. The
behavior of these complex eigenvalues underlies and characterizes the
mathematical mechanism of the Hopf bifurcation.
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The subject of complex eigenvalues brings up in me the emotionally
disturbing subject of imaginary and complex numbers. I can still feel a little of my
earlier anxiety. The episode started benignly enough. Our high school’s freshman
algebra class was studying how to solve quadratic equations, equations in which
the highest power of an expression was two. Told to work at the blackboard in front
of the class, I was given the problem of finding the two values of x that were the
roots of the equation, 5x 2 + 3x + 4 = 0. I had been taught to use the memorized
quadratic formula,
2
− b±
b −4ac
x = , in which a = 5, b = 3 and c = 4. I always
2a
calculated the square root part first and wound up with the expression,
9 − 80 = −71.
I can still feel the sinking feeling in my stomach as I looked at the
result. I anticipated the usual snide remarks and embarrassment as I contemplated
doing what I did not know how to do, take the square root of a negative number. Mr.
Kirby, the retired mechanical engineer who was my high school freshman algebra
teacher tried to help, but I did not trust him. It seemed to me that he had already
humiliated me in front of the class, several times. He asked, “… what number when
squared, multiplied by itself, would equal –1.” He then asked it another way: solve
the following equation for x: x 2 +1 = 0. Seeing something I could do, I wrote the next
line quickly x 2 = -1 and then, taking the square root of both sides, I wrote x = − 1 .
He then asked me what that meant. I answered by writing quickly, glibly and blindly
that that meant that − 1×
− 1=−1. He asked me to explain what that meant by
giving him an example from the real world. Not yet knowing about imaginary and
complex (combine real and imaginary numbers), I stood head down, ashamed and
silent, thinking that my smart friend Jerry Blau would get the answer immediately.
Mr. Kirby said he would go on with the class while I continued to stand in front of the
blackboard and thought about it. He told me to interrupt him when I was ready to
answer. Some classmates were smirking, others giggled aloud. They had seen him
do this to me before.
Mr. Kirby, a short, muscular man, an ex-marine with a military haircut and a
brusque manner, lectured that mathematical competence and obedience to
authority and class discipline were all of a piece. I asked him about mathematical
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creativity and he said that this class was certainly not about that. I disliked and
feared him. He seemed to feel (and wrote a note to my parents to the effect) that,
being “too arrogant” I needed to be “brought down a peg or two.” I had gotten the
best grades in the first two exams and was enjoying the role of after school tutor for
some of my friends. I suspect I was getting pretty egotistical. In class, I found myself
eagerly shouting out answers without holding up my hand, behavior that Mr. Kirby
met with his characteristic look of fatigued disgust. Twice I was thrown out of class
for my introjections. He then began to give me problems that I could not do, for
which I was not prepared. This left me standing at the blackboard until the end of
the hour, after all the rest of the students had solved theirs and sat down. On
parent’s night, Mr. Kirby told my father that I needed more “social and intellectual
discipline.”
Inspired and personally directed hard work and socially defined correct
behavior were not synonymous to this arrogant 13 year old who had already
brought chagrin to his mother, the conservatory classical piano instructor, with his
satirical pianistic jazzy composition called “How High the Moonlight Sonata.” I was
also a secret reader of the book on the top back shelf in my father’s library by Jack
Hanley called “How to Make Mary; A Gentlemen’s Guide to Seduction.” In Mr.
Kirby’s class, inspired by the book, I sometimes reached behind me, through the
crack in my desk seat, to caress the inside part of the long smooth legs and
sometimes moist panties of a well developed, tall and beautiful brunette girl behind
me. I was never caught and she pretended that nothing was happening. In fact, she
never talked to me outside of class. I felt then, vaguely, and now, more specifically,
that a content enriched, instinctually titillated and excited unconscious could lead
me to the solutions of intellectual challenges if it were both sufficiently indulged and
untrammeled, left alone in its work of being itself. Mr. Kirby did not see things that
way.
Since then, among my graduate and post-doctoral students in the
neurosciences, I have learned that the Mr. Kirby’s of modern American educational
practice have ruined generations of potential mathematicians and physical
scientists. Worse, they have created generations of very bright math phobics who
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run to other graduate fields such as biology and medicine and come to resist the
potentially humiliating incursions of new and potentially helpful abstract ideas and
operations from mathematics and physics into their fields. They do not want their
persecutory versions of Mr. Kirby to take up residence once again in their heads. I
can still feel his negative presence during long hours of struggle with the ego
deflating feelings of dumbness that an understanding of almost any new
mathematical concept requires of me. Holding Mr. Kirby’s voice off as long as I can
until, sometimes, the wonderful “aha!” experience arrives. I have tried to forgive him
since but forgetting him has not been possible.
It turns out that in the world of elementary, physically representative, real
numbers, the square root of a negative number has no meaning. Such a number
has understandably come to be called imaginary. Was this the answer Mr. Kirby
wanted? There was some conflict among mathematicians in the 17 th
and 18 th
Century about the arbitrary definition of −1
as an imaginary number. It was
symbolized by a letter, i, that is −1 ≡ i . The existence of i extended the range of
algebraic definitions so that a solution of the quadratic formula as above could be
found for the square root of a negative number. A further expansion of this idea was
to that of a complex number that can have both a real and an imaginary part. For
example, letting letters be generalized representations of numbers, a complex
number might be written, a + bi, real number a + real number b times i, the letters
such as a, b, c, d… symbolized real numbers. Consistent with membership in an
algebraic system, a + bi and c + di can be added and multiplied. This extension of
the real numbers into the imaginary realm permitted d’Alembert’s and Gauss’s
proofs (and many, more complete ones since) of the powerful Fundamental
Theorem of Algebra from which the faith derives about always being able to find at
least one solution to an algebraic equation. It was proven that any n th
algebraic equation (e.g.
has at least one real or complex root.
x
n
degree
n 1
+ x − + ... = 0 ) with real or complex coefficients always
Closer to an image that helps make intuitive connections with human born
again bifurcations, limit cycles and directional splitting is the geometric interpretation
of a complex number, let us now call it z. As above, algebraically, z is the sum of a
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real part a, plus b times the imaginary part, bi”; that is, z = a + bi. We can then set
up a geometric space to represent z by imagining a two dimensional plane with the
horizontal real axis extending from left to right, the usual x axis, and the vertical
dimension, called the imaginary axis, extending from bottom to top like the standard
y axis. These two axes, going from negative values to positive ones, left to right and
bottom to top, cross at the shared value of 0. Thus a and b can be visualized as the
rectangular coordinates of a point in the plane and the point locates the complex
number, z = a + bi. Since real parts and imaginary parts are like apples and pears
and for addition, like must be added to like, if two complex numbers, a + bi and c +
di are equal, then a = c and b = d and their sum is written (a + c) + (b + d) i.
Now that we’ve set up a point z on the plane, located with a complex number
at z = a + bi, we can then draw an arrow, called a vector, from the intersection of the
imaginary and real axis at 0 to this point z. Its length from 0 to z, 0z, we’ll call that
length ρ , is the size or amplitude-like modulus of the complex number, z = a +bi.
The angle this 0z vector makes with the real, 0a-axis, lets call this angle φ , is called
the argument of complex number z = a +bi. ρ is a length that can grow or shrink, φ
is an angle that can rotate. We imagine vectorial movement like that of a variable
length hand of a clock. This geometric explication of complex numbers prepares us
to visualize complex numbered eigenvalue solutions to matrices representing the
relevant equations that bifurcate to limit cycles and directional good and evil
splitting. ρ represents the dilatable clock’s radial amplitude of circular motion and
φ , the angle of vectorial turning from the 0a-axis.
The complex conjugate of the complex number, a + bi is the complex number
a – bi in which the sign of the imaginary part is reversed. Geometrically, this means
that a pair of complex conjugate numbers with the ρ ’s of both having below zero
values relative to the 0a-axis, that is negative real parts, could be imagined as the
points indicated by two same sized, mirror image, clock hands pointing at 8:00 and
10:00 o’clock. Note that the φ , the angle of vectorial deviation of the arrow pair from
the 0a-axis, turn in opposite directions in these mirror image moving clock hand
vectors. Without going deeper into the representation of the actions of the system in
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question (its differential equation) in the form of what is called its Jacobian matrix of
partial derivatives (a matrix representation of the differential equation indicating
orthogonal directional velocities of change of locations of the components of the
motion with respect to changing values of the control parameter), we know that
when the ρ of the matrix’s set of two complex conjugate eigenvalues is less than
zero, ρ < 0, the orbit representing the system, spirals into a stable fixed point. This
is analogous to going to the bottom of the parabolic attractor basin as described
above. Values of the invisible eigenvalues and their changes constitute the abstract
mathematical mechanisms underlying the observable dynamics of the system
observable physically.
The mathematical mechanism underlying the Hopf bifurcation of fixed points
into limit cycles (associated with bi-directional splitting that accompanies the
amphetamine transformation into limit cycle stereotypy of rigid ideas and equally
likely mirror image motions in the directions of good versus evil) is the crossing of
the systems real valued parts, ρ ’s, of its complex conjugate eigenvalues into
positive territory, ρ > 0. The mirror image of clock arrows is transformed from 8:00
and 10:00 o’clock to the clock locations of 4:00 and 2:00. At a Hopf bifurcation, a
pair of complex conjugate eigenvalues crosses the imaginary (vertical) axis such
that is real parts have positive value. In the orbit representing the motions of the
system itself, the fixed point disappears to be replaced by the action spiraling out to
an invariant circle. This is analogous to our manifold image of the disappearance of
the central attractive point and the sudden appearance of a small hill at the bottom
of a parabolic basic of attraction.. The new attractor is an invariant circular path
around the hill, with the spiraling out to the invariant circle being a two dimensional
picture of the disappearance of the bowl-bottom and appearance of a missing point,
hill top fixed point and a spiral flow to the path circling the hill. Underlying the
transition from a fixed point to a limit cycling, invariant circle, are a pair of mirror
image complex conjugate eigenvalues that turn in mirror image, we could say, good
versus evil, opposite directions. The Hopf bifurcating system inevitably has both.
The implications of this very abstract metaphor for the emergent limit cyclesplitting
style of spiritual transformation can be made deeper by considering the
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common practice of Rumi’s Mevlevi (and other) orders of Islamic Dervishes that
facilitate the onset and maintenance of their ecstatic states by an improvisational
dance which goes from rocking to irregular whirling. The Dervish teaching tales
place a symbolic emphasis on the power of the rotating wheel, the circling of the
heavenly bodies, the mill wheel and the millstone. As Rumi said, “The mountain of
the sun I’ll fashion to a mill. And as my waters run, I’ll turn thee at my will.” Note that
their work toward spiritual transformation results in neither the emergence of the
involuntary and rigid limit cycles of invariant circles or the associated divisive
internal eigensplitting of good self from evil other. The Sufi compass points to an
integrated field of divine consciousness, which contains the appearance of the
world’s multiplicity. In this profound unity, all humankind is perceived as one family.
The singular direction of all prayer, Salat, five times a day, at dawn, high noon,
afternoon, sunset and an hour after sunset, turns the entire world into a unified
directional field of prayer. At its center, the Islamic pilgrims wander round and round
the black cube of the ancient shrine of Kaaba,
This leaves one with the speculation that we started with: that the simple,
authoritarian rules of the amphetamine, roll-up and splitting religions may be
intrinsically more vulnerable to unpredictable breakouts into morally inconsistent
actions and that the righteously rigid limit cyclists are invariantly split into
ambivalence. In contrast, the more free form, chaotic turns of the entheogenic
dervish define us all as belonging to one unified ecstatic field.
Further Readings for Amphetamine Roll-Up And Splitting
Psychology and Religion. Carl G Jung, Princeton Univ. Press, N.J. 1938.
The Faith of a Heretic, Walter Kaufmann, Meridian, N.Y. 1959.
Nightmare Season. Arnold J. Mandell, Random House, N.Y. 1976.
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The Rabbinic Mind. Max Kadushin, Bloch , N.Y. 1972.
Coming of (Middle) Age. Arnold J. Mandell, Simon and Schuster, N.Y. 1978.
Introduction to Islamic Theology and Law. Ignaz Goldziher, Princeton Univ. Press,
N.J. 1981.
Tales of the Dervishes. Idries Shah, Dutton, N.Y. 1970.
Open Secret; Versions of Rumi. J. Moyne and C. Barks, Threshold Books, Putney,
Vermont. 1984.
Amphetamine Psychosis, P.H. Connell, Oxford University Press, Oxford, 1958.
Amphetamine Use, Misuse and Abuse. David Smith, Hall, Boston. 1979.
Long-term Administration of D-Amphetamine. David S. Segal and Arnold J. Mandell,
Pharmacology, Biochemistry and Behavior. 2:249-255. 1974.
Amphetamine Enhancement of Reward Asymmetry. S.D. Glick, L.M. Weaver and
R.C. Meibach, Psychopharmacology 73:323-327, 1981.
Hopf Bifurcation and Its Applications, Appl. Math. Sci. Vol. 19,. Springer-Verlag,
N.Y., N.Y. 1976.
Dynamics, The Geometry of Behavior, I-IV, Aerial Press, P.O. Box Office 1360,
Santa Cruz, CA 1982.
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. John
Guckenheimer and Phillip Holmes, Springer-Verlag, N.Y. 1983.
166
Psychiatric Aspects of Neurologic Disease. D. Frank Benson and Dietrich Blumer,
Grune and Stratton, N.Y. 1975.
Drives and Reinforcements. James Olds. Raven, N.Y. 1977
Neurobiology of Stereotyped Behavior. S.J. Cooper and C.T. Dourish. Clarendon,
Oxford, 1990.
Mathematics Unlimited—2001 and Beyond. B. Engquist and W. Schmid, Springer,
N.Y. 2000.
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CHAPTER 8:
FAITH AND RATIONALITY
It was my belief that, without subjective evidence of Holy Spirit Energy, the
rush of reconfiguring transcendent experience, some glimmering of grace no matter
how fleeting, an experience of intoxication with God, Martin Buber’s self
authenticating I-Thou encounter, the many good citizens of this world, without these
moments of illumination, must be attending church or temple to negotiate a better
now and hereafter. Attending synagogue or church without the promise of a
mystical high seemed like a superstitious rabbit foot rubbing for personal health and
safety and a sharing of propitious contacts for social and economic advantage. Why
else?
I have had the good feel of what Jews call Tzedakah, the sharing of supplies
by the haves for the betterment of the have nots. I have known the quiet calm of
human right action as in the Unitarian Universalist’s serving the needy, open and
flexible, intimate mindfulness of others and their needs. Considering E.O. Wilson’s
brand of brain herd biology of altruism gives me a warm feeling about the potentially
intrinsic goodness of man. But compared with the Jamesian brands of ecstatic
transcendence, minds blown in Sufi twirling, Orthodox Jewish chanting, rocking and
dancing, hands-in-the-air praying and hands-on-the-head healings of Wednesday
night Pentecostal services, the soberly serious social engagement and
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responsibility sermons of Reformed Judaism and the Unitarians as well as the 19 th
Century hymns and high I.Q. apologetics of some Presbyterian and Methodist
clergy, are like near beer. Formally equivalent but without the rush and the delicious
risk and promise of life long addiction.
National opinion polls have found my preference for churchly fireworks in
religious experience quite common. My Charismatic Christian sons are among the
many with a preference for and loving labeling of these kinds of houses of worship
as rock and roll churches. In a recent survey of Americans, 46% of respondents
claim to be twice born, Evangelical Christians. Perhaps unfortunate with respect to
their children’s academic and professional ambitions, 48% do not accept a
Darwinian view of biology. Fifty million American readers are now buying books with
plots taken from the Babylonian prophecies and anticipate the Rapture of Return
with weekly, joyful, mini-rehearsals. They include praying in tongues as the Spirit
moves them like Peter, John, James and the rest of the one hundred and twenty in
the upper room on the day of Pentecost.
Those of us with two or more available cable religious networks can, on any
given Sunday morning, choose a smiling, kind, Proverbs quoting, rational
Presbyterian liturgical stylist. In his seventies, standing tall with a full head of white
hair and in a quietly resonant voice, he delivers a sermon about seven ways to
avoid growing old. His list includes learning new things and continuing to work. His
spiritual proposal was about personal faith, always leaning on the Lord. On another
network, the three hundred pound, restlessly pacing preacher of the Cornerstone
Assembly of God Church of San Antonio, Texas, stood in front of large maps of Iraq
and the Middle East. He preached from Ezekiel about the refleshing of dry bones
and a return of all Jews to Israel. He said that contributions to his church over the
past year helped finance the return of 4000 Russian Jews to Israel. He reiterated
the promise that, when the return was completed, there would be a massive Islamic
attack on Jerusalem and “we will all rise up to Heaven” in an ecstatic
disappearance. Jews, as long as they accepted Jesus as their Savior, were
welcomed along on the ride. More then two thousand parishioners erupted into loud
applause along with shouts of “praise Jesus.”
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An inkling of something entirely different, neither human psychology nor
frenzy, was an unanticipated benefit of being at England’s Warwick University in
sabbatical residence in Math House #2. This large, round, many windows and black
boards, study with a small upstairs bedroom was one of the apartments for visiting
professors behind the Warwick Mathematics Institute in the English Midlands. I
attended a variety of churches and synagogues on the weekends. The perspective
that emerged for me at Warwick was that rabbinic Haggadah, inferences to be
drawn from imaginatively spawned narrative, isn’t the same thing as Halakhah, the
law dictated by Jewish legal tradition; that geometric insight and other intuitions
aren’t the same as mathematical proofs; that the mystical visions of the English
romantic poet and illustrator, William Blake, were not necessarily consistent with the
scientific observations and logical arguments of the contemporary Scottish
philosopher, David Hume. Paul Tillich wrote that the wisdom attendant to primary
spiritual experience that was without the unconditional character of sensible moral
obligation was not to be trusted without critical analyses. I learned that among High
Episcopal and Reformed Jewish English academics, God is not a hallucinogen, but
more like a spiritually based, social contract.
In his 1929 essay, Mysticism and Logic, Bertrand Russell noted mysticism’s
preference for: (a) Insight over discursive analytic knowledge; (b) Belief in the unity
of all things over oppositions or divisions in representational thought; (c) The denial
of the reality of time, even in the divisions of past, present and future; (d) Belief that
evil is unreal, manufactured by the innate divisiveness in some analytic intellects. In
modern brain hemispheric and other neuropsychological philosophies, these
countervailing descriptions of external observables can grow naturally out of the
brain’s abilities to maintain logically incompatible perspectives simultaneously.
Right-brain aesthetic holism in contrast with left-brain categorical analytics recalls a
popular example. Would one chose Blake or Hume to better explain how the time
dimensions of memory disappear with the scent of a past lover or the hearing of his
favorite music for lovemaking.
In the inevitable mix of primitive instinct with high purpose, the visiting
professors’ Math House #2 had an aura of infamy. It was the one in which, by the
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accidental intrusion of a campus security officer, the brilliantly eccentric Northern
California mathematician, Ralph Abraham, was famously arrested for pot smoking.
The campus officer told me that, late one night, thinking he had smelled fire, he
used his master key to make an unwelcome entrance. The incident became part of
the record in House of Commons hearings about the intellectual and moral decay of
English Universities. Apparently, even among English intellectuals, there were trivial
and politicized definitions of virtue.
Christopher Zeeman, the head of the Mathematics Institute was a worldclass
topologist who, among other things, demonstrated biological and socialpsychological
applications of Rene Thom’s Catastrophe Theory. I was invited as a
brain person and amateur mathematician, to see what might result from mixing me
with members of his fine mathematics faculty. In addition to learning some
bifurcation and lots of ergodic (statistical) theory, my chats with Christian and
Jewish mathematicians on Saturday and Sunday morning visits to the synagogues
and chapels of Oxford and Cambridge introduced me to an English intellectual’s
religious tradition. The spirit of C.S. Lewis was still very much alive. Surprising,
however, was that more than a few of these scholars had the elements of Christian
faith in full menu: virgin birth, incarnation, crucifixion, resurrection, original sin and
the promise of salvation. I was disabused of my belief that these elements of
Christian belief were incompatible with high mental capacity and intellectual
sophistication.
Yet, the spiritual climate of these English intellectual Christians were different
from today’s post, post Vietnam return of the religious themes of the turn of the
Twentieth Century, big tent revivalism and Billy Sunday’s brand of Christian patriotic
America. Today’s religious patriotism infuses George W. Bush’s Republican base,
National Security Adviser Condoleezza Rice’s after dinner hymns and Attorney
General John Ashcroft’s early morning bible study groups for his Assistant Attorney
Generals. Even the most religious of my English math buddies are without what
seems like adventitious baggage of today’s faith based Republicans: the belief in
the immorality and godlessness of teaching evolution in schools, what has been
called the massacre of the innocents in stem cell research and abortion clinics, the
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right to bear machine guns and the intrinsically venal sinfulness of a man’s
commitment in love of another man. Was the clustering of these apparently diverse
concerns the accidental result of a sociopolitical-religious short circuit, a classresentment-driven
spiritual split in geographic, socioeconomic and educational
class? Tim LeHay is selling millions of books, whole tables full at Wal-Mart’s, which
come packaged with these assumptions.
