Excellent
fractal images by FracPPCA Fractal Perspective on the Dynamic
Frontal Lobe: Hurst fMRI
The Tacit Infrastructure
of Scientific Ideas Underlying Concepts of Frontal Lobe Function
Hurst fMRI and Lessons
from Development: New Perspectives on Dynamic Brain Self-Organization
Unity Through Diversity: The Self-Organizing
Mind
Of principal interest are ubiquitous self-similar burst-within-burst
patterns, observed in membrane potential fluctuations, neuronal
firing patterns, episodes of fetal rapid-eye-movement sleep (REMS),
and traffic patterns both on expressways and over computer networks
such as the World-Wide-Web (WWW). Burst or clustering patterns
are, in fact, familiar to anyone who has driven in or observed
highway traffic from a passing airliner. Rush hour traffic slows
to a standstill due to the tendency of nearby automobiles to spontaneously
cluster, forming super clusters or jams, despite, and, as a result
of, the efforts of drivers to find some open road. From the perspective
of a passenger in an aircraft, individual vehicles appear to cluster
into larger groups, dynamically jumping from cluster to cluster
over a larger expanse of roadway. The pervasiveness of self-similar
burst within-burst patterns becomes quickly apparent when commuters
arrive at work and find slow access times on the internet. Although
the self-organized fractal burst patterns common to both situations
appear to be only epiphenomena, they place fundamental constraints
on traffic flow.
Biological systems, in sharp contrast, appear to thrive and grow
via self organized fractal burst patterns. These spontaneous,
impulsive, apparently random movements common to all embryonic
animals during development have long been enigmatic to researchers.
Recently, a spontaneous fetal behavior, nuchal atonia, associated
with REMS, the predominant in utero state, has been shown
to contain intrinsic recurrent fractal burst patterns. Developmentally,
bouts of spontaneous activity, such as limb movements or the occurrence
of Active (REM) Sleep or fetal breathing, exhibit self-similar
bursting analogous to traffic patterns, and appear associated
with the development of lungs, motor systems and the maturation
of the brain. Burst patterns appear to provide a means to synchronize
the linkage of movements in the fetus with other neural motor
events such as the ingestion of amniotic fluid, allowing long-
and short range coordination among different emerging behaviors.
In fact, as new evidence shows, developmental disorders resulting
from maternal separation or in utero exposure to drugs
of abuse, and autism appear to be associated with chronic alterations
in these normally stable bursting patterns.
Bursting patterns over a wide range of temporal periods can be
characterized by a small number of relatively simple statistical
measures, examples of which are the Hurst and Lévy exponents.
Hurst's rescaled range analysis provides information about correlations
among bursts over all times. On the other hand, the Lévy
exponent, alpha, is a measure of the probability distribution
of events, specifically the tail or the variance associated with
burst events. Concordance between both exponents has been demonstrated
for fetal behavior and WWW traffic.
Hurst analysis is also useful in functional brain imaging. It
is widely know that much of the variance in fMRI signals is due
to background fluctuations emanating from the subject and not
the scanner. Preliminary Hurst analysis of these task-related
fMRI fluctuations reveals that brain regions, such as the frontal
lobes, have self-similar patterns of fluctuations over multiple
time scales. Self similar fluctuations may also exist at faster
scanning times and higher spatial resolutions. Could this observation
of fractal fluctuations in cortical fMRI activity have any relevance
to current theories concerned with the neural implementation of
cognitive functions?
Recently, the variability of neural processes has become a point
of concern for cortical physiologists who want to precisely define
the characteristics of cortical populations or individual neurons
during cognitive tasks. Relationships between the variability
of individual neurons and populations of neurons have been found
to have much stronger correlations than previously thought, resulting
in the transmission of fluctuations of single units into the large
scale fluctuations of populations. Response fluctuations of single
neurons to repeated stimulation are mirrored in population fluctuations,
leading some researchers to conclude that the "noise"
in ongoing cortical activity is an integral aspect of cortical
function. Although the complete meaning of these findings are
not yet known, this burst-within-burst characteristic of brain
activity is reminiscent of the spontaneous bursting observed in
the behavior of fetal animals which enhances long and short range
neural-motor synchronization during development. Parsimoniously,
task-specific self-organization of functional relationships among
cortical regions in adults observed with Hurst imaging fMRI may
parallel the dynamics of fetal neural-motor development.