Surely higher-level theists would make today’s evil more subtle, abstract and
pervasive, perhaps involving inner life themes of envy, vengeance and aggression;
goodness implicating empathically made moral choices involving interpersonal
kindness and evidence of caring about the well being of others. My contact with
some English academicians taught me that even the mathematics of hard science
can be viewed as a gift of grace and belief in the possibility of a continually
emerging, Christ-centered, evolutionary process. Protestant philosopher
mathematician Alfred North Whitehead in his 1926 Religion in the Making, Catholic
anthropologist priest, Pierre Teilhard de Chardin in his The Phenomenon of Man
and the more modern process theologists of New York’s Union Theological
Seminary do not exclude Christ’s involvement in evolving science and other new
knowledge. They see Him participating in a spiritual evolutionary progress which
does not gather the barnacles of irrational ideas about the murder of less than
hundred-cell blastula or the psychoneurohormonally determined sexual partner
preference. They know about the ever-changing cultural and political appearances
of faux and real evil. Nonetheless, what I learned from my Christian and Jewish
friends at the mathematics institute was that, though the definitions of evil may
change, evil as a construct and spiritual mechanism is an apparently essential
component of the Christian experience. On Rosh Hashanah, even the reformed
Jews commit themselves to Teshuvah, making up for past evil deeds. The good
versus evil dichotomous view of man’s existence is true in the lives of Assembly of
God Fundamentalists of Georgia as well as the sophisticated Readers, Professors
and Dons of the high Episcopal churches and university chapels of Oxford and
Cambridge.
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Finding high-level mathematical thinkers at home in metaphysical surrounds
and metaphysicians diligently practicing mathematics are certainly not new. Some
instructive examples include, famously, the Pythagoreans, the 15 th Century Catholic
Cardinal Nicholas Von Cusa, who used geometric symbols to record his spiritual
philosophy, and the Talmudic-Cartesian style of argumentation of Nicholas de
Spinoza. This approach to an examination of metaphysical systems, sometimes
called mathematicism, exploits the machinery of the mathematical mind to evaluate
the consistency and completeness of thoughts, to create representative axiomatic
structures and to operate within them using syntactic calculus. The practice of the
rational dialectic of mathematicism, working for moral purity of heart, develops a
brain-somatic discipline much like the exercises of Yoga.
This approach flies in the face of the major premise of these essays, my
belief in the necessity of what William James and others have called the primary
religious experience in order to know God. Recall that my father’s favorite Jewish
mystic, Abraham Abulafia, said this experience gives birth to an activated mind that
can then immediately and completely inform the Spirit. Among the religious English
mathematicians, I learned that it doesn’t have to happen this way. One can
apparently think oneself to It. A well known example of a modern theistic Oxford
type, the Magdalene College English tutor and Don, C.S. Lewis, in his introduction
to St. Athanasius’s The Incarnation of the Word of God, wrote, “…I believe that
many who find that nothing happens when they sit down or kneel down with a book
of devotion, might find that their heart would sing unbidden while they are working
their way through a tough bit of theology with a pipe in their teeth and a pencil in
their hand…” In contrast, without my personal experiences with joyful
transcendence, the direct feeling of His presence, I would not have known about the
goals of his more analytic efforts. It was a struggle for me to use a rational mind to
share the meanings of the poetic ruminations in his BBC lectures, Mere Christianity.
This Reader from Oxford with two firsts in Latin and Greek followed by another first
in English Literature, described the world as “…enemy occupied territory…” the
omnipresence of the Good Power turned Dark Power of the Prince of Darkness and
the Christian as “…a man who is enabled to repent and pick himself up…”
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For C.S. Lewis, religious faith came from intellectual hard work. He was put
off by spirituality that arrived by thoughtless fiat. He rejected the idea of living in
simple and loving direct conversation with the God within, as described by Brother
Lawrence. Lawrence was described as the simple “great awkward fellow who broke
everything.” Lewis had little faith in what he perceived as the mindless spiritual
methodology of this selfless, silent, hard working Parisian monastery cook for a
hundred fellow monks who was also their dedicated smelly sandal repairer. Perhaps
reflecting his place in the British intellectual class system, Lewis wrote that
Lawrence’s conversations and letters in the brief pamphlet, Practice of the
Presence of God, “…full of truth… but unctuous and repulsive.” At the same time,
Lewis spoke of his own experiential evidence for God in Surprised by Joy in which
he admits, “I am an empirical theist. I have arrived at God by induction.” It is likely
that Brother Lawrence did not know and did not need to know the difference
between an inductive and deductive argument.
For most of my years, I have been a subject of Jamesian transcendent
experience, LSD expansive visions, Sufi moving meditation, long distance running,
Black Baptist shouting, Tantric orgasmic withholding, Yiddish Labovicher dancing,
Charismatic Christian Church rock and rolling, Hindi meditative rising Kundalini,
almost any ecstatic crisis inducing, God type. Recall that I am from a generation
that a Donovan song inspired to smoke bananas. I did not personally access
Brother Lawrence’s calm, work-a-day, devotional, quietly persistent, perspective
yielding, inner conversations with God until my sixth decade. The opportunity came
from my growingly severe, unfixably chronic, pain. The counter-intuitive insight and
helpful identification was gained from reading about Joseph de Beaufort’s
conversations with Brother Lawrence. Beaufort said Lawrence was born with the
name Nicholas Herman in 1611 and renamed Lawrence in honor of his parish
priest. As young soldier in the Thirty Years War of the 17 th Century, he was severely
injured. He was left with both sciatic nerves trapped between bone spurs and tissue
scarring from his early twenties. These injuries, involving the two biggest painconducting
nerves in the body, left him crippled in gait and in chronically severe
lower back and leg pain from which he would never be free. It was after this time
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and a few years of looking for God in what he called “wondering in the wilderness”
that he began his 40 years of monastery service as cook and sandal maker. He was
described as amazingly selfless and a “…gentle man of joyful spirit…” who
“…continually walked with God…not from the head but from the heart…” Doing long
hours of selfless work with such painful disabilities, how was it that he maintained
his joyful, loving and calm contact with God and his fellow man? How did he do it? I
found that, as with all miracles of God contact for me, it happened by itself.
I suffered my first testicular cancer in my thirties. I felt the little hard rock by
accident while scratching. It was on the left side. Surgical removal was followed by
a five-hour radical abdominal lymph node dissection that left me with incidental
abdominal sympathetic nerve damage, urinary hesitancy and ejaculating backwards
into my bladder. The tissue diagnosis was of embryonic cell carcinoma with
chorionic elements. The U.S. Armed Service Pathology Department’s statistical
book gave me 5% chance of living beyond two years. My second testicular cancer
occurred in my fifties and on the right, two little joined lumps found by my wife. It
was a seminoma with cure rate of 85% but requiring four weeks of almost daily x-
ray treatment. The combination of radiation induced blood vessel scarring (they had
to blast widely since my earlier lymph node dissection confused the usual
radiological anatomy), a pre-existing laterally curved spinal column and the arthritic
changes resulting from fifteen years of running over 10 miles per day with this kind
of back led eventually to the degeneration and collapse of several of the bodies of
my vertebrate pinching several leg nerves between bone spurs and radiationinduced
scarring. I have been in increasingly severe back and leg pain for fifteen
years.
It was in this way that I fell heir to both Brother Lawrence pain syndrome and
what I now think was his strong inclination to live in the Spirit, as far as possible
outside the concerns with his own mental and physical body. In my experience, this
led naturally to a decreased in my life long narcissistic preoccupations, diminished
my ego-driven achievement desperation, setting up a more comfortable inner
seating for conversations about and with God. The choice was between fully
embracing a God-oriented place for most of my daily existence or the chronic use of
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enough narcotics to eliminate complexity of thought, real interpersonal feelings and
hope for meaningfully creative work. The remarkable thing to me was that people
began to talk about my “improved disposition,” an increase in out-of-mypsychiatrist’s
office personal empathy and kindness as well as a significant
decrement in my overweening, ego-stoking ambitious and competitive urges. Any
return to the earth body of tense readiness to competitively succeed, protect with
ego defensive anger, fantasies of assertive sexuality, stand tall grandiose notions of
intellectual superiority, even getting up for scientific combat, was accompanied by
the return to this world of pain. Only lovingly detached, unpretentious, other
directed, quietly calm inner dialogue with Him was a place that I could live. This was
an inner land of still another kind of God than I had previously known. I could even
read and struggle with theological ideas thoughtfully, without referencing personal
mystical, psychopharmacological, Holy Ghost-mimetic, experiences. I could enjoy
the rational, social responsibility valuing, spiritual peace of a white Protestant
Sunday morning service. I could attend Reformed Jewish Friday night services
about man’s responsibility to man without restless boredom. No longer seeking the
feeling of God’s thrill, I could think about it, even without being in the state of my
father’s and Abulafia’s activated mind.
If I had been benefited with a classical language education beyond the high
school and early college Latin of Julius Caeser and Cicero or matriculated in an
academic theological seminary, I would have already studied, maybe even worn
out, the deeper aspects of what seemed like a paradox of the consonance of faith
and reason. I would have been familiar with the rhetorical argumentation in the
patristic Latin commentary on sacred texts by Tertullian and other Fathers of the
early Christian Church, the Talmudic discussions (the Mishna in Hebrew and
Gemora in Aramaic) of the oral Torah by the Rabbinate, the Muslim explication of
Koranic Islam in the oral tradition of the Hadith. Robert Wilken in his recent The
Spirit of Early Christian Thought was in no doubt about the harmonic relationship
between rationality and faith: “…by putting itself in the service of truth, faith enables
reason to exercise its power in realms to which it would otherwise have no
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access…” It is perhaps strange to come to this common knowledge so late, but I
came to my life with my forbearers and father’s magical, mystical biases.
My father had parodied what he thought was the “wasteful time” spent in
rational, Talmudic discussion. He said that is what Jewish men spent their time
doing to avoid physical work while sitting near the city gates. It was the women who
raised the crops and cared for the cattle and children. He had a favorite conundrum
satirizing the village gate discussions. Jewish males, after the age of thirteen,
accompany their morning prayer of commitment to loving and serving God with the
ritual of wrapping scripture embedded animal skin, tefillin, and winding them seven
times around the left arm, near the heart, and around the head, symbolizing the
mind. This contextualizes how my father made fun of a typical topic of these all
male Talmudic seminars: “If one had seven arms, would one wrap the tefillin once
around each appendage or seven times about one of them. If the latter is the case,
how would one chose which one.” In fact, there remains an on-going debate about
the order with which the embedded four passages from Exodus and Deuteronomy
should be arranged and inserted in the tefillin such that some compromising
orthodox Jews wear two types of tefillin, each representing one of the theoretically
justified orderings. I know now that there is an implicitly positive confirmation of a
jointly held faith and feeling of ethnic belonging achieved by such apparently
abstract discourse and argumentation.
In truth, I had not come to Warwick to explore the relationships between faith
and rationality using the cognitive style of mathematicism, but rather to be saved by
the mathematical miracles of the Brain God. Not unrelated to what C.S. Lewis saw
as a prominent characteristic of spiritual experience, “wonder,” and what Philip
Davis and Reuben Hersh in their 1981 book, The Mathematical Experience, spoke
of as “beauty” and “surprise.” I know about the attack of excitement that comes with
the sudden emergence of counterintuitive conceptual connections while exploring
new mathematical ideas. In energetic high, I start skip reading, underlining the book
frantically, jotting commentary on the margins, copying the relevant equations into
my notebook. Was this the same break through to a glimmering of grace, everything
beautifully in order and precious, that I experienced on LSD while sitting for hours
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inside Paris’s towering, echoing, Notre Dame Cathedral, hearing Latin chants in the
dank sweet smell of old church and chained, swinging canisters of smoking incense
as the pipe organ roared? Those realities that George Berkeley, the 1721 author of
Treatise Concerning the Principles of Human Knowledge, the theist whose name
was given to a mostly agnostic Northern California city, saw as grounded in the
spirituality of God’s infinite mind and broadcast as universal ideas through our
derivative, finite minds. Rational religion and mystical religion joined in faith by the
presence of implicit and universal mathematical structure
I spent about two years at a mathematics institute in France, Institute des
Hautes Etudes, IHES, sitting at the guru feet of the mathematical great and
metaphysician, Rene Thom. His mathematical pallet was breathtakingly broad, a
taste of what in past centuries was called natural philosophy and what seemed to
me to be about the unapologetic geometrization of the Intuitive God of the Mind.
Natalie Angiers, erstwhile mathematician, now reporter and atheistic hard ass,
writing in the New York Times, called Thom’s ideas the talk of “…an Emperor
without clothes…” The Kantian theme of the personal a priori status of an intuitive
geometry, an already in us representation of all that’s out there, was implicit in his
Catastrophe Theory research program and was published first in his classical
Structural Stability of Morphogenesis (1977) and made more overt in his later
(1990) Semiophysics.
To get a feeling for the rational-logical versus mystical-intuitive spiritual issue
in a mathematical context, consider the following: most of us remember the struggle
to unify the strange and difficult cognitive duality of the high school geometry
experience. On one hand, shapes and their relations and rearrangements could be
intuitively grasped, even manipulated; on the other hand, we were taught that these
mental images and the results of their intuitive transformations were not to be
trusted.
In mathematics, as in my belief in the fireworks of primary religious
experience, seeing is not necessarily believing. In my high school geometry class,
what was to be believed was what followed from the proper practice of the tightly
organized, Euclidean system of axioms, postulates and the derivative logical
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operations resulting in the surety of proofs. The unresolved tension about what I
believed from intuitive experience and what I was allowed to believe from the logic
of theorem and proof, perhaps not unlike my belief in the transcendent experience
over logical theological argument as Reality, continued throughout my life. For
example, many decades later at IHES, I saw the world class dynamical systems
theorist and differential geometer-topologist, Dennis Sullivan, use a projector to
display a computer-generated, intricate and beautiful, mathematical object, the well
known, computer screen saver, Mandelbrot set. It represents the control parameter
plane of the well studied complex analytic map, z → z 2 + c. Sullivan, pointing to a
small, discrete complicated little part of it that looked like a little version of the whole
of it, from a distance looking like a point, said, “An important Ph.D. dissertation is
waiting to be done on the question: is this (pointing to the little object) really there?”
In the audience of about a hundred professional mathematicians and one amateur, I
was the only one that laughed.
Historians of mathematics point to the successful generalization of Euclidian
geometry via its abstract axioms, postulates and logical operations to a new, not
naturally intuitable, almost nonvisualizable, non-Euclidean geometry (with the new
geometric axiom, parallel lines do meet at infinity), as evidence against the Kantian
idea of the intuitively accessible, a priori status of geometry. This served as an
example of where mathematics naturally resides, and argues in favor of the thought
control imposed by the modern set theoretic and logical rituals of mathematical
theorem and proof. Thom, in a hereditary-evolutionary biological argument
developed in Semiophysics, said “Objections raised to the Kantian apriority of
Euclidean geometry after the discovery of non-Euclidean geometries, and the
theories of twentieth century physics (restricted and general relativity, quantum
mechanics) appear to me to be irrelevant…they deal with …the infinitely small and
infinitely large…which lies outside the usual cognitive activity of ancient man.”
In my discussions with him, Thom found equivalence relations between
mental and real world objects and their behaviors. He described what he called an
abstract physicalist truth that describes a psychic universe, which, in turn, simulates
outside things and processes. Much like the transcendent experiential God I have
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experienced, seek and think I know about, Thom was not after the logical proofs of
geometry but rather viewed mathematical theorem and proof work as activity
derived from intuitive experience with geometric relations as the thought forms that
represented real Reality. Though a Field’s Medal winner in mathematics (recall that
it is the Nobel Prize in mathematics awarded every four years at the International
Congress of Mathematics) and for his life time, one of the most brilliant and fecund
mathematicians in the world, so many mathematicians admit that they got the seeds
of their life work from his throw away remarks, Thom, with a little smile and his eyes
twinkling, admitted to me with apparent pleasure that “I have never proven any
theorem in my life.” All his discoveries came from insightful moments of grace and
the courage to pursue them. Riding back from Paris late one night on a train that
didn’t stop at IHES’s town of Bures sur Y’vette, I watched him use the red
emergency phone to call the train’s engineer to stop the train suddenly for our exit. I
loved him, in part, because he had the courage to believe in and act on my kind of
intuitively realizable, experiential God.
In keeping with his characteristic style of generalizing mathematical systems
beyond their carefully defined specifics, Thom defined the concept of singularity
very broadly, speaking of them as distinctive and noteworthy things, points where
the usual or expected properties, laws and definitions fail, where smooth and
continuous processes become discontinuous. For Thom, these were the settings for
the unexpected and miraculous. He believed that his work and that of many others,
now and in the future, would indicate that the set of miraculous singularities were
finite, systematic, universal and describable. Most importantly for our purposes,
Thom believed them to be archetypal. It was through the structure of archetypal
singularities that he regarded inside and outside realities as mutually reflective.
I was blessed by hours of discussion with him during his car travels to
lecture around France. Thom often asked me to accompany him as he drove from
IHES to various branches of the University of Paris. He used these times to
exercise my geometrically flavored, mathematical intuitions. He used words to
create visualizable structures without the diagrammatic aid of a blackboard. He
used mental topological structures created by the properties of imagined motions,
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flows, which led to examples of some of his universal singularities that he claimed
could be found in all real physical, biological and psychological systems. For some
examples: One of his archetypal singularities was a boundary at x = 0 such that the
flow couldn’t spread from where it was in x ≥ 0 into x< 0 and was therefore like the
border, the membrane, between the inside and outside of a cell as well as the
hoped for sociopolitical functions of the Great Wall of China and the Maginot Line. If
we were to blow up the boundary line from two to three dimensions, R
2 →R 3 , the
straight boundary line becomes a cylinder for directionally organizing and
connecting flows as in blood vessels, oil pipes, cables and wires. Since production
and delivery need not occur at similar rates, temporary storage is required and may
take the form of a spherical blow-up in the vertical segment of R 3 leading to an open
bottle which may serve as a dead end storage branch of a network of connected
cylinders. In the conceptual reductionism of Semiophysics, Thom said, “…life is
essentially a question of embankment, canalization and the struggle to stem
dispersion.” These structures of mind and world are built and maintained.
Coagulation of blood is an example of a canalized fluid repairing gaps like a
tubeless tire. Thom considered apparent the problem of making something from
nothing, birth, that of finding the hidden sources: the bubbling spring emerges from
an unseen, underground network of canalized fluid flow converging on the apparent
source, birth being the invisible becoming visible. In contrast, a canalized flow
emptying into lake can represent disappearance as a flow.
Mathematicians from all over the world attended Thom’s 65 th birthday
celebration at IHES. His Field’s Medal winning work on the topology of differentiable
(smooth) manifolds, cobordism and related ideas, was mentioned frequently, and
great homage paid to him with respect to these areas of his work. However, in two
days of lectures of personal and professional tribute by the world’s great
mathematicians, his work relevant to Catastrophe Theory and Semiophysics was
not mentioned, even once. The form taken by mathematicians’ most severe
judgments is silence. As the New York Times’ Natalie Angier’s comments indicated,
this is not the time for the intuitive conduct of applied mathematics.
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It was upon Thom’s recommendation, that I spent the year in the
Mathematics Institute in Warwick, England. Using the Math House #2 s home base,
I made many trips to Oxford University and a few to Cambridge. It was in these
places that I learned first hand that belief in the Resurrection was not simply a
matter of socioeconomic class. I tried to schedule my trips to Oxford or Cambridge
to coincide with the weekend so I could hear the remarkably literate sermons at the
Universities college chapels. In these places, for hundreds of years, just because
one was a top-notch practitioner of mathematics or linguistics did not mean that the
Don did not have within him the full panoply of beliefs attendant to the Christian
God.
Maybe this easy combination of logic and Spirit derives from the character of
English mathematics. There are graduates with professorially enfranchising Masters
of Art Degrees in Mathematics from Cambridge University where the subject is
considered by many to be part of the culture of the humanities, closely akin to
philosophy and linguistics.. In the universities of United States, for example the
Massachusetts Institute of Technology, an academic degree of Ph.D. in
mathematics is seen by most faculty as an indication of the intellectual equipment
required for a life of scientific work in which disconfirmable experiments are the
ultimate criteria for knowledge. The field of pure mathematics (not ostensibly
relevant to the real world outside the mind) has itself evolved in this direction.
Recently, a physical scientist, a theoretical physicist, Edward Witten, was given the
mathematician’s ultimate award, the Field’s Medal. In American universities in
general, very few mathematics departments are in schools of the humanities. Most
are in the schools of science. This variation in bureaucratic, metaphysical, sorting
reflects our continuing struggle with the true nature of reality and the role of
mathematics in its knowing. The now emergent field of computer science removes
mathematics even further from intuition and Spirit. Difficult problems such as proofs
of theorems can be systematically examined for all possibilities quickly by trying
them out in what is now known as a computational proof. On the other hand,
pointing at this computation’s graphics, the theorem and proof, real mathematicians
can ask, is this really there?
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Mentioned briefly above was one of humankind’s beacons, Pythagoras, the
intellectual and spiritual progenitor of Plato. He taught the disciples of the
Pythagorean Brotherhood in Crotona, Italy, that reality at its deepest level was
mathematical thought. Studies there included philosophy, geometry, music and
astronomy, all at the service of achieving closer union with the Divine. Pythagoras
and his school, only his student’s writings remain, was said to be working at unifying
elements of the ancient tribal mystery cults with the observables of worldly events
through meditative, mathematical, philosophical mysticism. Knowledge was gained
through spiritual intuition made harmonious with formal systems of thought. As Plato
later said and as quoted by Thomas Heath in his 1921 History of Greek
Mathematics, about the study of the motion of stars, “…leave the heavens alone…”
because what one sees is only an approximation of the real and more perfect
mathematical structures involving points, lines and circles. To which Newton added
an elongated circle, the ellipse, and Nineteenth and Twentieth Century
mathematicians and physicists, the geometries of positively and negatively curved
space.