Extensive descriptions of self-similar patterns of behavior at
many levels of neural, behavioral and social organization point
to a new foundation upon which theoretical and experimental investigations
of brain function can be unified. Clearly, the key question for
the 21st century remains how cognitive neuroscience will, in the
words of William James "...steadily look after the irregular
phenomena (fractal fluctuations)" and incorporate these comprehensive
observations of the spontaneous self-organization of behavior
to "...renovate..." the sciences of mind. [Top]
Excellent
fractal images by FracPPCScience often progresses through the cross fertilization of ideas between very different fields of investigation. One example, the concept of "signal to noise ratio", originally developed and used in radar detection, is used widely in descriptions of activity in the nervous system. This concept is commonly used in cognitive neuroscience to describe, for example, how release of the neurotransmitter norepinephrine in frontal cortex reduces the spontaneous background neural "noise" and enhances detection of "signal" at the cellular level of a network (see for example Posner and Raichle, p. 224). Although this use seems appropriate for conceptualizing neural events and linking them to cognitive functions such as attention, it tacitly assumes that the spontaneous background "noise" is a random uncorrelated component of neural activity, much like the hiss of static between radio stations, which can be conveniently ignored in models of function or subtracted away from signal in experimental investigations.
Is this spontaneous background neural "noise" really
noise? The famous comparative neuroscientist Theodore Holmes Bullock
in his 1977 classic Introduction to Nervous Systems doesn't
completely neglect or deny a role for neural noise in brain function:
"There are certainly sources of meaningless fluctuation, including threshold fluctuations, the convergence of independent rhythms, spontaneous activity, and other events with good causes but no [apparent] signal value to the system. This is true noise, and it may be large or small in proportion to signals; highly structured, or quasirandom. Since it is defined not by its causes or structure, but its lack of meaning or value, it is difficult to identify in any given case. We rarely know the system well enough to be confident of what has no value. Heuristically it is better not to label unexplained activity or fluctuation as noise or uncertainty. Calling it something like "unexplained variation" may encourage the search for both causes and possible significance (p. 236)."
That this "unexplained variation" is not restricted to neurons, but could conceivably be an important element in the overall behavior of organisms is implicit in Bullock's words. Kenneth Roeder, a respected insect physiologist, makes this point passionately in his 1955 article "Spontaneous activity and behavior" describing his observations of a variety of insect behaviors strongly associated with spontaneous bursting patterns:
"Many years of teaching neurophysiology along Sherringtonian
lines, research on the insect nervous system in which spontaneous
nerve activity is practically universal, and the observation of
the unpredictability of most animal behavior have led me to seek
reconciliation of these apparently contradictory aspects of nerve
activity and behavior (p.362)."
Roeder went on to propose that spontaneous neural activity was
the major drive for the appetitive, or searching component of
insect behavioral patterns, as opposed to the prevailing view
of reflex driven behavior.
Recently, through the application of perspectives and methods
derived from and inspired by Benoit Mandelbrot's science of the
irregular, fractal geometry, the assumption that spontaneous background
"noise" in neural activity is orderless and uncorrelated
can no longer be categorically accepted as true. In passing, this
seems only a minor correction. However, this observation of unrevealed
structure in spontaneous background activity has profound implications
not only for the interpretation of many of the commonly used experimental
methods and statistical procedures in the sciences of the mind,
but also for the very way that the brain/mind is conceptualized.
[Top]
Just as sensory representations have been found to be more plastic than previously thought, associational areas such as the prefrontal cortex may have a more plastic organization when examined from the developmental perspective of vertically integrative spontaneous patterns.
One function of REM sleep may be to periodically present developmentally
invariant fractal patterns to the adult brain, facilitating the
consolidation of synaptic and large scale network changes.
These recurrent fractal patterns may be present in the phasic
activity of nightly REM sleep and become enhanced during sleep
pathology associated with PTSD as an intrensic brain mechanism
for healing.
Neural activity and communication occur over a broad spectrum of timescales.
Spontaneous fractal activity may underlie efficient searching strategies in animal behavior and human cognition.
Neural-organizational strategies based on spontaneous activity
during CNS development, may play a role in the organization of
dynamic functional connectivity in the CNS of adults.
The recurrent theme of this introduction, that order can exist
over a multitude of timescales in data that at first glance appears
to be only random fluctuations, cannot be fully appreciated in
the context of the current scientific paradigm that places greater
emphasis on a linear view of brain function in which independent
elements can be isolated and dealt with separately. When this
linear approach is used in complex interactive situations with
long-range correlations (e.g, the nervous system), the wider context
that lends these fluctuations their unity is obscured. Linear
approaches, to remain successful, must be cross-integrated within
the context of realistic, "practically universal" naturally
noisy or variable behavior.
The essential idea common to the wide range of examples given
is that the brain/mind lives as much at microseconds as at decades,
and only by expanding our point of view to encompass a whole range
of timescales in experimental or conceptual investigations of
neural behavioral processes, can we more accurately grasp its
functional organization. [Top]
Excellent
fractal images by FracPPCWhat is unique about the fractal geometry of nature? Why should neurobiology in general and cognitive neuroscience in particular take notice of a new geometric view of the natural world that places emphasis on invariant properties derived from measurements over many scales of time or space?
We are now living in an age of scientific and technological renaissance,
which, like the original Renaissance, is a time of accelerated
change and advances in many areas of human thought, particularly
in the sciences of the mind. Dr. Rhonda Shearer has proposed in
her 1996 book, The Flatland Hypothesis: Geometric Structures
of Artistic and Scientific Revolution, that the appearance
of new geometries, such as the development of geometric perspective
in art with the Renaissance, heralds the appearance of changes
in thought and social values, foreshadowing innovations in science
and art. Flatland, Edwin A. Abbott's 1884 book, referred
to in the title of Shearer's work, depicts a two-dimensional world
peopled by lines, squares and multi-sided polygons in a rigid
hierarchal society with no concept of higher dimensions. When
a three-dimensional sphere being appears as a circle in Flatland,
his very existance plants the perceptual seeds of a scientific
and cultural revolution. Shearer credits Abbott with the formal
insight later attributed to Thomas Kuhn, author of the Structures
of Scientific Revolutions, that scientific thinking is linked
to the individual's perception of conceptual change.
If a new geometric perspective inspires innovative approaches
to fundamental scientific questions, then the importance of adopting
and incorporating the fractal perspective should not be underestimated.
It is the contention of the author that without this well-rounded
viewpoint on the spontaneous aspects of brain and behavior at
all levels of description, the sciences of mind might become trapped
in a kind of conceptual and experimental "Flatland".
The concept is perhaps even more multi-faceted. The World-Wide-Web
(WWW) is possibly the best place to see and learn about fractal
concepts in general, and their applications in biology
and psychology or just to educate the eye to the endless
possibilities of fractal
forms.
The term fractal applies to objects in space or sequences of events
in time that possess a form of self-similarity: fragments of the
object or sequence can be made to fit to the whole object or sequence
by shifting and stretching. Fragments of fractal objects can be
exact or statistical copies of the whole. Mathematical fractal
objects visually convey the concept of self-similarity (see Fig
1).
However, only fragments of mathematical fractal objects can
be exact copies, and exactly represent the concept of self-similarity,
whereas the fragments of natural fractals are only statistically
related to the whole (see Fig 2).
Another way to think of fractals is in terms of clusters of points
or events in space or time. Self-similar clusters have smaller
clusters within larger clusters of clusters. Clouds, broccoli,
or the surface of the brain can all be visualized as clusters
of clusters in space. Phenomena with self-similar clusters in
time, such as electronic traffic on the internet, spontaneous
activity associated with REM sleep or neural spike trains appear
very similar when displayed side-by-side
(compare Fig 2 with Fig
1 and Fig
2).
The repetition of "self-likeness" in recordings of physical
or neurobiological phenomena is sometimes described in different
ways depending on the degrees of freedom present. In geometric
terms, if variation at different scales of measurement occurs
along one dimension, such as time, the phenomenon would be called
"self-similar". If fluctuations are present over two
dimensions, such as time and amplitude, they are termed "self-affine".
In this circumstance, the time series is statistically invariant
under a transformation that scales the time and amplitude dimensions
by different amounts. The nervous system is an especially rich
source of examples of fractal processes that can be described
as self-affine. In this essay I will use the terms "self-similar"
and "self-affine" interchangeably, for descriptive reasons,
although the reader should be aware that these terms are less
interchangeable in other contexts.
Self-similar clusters in time have unusual statistical ramifications
for data.
For example, the mean and/or the variance may grow with the sample
size, becoming for all practical purposes very large or infinite.
Mandelbrot, who first observed these strange statistical properties
in price changes, commented: "To anyone with the usual training
in statistics, an infinite variance seems at best scary and at
worst bizarre (p.338)." These curious qualities are the result
of large intervals or so-called "outliers" among the
clusters, which give plots of the distribution of interval times
long or heavy tails.
While not Normal or Gaussian, these distributions are related
through the stable Lévy
model, which represents a mathematical space of possible distributions
mapped by the convergence of mean and variance, sometimes referred
to as the "moments" of the distribution. For Gaussian
distributions, as the sample size is increased, the mean and variance
both converge. While in Lévy space, Gaussian distributions
are mapped to an exponent = 2, which is determined by the rate
of convergence of the tail of the distribution. Non Gaussian distributions
with a convergent mean, but non-convergent variance have exponents
in the range 1< < 2. Distributions such as the Cauchy lack
both a convergent mean and variance, and have 0< < 1.