It is perhaps not an accident that debates about evidence for the existence
and location of God and where the ideas and structures of mathematics live and
breath generally involves a stand off between those that believe that both are out
there and can be seen, like thoughtful, humanistic actions and caring service for
needful others, versus those that feel the phenomenology of both are projections of
the psychobiologically intuitive Brain God and can be felt like an ecstatic rush of
insightful illumination.
Further Reading for Faith And Rationality
Introduction of Comparative Mysticism. Jacques De Marquette, Philosophical
Library, N.Y. 1949,
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Mysticism and Logic. Bertrand Russell. Norton. N.Y. 1929.
Sefer shel Devarium (The Book of Words). Lawrence Kushner, Jewish Lights,
Woodstock, Vermont. 1998.
Semiophysics: A Sketch, Aristotelian Physics and Catastrophe Theory. Rene Thom.
Addison-Wesley, Reading, MA. 1990.
Mere Christianity. C.S. Lewis. MacMillan, N.Y. 1952.
Sacred Geometry. Miranda Lundy. Walker, N.Y. 1998.
Fractals, Form, Chance and Dimension, B.B. Mandelbrot, Freeman, San Francisco,
1977.
Non-Euclidean Geometry, H.S.M. Coxeter, University of Toronto Press, Toronto,
1957.
Fundamentals of Mathematics, Vol. I. H. Behnke, F. Bachmann, K. Fladt and W.
Suss, MIT Press, Cambridge. 1983.
Religion Explained, The Evolutionary Origins of Religious Thought. Pascal Boyer.
Basic Books, N.Y. 2001.
Catastrophe Theory. Alexander Woodcock and Monte Davis, Dutton, N.Y. 1978.
It Must Be Beautiful; Great Equations of Modern Science. Graham Farmelo, Granta,
London, 2002.
Neurobiological Barriers to Euphoria. Arnold J. Mandell, American Scientist 61: 565-
573, 1973.
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Brain Physics and the Respiritualization of Healing. Arnold J. Mandell, Bulletin of
the National Guild of Catholic Psychiatrists. 28:19-24, 1983.
Toward a Psychobiology of Transcendence, God in the Brain. Arnold J. Mandell, In
Psychobiology of Consciousness (eds. J.M. Davidson and R.J. Davidson). Plenum,
N.Y. 1980.
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APPENDIX
AN INTUITIVE GUIDE TO THE IDEAS AND METHODS OF DYNAMICAL
SYSTEMS FOR THE LIFE SCIENCES
Arnold J. Mandell and Karen A. Selz
Biological Scientists Can Understand and Use Ideas and Methods of
Nonlinear Science
A yield of advances in computer hardware and software is that even quite
difficult applied nonlinear mathematics can become accessible to experimentally
oriented biological scientists. Before this time, the development and analysis of a
particular set of nonlinear differential equations, describing the actions of a
neurobiological system in motion, involved decades of specialty training, rare insight
and many hours of highly skilled, trial and error computations by hand. Since the
idiosyncrasies of each nonlinear system were considered unique, the results of their
analyses were thought to concern only the particular nonlinear system being
studied. Often a shift in hypothetical mechanism meant starting the long and painful
process all over again. In addition, these findings were usually communicated only
to a small and arcane mathematical community in the form of dense theorems and
difficult to follow proofs, insurmountable language barriers to biological researchers
wishing to use them to better describe and understand their experimental
observations.
For today’s neuroscientist with a desktop computer, an inclination to program
and access to computer algebra and numerical software such as Maple,
Mathematica or MatLab, operational definitions and computational empiricism can
replace the theorem and proof continuity required to do old style applied
mathematics. For those of us without sufficient facility in algebraic manipulation to
easily follow the arguments of professional mathematicians, a computer algebra
program such as Maple serves as a delightfully accessible consultant with which to
“check out what the guy is saying". Those motivated enough to write their own data
generating or analytic programs in C, Fortran, Pascal or Basic (though not
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essential) can find easy-to-use algorithmic help in Cambridge University Press’s
Numerical Recipes series (Press et al, 1991).
The conceptual and communication gaps between applied mathematicians
and physicists and the bench practitioners of the neurosciences, that inevitably lead
one or the other, most often both, to surrender their deepest intuitions to jointly
shared images that are inevitably more simplistic, are no longer inevitable. With her
own hands on both the quantitative conjectural and experimental machinery, the
motivated practicing neuroscientist can honor her own insights, read about and
construct symbolic representations from her intuitions and do her own quantitative
theory. Computerized numerical techniques have become so powerful and
accessible that, even in academic settings, there is debate about whether
fundamental analytic tools, such as series expansions, should be taught in
undergraduate courses about differential equations.
The practice of “try it and see what happens", with the current name of
experimental, computational mathematics, is accessible to all. In addition to the
powerful general mathematical programs noted above, there exist several sets of
more specifically targeted software with the capacity to generate, portray and
quantify the behavior of nonlinear continuous and discrete abstract and real
dynamical systems. These often also include algorithmic modules that are useful in
tailoring new models and measures (see for examples, Parker and Chua, 1989;
Baker and Gollub, 1991; Nusse and Yorke, 1991; Sprott and Rowlands, 1991;
Sprott, 1993; Korsch and Jodl, 1994; Enns and McGuire, 1997). Learning from and
using this software, along with only a little programming in the high level languages
and computer algebra programs listed above, permit the non-mathematician
neuroscientist, willing to read in the literature such as that described below, to do
independent, cutting edge research in applied dynamical systems.
Described below will be the computational discoveries in abstract systems
and real neuroscientific data that have led to multiple contexts of quantitative
description. These include those that are: (1) Geometric and conserve metric
distances; (2) Topological and conserve relative positions but not distances; (3)
Single or multiple global quantitative descriptors such as scaling numbers or scaling
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number spectra; (4) Non-Gaussian distributions with heavy tails and correlations
reflected in their Hurst, Fano, Allan and Levy exponents; (5) Statistical dynamical
descriptions of trajectories of the system in their embedding space such as
Lyapounov exponents, Hausdorff-Mandelbrot dimensions, Sinai-Ruelle-Bowen
measures, and Adler-Weiss-Ornstein topological and metric entropies.
Characteristics which discriminate between experimental versus control
conditions in parametric computational and real physiological and pharmacological
experiments serve to generate and test ideas and imagery arising out of behavior
observed in both biological and abstract dynamical realms. New experiments can
be suggested by the implicative structure of dynamical systems theory as well as
neurobiological findings and intuitions. As examples, the sudden “switch” of manicdepressive
bipolarity syndromes may be a “bifurcation” in nonlinear dynamical
systems; the “noise” of the statistical physicist may be the “arousal” of the brain
stem-thalamic biogenic amine and reticular formation neurophysiologist; aspects of
“thought disorder” in the pathophysiology of schizophrenic patients may be an
entropic sequencing idiosyncrasy in the “symbolic dynamics” of a particular brain
system attractor; neuronal “bursting” may be the “intermittency” of a neurodynamical
system; a multiplicity of “discrete ion channel conductances” may be a single “global
scaling hierarchy” of conductances times. The number of published examples of this
fusion of ideas and methodology in the biological-relevant literature is already in the
several hundreds and Medline counts indicate is growing exponentially.
Representative samples of these are described below.
In addition to the technological advances in computational hardware and
software, the major scientific surprise making this new era possible is the discovery
of universalities, the finite set of behaviors characteristic of most, if not all nonlinear
systems, across most if not all of the specific equations or neural systems being
explored. This makes the emergence of semi-quantitative equivalence relations
between model and data not only possible but likely, even though we don’t now and
perhaps never will know enough to either write or solve completely the specific and
detailed equations for the biological system of interest. We neuroscientists need not
be apologetic for using these ideas and tools qualitatively and empirically. In fact,
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unanticipated results of analog and digital computer experiments were responsible
for most if not all of the discoveries underlying the current era’s revolution in applied
nonlinear mathematics
.
Modern Applied Dynamical Systems Emerged from Accidental
Computational Discoveries
A medical student named Herr, in his thesis research with the “radio
engineer”, Van der Pol (1926), was simulating cardiac electrophysiology with an
analog device which permitted real time, exploration of a full range of parameter
values long before there were fast enough digital processors to do so. Studying the
behavior of equations of a periodically, pace maker, driven, nonlinear triode
oscillator, Herr found orbital points that appeared to belong to two different periods
simultaneously thus violating the uniqueness of solutions of differential equation
theory. The Van der Pol relaxation oscillator equations, with their slow buildup and
sudden discharge of membrane potential are good models for the slow-fast
processes of repolarization and depolarization of Hodgkin-Huxley type equations
(Rinzel, 1985). Periodically driven, nonlinear differential equations of the Van der
Pol type are generally applicable to the multiplicity of dynamical regimes of neuronal
dynamics (Carpenter, 1979; Aihara et al, 1984; Chay and Rinzel, 1985) and, with
periodic and aperiodic driving and noise, can be made relevant to particular
mammalian neuronal subsystems in the context of clinically relevant global
electrophysiological phenomena such as Magoun’s (1954) brain stem evoked EEG
and behavioral arousal (Nicolis, 1986; Selz and Mandell, 1992; Mandell and Selz,
1993).
In the early 1940’s, using the pre-publication results of similar analog
computer studies (Levinson, 1949), the Cambridge mathematicians, Mary
Cartwright and Joe Littlewood (1945; McMurran, S., Tattersal, J.,1999) used
geometric methods to prove that the highly nonlinear, periodically driven Van der
Pol equations, depending upon one or two changing parameters, generated fixed
point (“homeostatic”), periodic (“cyclic”), subharmonic (“period doubling”),
quasiperiodic (“multiply periodic”), intermittent (“bursting”) and “deterministically
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random” patterns. We now know such phenomena to be universal characteristics of
bifurcation scenarios in nonlinear dynamical systems where bifurcation means
discontinuous changes in patterns of behavior (dependent variables) resulting from
smooth changes in parameters (independent variables). Alerted to their presence in
computer experiments with biologically relevant nonlinear differential equations,
these phenomena have since been found in time series from patch clamped
membrane channels, single neurons, neuronal networks, neuroendocrine systems,
brain waves and patterns of behavior in animals and man (see below). Cartwright-
Littlewood found that the inner and outer edges of the domains of attraction (all the
initial values that eventually wind up in the attractor—the limit set of all bounded
solutions) of two different sets of subharmonic periods for the same parameter
settings were interlaced at many scales in what is today called a fractal basin
boundary. It was in this way that the specific values of the end state are understood
to be indeterminate since the starting values in the fractal basin boundary are
impossible to isolate and specify with adequate experimental precision.
Similar biologically-relevant analog computer discoveries about the Van der
Pol and comparable periodically forced, dissipative (energy utilizing) Duffing
equations (Zeeman, 1976) were made in the early 1960’s by electrical engineer,
Yoshi Ueda (1992), but his thesis director, Chihiro Hayashi of Japan’s Kyoto
University, was sufficiently disturbed by this evidence for the existence of bounded
solutions (attractors) that were neither fixed points (equilibria) nor periodic orbits
(cycles), the only ones known at the time and therefore “strange,” that he refused to
let Ueda publish his findings until he did so as an independent investigator in the
1970’s.
In the early 1960’s, Edward Lorenz (1963), a meteorologist and student of
the Harvard mathematician and dynamical systems pioneer, George Birkoff (1922),
was computing the output of a very reduced subset of Saltzman’s differential
equations for predicting the weather (1962). Lorenz found that numerically
integrated trajectories manifested unpredictable times and directions of motion
between the two spiral orbits of what has come to be known as the Lorenz attractor.
Very small differences in starting values led to widely diverse final values, and, just
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as importantly, far apart initial values could be found close together in the limit set.
This behavior was called “sensitivity to initial conditions” by David Ruelle (1978;
Ruelle and Takens, 1971). It is noteworthy, however, that over a range of values of
the parameters, the overall pattern of the orbits of the Lorenz attractor results in
characteristic geometric pictures as well as invariant statistical descriptors.
Qualitative and quantitative global similarities were gained while specific solutions
were lost in these “strange attractors” of nonlinear systems. Analog computer
simulation of a simpler set of equations inspired by nonlinear chemical reaction
kinetics led to the discovery by Rössler (1976) of another early and generic strange
attractor combining sensitivity to initial conditions and characteristic geometries and
measures.
It was the Russian mathematicians, A.N. Kolmogorov (1957), Sinai (1959)
and V.I. Arnold (Arnold and Avez, 1968), the French mathematicians, Rene Thom
(1972) and David Ruelle (1978) and the U.C. Berkeley mathematicians, Steve
Smale (1967) and his student, Rufus Bowen (1975), and their associates who
gathered together these and other related computational discoveries and embedded
them in a qualitative theory of nonlinear differential equations, using a variety of
formalisms, including point set and differential topology, geometry, analysis and
ergodic (having an invariant statistical description) measure theory that formally
established the fundamentals for research in nonlinear dynamical systems. Here a
dynamical system refers generally and simply to the components and nonlinear
processes (transformations) that move points (values) in discrete (“map”) or
continuous (“differential equation”) time around in an appropriately defined space.
The phrase, “nonlinear transformation” in this context does not imply easily solvable
curved functions, such as those representing the sigmoid kinetic or threshold
functions of enzymes and neuronal networks or those that smoothly log transform
the amplitudes of auditory or other sensory modalities in man, but rather allude to
expressions containing products, powers and functions of the computational and/or
3
experimental variables , such as xx , ( x i
or si n( x)
.
x i
1 2
)
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As noted above, the cross-disciplinary cohesiveness of such a vaguely
defined field occurred as the result of the unanticipated discovery of a relatively
small set of nonlinear phenomena, universalities, that implicated many fields of
mathematics, from differential geometry to number theory, and were found in a
broad range of physical and biological realizations, from turbulent plasmas and
chemical and enzymatic reactions to neuroendocrine hormone release patterns. It is
perhaps counter-intuitive but, whereas linear systems can generate an infinite
number of solutions locating points anywhere the person writing the equations
wants them to go, nonlinear systems are generally restricted to a finite set of global
dynamics and these emerge on their own from the intrinsic dynamics of the system.
Trying to make these systems follow orders, not unlike finding the most clinically
effective dosage range of a psychopharmacological agent, require the empiricism of
trial and error experiments.
A second class of computational accidents involving nonlinear systems
resulted in unanticipated coherence rather than unpredictable disorder. Using one
of the early “high speed” digital computers at Los Alamos, MANIC I, Enrico Fermi
with Pasta and Ulam (1955) attempted to obtain a many-body statistical
thermodynamic equilibrium analogous to heat generated noise by coupling 64
particles together with nonlinear springs. They found only a few low period modes
that oscillated indefinitely. Instead of equidistribution of the energy into 128 degrees
of freedom (64 locations × 64 velocities in 128 dimensional phase space), they
found it gathered up into only few coherent modes. Although the relevance to
biological science of nonlinear multifrequency coherence is a bit off from our focus,
it is worthwhile noting that a recent (Karhunen-Loeve) decomposition of the alpha
band of the resting alert human EEG revealed only three dominant temporal–spatial
modes: anterior-posterior, rotational and standing (Friedrich et al, 1991) and “few
frequency coherence” is a frontier of inquiry in brain wave research.
A heterogeneous collection of coupled nonlinear elements in the form of
widely distributed, multi-location, multifrequency systems such as cross-cortical,
brain stem-thalamic-cortical and interconnected spinal motor neurons can generate
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coherent activity. This temporal and phase coherence plays an important role in
current theory of sensory-associative-motor integrative function, how distributed
attributions come together in the brain representation of a single object or process,
in the context of the so-called “binding problem” (Singer, 1993; Bressler, 1995;
Nicolelis, 1995; Schiff et al, 1997). Diffusely distributed neurochemical variables
have been invoked. For example, the role of metabotropic glutamate receptors in
driving the synchronization of interneuronal networks has been suggested as a
mechanistic model (Whittington et al, 1995). The objects of relevance to the
discovery of Fermi-Pasta-Ulam are studied as the nonlinear physics of
nondissapative wave processes and are called solitons (Zabusky and Kruskal,
1965). They have been invoked to model nerve conduction and information
transport in brain (Scott, 1990).
A third counter-intuitive set of accidental computational findings is in an area
of research called symbolic dynamics which involves the universal parameterdependent
coding language and capacity of nonlinear systems. In the early 1960’s,
a group around Stan Ulam at Los Alamos (Cooper, 1987) used one of the early
“high powered” computers, MANIAC II, to iterate (letting the output of the action of a
discrete time function serve as its input the next time around) simple equations they
called “maps.” These reduced dimensional objects shaped like tents, sine functions
and parabolas can be extracted from and represent the behavior of higher
dimensional, nonlinear differential equations (see Devaney, 1989; Schuster, 1989 or
Moon, 1992 for intuitive descriptions). They varied a parameter, such as the height
of the tent or parabola, to systematically change the period and/or phase (order) of
the symbol sequence (Metropolis et al, 1973). Normalizing the range of values of
the output to [0,1] and transforming the series of values into a binary code, L ≤ 0.5
and R > 0.5, they found an invariant, one parameter dependent, progression of
ordered periods, R, RLR, RLRR…RLLRL…, in all such single maximum maps. This
“U (universal)sequence” has also been found as singly or multiply present in a
variety of real systems, including complicated chemical reactions (Simoyi et al,
1982; Coffman et al, 1986). This means one can “dial” the parameter to generate
“words” of sufficient computational complexity to serve as a language. These
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computer experimental findings had already been anticipated in a remarkable
mathematical proof by Sharkovskii (1964).
The dynamical richness of these simple, single maximum, one dimensional
maps was computationally explored in the context of ecological and epidemiological
issues in the classical studies of Robert May (1976). It has been possible to relate
the individually characteristic L, R sequence behavior of human subjects on a
computer task to a unique parameter of a tent map generating those sequences
which predicted age and discriminated subclinical obsessive compulsive from
borderline syndromes (Selz and Mandell, 1993). The dynamical entropy of
unstructured L,R behavior also discriminated a population of schizophrenic patients
from normals (Paulus et al, 1996). More generally, parameter dependent dynamical
coding, built into the universal behavior of its constitutive equations, is a mechanism
with which a nonlinear dynamical system, such as nerve membrane equations as
above, or in the aggregate, the middle layer of a completely connected neural
network, can encode, Morse code-like, messages (Paulus et al, 1989).
Bifurcations in Biologically Relevant Dynamical Systems
Bifurcations, “splitting into (two) branches,” are observed over a smooth
change in control parameter(s) (independent variables), as a discontinuous and
qualitative change in the dynamical (time-dependent) pattern of the observable
(Guckenheimer and Holmes, 1993; Wiggens, 1990; see Strogatz, 1994, for a
particularly intuitive description). Qualitative here means how the dynamics of the
trajectory appear as a geometric-topological (relative shaped not necessarily sized)
pattern in phase space. In such a space, the orbital points are located along the x-
axis by their value, x at time t, and along the y-axis by their time rate of change at
that t, dx . To visualize a representative phase portrait in the plane, start by
dt
imagining the pattern made by mass hanging on a linear spring at rest as
dx
represented by a point centered at x = 0, y = = 0. When perturbed from rest, the
dt
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phase portrait of the motion of this “harmonic oscillator,” is composed of a
(continuous) series of points representing its location, graphed along x , its rate of
motion graphed along , y =
dx dx
. x and
dt dt
co-localize the circular orbit as it speeds
up and slows down while it bobs up and down. The transition from a fixed point (the
mass at rest) to a circle (the bobbing mass), a bifurcation in phase space, results in
the loss of topological equivalence. That is, the phase space geometries before and
after the bifurcation cannot be smoothly distorted into each other. Continuity and
connectedness of the space is lost. For topological equivalence, stretching, bending
and warping are allowed but not tearing apart and/or gluing together. Following the
bifurcation of a fixed point into a circle, even limitless shrinking of the ring leaves a
hole. The appearance or disappearance of an equilibrium fixed point (called a
“saddle-node” bifurcation), splitting into two (“period doubling” bifurcation), its
exploding into a circle (“Hopf” bifurcation to a limit cycle), a circle splitting into two or
more incommensurate cycles (“secondary Hopf” bifurcation) and these multiperiodic
(“quasiperiodic”) dynamics breaking down into a recursive spirals (“homoclinic
bifurcation to chaos”) are among the common bifurcations in nonlinear dynamical
systems, and all of them have been observed in many neurobiological settings.
In the forced-dissipative (energetically driven and energy consuming)
dynamical systems relevant to the neurosciences---this characteristic contrasts with
the dissipation free momentum of the classical mechanics of astrophysical bodies---
there are four “most generic” bifurcation scenarios as a parameter changes that
may, but need not, lead to chaos (see below for definition) (for early and physically
oriented treatments see Eckmann, 1981; Ott, 1981; Berge′ et al, 1984, Kaneko,
1983). These scenarios are: (1) Fixed point or cycle splittings into twice-as-long
period lengths 1→2→4→8→16…called the subharmonic or “period doubling route”;
(2) The transformation of fixed points to one and then more periodic orbits, multiple
independent (nonharmonic, incommensurate) frequency oscillations, their mode
lockings and then breakdown called the “quasiperiodic route”; (3) Fixed point or
cyclic equilibria metamorphosed into irregular bursting patterns called the
“intermittency route”; and (4) In the context of quasiperiodic dynamics, adjacent
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nonharmonic frequency encoding parameter spaces fusing, resulting in new periods
that are the sums of their adjacent ones: period 2 + period 3 = period 5, in what is
called the “period adding route”. Technically precise classification of bifurcations
involve much more careful definitions and well studied technical constraints
involving such issues as the symmetries and dimensionality of the system of
observables, how many control parameters (“codimensions”) are required to
reasonably realize the bifurcation and the particular way the fixed points of the
system become unstable, all of which are directly explorable when the equations
are known or can be hypothetically inferred from the qualitative behavior of real
data.