A unique property of distributions that fit the Lévy model
is vertical convergence or "convolutional stability."
Intervals from separate stable distributions like the Gaussian
can be summed into larger stable distributions. Stable distributions
are therefore self-similar over different sample sizes.
Lévy exponents provide one way to categorize self-similar
clusters or bursts that have non-convergent moments. They have
also proven useful in understanding the possible origins and roles
of spontaneous fluctuations in developmental systems, as well
as providing a measure of their pathology.
In the following section, a "cross fertilization" between
the concept of vertical convergence from the Lévy model
and many observations of self-similar spontaneous fluctuations
at different levels of the nervous system provides a "conceptual
bridge" linking cellular and behavioral processes. [Top]
Excellent fractal images by
FracPPCAlthough the fractal model is still in dispute and awaits the development of new experimental techniques to support or refute its claims, a large body of experimental data and simulations support the model, as well as new findings that the conformational states of proteins exist over a wide range of time scales. A large number of equal energy conformational states provide flexibility, allowing local fluctuations in membrane fluidity, ion species, and influences from the cell's protein skeleton to self-organize the channel structure into open and closed states. Dr. Liebovitch's work promises to provide new insights into elements of the cell that are critical in neurotransmisson. [Top]
Many early observations of firing patterns throughout the visual
system suggested the presence of long-range correlations; however
many of these recordings were brief and the mean and variances
were calculated over short observation windows. In many cases,
under Gaussian assumptions, occurrences of high variability were
excluded. Teich describes how, in a recent report by Softky and
Koch, about firing patterns in visual cortex neurons, the investigators
state that data segments were selected for "constant firing
rate" to suppress nature bursting and variability, which
they considered noise.
Dr. Teich's work strongly supports the existence of spike train
patterns from numerous regions of the CNS which show fractal characteristics
over many orders of time just as neurotransmitter release, or
ion channel currents. If self similar patterns of activity enable
vertical convergence of fluctuations over diverse levels of neural
organization, how is this adaptive for organisms?[Top]
For example, a recent analysis of the search patterns of albatrosses,
recorded electronically while they foraged in the south Atlantic
over a period of three months, demonstrated spatial and temporal
fractal organization. The complex patterns observed seemed to
allow the birds to efficiently search a large region of open water
and were modeled successfully as Lévy walks. In addition,
many features of the earth's surface and environment are irregular
fractals in space or time. If vertical convergence of self-similar
spontaneous activity at cellular and neural levels is a normal
functional aspect of the brain which results in efficient searching
strategies, it would provide an adaptive advantage to organisms
in a world filled with fractal characteristics.
Roder was aware of the connection between spontaneous activity
in nerves and in behavior; however he lacked the concept of fractals
and the tools to measure and compare fractal patterns:
"The circumstances of spontaneity in nerve elements are strikingly
analogous to those of kinesis or appetitive [search] behavior.
However, it is very difficult to determine whether or not a causal
or homologous relationship exists between these two types of biological
activity (p. 368)."
I am sensitive to Roder's predicament, having also observed visually
the relationship between spontaneous patterns of twitching in
baby rats and behavioral state development and was unsure scientifically
how to explore the connections between these phenomena. In the
next section I will describe the methods I was able to apply in
order to investigate the relationship between spontaneous activity
and the structure of REM sleep in developing animals.
In our 1996 chapter, we also proposed a developmental hypothesis
concerning the fractal nature of rapid eye movement (REM) sleep
over the life span of a mammal, from fetal to adult life. In order
to investigate further this developmental fractal hypothesis of
REM sleep, I needed to study the occurrence of this state in fetal
and newborn animals. I chose to study nuchal atonia or the loss
of nuchal muscular tone in the primary anti-gravity muscles of
the neck which is a key indication of the onset of REM sleep in
many animal species. Nuchal atonia becomes apparent when anyone
tries to sleep in an airplane seat; as one goes into REM sleep,
nuchal atonia occurs and one's head drops, usually waking him
or her up. Nuchal atonia can be studied by recording the electromyographic
(EMG) activity of the neck muscle, which is much lower during
periods of REM sleep. I was able to obtain neck EMG recordings
from fetal sheep recorded over the last 13 days of in utero
life before birth. I also made recordings of neck EMG in baby
rats, while they slept in a warm incubator.