We note a few examples from the wide variety of bifurcating systems that
can be found in the biomedical literature of interest for the biological sciences. With
substrate input rate as the bifurcation parameter, the phosphofructokinase regulated
glycolytic cycle in yeast extract was found to change among steady state, periodic
and period doubling (subharmonic) regimes (Boiteux et al, 1975). Transitions
between steady state, oscillatory and chaotic patterns have been reported in variety
of physiological measures in man including respiratory rhythms and circulating
blood cell concentrations over time (Mackey and Glass, 1977; Glass and Mackey,
1988 ) and models of dopamine cell dynamics (King et al, 1984). Flow rate
parameter sensitive periodic, bursting and chaotic behavior has been found in a
peroxidase-oxidase system (Olsen and Degn, 1977). A brain enzyme, substantia
nigral dopaminergic tyrosine hydroxylase, manifested different saturation and
fluctuation patterns, including bursting and periodicity, in experiments in which low
(physiological) levels of tetrahydrobiopterin cofactor were the bifurcation parameters
and adrenergic drugs were used as modulators (Mandell and Russo, 1981).
All four of the generic bifurcation routes to chaos, period doubling, changing
multifrequency (quasiperiodicity), period adding and bursting (called “intermittency”)
were observed in self-sustained oscillations induced in the neural membranes of
space clamped, giant squid axons that were immersed in a 550mM NaCl, and
electrically stimulated over changing amplitudes and frequencies (Aihara et al,
1986; Takahashi et al, 1990). With external stimulus current level as the control
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parameter, the R15 cell of the abdominal ganglion of the Aplysia demonstrates
transitions between bursting and periodic modes as well as period doubling, a
signatory period 3 and the Lyapounov characteristic exponent evidence (see below)
for the discontinuous onset of chaos (Canavier et al, 1990). Manipulating feed back
delay, the human pupillary light reflex will bifurcate into regular oscillations (Milton
and Longtin, 1990). A transition between a regime of irregular discharging to
oscillatory bursting behavior was induced in basal forebrain cholinergic neurons by
neurotensin (Alonso et al, 1994). Sympathetic nerve discharge in decerebrate,
ventilated cats demonstrated transitions between periodic, multiple periodic
(quasiperiodic with changing ratios to the ventilation frequency) and subharmonic
behavior in response to inferior vena cava occlusion, vagotomy, aortic constriction
and spinalization (Porta et al, 1996). Period adding bifurcations were induced by
changing calcium concentrations or the addition of a potassium channel blocker in
the “pacemaker” formed when (rat) sciatic nerve is chronically injured (Ren et al,
1997). Changing levels of the L-type calcium channel antagonist, verapramil, alter
the pattern of vasomotion of rabbit ear arteries among sets of multiple independent
periods, “quasi-periodicity,” mode locking and chaos (De Brower, 1998). At critical
intensity and frequency, flicker visual stimulation of the salamander generates a
pharmacologically modifiable period doubling bifurcation in their ganglion cells (one
spike for every two flickers) which is also seen subjectively and in occipital lobe
evoked potentials at critical frequencies in bright, full-field flickered humans (Crevier
and Meister, 1998).
Qualitative and Quantitative Universality in Nonlinear Dynamical Systems
“Universality” (see above) entered the parlance of physics in the context of
the statistical mechanics of phase transitions near their critical points (Stanley,
1971; Stauffer, 1985; Yeomans, 1993) and has come to refer to the finite set of
transitions and quantities common to nonlinear systems arising in their
neighborhoods. A common physical example is the triple point of water-ice-steam
on the temperature-pressure phase plane where a small change in temperature or
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pressure leads to a global qualitative change in physical state. Analogously, the loss
of topological equivalence occurs at the fixed point that, for examples, splits into two
or explodes into a cyclic orbit in phase space. The same critical point behaviors and
quantities occur in a wide variety of specific processes and their equations, and they
are independent of the way the trajectories first arrived in the fixed point
neighborhood. Once the system enters the regime of critical behavior, the predictive
significance of its dynamical history is lost. This may also be the case for emergent
psychiatric disorder (Mandell et al, 1985; Mandell and Selz, 1992; Ehlers, 1995;
Paulus et al, 1996; Huber et al, 1999).
There are diagnostic patterns of behavior when a nonlinear system is in a
neighborhood of a potential bifurcation. They include sudden and/or large jumps
resulting from a small change in experimental conditions, the appearance of big
baseline fluctuations (anomalously large variance), the lengthening of the time
required to relax following evoked or spontaneous perturbation (‘”critical slowing”),
the same global change in state occurring at different values of the parameter when
increasing versus decreasing a parameter’s value (“hysteresis”), the existence of
some range of values of the observable that cannot be attained by manipulation of
the parameter (“inaccessibility”) and the availability of two or more distinct states in
the same parameter neighborhood (“modality”) (Thom, 1972; Arnold, 1984; Gilmore,
1981). It is perhaps relevant to polydrug psychopharmacology and clinical
management that the higher the co-dimension (the greater number of effective
parameters being manipulated), the greater the accessibility and control of selected
state stability becomes with respect to difficult to obtain behaviors. Examples of the
potential advantages of simultaneous manipulation of multiple influences have been
developed for affect disorder and anorexia nervosa (Callahan and Sashin, 1987).
As evidence for the independence of critical behavior from specific history,
the qualitatively universal bifurcations along the four canonical routes to chaos
manifest dimensionless ratios of parameter and phase space geometries between
bifurcations. These ratios are quantitatively universal. The formalisms that rescale
the distances from fixed points in parameter and observable spaces result in the
same picture across scale, a dilatational symmetry (also called self similarity or
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affinity). They are called renormalization group equations, and, with respect to
prediction, they replace any or all of the original specific predictive equations for the
particular system under study (Cvitanovic′, 1989). Whereas the U sequence and
critical point behaviors are manifestations of qualitative universality, these scaling
numbers are manifestations of quantitative universality. We discuss them here
because their omnipresence in computationally realized differential equations as
well as physical and chemical experiments along with their quantitative specificity
(with values in all systems as “constant” as π) constitute a most persuasive
argument for the substantiality of modern dynamical systems approaches to brain
and other biological research.
The physical and physiological requirements for manifestations of these
universal bifurcation scenarios can appear to be remarkably minimal. In physics, for
example a full panoply can be observed in a “dripping faucet” (Shaw,1984).
Similarly, a small piece of extirpated and perfused myenteric or femoral artery will
demonstrate these transitions in vasomotion spontaneously and almost
independent of flow rate (Stergiopulos et al, 1998).
Feigenbaum discovered that in dynamical systems manifesting a series of
period doubling bifurcations, the ratio of the parameter value at which the next
period doubling bifurcation occurred relative to the last one ≈
1
… and the ratio
4.
6692
of the magnitude of the spawning point to the one spawned ≈ 2.5. (Feigenbaum,
1979). By “rescaling” distances along a parameter value (see below) using what is
called a “universal renormalization operator” the geometric situation around each
bifurcation point (though of different absolute “size) remains relatively the same. In
intermittent systems, burst length varies as the inverse square root of the distance
of the value of the parameter from that value that elicited the fixed point (Manneville
and Pomeau, 1980). The universal characteristic of the third common parametric
route to chaos, quasiperiodicity, is that the ratio of independent frequencies found
most resistant to mode locking and breakdown into chaos is, ω i
= 1.618… the
ω
i+1
number to which the ratio of adjacent Fibonocci numbers converge (1, 2, 3, 5, 8, 13,
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21…) (Shenker and Kadanoff, 1982). Similar quantitative scaling properties were
also discovered in the parametric period adding route (Kaneko, 1983).
All of these scaling numbers have been found in experiments and in
remarkable agreement with theory. Examples have been discovered in electronic
circuits, hydrodynamic and mercury flows, acoustic systems, laser dynamics and
oscillating chemical reactions (see Cvitanovic′, 1989, for representative list of
references). Whereas qualitative evidence for all of these bifurcation scenarios have
been found in brain relevant experiments, there is yet to be a bifurcating
experimental biological system with adequate precision across a sufficient range of
magnitudes such that quantitative universality could be demonstrated across a
sufficient range of values to be convincing. We remind ourselves that in order to
establish a Fiegenbaum number, each period doubling bifurcation of the several
required necessitates about a five-fold improvement in the experimenter’s ability to
specify the control parameter.
Using Invariant Measures of Dynamical Neurobiological Systems
Before the modern era of dissipatively forced (energy utilizing) dynamical
systems research, the known attactors of an experiment’s initial values resulted
from their convergence onto either a fixed point or a limit cycle. An attractor can be
regarded as a set which remains in bounded space and to which all orbits in this
neighborhood converge (Milnor, 1985). Since by the rules of differential equations,
orbits are required to be both smooth (graphable without lifting the pencil) and
unique (different trajectories don’t intersect since the point of intersection would no
longer be unique), the foundational Poincare-Bendixon theorem says that any such
orbit confined to a two dimensional phase space that doesn’t converge to a fixed
point must, no matter how long it irregularly wanders, must, eventually intersect with
itself and then go around the same route again in a (perhaps very long) cycle. In
most neuroscience research as well, we have generally regarded our data as
manifesting either tolerable (or intolerable) fluctuations around mean values (fixed
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points) or more or less regular cycles. We analyze our “fixed point” data using
quantities such as the mean and variance of distributional statistics and the cycle
data using the amplitude, frequency, cycle length and phase of trigonometric
functions. In central tendency-oriented research, rare, very high amplitude events
have usually been considered aberrations and tossed, and imperfect periodic
behavior is treated by “cosiner analysis” as regular cycles contaminated by
measurement or system noise. Whereas technically, chaotic dynamics must live in
dimension greater than two (for orbits to be more than a fixed point or limit cycle,
able to snake around without necessarily intersecting ), the Lorenz attractor has
dimension just a little over two, our difficulties with establishing the “true”
physiological dimension of real biological observables (see below) makes such a
consideration more theoretical than practical.
The orbits of a forced-dissipative dynamical system in a parameter regime
engendering chaos, converge onto an attractor which is neither a fixed point nor a
limit cycle, thus the origin of the name “strange attractor” (Ruelle and Takens,
1971). It was James Yorke that first named these dynamics “chaos” (Li and Yorke,
1975). The necessarily statistical properties of the chaotic orbits on strange
attractors follow from the generic characteristics of their motions (see Shaw, 1981
for a still conceptually current, non-mathematical treatment). These kinds of
statistics are studied in a research context called the “ergodic theory of dynamical
systems” (Ruelle, 1979; Eckmann and Ruelle, 1985). Ergodic is a word used to
characterize a system with (or without) a particular condition placed on its statistical
measures: the existence of an invariant measure which is undecomposabile into
two invariant measures and, equivalently (though not obviously) one in which the
time average equals its average in the geometric space into which it is embedded.
One may arrive at the same ergodic measure from studying a single very long orbit
or from summing across many individual but shorter orbits. This ergodic
equivalence is made possible due to the definitional existence of at least one
invariant statistical measure and the dynamics of the system which ideally include a
uniformly, sequence disordering process called “mixing” (see below).
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Of course, most real biological dynamics are not uniformly mixing and so are
non-ergodic, but we shall see that the ways they fail to be ergodic (and thus remain
in the conceptual context of ergodic measures) are descriptively useful (Mandell
and Selz, 1997a). The emergence of many statistical approaches to characterizing
these motions have been accompanied by the expected controversies about which
is best or correct (see below) and have been applied to the problem of diagnosis
and clinical discrimination in a variety of neuroscience settings. In ideal abstract
chaotic dynamical systems called Axiom A (Russians called them “C systems”),
where most mathematical theorems are proven (Smale, 1967), all these measures,
if properly computed, are equivalent. In real life, as in the related case of ergodicity,
they are not, and since no single one is complete, the more (incomplete) measures
we use in our studies along with interest in the way that they differ, supplies more
useful information about the system. Though researching and elucidating the most
reliable and valid ways of computing these measures are a valuable goal, the
current debates focused on the superiority of a single particular measure,
constructed in a particular way in relationship to issues of insoluble absolutes like
“randomness” versus “deterministic chaos may not be particularly valuable for
uncovering new characteristics and potential mechanisms underlying a specific set
of real neurobiological observables.
Emphasizing diversity and relevance to the clinical biological sciences, we
note that quantifying patterns in ergodic (non-ergodic) measures have aided: the
discrimination between normal and abnormal opticokinetic nystagmus in neurology
patients (Aeson et al, 1997); localizing a two year old subcortical stroke in an EEG
of a patient with no other signs or neurological findings (Molnar et al, 1997); the
diagnosis of early (not late) multiple sclerosis, as a nonspecific long tract disorder,
in patients with mild optical neuritis using cardiac rate dynamics (Ganz and
Faustman, 1996); seizure prediction from minutes to hours before the event in
which subthreshold, pre-phase transition spatial diffusion and oscillations in
characteristic changes in these measures can be found (Martinerie et al, 1998;
Elger and Lehnertz, 1998; Pign et al, 1997; Iasemidis et al, 1990); using these
measures on the EEG to differentially predict hereditary predisposition to alcoholism
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(Ehlers et al, 1995); indicating the presence or absence of septic encephalopathy
(Straver et al, 1998); using time series from jejunal manometry to discriminate
objectifiable somatic from psychological conversion related irritable bowel syndrome
(Wackerbauer et al, 1998); analyzing time-dependent patterns in plasma hormone
levels to discriminate between the presence or absence of a functioning tumor
(Hartman et al, 1994, Mandell and Selz, 1997a); automated differentiation of ataxic
from normal speech (Accardo and Menulo, 1998); and discrimination of
temporomandibular joint dysfunction from normal patterns of chewing motions
(Morinushi et al, 1998).
Styles of Orbital Motions in Chaotic Dynamical Systems
In chaotic dynamics, in various specific ways, an initial hypothetical handful
of points lined up along the trajectory and acted on over time by the nonlinear
differential equation (“operator”), get out of order in an unpredictable way. Here the
hypothetical handful can come from a statistical aggregate of initial conditions or
from a single recursive orbit studied over long times. As noted above, ergodic
theorists call this getting out of order “mixing” and how and to what degree this
happens consumes many mathematical theorems but for purposes of brain
research, it can be best described using a variety of statistical measures. For
example, visualizing the Lorenz attractor (see above) as a butterfly in phase space,
the points get out of order because as they spiral out (“stretching”) to the edge of
one wing and return (“folding”) to the unstable fixed point on the butterfly’s body
whence they either jump to some place on the other wing to spiral to its edge or
return to the same wing to spiral out again. Which one of these is chosen is
exquisitely sensitive to very small changes in where the trajectory started and very
small fluctuations in where it returned to the unstable fixed point on the butterfly’s
body. In fact, specification of these locations is beyond the precision of any real,
thermodynamically vulnerable system.
Chaotic trajectories on the Rössler attractor (see above) wind out (“stretch”)
to the edge along the inside of a conch shell in phase space and then are mapped
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back (“fold”) into the spiral unpredictably somewhere in a mixing mechanism that
has been called “displaced reinjection.” In the slow-fast oscillations of the forced van
der Pol in the chaotic regime, points in the slow phase (“repolarization”) jitter around
and step on each other’s heels, getting out of order while waiting on the ledge
before jumping (“depolarization”) to the next slow phase (“repolarization”) at some
unpredictable time, thus generating a variably irregular series of interspike intervals.
Stretching and folding are also responsible for getting points get out of order
in the single maximum map of the unit interval (studied for universal qualitative and
quantitative properties by May and Feigenbaum and others as described above).
With increases in parameter values, the parabolic hill function onto which the unit
line has been stretched gets steeper, more stretched. Mapping points on the hill
back onto the straight line of the unit interval results in what amounts to the line
folding back on itself. This stretching and folding eventually fills the line with points,
but their sequence, from end to end, gets shuffled like a deck of cards.
As described more generally above, points that start as neighbors may get
separated (“divergence along the attractor”) and those that start at a distance from
each other may be thrown together (“compression back onto the attractor”). These
expanding and folding motions that characterize the chaotic behavior on strange
attractors have been likened to the actions of a taffy puller (Rössler, 1976). It is in
this way that nearby points can separate without leaving the attractor. It is also the
case that once indistinguishably close but then separated points may be
compressed together again generating new, temporary (unstable) cycles of all
possible period lengths. These unstable fixed points may be the most important
feature of chaotic systems from the standpoint of new ideas about brain
mechanisms (Pei and Moss, 1996; So et al, 1997). This aggregation of unstable
loops can occur from points fluctuating away and back to the attractor as well as
during the crowding of points at the turns after their stretching out on more linear
parts of the flow. Under the mixing flow of a chaotic dynamics, it is also true that a
single point eventually explores the entire attractor, no attractor location is
inaccessible to it.
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Although counter-intuitive when expressed in words, the trajectories that one
sees in the graphics of chaotic attractors result from the actions along the “unstable”
directions of the stretching distortion; the actions in the otherwise invisible stable
directions “iron down” the points onto this unstable manifold (n dimensional abstract
surface).
As might be expected from this set of characteristic motions, the diagnostic
triad of chaotic dynamical systems are: (1) Sensitivity to initial conditions—tiny
distances between starting points are magnified and large distances between
starting points are reduced under the stretching and folding actions of the system;
(2) The presence of a theoretically infinite but countable number of unstable
periodic orbits of theoretically all period lengths—points in phase space can be
viewed an attractive-repellers, visited and left by the orbits recursively as the
dynamics proceed; and (3) Indecomposability—the attractor is not separable into
isolated regions and no points escape (see Devaney, 1989, for one of the clearest
definitions). Of particular relevance to information encoding and transport by brain
mechanisms, it is important to visualize that new information in the form of unstable
periodic orbits is being created as well as destroyed by the dynamics. The
logarithmic rate of formation of these new orbits is computed as the system’s
topological entropy (see below).
Assuming the real neurobiological system under study is behaving in these
ways (and often much has to be done to help justify such a claim), the observables
take the form of an irregular and/or episodic time series of amplitudes, as in
repeated sample, neuroendocrine studies of plasma hormone levels (Veldhuis and
Johnson, 1992) or a sequence of times between events as in neuronal interspike
intervals (Katz, 1966; Perkel et al, 1967). These time or time-sequence series are
generally studied from three relatively distinct yet complementary quantitative
perspectives: (1) As stochastic (“random”) processes with various amounts of
sequential dependency (autocorrelations) and scale (sample length) dependencies;
(2) As “deterministic” smooth or discrete, vectorial geometries in phase space
following reconstruction and/or embedding of the series as phase portraits or return
maps; (3) As information generating and transporting, topological (about relative
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nearness and sequential order not absolute distances), symbolic dynamical
processes which as either (1) or (2) can be analyzed with respect to its various
entropies, algorithmic complexities and word content and syntax. A variety of
techniques aimed at deciding between the relevance of one or another of these
underlying assumptions (such as series and Fourier phase shuffling to destroy
statistical autocorrelations and vectorial continuities but leaving the probability
density distributions intact ) may at times help emphasize one or another of these
orientations in the analyses (see Ott et al, 1994 for a collection of articles on this
topic).
Nonconvergent Distributions and Power Law Scaling in Biologically Relevant
Time Series
The statistical distribution with which most of us are familiar is the Gaussian
which can be generated by summing and averaging a series of independent
random events. The average behavior head/tails probabilities observed by one
person flipping a fair coin for a very long time or by many people flipping similar
coins for shorter times converges upon the invariant measure of 0.5. The variance,
“second moment” in the distribution of a population of coin flipping sequences will
be finite and computable. In a graph of this distribution, the tails will converge to the
x axis in a Gaussian exponential manner. The longer or the more numerous the
“sample” series of observations, the closer they will approximate the “ergodic”
invariant measures representing the true “central moments” of the behavior of this
“population” of fair flipping coins. Since the coins are not changing their relevant
characteristics over the time of observation, we say that the series is not time
dependent but instead is “stationary.” Computation of correlations over increasing
lags to determine how much and for how many flips the sequences continue to
resemble themselves yield an exponential decay with a single characteristic
correlation length. This reflects the existence of a finite variance from which its
amplitude is derived and serves as the single characteristic temporal scale of the
random process.
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Before describing the relatively new set of measures of biological processes
designed to find and quantitate what are assumed to be relatively sample size
insensitive, distributionally nonconvergent and multiply correlated processes that
are without a single time or space scale, we should remind ourselves that there is
already much more apparent “order” in a generically random situation than our
intuitions would lead us to believe. For example, if we keep cumulative scores in a
competition between heads and tails and determine the distribution of trials between
those in which the number of heads and tails are even, we will get periods between
zero crossings of many lengths with very short ones and very (very) long ones
being most statistically prominent. The distribution of these wavelengths is shaped
like a symmetrically fat-tailed, bowl (Feller, 1968). As another illustration, expected
runs of heads or tails in this Gaussian random task are longer and more frequent
than we might suspect. It has been proven that the expected run length grows with
n coin flips (as an order of magnitude estimate) like the logarithm (for a fair coin,
base 1/p = 1/0.5 = 2) of n. For example, in 512 ( e.g. 2 9 ) tosses, we cannot report a
run of 9 heads as a evidence for a biased coin or the sign of some deterministic
coin tossing mechanism (Erdos and Renyi, 1970). If we had a 0.6 head biased coin,
then the observation of a run of 13 heads couldn’t dissuade us from a random
mechanism!
Unlike our random coin task, the variances of many, perhaps most, time
series of biologically-relevant events, do not tend to converge onto a limiting value
as sample size, n, grows, but rather continue to increase (or decrease) with n in a
scale invariant manner. Instead of “regressing to the mean” with increasing sample
length or number, the likelihood of a larger deviation than previously observed
increases with n. Analyses of inter-event intervals reveals a multiplicity of
characteristic times. One interpretation of these finding might be that this represents
evidence for the inherent “nonstationarity” of biological mechanisms as reflected in,
for examples, the frequency of saccades concomitant with ceaselessly shifting foci
of visual attention (Steriade and McCarley, 1990), or our inability to not think of
“white bear” when so instructed (Wegner, 1994). Hermann Haken, the father of
laser-inspired “synergetics,” has said that biological mechanisms are not in a steady
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state for very long, spontaneously and irregularly jumping from one unstable
dynamical state to another (1997). This suggests that meaningful tension between
experimental sample lengths long enough to minimize statistical error and short
enough to be stationary may be, for the biological sciences, more apparent than
relevant.
The studies reviewed below exploit measures arising from the view that the
noisy statistics of nonstationarity in biological processes are not a sign of
measurement error, but rather evidence consonant with the statistical physics of
nonequilibrium states and phase transitions (Stanley, 1971; Stauffer, 1985;
Yeomans, 1993). Very high amplitude fluctuations and multiple, up to infinite,
correlation lengths are characteristic of the normal, on-going biological dynamical
behaviors, which are apparently without characteristic amplitude and time scales.
From this point of view, if most or all information is widely distributed in the brain
(e.g., serial order of visual tasks involving motor cortical neurons, Carpenter et al,
1999) ) then the “binding problem” (see above) could also be solved by multiple, up
to infinite spatial and temporal correlation lengths in place of the current theories of
monofrequency resonances (Singer, 1993). Hierarchical neurodynamical
mechanisms communicating across many mechanistic temporal and spatial scales,
brain information transport analogous to the energy cascade of hydrodynamic
turbulent velocities (Tennekes and Lumley, 1972), would be likely in the parametric
vicinity of incipient bifurcations and phase transitions.
Three closely related techniques for quantifying the systematic changes in
average fluctuation amplitudes with n (scale, sample length) involve a “power law,”
linear slope relationship between the logarithm of an index of variability and the
logarithm of sample segment sizes. These easy, yet powerful methods were
brought to experimentalists’ attention by Benoit Mandelbrot (Montroll and Badger,
1974; Mandelbrot, 1983; Fedor, 1988; Bassingthwaighte et al, 1994; Liebovitch,
1998). To estimate the exponent in Hurst rescaled range analysis, we compute the
standard deviation and the range of the deviation of the running sum from the mean
on sequential subsamples of increasing size. The Hurst power law exponent is the
slope of the straight line formed by graphing the logarithm of the subsample length
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along the x axis and the logarithm of the ratio of the range to the standard deviation
on the y axis. An independent random system has a Hurst of 0.5. If a sequential
increase or decrease in an amplitude or inter-event time tends to be followed by a
change in the same direction, the Hurst > 0.5. If an increase in the measure tends
to be followed by a decrease, then Hurst < 0.5. Computation of the Fano factor
(power law exponent) exploits the same general strategy using the variance/mean
in place of the range/variance and counting the number of events (such as single
neuron discharges or heartbeats) in time windows of increasing length, generating a
similar log-log graph. There is a relatively long history of the use of spike-number
variance-to mean ratio in studies of response variability in visual cortical neurons
(see Teich et al, 1996 for a review). The Allen factor (power law exponent) tends to
reduce the influence of local trends by a computation of the variance of the
difference between the number of events in two successive time windows divided
by twice the mean number of events in the window.
Each system’s invariant logarithmic slope across sample segment sizes
takes the place of its missing finite variance in characterizing experimental data in
which the distributional tails do not converge (or do so very slowly) to the x axis.
Recent approaches to these measures in the context of stochastic analysis of DNA
sequences, but also applied to normal and pathological cardiac inter-beat intervals
and gait interval sequences, have dealt with the influence of non-stationarity due to
apparent trends in the data on α-equivalent indices by local mean-normalization of
the fluctuations at each window size (Peng et al, 1993; Peng et al, 1995; Hausdorff
et al, 1995).
The rate of decay of the densities in the tails of the probability distribution as
they approach extreme values along the x axis, called the Levy exponent when
represented in Fourier space (technically, as a “characteristic function” of the
probability distribution) (Shlesinger, 1988; Shlesinger et al, 1995), can also be
computed directly on the distribution by fitting the tails with a two parameter curve
quantifying their “fatness” and rates of decay (Mantegna, 1991). We can speak of a
Gaussian tail as having an exponential decay rate representable by α = 2 implying
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finite variance. A tail with a nonconvergent decay rate of 1 < α < 2 indicates nonfinite
variance in the data such that the usual “normal curve” derived, standard
deviation dependent tests of statistical significance are without meaning. α < 1
indicates the data is without a consequential mean and will require the use of
interquartile measures to locate the center of the distribution (Adler et al, 1998).
Recalling that the Hurst, Fano and Allan indices are invariant across sample
segment size, we remind ourselves that, as is the case in the finite mean and
variance, α = 2, Gaussian, any of the other “α tails” also retain their value (“shape”)
across all partitions that might be used to sort and sum the observable. This
property is called convolutional, α, stability. In passing it should be noted that the
last outpost of convergence of a probability density distribution with α = 2 is called
“log-normal,” in which the tails along the x axis are “pulled in” by the variable being
plotted as its logarithm.
A Hurst exponent of > 0.5 in the data is associated with a Levy exponent of <
2.0, and both would be indicative of a process in which the characteristic style of
change, rather than decay with some finite correlation length, would persist across
all time. Using a bursting neuron as a generic example, a short interspike interval
would, on the average, be followed by another short one and a long one by another
long one, and this behavior, unlike our fair coin flipping sequence of observables,
would not become uncorrelated with itself even over infinite time. Another way to
represent this infinite, innumerably lengthed, correlation property is via its implicate
frequency (inverse wavelength) content by computing its best fit assortment (along
with their densities) of a range of short to long sine waves forming the Fourier
transformation of the correlation function. The condition of correlated fluctuations
across many measured temporal scales yields yet another power law slope when
graphed as the logarithm of its range of frequencies, f, plotted along the x axis,
versus their corresponding amplitudes squared, powers, plotted along the y axis.
Naming this spectral power law exponent β, the system’s characteristic scaling law
is usually expressed as 1 (Fedor, 1988; Hughes, 1995; Shlesinger, 1996;
β
f
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Liebovitch, 1998).
We see that the Hurst exponent, Fano and Allen factors, Levy exponent and
power spectral scaling exponent are kindred statistical descriptors. They are most
usefully applicable to systems with distributions that fail to be Gaussian or
asymmetrically Poisson, the latter from random data sequences with only positive x
values, thus backed up toward zero by a minimum inter-event interval or amplitude.
These time series are sequentially dependent, not conventionally stationary, without
finite central moments and with self-correlations that don’t demonstrate Gaussian
exponential decay with sample length or time. The following are some examples of
the use of these measures in studies of biological dynamics. .
Examples of Biological Data with Divergent Distributions and Power Law
Scaling
A paradigm challenging group of experiments involved models and measures
of the distribution of characteristic open and closed times of membrane ion
conductance channels. The usual approach to this problem assumed the existence
of a small set of distinguishable channel types that were reflected in discrete
conductance events with a small set of characteristic open and closed times. The
distributions of each of could be fitted with its own, Markov process derived,
exponential. With technical advances and improved temporal resolution, more
characteristic times and their associated α = 2 exponentials were reported with as
many as three not being unusual. Liebovitch (and Sullivan,1987; 1989) used
analogue to digital transformation of current recordings from the unselective corneal
epithelial channels and voltage dependent potassium channels in cultured mouse
hippocampal cells at temporal resolutions ranging from 170 to 5000 Hz and found
similarly shaped, α < 2, nonconvergent distributions across temporal scales. This
led these investigators to suggest that, related to the >16 recorded magnitudes of
characteristic times, from picoseconds to months, in autonomous protein motion
(Careri et al, 1975; Gurd and Rothgeb, 1979), that there was an “α stable” hierarchy
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of lifetimes of states, observable at almost any temporal resolution that methods
would allow.
Early and representative studies comparing the fit of the data with
hierarchical scaling functions versus a sum of a small number of Markovian
exponentials included studies of a calcium activated potassium channel in human
fibroblasts (Stockbridge and French, 1989) which yielded evidence to support both
models, as did studies of membrane conductances in corneal epithelial cells by
another group (Korn and Horn, 1988). In a systematic comparison of scaling and
Markov exponential modes of the gating kinetics of GABA activated chloride
channels, acetylcholine activated end plate potentials, calcium activated potassium
channels and fast chloride channels (McManus et al, 1988), it was found that the
latter fit the data best in most experiments. Similar results were reported in studies
of the glutamate and delayed rectifier potassium channel with respect to
distributions of open and closed times (Sansom et al, 1989). Space does not permit
a systematic account of the continuing debate and conflicting studies about these
representations and the implicit biophysics of discrete, finite versus continuous,
hierarchical channel event heterogeneity. It is interesting that recent experiments
making use of Hurst rescaled range analyses of time series of whole cell membrane
voltage fluctuations (without the assumptions and current renormalizing procedures
associated with patch clamping) have yielded additional evidence for multiply
correlated, Hurst > 0.5, α < 2 power law behavior of what some might regard more
generally as a protein relaxation time mediated hierarchical array of ion
conductance behaviors (Liebovitch and Todorov, 1996).
Following the discovery of (very) subsaturating (“far from equilibrium”) rat
brain levels of the common cofactor for tyrosine and tryptophan hydroxylases,
tetrahydrobiopterin (Bullard et al, 1978), studies of amino acid substrate saturation
functions and time courses determined at these low, physiological co-reactant
concentrations manifested patterns of hierarchical multiplicity and discontinuities
suggestive of bifurcations and time-dependent fluctuations with fractional
(hierarchical) time scaling exponents that were sensitive to psychotropic drugs
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(Mandell and Russo, 1981; Knapp and Mandell, 1983; Russo and Mandell, 1984a;
Russo and Mandell, 1986). Similar bifurcating and power law kinetics were found in
receptor-ligand binding systems (Mandell, 1984) which were confirmed by more
recent studies of diffusion-limited binding kinetics with receptors immobilized on a
biosensor surface (Sadana, 1998). Hierarchical kinetics have also been reported in
time courses of drug and metabolite levels (Koch and Zajcek, 1991), tissue tracer
washout studies (Beard and Bassingthwaighte, 1998), carrier mediated transport
processes (Ogihara et al, 1998), general pharmacokinetic functions (Macheras et al,
1996) and biochemical networks (Yates, 1992). It is likely that bifurcating and
hierarchical, power law kinetic functions will be studied more commonly in the
chemical literature in general (Shlesinger and Zaslavsky, 1996; Berlin et al, 1996)
as well as applied to a variety of protein-mediated biological functions (Dewey,
1997).
The first demonstration of and stochastic model for nonconvergent
distributions of interspike intervals of a single neuron was by Gerstein and
Mandelbrot (1964). Though rich with possibilities, it has been only very recently that
additional work from this point of view has been published. This is likely due to the
fact that most neuroscience oriented statistical packages, with rare exceptions, are
without techniques for computing descriptive parameters for these divergent
probability density distributions. This has not been the case for economic time
series, download STABLE from http:///www.cas.american.edu/~jpnolan. Recently,
applications of the Fano and Allan factor as well as power spectral scaling
exponents to observed and shuffled series of spike counts and interspike intervals
in the auditory and visual systems (including spatial and/or time resolved single unit
recordings in retinal ganglion, lateral geniculate and lateral superior olivary cells as
well a auditory nerve fibers) demonstrate the characteristic behavior of
nonconvergent, hierarchical stochastic systems (Teich, 1989; Teich et al, 1990;
Lowen and Teich, 1992; Kumar and Johnson, 1993; Kelly et al, 1996; Teich et al,
1997). These statistical techniques are well suited to the characterization of the
irregularly intermittent bursting patterns generic for activity in single neurons as well
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as in nonlinear equations representing them and other brain processes (Mandell,
1983).
An early study of power spectral scaling in the EEG reported alpha band
fluctuations that extended a 1 , β ≈ 1 pattern to 0.02 Hz (Musha, 1981), as did
β
f
other applications of the log-log power spectrum to the EEG in man (Hu and Hu,
1988; Prichard, 1992). This power law scaling led naturally to the suggestion that
the range of frequencies available in the electromagnetic signal from the calivarial
surface extends far beyond those currently appreciated and may be available for
study using relatively noise free recording techniques such as the
magnetoelectroencephalogram (Mandell and Selz, 1991). A not surprising range of
intrinsic correlation lengths reflected in Hurst > 0.5 and/or Levy exponents < 2 have
been reported in lamb fetal breathing patterns (Szeto et al, 1992). The exponent
has been shown to be sensitive to maternal alcohol intake in humans (Akay and
Mulder, 1998), rat neonatal motoric activity (Selz et al, 1995), and nuchal atonia
duration sequences (associated with putative intra-uterine REM sleep) in fetal
sheep (Anderson et al, 1998).
Sequential amplitudes in 1 Hz stimulated soleus spinal cord H-reflex
demonstrated a 1 , β ≈ 0.83 in control subjects and, reflecting the decrement in
β
f
correlations, by 0.31 in patients with losses in supraspinal input from spinal cord
injury (Nozaki et al, 1996). Whereas the sequences of fixation times in eye
movements of normal control subjects reading difficult material demonstrated an
exponentially decaying distribution, those of schizophrenic patients demonstrated a
power law tail, consistent with more sequential correlations (Yokoyama et al, 1996).
This finding may be related to the appearance of velocity arrests, runs of sticky fixed
points, in a spatially oscillating target task, called “smooth pursuit eye movement
dysfunction” in schizophrenic patients which has been modeled as a parametric
disorder in a periodically driven nonlinear dynamical system (Huberman, 1987). The
“short time fractal dimension” has been used to discriminate acoustic signal
transformations from the speech of normal subjects and ataxic patients (Accardo
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and Mumolo, 1998). Spontaneous changes in the apparent syllabic sound made by
regularly presented, word-like auditory stimuli emerge irregularly, the duration of
perceived sameness demonstrating a power law distribution of “dwell” times (Tuller
et al, 1998). The same kind of power law distribution of characteristic “brain times”
can be found in studies of gait cycle durations in normal walking (Hausdorff et al,
1996) with a decrease in this locally detrended, α-like index compared with controls
(0.91±0.05) in patients with the basal ganglia disorders of Parkinson’s (0.82±0.06)
and Huntington’s (0.60±0.04) Diseases (Hausdorff et al, 1998). Hurst > 0.5 has
been speculated to more accurately quantitate the fundamental time structure of
cells that was previously called circahoralian (ultradian) intracellular rhythms
(Brodski, 1998).
Reconstructions of Time Series as Orbital Geometries
Rene Thom (1972), extending the ideas of Poincaré and D’Arcy Thompson
(1942), argued that experimentally useful, intuitive connections between the
qualities of biological processes and the quantities of an explicit (equations known)
or implicit (equations unknown) dynamical system could be best achieved through
the use of graphic representations of their geometric and topological forms. Notably
successful examples can be found in the work of Thom, Arnold (1984) and Zeeman
(1977), who were inspired by “caustics” (the shapes made on surfaces by the
coincidence of reflected or refracted light rays) and Whitney’s representation of
parametric manifolds (surfaces) by the shadows they would make on a plane when
back lit (Whitney, 1955). This led to a small number of qualitatively predictive,
number-of-independent-parameters dependent shapes, such as “folds” “cusps” and
“wavefronts.” Experimentally crossing the values of these independent variable
forms at their singular boundaries successfully predicted discontinuities in the
otherwise smooth alterations in the dependent variable; i.e. bifurcations
(“catastrophes”) in the behavior of the observable. This approach was best suited to
the study of systems with many independent variables and one dependent variable
that could be mapped on the axis of the latter to represent a continuum of
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operationally defined “energy states.” Smooth changes along the path of the
nonlinear parameter manifold generated discontinuous changes in energy levels
indicating states of the observable. Crossing a wrinkle in an “independent variable”
(some call it “order parameter” to indicate its emergence rather than availability for
predictable manipulation) such as the nonlinear parameter surface of the
countervailing influences of survival fear and financial cost, may lead to a bifurcation
in behavior from peace (“low energy”) to war (“high energy”) (Zeeman, 1977).
In a similar geometric spirit but dealing with nonequilibrium systems in
motion, the conditions such that one could “smoothly” embed a trajectory like a
continuously recorded EEG record, a complicatedly coiled snake into a three or
higher dimensional box without loss of its essential dynamical or statistically
measureable properties, was settled by Whitney in what is now referred to as the
“embedding theorem” (Whitney, 1936). Starting with a tangled knot of overlapping
vectorial orbits with apparent “non-invertable points” (given a point, one cannot
chose among or between the more than one point that it apparently came from), it
can always be unwrapped into a non-crossing trajectory satisfying uniqueness when
reconstructed in a box of a little more than twice the parameter-determined
dimension of the original space of observables.
A common technique for the spatial reconstruction of the output of a
dynamical system is called a “time delay embedding.” This approach, first
suggested by Ruelle (1987, pg. 28) replaced the value, x, versus the time
derivative, dx , phase portrait plot described for a continuously perturbed bob on a
dt
spring above. A sequence of observables over time, in, for example, three
dimensional “phase space” (Packard et al, 1980; Takens, 1981; Sauer et al, 1991),
is depicted by a curve representing the system’s trajectory at times t 1 , t 2 , t 3 , by
sliding one-by-one down the series and plotting each p 1 , p 2 , p 3 , location with
respect to each other along the x, y and z axes respectively. The choice of time
interval between the points, the delay, can be delicate and usually some standard
fraction of the decay time of the sequence’s autocorrelation length, “the decay time
of mutual information” is chosen. There are many technical considerations,
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including those involving the choice of the embedding space vis a vis the “true”
dimension of the attractor. This becomes an issue when, for example, the attractor
shrinks over time to some subspace of the initial embedding (Liebert et al, 1991 and
references therein).
If we imagine the process of time series reconstruction to inscribe an
attractor’s untidy ball-of-string of recurrent trajectories in three dimensions, we can
then, by making the z-dimension a constant value, cut the ball with a two
dimensional plane, a “Poincaré surface of section.” This could yield a roundish
cloud of discrete points on the x,y plane and t n-1 – t n would be the time between two
piercings of this surface. It has been proven that almost any cut, as long as it is
made transverse to the direction of the orbital trajectories, is equally valid and useful
for further analyses (Oseledec, 1968). If the original embedding and subsequence
section was in high enough dimension to allow invertability, we might have enough
(trial and error) knowledge to be able to write a discrete equation, a “return map,” f,
f
that would move one point to the next on the plane as (x,y) t ←→ ⎯ (x,y) .
What can sometimes be case with real systems (Coffman et al, 1986), is that
reducing the geometric reconstruction still one dimension further, accepting noninvertability,
ironing down the points in the plane onto the x axis line (normalized to
[0,1]), and plotting the values at x t against x t+1 (“mapping the unit interval to itself”),
can generate points in the general shape of a parabola with dynamics representable
by the same family of one parameter, single maximum discrete equations that
generated May's sequence of bifurcations, Feigenbaum’s scaling and Metropolis,
Stein and Stein’s (and Sharkovskii’s) U sequence (see discussions of qualitative
and quantitative universalities above).
Although sometimes a significant change in brain system physiology, such as
penicillin-induced epileptic neuron spiking activity is revealed simply by a change in
the graphic appearance of suitably embedded time series data (Zimmerman and
Rapp, 1991), more often statistical measures made on the geometric dynamics of
the points on the attractor are required.
n−1
t n
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Orbital Divergence Characterizes Expansive Dynamics on Biological
Attractors
In the dynamical world of equilibria (fixed points in phase space) and
periodic cycles (fixed points of a return map), a common concern involves their
stability. What happens if an adventitious jiggle moves the orbit a little distance
away from the fixed point? Would the wind wiggled suspension bridge start to flap
with increasing amplitude or would it damp back down quickly to its stable state. A
“Lyapounov functional,” L, is constructed which can be visualized like a smooth
potential bowl around the fixed point such that any L stable solution that starts at its
bottom tends to stay there or is asymptotically L stable if the solution converges to
the fixed point at the bowl’s bottom as t → ∞ . If the point is not L stable, it is L
unstable.