This data required a method to detect and measure self-similar
clustering patterns allowing comparisons between different data
sets. One method called Hurst analysis appeared simple to program,
robust for small sample sizes, and allowed comparisons across
data sets.
Hurst Rescaled Range Analysis, or simply Hurst analysis, was developed
by the British hydrologist Harold E. Hurst, during the engineering
of the high Aswan dam in the 1950's, to determine from historical
records if the yearly flows of the Nile were random or clustered
from year to year. Hurst reasoned if high and low Nile flows were
random over successive years (i.e., did not cluster in sequences
of high or low years), then the reservoir size estimate could
be based on the average of the recorded flows. On the other hand,
if years of large Nile flows were not independent, but clustered
or demonstrated serial dependency over successive years, then
the "memory", or carry over between a series of wet
years, could create a situation where reservoir capacity would
have to be larger than the estimate based on the mean. Hurst examined
800 years of Nile flows recorded at the Roda gauge and determined
they were not random but tended to cluster in runs of high or
low years, supporting the need for a larger reservoir. He also
examined 837 records of other natural phenomena (e.g., annual
river levels, rainfall, temperature and pressure records, tree
rings, varves, sunspot activity) finding non-random positive correlations
in most of them.
Hurst's method provides information about correlations among each
burst or cluster event in a time series (see Fig 3). For collections
of events of different sizes, the difference between each event
and the average of events is obtained and successively added to
a cumulative sum. For example, for a total of 256 events, cumulative
sums would be generated for subsets of size 4, 8, 16, 32, 64,
128, and 256. A normalized range is then obtained by taking the
difference between the maximum and minimum values that the cumulative
sum attains (for each subset), and dividing by the standard deviation
of events for each subset. The average normalized range for each
subset is then plotted against the size of the given subset on
a log-log axis, and the slope of the relationship is calculated,
yielding the Hurst exponent. When this exponent, called simply
"H", is > 0.5, positive correlations exist among
all events and self-similar burst within burst patterns abound.
The "memory" in the clustering over time is termed "persistence",
as in the reoccurrence of high river levels seen in the Mississippi
river over the last decade. However, H = 0.5 indicates randomness
and a lack of correlation between burst events, and no long correlations
are seen. Many of the natural phenomena that Hurst examined had
values of H = 0.75. My own investigations of spontaneous activity
in fetal and neonatal animals found H ranged from 0.70 to 0.80.
[Top]
Hurst analysis has, over the last five years, been applied more frequently in engineering and neuroscience. Hurst analysis also seems useful in fMRI analysis, to characterize the spatial distribution of self-similar fluctuation patterns in the brain. Applications of Hurst analysis to internet traffic have been particularly insightful, shedding light on how burst-within-burst patterns arise in complex networks, which may supply ideas for visualizing developmental and cognitive processes self-organized by spontaneous bursting.
Would it be possible to use highways and freeways more efficiently
by keeping traffic flow in a regime of maximum flow for example?
As it turns out the characteristics of this regime of traffic
flow are complex and poorly
understood. In fact, work using simplified models of traffic
flow indicates that, in situations where the highway is filled
with cars, usually on holidays, fractal clusters of traffic jams
of all sizes are more likely to occur. When one driver breaks
too soon or too late, an avalanche of stops and starts is propagated
for great distances over the highway, resulting in long-tailed
distributions of and long-range correlations in inter-car-intervals.
Traffic tends in general to self organize into a critical state,
where small fluctuations can lead to traffic jams of all sizes.
Steps taken to reduce jams, for example, mandating automatic car
following systems as in simulations, push traffic closer to the
critical point where jams are more likely to occur. Traffic jams
could be considered a one dimensional model of vertical integration,
in which all cars are effectively locked in one huge spontaneous
fluctuation. Traffic jams in developing brains may lead to the
enhancement of synaptic connections, sparing of axons, and synchronizing
twitches that allow distant regions of the unconnected organism
to link and coordinate gene expression and motor development.
Traffic jams on the internet are more complex, because of their
multdimensional nature (literally a many-dimensional spider web)
and rich connectedness, and therefore are more relevant to developing
organisms and the brain. Information travels over the internet
in the form of packets with a wide range of sizes. As a result
the occurrence of bursts is more pronounced, because this form
of traffic spans vastly different timescales, from microseconds
to seconds and minutes. These bursting patterns have statistical
self-similarity and are in some cases very similar to bursting
patters from biological systems (see Fig.2). The fractal bursting
patterns originate in the complex interactions between actions
of file transmission, caching systems, user choice, file size
distributions and user "think times". Recent
fractal models of internet traffic have been successful in
modeling many of its characteristics, including self similarity,
long-range correlations, and clustering that results in long-tails.