The modern study of nonlinear systems have produced another kind of
stability issue with a similar appellation yielding other direction specific indices, the
Lyapounov characteristic exponents, λ (Oseledec, 1968; Eckmann and
Ruelle,1985; Ruelle, 1990; Ott et al, 1994). In this context, the instability is not one
of perturbative escape from a fixed point, but of the average rate with which the
(theoretically infinitesimal) distances among a handful of points representing a set of
initial conditions (each a precision limited, hypothetical repetition of the same
experiment), are being stretched apart by the expansive action of a strange attractor
system. In three dimensions, one can envision a ball of initial conditions being
elongated along the unstable direction and ironed down from both sides along the
stable direction over time, transforming the ball into an ellipsoid and then into a
(recurrent) curve. In simplest terms and thinking about a one dimensional scalar
time series, the Lyapounov exponent reflects the multiplicative average (logarithmic
addition) of the sequence of slopes of the series of straight lines connecting the
points. An average slope of > 45 ° is expansive such that a linear distance on the x-
axis is increased when mapped onto the y axis. A slope of < 45 ° is a contraction
mapping reducing the linear distance of the x-axis when mapped to the y axis.
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Rössler’s generic chaotic system (see above) moving recurrently in a three
dimensional box can be orthogonally decomposed into three directional motions in a
moving frame, each with a signatory sign of λ . The unstable direction of expansive
stretching is characterized by some number > 0, λ( + ), the stable direction of
contractive folding, some number, < 0, λ( − ), and the neutrally stable direction of
recurrence, λ( 0 ). For The “Lyapounov spectrum” of the Rössler attractor is
[ λ( + ), λ( −), λ( 0 )] (Shaw, 1981). An n-dimensional dynamical systems has n onedimensional
Lyapounov exponents, and it is sometimes the case in relatively noise
free, finite semi-stationary data lengths of the neurosciences, that a λ > 0 can be
shown to exist for a second one, in a dynamical situation called “hyperchaos” by
(Rossler, 1979). For example, two and sometimes three λ( + ) have been reported in
the flows on the EEG attractor of normal alert subjects (Gallez and Babloyantz,
1991). The presence of measurement noise, the finiteness of neurophysiological
sample lengths as well as the relatively small expansive actions in some directions
in the chaotic attractors of brain dynamics lead to the finding that most often, only
one “leading Lyapounov exponent,” λ( + ), is reliably computable (Sano and
Sawada, 1885; Wolf et al, 1985; Eckmann et al, 1986).
A counter-intuitive fact about the stability of a dynamical system when a
decrease in the value of λ( + ) is observed such that λ( + ) → λ( 0 ), is that this more
neutral stability augers a global bifurcation (Guckenheimer and Holmes, 1983). A
small perturbation does not change the global dynamics of an already expanding
and contracting (called “hyperbolic”) dynamical system, it will maintain the style of
its motions. However, when λ( + ) → λ( 0 ), a velocity changing perturbation evokes a
bifurcation to a new dynamic in what is called “loss of hyperbolic stability.” The best
examples come from the observations of this kind of change in the EEG predicting
the onset of epileptic seizures in patients with focal or temporal lobe epilepsy
(Iasemidis et al, 1988,1990; Iasemidis and Sackellares, 1996 ).
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The number and variety of algorithmic strategies for computing Lyapounov
exponents that are applicable to real data divide naturally into those that compute
directly the average rate of separation of neighboring points from the “fiduciary”
orbit, as observed on the reconstructed attractor, from which only the largest λ can
be obtained (Wolf et al, 1985), and a variety of techniques based on assumed
model maps of the unknown flow along which the sequential products of the local
derivatives are computed. The logarithms of the straight line slopes of the sequence
of directionally decomposed local tangent vectors multiplied, yield as many
Lyapounov exponents as directions (Sano and Sawaka, 1985; Eckmann et al, 1986;
Geist et al, 1990). The techniques of regularization by which these model processes
approximate the unknown flow include those with least squares, linear fit
assumptions (Eckmann et al, 1986; Sato et al, 1987; Buzug et al, 1990), more
detailed fits involving polynomial expressions in higher powers (Briggs, 1990; Brown
et al, 1991; Bryant et al, 1991) and techniques such as “singular value
decomposition” which decomposes the flow into orthogonal components before
computing the logarithmic rate of divergence of nearby points on each of them
(Stoop and Parisi, 1991). A clever check on the Lyapounov number obtained is to
study the flow backwards so that, for example, some rate of separation of points in
the forward direction would approximate the rate of convergence in the time
reversed data (Parlitz, 1992).
Among the sources of spurious Lyapounov exponents are sample lengths
that are too short and/or too measurement-noisy to compute a statistically stable
average, embedding dimensions that are too high or low and attractors (many of
physiological relevance) that have geometric features such as sharp corners or tight
folds as in the Rössler (where points gather) or delicate boundary points such as
those on the body on the Lorenz butterfly (see above) where very small distances
determine whether the orbit makes big jumps to the right or left wing leading to
uncharacteristically large separations. This “nonuniformity” in the rates of expansion
and contraction in the dynamics over the attractor, a source of error in computations
of statistical indices of the average behavior, becomes a useful tool in
characterizing individual differences in sets of neurobiological data ranging from
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brain enzyme kinetics (Mandell, 1984) and single neuron firing patterns (Selz and
Mandell, 1991) to human psychomotor and cognitive behavior (Selz, 1992; Selz and
Mandell, 1993).
The Leading λ( + ) of Some Biologically Relevant Time Series
An early application of a simplified form of leading Lyapounov exponent to
brain data involved the computation of the one dimensional averaged slope of in
vitro studies of psychopharmacological drug and peptide effects on time series of
catecholamine and indoleamine biosynthetic enzyme activities studied at
physiological, far-from-equilibrium reactant concentrations (Russo and Mandell,
1984b; Knapp and Mandell, 1984). A contemporaneous study also suggested the
influence of differences in initial conditions for pharmacokinetic equilibrium times in
drug binding kinetics by proteins (Bayne and Hwang, 1985).
The most extensive applications to the clincial neurosciences of the
Lyapounov measure of the exponential divergence of orbital points has involved
reconstructed brain wave attractors from the intracranial or scalp recordings of the
EEG (Duke and Pritchard, 1991; Dvorak and Holden, 1991; Jansen and Brandt,
1993). Space prevents us from surveying more than a small representative set of
the studies (Jansen, 1996). It should be noted, however, that this is an area in
which “state of the art” research has grown quite complicated and somewhat
controversial with respect to technical issues. The choices of the digitizing
frequency of the smooth record, the dimension of the embedding space and time
delays continue to be debated in the context of numerical computations of λ and
dimension measures (Mayer-Kress, 1986; Ott et al, 1994).
Controls for the implicitly required statistical discrimination between
“randomness” and “deterministic chaos” consist of sequence and (Fourier) phase
randomization generating “surrogate data” which conserve the probability
distributions and destroy the correlation properties and attractor geometries (Sauer
et al, 1991; Ott et al, 1994). Since neither bring with them any connections with
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known or explorable brain mechanisms, one might argue that at this early stage of
the work it would be more desirable to simply report the quantitative findings,
leaving unanswerable questions about ultimate causality for later discussion (see
below).
The first EEG λ( + ) was reported in a patient with epilepsy (Babloyantz and
Destexhe, 1986) which was confirmed by others (Iasemedia et al, 1988; Frank et
all, 1990). An important study of simultaneous time series from 16 subdural
electrodes placed in the right temporal cortex of a patient with a right medial
temporal lobe epileptogenic focus demonstrated that a decrease in a single lead’s
λ( + ) reliably anteceded and localized the first signs of the incipient seizure. The
rest of the leads followed with similarly decreased positivity in their leading
Lyapounov exponents associated with spatially coherent patterns of behavior. In
addition, the averaged value of the leading Lyapounov exponents in the 16 leads
increased post-ictally over the averaged values of λ( + ) in the pre-ictal state
(Iasemidis et al, 1988,1990). These findings, including seizure anticipation for 25
minutes, were confirmed using intracranial recordings in 16 patients with temporal
lobe epilepsy (Elger and Lehnertz, 1998). The para-ictal decrease and post-ictal
increase in λ( + ) found in patients with focal temporal lobe seizures was confirmed
more generally in left and right pre-frontal-to-mastoid EEG recordings made before,
during and after electroconvulsive shock treatment of psychiatric patients (Krystal
and Weiner, 1991). Pre-ictal changes were also found six minutes before seizure
onset from scalp EEG recordings in 17/19 patients with chronic focal epilepsy
(Martinerie et al, 1998). The most exciting potential application of this approach is
its use, in real time, for the prediction and prophylactic treatment of incipient
seizures, minutes to hours before the event, in place of or augmenting long term
drug management (Iasemidis and Sackellares, 1996).
There is a growing literature about leading Lyapounov exponent(s) in the
reconstructed attractor of the EEG associated with a variety of normal and
pathological human behavioral states. For examples, two and sometimes three
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λ( + ) were reported in awake relaxed subjects and were lost in deep sleep (Stage
IV) and coma (advanced Jakob-Creutzfeld disease), suggesting that level of
consciousness correlated positively with amount of orbital divergence (Gallez and
Bablioyantz, 1991). A “pathologically low” leading λ( + ) was also found to be
characteristic of the EEG of patients with Alzheimer’s syndromes (Jeong et al,
1998). Technically defined sleep stages (I, II, III, IV, REM) were found to correlate
well with the values of the leading λ( + ) of the EEG in normal subjects (Fell et al,
1993; 1996; Pradhan and Sadasivan, 1996). EEG recordings during problem
solving sometimes, but not always, demonstrated a relationship between values of
λ( + ) and the kind or amount of load of the task (Micheloyannis et al, 1998;
Popivanovov et al, 1998; Meyer-Lindenberg et al, 1998). Both emotionally positive
and negative videos increased the value of the leading λ( + )(Aftanos et al, 1997) as
did computer generated music with sounds that exploited a “pleasing” hierarchical,
1/f but not an “unpleasant” 1/f 2 frequency spectrum (see previous section about
power law scaling) (Jeong et al, 1998). The EEG theta rhythm of “day dreaming”
manifested a lower λ( + ) than the “relaxed alert awake” alpha rhythm (Roschke et al,
1997). Relationships between the Lyapounov spectra demonstrated both regional
independence and task-related dependence in the magnetoencephalography record
in man (Kowalik and Elbert, 1995).
These and other studies suggest that divergence rate of orbits on a
geometrically reconstructed attractor is a subtle measure, which can be quantified
as a continuous variable and which has been found to be useful in a variety of
neuroscience-related, experimental contexts. The range includes the
characterization of the discharge pattern of a single somatic or renal sympathetic
nerve fiber (Gong et al, 1998;Zhang and Johns, 1998); quantifying the results of
perturbing autonomic nervous system activity, for examples, exercise, atropine and
propranolol decrease λ( + ) in the cardiac interbeat interval attractor (Hagerman et
al, 1996) and interference with the function of the baroreflex or clonidine alters the
λ( + ) in the blood pressure attractor in man and animals (Wagner et al, 1996;
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Mestivier et al, 1998); and predicting defects in visual learning functions from
decreases in the λ( + ) of the cardiac interbeat interval attractor in patients with
multiple sclerosis (Ganz and Faustman, 1996).
We recall that on theoretical grounds (Guckenheimer and Holmes, 1983), a
decrease in the positivity of λ( + ) → λ( 0)
in a delay coordinate, geometric
reconstruction of a time series of observables may auger an incipient global
bifurcation in the system’s dynamics. As reviewed above, this has turned out to be
the case in several studies of the EEG and electrocorticogram in epileptic patients.
Futher research will be required to see if this idea has substance more generally for
predicting “catastrophic” changes in other brain-related systems.
Power Law Scaling of Orbital Geometries in Time Series Reconstructions
Benoit Mandelbrot’s book in its first incarnation was derived from his lectures
at College de France in 1973 and 1974 and was called Les Objets Fractals: Forme,
Hasard et Dimension (Mandelbrot, 1975). This essay was translated into English as
Fractals, Form Chance and Dimension (Mandelbrot, 1977). Later expanded and
reworked editions displayed another title, The Fractal Geometry of Nature
(Mandelbrot, 1982) but the deep conceptual, sometimes poetic fusion and confusion
generated by the apparent identity among the objects of his first title remains.
“Fractal,” along with “chaos” and “strange attractor” are among the most widely
familiar new words in modern dynamical systems research. Fractal is the most
difficult to rigorously define and is commonly misunderstood due to the evocative
yet dream-like cognitive condensations provoked by the first title and its reflections
in Mandelbrot’s prose. A common conceptual confusion is exemplified by the
assumed relation between “fractal time event distributions” of the cardiac interbeat
interval and the “fractal like” anatomy of the purkinje network of the cardiac
conduction system. Data from both contexts are often shown juxtaposed in the
same illustration as though their relationships were obvious (Goldberger et al, 1990;
Goldberger, 1996; Liebovitch and Todorov, 1996). “Fractal times” and “fractal
224
geometries” are not related to each other essentially, either in the mathematical or
physiological domain, but are often made vaguely equivalent on the basis of their
lexical similarity.
An experimentally meaningful relationship between fractal statistics (hazard),
dynamical fractals (dimension) and fractal geometries (form), has to be proven on a
case by case basis and not assumed from their common designation. Among the
informal attempts to do this have been those that involve the branching pattern of
nerves and the associated reductions in their diameter-dependent characteristic
conduction velocities yielding a multiplicity of “arrival times.” There is, however, a
more central idea common to these concatenated meanings of fractal: the
statistical, dynamical and geometric expressions of “scaling,” a word which is not
mentioned in Mandelbrot’s book titles. The cluster of theories, theorems and
methods associated with the idea of scaling (and renormalization) have led to Nobel
Prizes for Flory (1971), Wilson (1975) and de Gennes (1979) and the (equivalent
mathematical) Field’s Medal for McMullen (1994). There is speculation that the last
two awards were supported by the inspiration and interest given their research by
Mandelbrot’s intuitions and books.
Scaling laws take the place of (unknown causal) physical laws by indicating
the proportion by which observables of a system can be changed in relationship to
each other such that some statement about them, “this varies with that,” still holds.
In a cross species comparison, as the average weight of a mammalian body, called
lb, increases, the skeletal weight, called w, increases at an exponentially greater
rate: w goes like lb 1.08 where lb 1.0 would indicate that they grew across species at
the same rate. Plotting log (lb) on the x axis and log (w) on the y axis in a log-log
plot results in a straignt line with a slope that indicates the power law scaling
relationship between body weight and skeletal weight across mammals. The slope
of the scaling exponent of 1.08 is a little over 45 ° = 1. In contrast, the metabolic rate,
r, goes like (lb) 0.75 , r ≈ (lb) 0.75 . Larger animals (relative to their weight) have lower
basal metabolic rates (Schmidt-Nielsen, 1984). We don’t completely know the chain
of intervening mechanisms that relate these variables to each other but we do know
225
invariant scaling laws that describe their relationships within some limits on the
range of values.
In describing the functional size, radius of gyration, R g , of a polymer such as
a polypeptide, composed of N monomers, assume each of the amino acids to be
the same and that they are in a “good” hydrophobic solvent that didn’t stick the
polymer together in a fold. Flory (1971) found a scaling law for certain broad classes
of polymers and solvents, R g ≈ αN ν , where the exponent, ν = 3/5, was universal, N
indicated the number of monomers in the chain and the value of “pre-factor” α
depended upon the particular monomer and solvent chosen. Log R g plotted against
log N has a “power law” slope of 0.60. For an equally static but less physical
example, there is the well known Zipf law of “vocabulary balance”(Zipf, 1949). First
reported for the 260,450 words of James Joyce’s Ulysses, the slope of the log of the
rank of the words found (ordered from most to least along x) plotted against the log
of their frequency (along y) results in a power law that is (generally) true for other
collections of words and in other languages.
An accessible example of a dynamical scaling law arises in a two
dimensional lattice model of a forest which is to be set on fire with probability p
independent random single tree ignitions. At some critical p, p c , the fire sweeps
through the entire forest (“percolates”) and the correlation length of the connected
clusters grows as |p-p c | -γ with a universal scaling exponent, γ = 4/3, for all Monte
Carlo, two dimensional percolation problems (Stauffer, 1985; Grimmett, 1989).
Mandelbrot’s scheme for the power laws that compose his fractal geometry
of dynamical objects is a measure made on the pattern of occupancy in the
embedding space by the reconstructed orbits of an attractor. It is, generally,
mass ≈ length D 0
in which D (the subscript that of the “capacity dimension”) is not the
0
whole number of Euclidian dimensions, d, of the space in which the orbits are
embedded. After Hausdorff’s “convergence of external and internal measures”
(Hurewicz and Wallman, 1948), the (capacity) fractal dimension D is also defined
as being larger than its topological dimension and smaller than its Euclidian
embedding dimension. Graphing a time series on a plane one can think of its
0
226
topological dimension as that of a line equal to one. If each time step had the
largest up or down amplitude as possible, its fractal dimension would approach (but
not reach) that of the embedding plane, Euclidean d = 2.
The D 0 of the one dimensional Richardson technique (Mandelbrot, 1967) can
be computed by covering the one dimensional surface of a time series with a
number, #, of line segments of several orders of magnitude range of lengths, l
.Graphing log(l) along the x-axis and log #(l) along the y-axis yields a negative linear
slope, -s. As defined, 1- s = D 0 noting that (-(-s)→+s) such that 1 < D 0 = 1+s < 2.
Strain differences and peptide and psychotropic drug-induced changes in D 0
computed in this way were found in time series of fluctuations in rat brainstem
tyrosine and tryptophan hydroxylase activities under far-from-equilibrium coreactant
concentrations (Mandell and Russo, 1981; Knapp et al, 1981; Knapp and
Mandell, 1983; 1984). Systematic influences of stimulant drug dose on D 0 were
found as well in these systems (Mandell et al, 1982). This simple measure, made
directly on the “roughness” of the graph of a one dimensional time series rather than
on its orbital reconstruction, has been used to discriminate the pattern of
fluctuations in daily mood scales in normal subjects and mood disordered patients
(Woyshville et al, 1999). These findings confirmed dimensional scaling exponents
on higher dimensional embeddings of similar time series in mood disordered
patients (Gottschalk et al, 1995; Pezard et al, 1996). Due to the ease and rapidity of
its computation, techniques involving D 0 on one dimensional time series are
currently in development as possible real time epilepsy predictors when analyzing
the output of a large number of EEG leads simultaneously.
If M(ε) is the minimum number of d-dimensional cubes of side ε required to
cover the d-dimensionally embedded attractor, plotting a logarithmic range of rulers
of length ε (as ε→0) along the x axis and a logarithmic range of number of cubes,
M(ε), each of corresponding ε-edge size, along the y axis, results in a negative
(more smaller M(ε) ‘s and fewer bigger M(ε) ‘s) power law slope D 0 . Here the
numbered covering cubes, M(ε), are those in which the probability of containing at
least one point (its “probability density measure,” often called µ) is not zero. We
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note that changing the ratios of the numbers of cubes that are dense in point
probability to those that are sparse would not influence the value of D 0 . This helps
differentiate D 0 from other dimensions and, as noted above, D 0 as a maximal
estimate of the fractal dimension, is called the capacity dimension and by
convention the scaling law is written
M
( ε)
D
≈ ε
− 0
. More specifically, D0 is calculated
by repeatedly dividing the d-dimensionally embedded phase space into equal d-
dimensional hypercubes and plotting the log of the fraction of the hypercubes
containing data points versus the log of the (normalized) linear dimension (“length
scale”) of the hypercubes. The slope fitted to the most linear part of the slope
(usually the middle 50%) indicates the capacity dimension. D 0 is computed for
increasing embedding (and cube) dimension, d, until it achieves an asymptotic
plateau, it “saturates”. This is but one of a range of geometric scaling exponents,
“dimensions,” that are currently being computed (Farmer et al, 1983; Grassberger
and Procaccia, 1983; Meyer-Kress, 1986; Theiler, J. (1990); Gershenfeld, 1992; Ott
et al,, 1994).
Although still subject to debate, convention has it that the sample length
required to determine this most primitive of dimension computations goes like
D
10 0
(e.g. a dimension of 2.45 requires a sample length of at least 282 points).
Assuming robust findings using D0 as indicated by non-parametric tests of
significance in test-retest, before and after, drug treatment designs, this arbitrary
criteria sounds more like ritual than meaningful help for the clinical neuroscientist
with (say) 100 spinal fluid hormone and metabolite samples painfully and
laboriously collected from a patient’s indwelling catheter over 48 hours. In the
context of real data (and not numerical studies of differential equations), we are
dealing with empirical findings that must find their meaning (or lack of) in the context
of questions about issues in the neurosciences, not in abstract questions such as
those about the number of dimensions that an unknown differential equation would
require to represent the data (Broomhead and King, 1986). In a similar arbitrary
spirit, a system manifesting a D 0 > 5 is considered not discriminable from a random
process; e.g. the difference between D 0 = 5 versus D 0 = 7 (though perhaps
statistically significant) is thought to be without meaning. Since in neurobiological
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research, “random” (if it doesn’t mean measurement error) indicates unknown
degrees of freedom, this D 0 > 5 rule is also without relevance for brain research.
D 1 is called the “information dimension” and is computed by counting the
number of ε-cubes, M(ε), it takes to cover the points constituting some fixed fraction
of all of the points of the set of orbital points on the attractor and can be regarded as
the “core dimension” (without the outliers) of the set. The counterintuitive finding is
that D 1 is nearly constant across a range of fixed fractions that are less than the
whole measure (Farmer et al, 1983). The invariance of D 1 can even be taken to the
extreme by computing the D
M
lim ln ( ε )
ε ln( ε)
1
=
→o
around (typical, not all) single points. In
this context, D 1 is called the “pointwise dimension” or “singularity exponent” and, as
might be anticipated, its value is usually less than that of D 0 .