In these models, the superposition or cumulative counting of many
packet trains with different short- and long-time characteristics
generates self-similar traffic traces, with Lévy stable
characteristics. Recent findings by the cognitive psychologist
D.Gilden assert that fractal time variation is a basic element
of human judgment and decision making, suggesting that the user,
and his or her fractal processes may supply some of the fractal
bursting present in the WWW.
Again applying these described conceptual models to developmental
processes, we can see spontaneous activity at the cellular level
as an avalanche within an ion channel causing it to shift into
an open state, this in turn tips the fluctuating membrane potential
into an avalanche of depolarization which then may trigger an
avalanche of acetylcholine packet release that triggers a muscle
contraction experienced as a spontaneous twitch. Like traffic
on the WWW, this cascade of avalanches spans vastly different
timescales, from microseconds to seconds and minutes.
Another aspect of this conceptual model involves the idea of horizontal
integration. Developing tissues and networks must reach a critical
level of connectivity in order to propagate fluctuations to the
next level of developmental organization. An example of this may
be the onset of spatial learning in young rats, which is individually
variable, appearing when hippocampal pathways have reached a critical
density. Another example that illustrates this important concept
is the development of nuchal atonia, the marker of REMS.
Developmentally, during the early third trimester in fetal sheep,
eye movements, nuchal muscle activity, breathing movements, and
desynchronized brain electrical activity (EEG) are almost continuous.
This may represent vertical integration of cellular spontaneous
activity without much horizontal integration within the brain
or among motor systems of the animal. Nuchal atonia first appears
with differentiation of EEG into synchronized high voltage slow
waves and an increased mean amplitude of desynchronized EEG beginning
near the midpoint of the third trimester. At this time continuous
or "tonic" nuchal activity starts to break up into periods
of atonia, signifying the emergence of horizontal integration
and neural motor connections. This reorganization of continuous
nuchal activity into alternating periods of atonia and activity
appears analogous in a general way to the emergence of spontaneous
electrophysiological responses in early chick muscle or coordinated
sequences of muscle contractions seen during later spontaneous
motility in the maturing embryo. In addition, from about midpoint
of the third trimester on, spontaneous changes in fetal movements
(e.g., rapid irregular fetal breathing, mouth and tongue, diaphragmatic,
and isolated body twitches and generalized bursts of limb movements,
changes in tracheal and arterial pressure) become correlated with
most of the criteria of REMS in adult sheep. The modeling
of systems with bursting patterns over many time scales, such
as WWW traffic patterns, may provide general models for neurodevelop
mental processes as well as be helpful in understanding normal
ontogeny and developmental disorders such as autism, or the presence
of REMS disturbances in posttraumatic stress disorder (PTSD).
[Top]
Excellent fractal images by
FracPPCAccording to some estimates, the mammalian fetus spends as much as 50 to 70% of in utero life in the state of active sleep also known as REM or paradoxical sleep. Many have hypothesized that this state is fundamental in brain development and that it provides an internal source of stimulation during the self organization of the developing brain. Although the mechanisms by which REM sleep may accomplish this role are for the most part unknown, I have proposed that the correlated bursting nature of REM, or Active sleep as it is sometimes called in the fetus and newborn, provides an invariant stable Lévy temporal framework in which cortical and subcortical networks can organize and consolidate changes. In support of this contention, we have found that the structure of REM sleep is highly correlated and has recurrent fractal structure during the last trimester in fetal sheep, as indicated by Hurst analysis. Also, analysis of the Lévy stable characteristics of nuchal atonia sequences demonstrate distributionally invariant activity. This finding is striking in that it demonstrates that REMS processes are not random, as Allan Hobson has proposed, and supports the findings of Edward Evarts who observed almost 30 years ago that spontaneous discharges of motor cortex neurons during REM sleep in adult monkeys were similar to stable Lévy distributions due to an excess of short and long interspike intervals. Stable Lévy processes my have great significance in understanding the relationship between REMS and neural plasticity.