The scaling exponent that is both sensitive to point densities and easiest to
compute from real data is the “correlation dimension,” D 2. Here, analogous to the
relationship between the amplitudes of the variance and the correlation function in
conventional statistics, the measure squared is of interest for the computation of D 2 ,
e.g.
M ( ε )
2
I( 2, ε ) = ∑[ µ ( Ci
)] (see below for this use of measure µ on sum Σ of cubes Ci).
i=
1
The selection of D 2 as the fractal measure dominates the studies that invoke scaling
exponents to quantify the distributions of points on the attractor as reconstructed
from time series in the neurosciences (Grassberger and Procaccia, 1983; Mayer-
Kress, 1986; Ott et al, 1994). Several sets of programs are available for its
computation (for example, Sprott and Rowlands, 1991). Generally, a correlation
sum (“integral”, R(ε) ) is computed from a starting point by counting all subsequent
point pairs with distances between them less than ε as ε→0 and plotting
D
2
lim ln( R( ε)
= . D 2 is computed for increasing embedding (and therefore
ε → 0 ln( ε)
hypercube) dimension, d, until D 2 achieves an asymptotic plateau, it “saturates”
(Ding et al, 1993).
It is generally the case that D 0 ≥ D 1 ≥ D 2 (Farmer et al, 1983).
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In his statistical explorations of experimental results in hydrodynamic
turbulence, Mandelbrot (1974) called attention to the need for a multiplicity of
characteristic scaling exponents, a range of values for each exponent and their
sensitivity to orbital point density distributions (the latter called the Sinai-Ruelle-
Bowen or natural measure (Eckmann and Ruelle, 1985)). These needs grew out of
the intrinsic heterogeneity in the time dynamics and the nonuniform point
distributions in phase space of orbitally divergent, real physical systems. Even with
relatively uniform orbital point distributions, it is intuitively obvious that as ε → 0, the
smaller ε- cubes are over-represented and larger ε- cubes are under-represented in
the M(ε) computation (Farmer et al, 1983). For a concrete example, the fraction of
the total number of cubes containing say 75% of the points would obviously
decrease as the ε-lengths studied gets smaller. Normalizing the D i measures with
respect to point densities would correct for this systematic distortion. In addition, the
non-systematic influence of real system heterogeneity and non-uniformity in both
time and reconstruction space distributions makes the need for relating the D i
measures to the natural measure even more pressing.
The derivation of many separate scaling exponents, as well as global
generalized exponents and the incorporation of point densities in their computation,
has been approached by a kind of method of moments (Renyi, 1970; Grassberger,
1983; Hentschel and Procaccia, 1983; Halsey et al, 1986; Mayer-Kress, 1986; Ott et
al, 1994). We outline the general arguments here so that the reader will be
generally familiar with the ideas and terms, not to serve as a definitive summary. It
is a complicated area and the reader will find the required detailed descriptions in
the references. .
We recall that with respect to a statistical distribution, the first moment is the
mean; the second moment, σ 2 , the variance; the third moment, σ 3 , the distribution’s
asymmetry, the skew; and the fourth moment, σ”, its relative peakedness with
respect to the probability mass in the tail, called the kurtosis. In these moment
computations of an observable
q
x i
’s deviation from the mean, |x − x| , the value for
q accentuate particular regions of the density distribution. Similarly, the q’s of the
i
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“generalized dimensions,” D q , emphasize different aspects of the relative point
density that are assumed to be uniform in the computation of D 0 . We recall from
above that the power law slope constituting D
ln M( ε)
lim
ε ln( ε)
0 = →o
. If we emphasize the
component of the probability (measure, µ) or, equivalently, time spent by the orbit in
cube i, µ( C i
) instead of simply the number of cubes occupied by any points, M(ε),
along with the different length scales of the cube as ε→0 we have a generalized
dimension. A common expression for the generalized dimension includes the
fractional pre-factor in q written so as to make things come out right:
M ( ε )
Iq
D = 1 lim ln ( , ε)
q
, where Iq ( , ε ) = ∑[ µ ( C i
)]
q −1ε
→ 0 ln( ε)
i=
1
the dominance of the higher probability cubes,
q
. The higher the q, the greater
µ( C i
). To see how this q-induced
separation in emphasis might work, if the ratio for q = 2 between the probability
containing cubes 0.25 and 0.05 is 25, their ratio for q = 3 is 125. For q = 0, the
scaling exponent is the capacity dimension. This result of the actions of a changing
q has been analogized to the way changing temperature in a thermodynamic
system evokes different aspects of its behavior.
The “multifractal formalism” generally begins by determining the statistical
densities over a range of scale lengths by one means or another including wavelet
transformations across wavelength scale (Arneodo et al, 1988). These densities by
scale are then systematically raised to a range of q exponents. Since q, and
therefore D q , can vary continuously, functions are created that shows how D q varies
with q. These are then further transformed, resulting in a single maximum parabolic
curve whose shape and size is sensitive to the conditions of the experiment (Halsey
et al, 1986). Generalized dimensions decrease as q increases. A unique
neuropsychopharmacological application of the multifractal technique to a study of
the behavioral influence of increasing amounts of cocaine on the time-dependent
patterns of spatial exploration, temporal-spatial fluctuations, in rats, demonstrated a
global splitting in the parabolic distribution suggestive of a cocaine-induced global
phase transition, not unlike the well-known, dose-dependent, amphetamine-induced
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shift from hyperactivity to motor stereotypy (Paulus et al, 1991). Studies that
followed demonstrated that “q-moment” distributions of heterogeneous scaling
exponents and their relative statistical weightings were useful in making subtle
discriminations between effects of psychopharmacological agents and behavioral
(isolation) influences on animal behavior as well as patterns of simple psychomotor
behavior in normal subjects and schizophrenic patients (Paulus et al, 1994; 1996;
1998; Krebs-Thomson et al, 1998a; 1998b).
Fractal Scaling Measures on Reconstructed Time Series from
Biological Dynamics
Publications involving the applications of various D measures, particularly D 2 ,
to brain-relevant times series number in the hundreds and are growing
exponentially. The following constitutes a brief review of a representative set of
empirical findings. In doing so, for the reasons discussed below, we ignore what
some might consider the rather abstract and philosophical issue of “determinism”
versus “randomness” or “error” (Sugihara and May, 1990; Casdagli, 1991; Wayland
et al, 1993; Kaplan and Glass, 1992; Kaplan, 1994) since this question is relatively
unproductive with respect to generating new neurobiological insights, novel
experiments or new quantitative approaches to brain dynamics. In addition, as
noted in the final section, this discrimination may not even have definitive theoretical
meaning in that the conduct of much of the rigorous mathematics about
“deterministic dynamical systems” involve Markoff partitions and matrices which are
also the generic operators of formal probability theory (Sullivan, 1979; Kolmogorov,
1950). For example, N-dimensional non-linear Markoff processes can be shown to
capture the dynamics of multidimensional neurobiological processes such as the
EEG (Silipo et al, 1998).
We have also ignored the related issue of the presence or absence of “low
dimensional structure” (Theiler and Rapp, 1996; Rapp, 1995) which, from the
authors’ point of view, resulted from an unfortunately concrete interpretation of the
word “dimensions.” With respect to experimental brain data, dimensions are defined
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most relevantly by their computational procedures and what are computed are
empirical scaling exponents describing real observables as limited by the precision
of the observations, their resolution and series lengths (Smith, 1988; Eckmann and
Ruelle, 1992). The “correlation integral,” the probability that two vectors chosen at
random from the phase space reconstruction lie within “r” distance of each other,
not unrelated to the phase randomization controlled, D 2 measure, yields statements
about amount of “nonlinearity” (not accountable by the linear regressively
capturable component of the power spectrum), which are also difficult to translate
into experimentally or theoretically useful concepts (Casdagli et al, 1997). These
efforts contrast with a more direct attempt to establish a spiking neuron system’s
dynamical “dimension” using trial and error prediction in which “dimension” was
defined as the number of potentially physiologically relevant variables required to
make the predictive equations fit (Segundo et al, 1998).
Computations of scaling exponent descriptors of orbital point distributions on
reconstructed attractors of the brain sciences have proven to be most useful as
atheoretical, empirical techniques discriminating experimental, clinical and/or
treatment conditions with various approaches to statistical significance. In this
regard, one can say that D 2 is often found to be superior to central tendency
oriented statistics in making these discriminations. Dimension and correlation
integral descriptors appear least useful when dealing with global issues such as
chaos, randomness, linearity and the “underlying dimensions” of (unknown)
differential equations. We discuss below the possibility that the failure to find chaos
in the more recent EEG studies (Theiler and Rapp, 1996; Prichard et al, 1996) may
be because the EEG attractor is better characterized as a “strange nonchaotic
atttractor” with orbital patterns manifesting fractional scaling exponents but no λ( + )
(Grebogi et al, 1984; Mandell and Selz, 1993).
The relatively subtle influence of high altitude (Mt. Everest) oxygen
concentrations was not seen in the central moments of the cardiac interbeat
intervals, but the D 2 of the attactor was reduced significantly (Yamamoto et al,
1993). The latencies and amplitudes of the visual evoked potential failed to
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discriminate normal subjects from those with early glaucoma, but the reconstructed
attractor of the steady state visual cortical response to full field flicker demonstrated
a statistically significant decrease in D 2 (Schmeisser et al, 1993). Marginal
qualitative differences in optokinetic nystagmus were quantitatively significant when
studied as the D 2 of the attractor’s points in patients with vertigo compared with
controls (Aasen et al, 1997). Reconstructions of maximum velocity waves from
Doppler studies of middle cerebral artery hemodynamics (using phase random
“controls”) demonstrated an increase in D 2 (and a decrease in λ( + ) correlated with
age in an adult population (Keuner et al, 1996; Vliegen et al, 1996). D 2 served as a
sensitive descriptor of functional changes in the EMG from the surface of the biceps
muscle, increasing with muscle load and rate of flexion and extension and
decreasing with muscle fatigue (Rapp et al, 1993; Nieminen and Takala, 1996;
Gupta et al, 1997), suggesting its use in suspected early myotonic dystrophies and
myasthenias. Reconstructed time series of stomatognathic motions in high school
students with temporomandipular joint syndromes compared with those with
malocclusion revealed a specific decrease in D 0 in the plane of horizontal motion in
the former (Morinushi et al, 1998). Time series of plasma growth hormone levels in
acromegalic patients with functioning pituitary adenomas manifested a statistically
significant increase in D 0 when compared with age-matched controls (Mandell and
Selz, 1997) which corresponded nicely to the reduction in “approximate entropy”
(Pincus, 1991a) computed on this same data set (Hartman et al, 1994). On the
other hand, comparative in vitro studies of growth hormone release patterns in
normal rat pituitary cells and their neoplastically transformed relatives, the GH3
strain, demonstrate a decrease in D 0 in the latter (Guillemin et al, 1983; Mandell,
1986).
The number of examples of the use of D 2 on orbital point geometries in
explorations of physiological and pharmacological regulation are increasing. The D 2
of respiratory rhythms is higher with intact vagal afferents than without (Sammon
and Bruce, 1991). Histamine induced an increase in D 2 in the attractor point
distribution of rabbit ear artery vasomotion, attributed to calcium-activated
membrane potassium channels in that TEA prevented and reversed the change
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(Edwards and Griffith, 1997). The role of central and autonomic innervation in
cardiac interval dynamics has been explored using D 2 in various ways. For
examples, the transplanted heart rhythm in man has a lower D 2 than that of the
normal heart (Guzzetti et al, 1996) and general anesthesia and cholinergic (but not
β-adrenergic) blockade decreased multisystem D 2 in a series of multiparameter
(respiration, mean blood pressure and heart rate) studies in piglets (Zwiener et al,
1996; Hoyer et al, 1998).
The activities of single and aggregates of neurons are being described and
differentiated by the D 2 of their interevent interval attractors. Early and important
studies related to both neuronal and field electrical activity indicated their promise
(Rapp et al, 1985; Zimmerman and Rapp, 1991). The olefactory bulb demonstrated
spatially uniform scaling dimensions that changed with event-related perturbation
(Skinner et al, 1990). An iron-induced spiking focus in the rat hippocampus in vivo
manifested the same decrease in D 2 as it did in the kindled in vitro hippocampal
slice (Koch et al, 1992). D 2 also differentiated among characteristic single unit time
series in norepinephrine, dopamine and serotonin neurons (Selz and Mandell,
1991) and among A8, A9 and A10 dopamine neurons (Selz and Mandell, 1992).
Attractors reconstructed from single unit interspike intervals in the substantia nigra
pars compacta and the auditory thalamus manifested discriminatable values for D 2
in neurons recorded by the same electrode (Celletti and Villa, 1996) and changes in
state manifested in patterns of subthreshold oscillations in single neurons in the
inferioir olivary nucleus could be characterized using this index (Makarenko and
Llinas, 1998).
D 2 reliably discriminated between states of arousal and between the
multiparameter (eye movements, neck muscle tone, EEG stage) defined EEG
stages of sleep (Bablyoyantz, 1986; Rapp et al, 1989; Ehlers et al, 1991) with non-
REM having a lower D 2 than REM. D 2 of the EEG record was selectively reduced in
Stage II and REM in schizophrenic patients compared with controls (Roschke and
Aldenhoff, 1993), this difference was made more prominent by treatment with the
aminodiazopoxide, lorazepam (Roschke and Aldenhoff, 1992). In the waking state,
235
higher EEG D 2 values were frontal in schizophrenic patients and more central in
controls (Elbert et al, 1992). The D 2 computed on the EEG during Stage IV (“delta”)
sleep was sensitive to acute sleep deprivation and recovery, but demonstrated
compensation (Cerf et al, 1996). Non-alcholic children of alcoholic parents
manifested lower values for D 1 in their EEG attractors than the children of a normal
control group (Ehlers et al, 1995). Higher I.Q. correlated with EEG D 2 in most leads
in the resting state but not during a visual imagery task (Lutzenberger et al, 1992).
These differences also correlated with individual differences in task performance in
a perceptual pattern predictive task (Gregson et al, 1990) and with a working
memory task load with regional differences most marked in the right fronto-temporal
cortex (Sammer, 1996).
Peripheral nerve stimulation in the earlobe and trapezius muscle induced
increments in D 2 in the EEG of specific brain regions (Heffernan, 1996). Memory for
but not induced pain increased EEG D 2 in chronic pain patients but not in normal
controls (Lutzenberger et al, 1997). Using contingent reinforcement of brain wave
modes by hypothalamic, but not cerebral hemispheric, stimulation reduced D 2 in the
EEG (Mogilevskii et al, 1998) resembling the changes accompanying defensive
reflex conditioning in the rabbit between the early and late stages of the process
(Efremova and Kulikov, 1997). Difficult to diagnose “periodic lateralized epileptiform
discharge” syndromes have apparently yielded to D 2 computations (Stam et al,
1998). In equally problematic “atypical seizure” syndromes in children, D 2 computed
on the autocovariance functions of 200 Hz digitized EEG records from multiple
channels demonstrated characteristic changes (Yaylali et al, 1996).
Unlike computing a reliable leading λ( + ) on a point set of a time series
reconstruction denoting the “sensitivity to initial conditions” requirement for the
diagnosis of chaos (and a potential for change such that a decrease in the positivity
of λ( + ) → λ( 0)
may auger a nearby bifurcation), the presence of a fractional scaling
exponent,
D i
, does not in and of itself implicate a chaotic dynamical state. A nice
example of a nonchaotic dynamic with
λ = 0 that has a fractional scaling exponent,
D = 0.538, is the “Feigenbaum” point where the above noted “infinite” series of
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period doubling bifurcations accumulate (Grassberger, 1981). This is a dust-like
region, which when endlessly dilated looks like the same dust. Some
mathematicians call these objects “Lebesgue points” because even though at low
magnifications when they look rather solid, they are not. Composed of points, they
have topological measure zero (a line has measure one) and non-integer fractal
dimension. These λ = 0 , D ≠ Integer, period doubling accumulation points can be
found in a wide variety of attractors, though in each case the parameter space in
which they are located is so small (in point set topology also called “Lebesgue
measure zero”) that they are very difficult to locate and therefore have little chance
of being physiologically significant.
This constrasts with a relatively new category of dynamical systems which
promises to be important in studies of the nervous system. These are ones that are
driven by two or more independent frequencies (called quasiperiodic driving). We
found them to be relevant to brain stem, thalamocortical neurophysiology of
perceptual processes and states of consciousness. They have the properties,
λ = 0 , D 0 and D 1 ≠ integer and a characteristic scaling “spectral distribution
function” (see below). They have been named “strange nonchaotic attractors”
(Grebogi et al, 1984; Romeiras et al, 1987; Ding et al, 1989). In addition, the
strange nonchaotic behavior of these quasiperiodically-driven, nonlinear oscillators
has positive (>0) measure in parameter space and thus is of potential physiological
significance. A good demonstration of a multiple frequency driven strange
nonchaotic attractor can be found and manipulated in the software package of
Nusse and Yorke (1991).
The neurobiological substrate for this system is the brain stem neuronal
modulatory driving of on- going thalamocortcal oscillatory brain waves (once called
“recruitment waves” in the 7-14 Hz, θ to α, day dreaming to quiet alert range) and
as perturbed by multifrequency driving in what was once called “reticular formation
arousal” are realized as dominant EEG modes and associated states of perceptual
acuity and consciousness (Moruzzi and Magoun 1949; Moruzzi, 1960; Klemm,
1990; Steriade and McCarley, 1990; Contreras et al, 1997). In addition to intrinsic
237
multiply periodic and aperiodic oscillations of thalamic and cortical cells and their
recursive, feedback coupling, the brain stem manifests more than two orders of
magnitude of “independent” neuronal driving frequencies ranging from serotonin
discharges at 1 Hz, cortically direct dopamine and norepinephrine neurons in the
10-50Hz range and mesencephalic reticular neurons discharging as fast as 100 to
200 Hz. The “thalamocortical brain wave oscillator” as their target has been a fixture
in global state neurophysiology since the 1940’s and 1950’s and is of great current
interest (Fessard et al, 1961; Bazhenov et al, 1998). We have explored the
relationships between strange nonchaotic dynamics and brain-stem neuronal and
thalamocortical physiology from the standpoint of neuronal coding and the
properties of the EEG attractor. (Mandell et al, 1991; Mandell and Kelso, 1991;
Mandell and Selz, 1992; 1993;1994;1997a). We found that the EEG attractor could
be characterized by the diagnostic triad identifying strange nonchaotic attractors:
λ = 0 , D 0 and D 1 ≠ Integer, and a signatory power spectral distribution in which the
number of peaks, N, with amplitudes greater than ϖ, N(ϖ ), went as ϖ -α , 1 < α < 2
(Romeiras et al, 1987; Mandell et al, 1991). In addition to being consistent with
known multifrequency, brain stem driving of thalamocortical oscillations, the EEG as
a strange, nonchaotic attractor is intuitively appealing in that it has the necessary
mechanisms for the power law scaling of a wide range of characteristic times (D 0
and D 1 ≠ Integer) from picosecond fluctuations of neural membrane proteins to the
decades of bipolar phenomena and since λ = 0 , the orbital points don’t tend to
“mix”(get out of order) on the attractor, thus protecting the fidelity of sequence
dependent brain information transport (Berns and Sejnowski, 1998).
Entropies, Unstable Periodic Orbits and Shadowing; Short Time Series
Can Discriminate Experimental Conditions in Studies of Biological Dynamics
We avoid the temptation to deal with the deep analogy between
thermodynamic entropy (Clausius, 1897) and information theoretic entropy
(Shannon and Weaver, 1949), constraining our discussion to the context of an
operational equivalence (in healthy systems) between gain of information and
238
decrease in entropy in brain-relevant dynamical systems. As we shall see, certain
pathophysiological processes appear to manifest themselves as reductions in
background or “resting” state entropy which then limits its supply with respect to
information gain and/or transport. Relationships between “physical” thermodynamic
observables, such as changes in heat capacity or temperature dependence of
kinetic constants, and information-transport driven, neurotransmitter evoked
conformational changes in neural membrane proteins may someday come together
in an experimentally productive way (Hitzemann et al, 1985; Zeman et al, 1987;
Borea et al, 1988), but they are beyond the scope of this paper.
The idea of taming the orbit of an expanding flow (with at least one λ( + )) by
partitioning the geometric space supporting its actions, its “manifold,” and then
labeling each box so that its trajectory is representable by a symbol string of box
indices is the way “symbolic dynamics” are applied to dynamical systems. Symbolic
dynamics arose in pure mathematics in the context of obtaining a one-to-one,
topological (sequence not distance preserving ) representation of a difficult to
characterize system of “geodesics on surfaces of negative curvature” (Hadamard,
1898; Morse, 1917; Morse and Hedlund, 1938). Geodesics here are the shortest
lines in this curved, non-Euclidean space in which nearby lines spread apart and far
away ones came together with (in Euclidian space) parallel lines meeting at infinity.
Remarkably, symbolic dynamic encoding of the motions on this abstract
manifold of negative curvature also capture how uniformly divergent (and
convergent), “hyperbolic” chaotic systems, such as brain systems, behave in
Euclidean space, an intuitive similarity about which Poincare experienced his
famous vacation bus trip epiphany (Stillwell, 1985). It should also be noted that
encoding neural spike trains in one dimension for symbolic dynamical comparisons
of sequence structure and recurrances, “favored patterns” has been developed
independently of orbital dynamics on manifolds (Dayhoff, 1984; Dayhoff and
Gerstein, 1983a; 1983b). A similar approach has been used to characterize firing
patterns and their response to acupuncture in dopamine neurons in the substantia
nigra and hypothalamic neurons (Chen and Ku, 1992).
239
For real neurobiological data, a time series and its n time delays are first
reconstructed as a trajectory in an n+1 dimensional geometric embedding space
and, following partition of that geometric space into n+1 dimensional lettered boxes
(the choice of partition being a sensitive step), what was once an orbit has become
a sequence of symbols. Dynamical systems in geometric space become symbolic
dynamics in sequence space. It was Kolmogoroff (1958) who first applied
Shannon’s ideas of entropy and information (Shannon and Weaver, 1949; Khinchin,
1957) to the quantification of these dynamical system’s telegraphic messages as
discrete, “stochastic” (random, probabilistic) output. Kolmogoroff turned to Shannon
entropy,
−∑ p i
log p i
(where p = 1/n and n = number of possibilities) to decide the
question whether a dynamical system that naturally partitioned into a two or three
box system per unit time had the same entropy. His answer was no, that –3(1/3 ln
(1/3)) = 1.098 > -2(1/2 ln (1/2) = 0.6931 loge and in computer relevant log 2 , 1.5850 >
1.0 (Kolmogorov, 1959). Entropy increases with possibility.
Nonlinear differential equations representing brain-relevant expanding
dynamical systems replace Shannon’s linguistically weighted and serially ordered,
Markoff-dependent random number generator of probabilistic language. As noted
above, in the case of the Sharkovskii sequences (Sharkovskii, 1964; Metropolis et
al, 1973; Misiurewicz, 1995), a small change in the single parameter of an entire
class of single maximum maps generating motions that are coded from their
position at the left or right of center of the unit interval, alters and determines
precisely the periodic output such as {1,0,0,1,0,1,1,0,0,1,0,1…) of its binary
message. In higher dimensional examples such as the Rössler and Lorenz
systems, one can visualize the joint actions of λ( + ) and λ( − ) moving the trajectory
so as to both enter, “create,” new boxes and generate new letters as well as visit old
ones, unstable fixed points, thus forming unstable periodic orbits. The latter, one of
three diagnostic features of chaotic attractors (see above), can also be seen as
resulting from the “coarse-grained” imprecision of real world neurobiological
measurement such that two points that are brought close to attractive-repelling
points are, within measurement error, recorded as having the same value.
240
Problems of measurement precision, amplified by the expansive actions of
systems that are sensitive to initial conditions, yield parameter sensitive entropies of
two (mathematically) fundamental kinds called topological and metric entropies, h T
and h M , proven to be the upper and lower bounds of any estimate of the entropy in a
uniformly expanding and/or equidistributed system (Adler and Weiss, 1965).
Measures of entropy, as “missing information related to the number of alternatives
which remain possible to a physical system” (Boltzmann, 1909), “index of
probability” (Gibbs, 1902) or the “amount of uncertainty associated with a finite
scheme” (Khinchin,1957) are obviously sensitive to the partition rules and its
fineness of the grain. The most theoretically defensible partition is called the
“generating partition” in which no box contains more than one point. Comparisons of
control and experimental data can be differentially sensitive to partition construction,
so that if a generating partition is not practicable due to sample length or dense
curdling in the point distribution, some arbitrary choices have to be made. These
have included naturally renormalized variational partitions, such that in one
dimension the boxes are defined by ±1, ±2, ±3,…standard deviations, or quartiles or
quintile, above and below the mean and in n dimensions. Partitions have also been
constructed and used to described drug effects on rat exploratory behavior by
sequential partitioning along the dimension of the highest remaining variation (after
the previous partition) called the “KD” partition (Paulus et al, 1991). Partition
strategies to capture entropic measures on serial ordering (Klemm and Sherry,
1981; Strong et al, 1998) can grow from knowledge or hypotheses about the
physiological sources of temporal irregularities and discontinuities in brain dynamics
including characteristic interval(s) of refractoriness, relaxation times of the inhibitory
surround, correlation time in dendritic tree summation, the time course of reciprocal
inhibition and its decay and chemical influences such as the synaptic half-life and
time of action of inhibitory influences such as GABA on cell firing.
The logarithmic growth rates of occupancy of new symbolically indexed
boxes or, equivalently, the growth rates of visitations to old ones generating
unstable periodic orbits, are called topological entropies, h T . They record new
happenings, the growth rate of the diversity of orbits, and not how likely with respect
241
to box occupancy densities they are likely to occur ( Adler et al, 1964; Alexeev and
Jacobson, 1981; Cornfield et al, 1982; Ornstein, 1989; Ruelle, 1990). The close
relationships in real brain observables between the appearance rate of new
symbols or new unstable periodic orbits, h T , and log λ( + ), reflecting the rate of
divergence from the next expected value generating a new, unexpected value, is
not surprising. In fact, a maximal estimate of the entropy of a dynamical system, h T
= log λ( + ) whereas the largest value that h M can attain is log(#of states). A great
deal of substantial mathematics has gone into proofs that similarities (“equivalence
relations”) and differences between dynamical patterns are robustly indicated by
differences in h T and h M (Adler et al, 1977; Adler and Marcus, 1979).
If the sum of the densities in each j box were normalized so as to sum to 1.0,
such that each is a probability, p j , then - Σ p j log p j represents the metric entropy,
h M . h M was first described in the dynamical context by Kolmogorov (1958;1959).
The sum having a –1 prefactor converts the negative log of < 1 to a meaningful
positive value in the expression. h M is maximal for the equidistributed, uniformly
expansive, C or Axiom A systems (see above). As noted above, generally h T = the
maximum estimate of the entropy and h M the minimum estimate (Adler and Weiss,
1965). h T = h M in uniformly hyperbolic systems (Bowen, 1975) and the difference,
|h T – h M | is an index of non-uniformity found useful in discriminating among classes
of single neurons from their discharge patterns (Mandell, 1987; Selz and Mandell,
1992; Mandell and Selz, 1993; Mandell and Selz, 1997a). These measures applied
to temporal and spatial patterns of rat exploratory behavior have been used to
discriminate among stimulant drug effects (Paulus et al, 1990; Paulus and Geyer,
1992). Similar computations involving the symbolic dynamics and disallowed
transitions have been used to study the complexity of the the EEG (Xu, 1994) in
which both extremely low (fixed point, periodic) and high (Gaussian random)
entropies are seen as manifesting low “complexity as a function of the diversity of
the available patterns of behavior (Crutchfield and Young, 1989a).
Before describing the simple but definitional matrix operations for h T and h M
below which might seem forbidding to those “not up on their linear algebra,” we note
242
that procedures such exponentiation of a matrix can be carried out automatically
using computer algebra programs such as Maple or for data processing available as
computational modules in MatLab.
One of the techniques for the computation of h T involves determining the
logarithm of the asymptotic growth rate of the major diagonal (“trace”) in the
transition matrix symbolically encoding the trajectory which would therefore count
the “self visitations” of each indexed boxes as the dynamics proceed. This involves
setting up a transition incidence matrix, each box scored for a disallowed, 0, or
allowed, 1, transitions and the matrix is exponentiated t times with the logarithm of
the asymptotic growth rate of the sum of the diagonal values serving as a (leading
eigenvalue) estimate of h T . More technical considerations involving the Frobenius-
Perron theorem guaranteeing the existence of such an logarithmic index of new
information generation rates, even in random matrices (Seneta, 1981), will not be
discussed here.
We have found that computing h T in this way is empirically useful for difficult
to obtain or only transiently stationary brain data series. Even with relatively short
samples lengths, if one is willing to make the pragmatic assumption of “temporary
stationarity” or “things as they are right now will, for the sake of argument, go on
forever” (perhaps the best we can do with intrinsically transient brain phenomena)
then this “freeze framed” representation of reality yields an asymptotic measure on
relatively short sample lengths since they are computationally infinite. A similar
approach to h M , requires repeatedly exponentiating a Markoff matrix constructed
from relatively short samples and generates the probabilistic (eigenvector) “dual” of
h T. h M computed in this way serves as a useful quantity, h M called by some the
Kolmogorov entropy in comparisons of control and experimental conditions of the
same sample lengths. Systematic decreases in h M (“Kolmogorov entropy”) have
been shown to accompany increasing “depth” of sleep using standard sleep staging
techniques (Gallez and Babloyantz, 1991) and increases in h M were associated with
both positive and negative emotional states induced by movies (Aftanas et al,
1997).
243
h T and λ( + ) have been analogized to what is called algorithmic complexity,
which quantifies a computer algorithm’s minimal representation of a symbol
sequence as it grows longer (Chaitin, 1974; Bennett, C.H., 1982; Nicolis, 1986;
Rissanen, 1982; Crutchfield and Young, 1989b). Examples of applications of a
pseudocomputational compression scheme have quantified differences among
protein sequences (Ebling and Jimenez-Montano, 1980), discriminated therapistdirected
“transference” manifestations in verbally encoded processes in
psychotherapy (Rapp et al, 1991), characterized neural spike train patterns in a
penicillin kindled spike focus (Rapp et al, 1994), differentiated among spike
sequence patterns of biogenic amine families of brain stem neurons (Mandell and
Selz, 1994) and as a sample length-dependent rate, in content-free, mouse driven
computer tasks differentiated borderline from obsessive-compulsive personality
patterns (Selz and Mandell, 1997).
Computation of lexical complexity is a good example of this approach. This
procedure recursively surveys the sequence of symbols for the longest word, where
“words” are subsequences that appear at least three times if they contain two letters
or at least twice if they contain more than two letters. Upon finding a longest
repeated word, the compression algorithm replaces all occurances of this word with
a single distinct (new) symbol and looks again for the longest repeated word in the
modified sequence. When the sequence cannot be further recursively compressed,
there may remain identical adjacent symbols in the sequence. These are coded as
the symbol raised to the power of the number of its adjacent occurances. This
exponent cannot exceed five because six adjacent identical symbols would be two
occurances of a three letter word. The numerical value of the lexical complexity is
simply the sum of the number of distinct symbols and the (sum of the) logarithm of
the exponents of the symbol sequences (Ebling and Jimenez-Montano, 1980).
A clear account of algorithmic and lexical complexity in relationship to other
measures of “complexity” in the context of brain relevant research data can be
found in Rapp and Schmah (1996). The relationship between thermodynamic and
ergodic, measure theories in relationship to forced-dissipative dynamics and the
244
role of self-intersection on manifolds in this new source of irreversibility (with a
resulting “arrow of time”) is developed in Mackey (1992).
As noted, the skeleton which configures attractors is composed of unstable,
“saddle” fixed points, each of which attract (iron down) the trajectory along one
dimension and repel or spread it out along another. Systems fulfilling the criteria for
a chaotic dynamical system have the property of a countably infinite number of
unstable periodic orbits composed of these unstable fixed points. Depending upon
parameters, the orbital points can pull up their tails to be discrete with respect to
each other or spread along the unstable direction to connect smoothly with others
along a curve such as a saddle cycle. Parametric control of the strengths and
structures of the saddle point skeleton of typical attractors can be used to change
both the rate of generation of novel symbols as well as recurrances to old ones in
the symbolic dynamics generating a brain dynamical system’s lexagraphic products
(Bowen, 1978; Alexeev and Jacobson, 1981)).
Using a variety of techniques to algorithmically register “return times,”
experimental condition-sensitive “saddle orbits” composing unstable periodic orbits
have been demonstrated in geometric reconstructions of real data series generated
by a 40+ component chemical reaction (Lathrop and Kostelich, 1989), in response
to natural stimuli in the time dependent behavior of the crayfish caudal
photoreceptor (Pei and Moss, 1996) and in the interburst interval sequences
recorded in hippocampal slices of the rat (So et al, 1997; So et al, 1998). If the
reader uses the software listed above to simulate the time evolution of one of these
attractors of abstract or real systems , she will learn that a remarkably small number
of points, a very short time sample, will outline, “shadow” (Bowen, 1978), the
complete array of unstable fixed points before filling in the attractor. It is tempting to
speculate about the potential nervous system relevance of this dynamical
anticipation of the attractor’s recognizable geometry, as well as a precis of what the
symbolic dynamics are going to say occurs many time steps before filling in the
attractor and its asymptotic message. Values of the measures made on the early
unstable periodic orbit arrays such h T , h M and λ( + ), resemble very closely those
245
made on their attractors when they were much more densely filled (Lathrop and
Kostelich , 1989). Bowen’s “shadow lemma” in support of a thin film of points over
the skelton of unstable fixed points of attractors is the fundamental reason that short
sample length time series can often discriminate between control and experimental
conditions in brain research studies.
Another recently implemented entropy, called “approximate entropy,” is
exploiting the underlying unstable fixed point skeletal shadowing principle in
expansive dynamical systems to find statistically significant differences between
control and experimental results in reasonably short, physiologically realistic,
sample lengths (Pincus, 1991; Pincus et al, 1991). This algorithm is somewhat
derivative of those involved in the computation of the correlation dimension (see
above). Instead of computing across a range (and taking the limits) of embedding
dimensions, d, and sequential paired-vectorial distances, ε, it empirically tailors and
fixes them to compute a “logarithmic likelihood” that points remains close through
incremental change in the time series. The “approximate entropy” is not easily
relatable to either h T and h M. One is tempted to predict that this geometrically
oriented algorithm might be fooled into a postive entropy diagnosis if applied to
strange, nonchaotic dynamical systems with fractal dimension but no λ( + ) -related
mixing. Since sequence position is conserved in this computation, two
simultaneously studied (“multiparameter”) systems can be examined for their mutual
coherence as the “cross approximate entropy.” Among the interesting findings from
applications of this index to neuroendocrine studies are an increase in approximate
entropy in LH and FSH secretory patterns with age in both sexes, perhaps
quantitatively heralding menopause (Pincus and Minkin, 1998) and decreased cross
approximate entropy, a decrease in regulatory coupling between ACTH and cortisol
secretion patterns in patients with Cushing’s syndrome (Roelfsema et al, 1998).
Among the many of other empirically derived entropies, one is called “power
spectral entropy,” which is equivalent to the normalized variance of the distribution
of frequencies in a power spectral transformation of a time series (Farmer et al,
1980). This has been successfully applied to brain enzyme and receptor fluctuations
246
(Russo and Mandell, 1984a; Mandell, 1984), and, more recently, to multiple
simultaneously EEG leads which demonstrated focal increases in epileptic patients
(Inouye et al, 1991; 1992). An entropy derived from the quantification of the failures
in temporal forecasting of EEG signals increased in the fronto-temporal region with
drug treatment in patients with Alzheimer’s syndrome (Pezard et al, 1998).
With respect to their implications for the clinical neurosciences, changes in
dynamical entropy in behavior of brain dynamical systems has been regarded in two
general ways: (1) Since representation of information requires the resolution of
relevant ambiguity, a nonrelevant and global reduction in the dynamical entropy of a
brain system (Stage IV sleep EEG slow waves, neuronal fixed point or regularly
periodic activity, extrapyramidal motor tremor, fixed paranoid or obsessional
mentation, the actions of some anxiolytics and antipsychotics ) reduces its potential
for information encoding and transport. In contrast, “arousal” induced increases in
the measures of entropy in brain wave and neuronal discharge patterns (pre-task
warning signals, motivating conditions, stimulant drugs) are associated with
improved psychophysical receptive and discrimination functions, learning rates and
memory. (2) Regarding as potentially pathophysiological both of the two extremes
of entropy generation, fixed point and periodic behavior as the lowest and fair coin
flipping, “Bernoulli” randomness as the highest, another descriptor, “complexity” is
defined as maximal (optimal) midway through the entropy range, making a new kind
of parabolic entropy curve (Bennett, 1986; Crutchfield and Young, 1989a).
In analogy with an optimal amalgam of periodic rotations and coin flips, in
higher dimension, the most meaningful maximum complexity of real, nonuniformly
expansive processes may derive from a multiplicity of measure invariants,
symmetries, of the system such as the growth rate of unstable periodic orbits,
divergence of the tail of a density distribution and specifiable linguistic variables
such as word length and redundancy. The more symmetries, the more potential for
complicated information encoding and transport with the maximum complexity
located midrange in each one. We have pursued the hypothesis that entropy is a
conserved property in the healthy brain and that complementarity in other statistical
measure mechanisms make that possible. For example, in uniformly expansive,
247
idealized systems, topological entropy has been proven be equivalent to the product
of an index of expansion and the dimension of the support such that an increase in
expansiveness , λ( + ), is compensated by a decrease in D 0 leaving h T invariant
(Manning,1981). This relationship has also been found in the behavior of some
nonuniformly expansive neuroendocrine, neuronal and human behavioral systems
(Mandell and Selz, 1995; Smotherman et al, 1996; Mandell and Selz, 1997a;).
Is Randomness Versus Determinism a Productive Question for the Biological
Sciences? Are There Better Ones?
Measures made on realistically nonuniformly expansive behavior of
dynamical systems emerging from nonlinear differential equations and that arising
from a variety of non-classical random walk models overlap such that making what
may be more a metaphysical discrimination at this point is labor intensive,
contentious and unproductive for generating new experimental work in the
neurosciences. It is important to note that random walk theory and computation has
matured to such an extent that almost any “nonlinear dynamical behavior” can, with
respect to statistical measure, be modeled using one of many varieties. For
examples, power law distributions in continuous time random walks (times of
movement are also randomly chosen) , random walks with traps (temporarily
immobilizing the trajectory like unstable fixed points), random walks in random
environments, time of passage of ants in a labyrinth and Levy leaps and local
diffusive exploration (looking for a wallet) among many others can represent much
of the irregular behavior we observe in the brain (Shlesinger et al, 1982; Montroll
and Shlesinger, 1984; Hughes, 1995; Klafter et al, 1996). On the other hand,
(Markoff) partition of the sequence and a probabilistic style of analysis of nonlinear
dynamical systems has been a major strategy for description and quantification
from the field’s beginnings (Parry, 1964; Adler and Weiss, 1967; Bowen, 1970;
Lasota and Yorke, 1973). The issue of randomness versus determinism remains
current although many if not most properties of deterministic dynamical systems can
248
be simulated with a suitably constructed random process and all of our random
number generators are deterministic.
This theoretical blind alley is reminiscent of the decades lost partialing out
causal attributes of nature versus nurture before knowledge of dynamical influences
on nucleotide dynamics was available. It is perhaps unfortunate that for finite length
real data, “house keeping requirements” (Ruelle, 1990; Rapp, 1993;1994) and
“warnings on the label” with various random sequence, random phase controls
(“surrogate data”) have become so intimidating to those of us in the early stages of
exploring the use of these theories and methods in the brain sciences. Currently the
“controls” are more relevant to abstract statistical processes and what can be said
about them rather than generating and addressing new claims and the controls for
them related to quantitatively oriented, experimental brain physiology.
Statistical caveats have arisen to retard the emergence of potentially
important and robust neurophysiologically-relevant phenomena. For example, a
recent well conducted and analyzed study of the influence of low doses of ethanol in
32 normal male subjects, which honored almost all of the current analytic rituals
including sequence and phase randomized surrogate data and searches for the
continuity features of deterministic dynamical systems such as time asymmetry,
concluded that the drug “reduced the evidence for nonlinear dynamical structure” in
the brain (Ehlers et al, 1998). Though honoring the currently popular statistical
rituals, what appears to be missing here are suggestions for new neurobiological or
mathematical intuitions that will lead to the design of the next experiment.
We now see that it is now possible to use these new ideas and methods to
ask and at least partially answer more specific questions relevant to the clinically
oriented neurosciences such as: whether increases in lithium-induced
expansiveness and mixing in the dynamics of brain enzymes, neurons and behavior
help explicate a mechanism of de-coherence in bipolar disease (Mandell et al,
1985); do these approaches to membrane conductance fluctuations suggest a new
way to think about ion channel dynamics (Liebovitch, 1990); can alcohol-induced
changes in statistical dynamics of the EEG predict genetic predilection in males to
249
alcoholism (Ehlers et al, 1995); do these approaches suggest a new neural
dynamical mechanism for the actions of anticonvulsant drugs (Zimmerman et al,
1991); can these measures made on non-verbal, psychomotor tasks yield a nonintrusive
measure of personality and character (Selz, 1992); can these approaches
to deviant patterns of psychomotor sequencing in schizophrenics give us some
insight into potential (cerebeller-basal ganglia?) mechanisms of the thought disorder
in schizophrenia (Paulus et al, 1994); does cocaine induce new patterns of behavior
that conserve pre-treatment entropy in developing animals (Smotherman et al,
1996); will these quantities applied to objective gait observables supply early
diagnoses and quantification of clinical course in patients with extra-pyramidal
disorders or taking anti-psychotic medication (Hausdorff et al, 1998); can these
transformations of time series on the EEG give us an early diagnostic approach to
Alzheimer’s disease (Jeong et al, 1998) or a new acute preventive pharmacological
approach to patients with psychomotor and partial seizures (Iasemidis et al, 1990).
To end where we began: We think that if neuroscientists “did their own”
nonlinear dynamical theory and analysis, shaped and tailored by intuitions growing
out of their own experimental work and thinking, abstract and philosophical
questions about what is determinism and what is random would retreat in favor of
new specific ideas and experiments about brain dynamical mechanisms and their
pathophysiology. From the studies reviewed here, it appears that a robust move in
this direction in the brain sciences is well underway.
250
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