These findings suggest that early stress results in disruptions
of vertical integration of fractal patterns, leading to fewer
numbers of spontaneous episodes, apparently the reverse of EM
changes in autism. What are the implications of these findings
for child abuse, drug abuse and PTSD? Recent findings by many
researchers, including the author, support the contention that
stress or abuse in early life induces functional hemispheric asymmetries
and disrupts the structure of REM sleep resulting in PTSD, predisposing
one to addictive and self-defeating behaviors resulting from the
lifelong effects of impaired interhemispheric integration. The
application of fractal concepts and developmental perspectives
will increase our understanding of the specific cognitive and
functional nature of these disorders. In the following sections,
I will consider the functional organization of the frontal lobes
in light of these same perspectives.[Top]
Einstein
Hurst fMRI images, at first glance, look like PET images. However,
PET images are the result of the collection and averaging of many
positron emissions, and do not contain information about fractal
time patterns during the collection. Hurst images portray the
self-similar time structure of spontaneous Blood Oxygenation Level
Dependent (BOLD) fluctuations which may contain a wealth of information
relevant to functional connectivity among multiple regions of
cortical activity. Although the interpretation of H in this context
is not yet clear, common fractal characteristics may reflect functional
relationships. Whether the observed clusters of persistent H values
originate primarily from neuronal or other physiological processes
(e.g. respiration, pulsatile flow, vasomotor oscillations) or
a combination remains to be investigated.
H clusters of persistent self-similar fluctuations over the frontal
regions are consistent over slices (see fig 4).
Figure 4. Top: Pixel-wise Hurst exponents in two adjacent axial planes through the frontal lobes at the level of the caudate; Middle: Hurst images displayed with sd=1.5 Gaussian Smoothing to enhance H cluster morphology; Left : Superior slice; Right : Inferior slice; Bottom: T1 slices approximating the plane of section.
However, H clusters seem to change in size in the same subject over a period of weeks (data not shown). Because we are highly confident that our motion correction algorthim has removed almost any frame to-frame movement, we believe these images could be displaying the cluster patterns of persistent large-scale cortical networks activated by the contingencies of the background task requiring the inhibition of movement (recently proposed by Bressler). We also believe that these images, becaused they have no inherent anatomical bias, can hint at the true spatial-temporally distributed nature of functional connectivity.
Hurst fMRI imaging is very different from traditional imaging
approaches which attempt to subtract background from task, and
then average within a subject and across subjects to find "invariant
networks of task-specific activation." In contrast, Hurst
fMRI can discern self-similar patterns in time within the same
individual, and determine how they change under different task,
drug or developmental conditions. The "background" is
the signal. The correlation of other physiological fluctuations
(e.g., respiration, pulsatile flow, vasomotor oscillations) with
Hurst fMRI time series is currently under investigation.
Fractals teach us that spatial-temporal patterns cannot be fully
described within a limited range of measurements, and therefore
it should not be expected that brain "circuits" underlying
cognitive concepts such as working memory are functionally organized
in absolute spatially localized regions. In the next section I
will comment on the apparent influence of "absolute"
anatomical concepts in recent views of frontal lobe function,
followed by a return to a discussion of the Hurst fMRI perspective
on self-organized brain function.[Top]
"Thus, at one level the dorsolateral and ventrolateral
frontal cortical regions may be broadly [spatial and non-spatial]
and each dedicated to a particular type of class of cognitive
process. Within these gross anatomical regions, however, [spatial
or non-spatial] fields may exist, each dedicated to a particular
informational domain (p.1338)."
This statement at first seems to support the idea of a kind of
fractal nesting of function in the frontal lobes. However, in
fact, it highlights the bias inherent in mapping function to spatial
absolutes. In some respects this controversy appears to be linked
to the mere existence of two anatomically distinctive regions.
If three distinctive regions were present would this be reflected
in tripartite models? Is theory canalizing the perception of experimentors
towards absolutes of anatomy and blinding us to the true dynamics
of function?
In the author's opinion, without the new perspectives afforded
by the application of fractal concepts in neurobiology, the sciences
of mind are currently at the same stage of development that the
science of physics was at the end of the 19th century. The central
conceptual problem then, as now, involved unexamined concepts
of absolute space and time. As David Bohm, the world renowned
physicist, describes in his last and most insightful book with
F. David Peat about the psychology of science "Science, Order,
and Creativity":
"Even a physicist as original as H. Lorentz at the turn of the century continued to use these concepts in an effort to explain the constancy of the velocity of light, irrespective of the speed of the measuring apparatus. Newtonian notions of relative velocity suggested that the measurement of the speed of light should yield an experimental result that depended upon the speed of the observing apparatus relative to the light source. For example, if the apparatus moves rapidly toward the source of light, it would expect to register a higher speed than if it moved away. However, no such effect was observed during very careful measurements. Lorentz, in an effort to retain the Newtonian concepts, proposed an ether theory, in which the anomalous results on the measurement of light were explained by actual changes in the measuring apparatus as it moved through the ether. Lorentz was therefore able to explain the constancy of the velocity of light, independent of the relative speed of the observer, as an artifact produced by the measuring instruments themselves, and there was no need to question the fundamental nature of Newtonian ideas (p. 21)."
Much of current neurobiology and cognitive neuroscience is based on theory driven experiments, and therefore lives and dies by what Bohm called "the tacit infrastructure of scientific ideas." Bohm states that these ideas are essential to scientific progress but can become rigidified and unquestioned:
"...[A] scientist possesses a [tacit infrastructure of
knowledge and skills] which are at his or her "fingertips."
These make day-to-day research possible, allowing concentration
on the main point of issue without the constant need to think
about the details of what is being done. Most scientists, for
example, carry out their research by using experimental techniques
or applying established theories that were first picked up in
graduate school. In this way a physicist may spend a decade investigating,
for example, the internal structure of metals without ever needing
to question this tacit knowledge in a basic way (p.20)."
Because of the dynamic nature of knowledge and the rapid pace
of discovery in the sciences of the mind, this fact of intellectual
life is extremely germane to ideas about task-related or spontaneous
spatial-temporal patterns of brain activity, particularly in the
functional analysis of the frontal lobes. In light of the large
variability associated with spontaneous self-organized spatial
temporal cortical patterns, observed with Hurst fMRI, would an
experimental program dedicated to delineating "modality specific
fields" within "processing specific fields" be
useful? It would greatly benefit the sciences of mind to reflect
on the subtle conceptual mind sets created by the use of terms
such as "circuits", "brain regions", "cortical
areas", and "networks", because of their tendency
to promote persistence in preestablished entrinched, perhaps unexamined
concepts of space and time in the brain/mind.
Could the developmental concepts of vertical and horizontal integration
of spontanous fluctuations across levels and time scales provide
insights into dynamic spatio-temporal cortical patterns? In the
abstract I also proposed that task-specific self-organization
of functional relationships among cortical regions in adults observed
with Hurst imaging fMRI may parallel the dynamics of fetal neural-motor
development. During development, spontaneous perinatal behaviors
appear to correlate many neural motor systems of the body. This
has been elegantly demonstrated by Scott Robinson at the University
of Iowa, who has demonstrated that spontaneous movements enable
coordination of limb movements in rat fetuses. If we view the
adult brain as a complex developmental system, much more richly
interconnected than the WWW, constantly updating, losing neurons,
strengthening synaptic connections under local and global changes
in neurotransmitters and hormones, and undergoing behavioral state
changes, how might spontaneous activity create new order?
Hurst fMRI may provide a unique perspective on the self-organization of functional connectivity associated with the development of cognition. Are H clusters developmentally stable or are they more dynamic in childhood, becoming more stable with learning and maturity? How would drugs that enhance attentional function such as Ritalin alter H clusters in children or adults? We have very preliminary observations that Ritalin in adults alters H cluster morphology, making it more diffuse and less persistent. The study of the spontaneous self-organization of frontal lobe function during childhood and later life might provide new insights into the dynamics of spatio-temporal cortical patterns, and provide a interesting comparison with traditional fMRI and PET images of the same individual under similar task conditions.[Top]
Excellent fractal images by
FracPPC"The emergent paradigm of self-organization permits the elaboration of a vision based on the interconnectedness of natual dynamics at all levels of evolving micro- and macrosystems. From such an interconnectedness of the human world with overall evolution springs a new sense of meaning."
Like an integrated, highly functional brain/mind, we must self-organize to consider multiple levels of space and time and to balance reductive experimental approaches with a global overview of the task at hand, in order to bring meaning to our explorations. Our world society appears to be experiencing a core social fragmentation, accelerated by the soulless march of materalism, marked by escalating drug addiction and suicidal behavior among our youth. Science reflects this fragmentation, despite attempts by individuals like Erich Jantsh and others to create a bridge between the fragments.
As scientists, we have an obligation to aid our society and
attempt to provide a meaningful and coherent view of our world
and our brain/minds to our fellow man.
In this introduction I have attempted to present in a diversity
of levels of neural function and organization a unified theoretical
and experimental approach to the brain/mind in hopes of creating
a coherent natural integration of fractal views of nature with
evolutionary, developmental and cognitive approaches. The larger
perspective offered by fractal concepts to the sciences of the
mind suggest that it is worth considering, and would, as James
said, "renovate [our] science....and when [cognitve neuroscience]
is renewed, its new formulas often have more of the voice of the
[fractal] exception in them than of what were supposed to be the
[absolute] rules." [Top]
Carl M. Anderson, Ph.D.,
Consolidated Department of Psychiatry,
Harvard Medical School,
Brain Imaging Center,
McLean Hospital,
Belmont, MA 02178 USA
carl_anderson@hms.harvard.edu