From Complexity to Creativity -- Copyright Plenum Press, 1997

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Part I. The Complex Mind/Brain

CHAPTER 3. A MODEL OF CORTICAL DYNAMICS


CHAPTER THREE

A THEORY OF CORTICAL DYNAMICS

3.1 INTRODUCTION

    The psynet model, as outlined above, is an abstract model of the dynamics of mind. It is not a model of the dynamics of the brain. This is an important distinction, both methodologically and conceptually. Mind is not brain; mind is, rather, a collection of patterns emergent in the brain. Mind is a mathematical entity; a collection of relations, not an actual physical entity.

    It is clear that, in modern mind theory, psychology and neuroscience must proceed together. But even so, given the tremendous rate of change of ideas in neuroscience, it seems foolish to allow one's psychological models to be dictated too precisely by the latest discoveries about the brain. Rather, one must be guided by the intrinsic and elegant structure of thought itself, and allow the discoveries of neuroscience to guide one's particular ideas within this context.

    An outstanding example of this point is neural network modelling. Neural network models take an idea from neuroscience -- the network of neurons exchanging charge through synapses -- and elevate it to the status of a governing psychological principle. Many notable successes have been obtained in this way, both psychologically and from an engineering point of view. However, the problem of connecting the states and dynamics of neural networks with mental states and dynamics has never really been solved. Neural networks remain a loose, formal model of the brain, with an uncertain, intuitive connection to the mind itself.

    The advantage of neural network models is that they allow one to import the vocabulary of dynamical systems theory into the study of the brain. One can talk about attractors of various kinds, attractor basins, and so forth, in a rigorous and detailed way. The idea of thoughts, memories and percepts as attractors is given a concrete form. However, in a sense, the representation is too concrete. One is forced to understand deep, fuzzy, nebulous patterns of mind in terms of huge vectors and matrices of neural activations and synaptic weights.

    In this chapter I will present an alternative approach to brain modelling, based on the psynet model. Instead of looking at the brain and abstracting a model from brain structure, I will look at the brain through the lense of the psynet model and ask: What structures and dynamics in the brain seem to most naturally give rise to the structures posited in the psynet model? The resulting theory is quite different from standard neural network models in its focus on overall structure. And it is different from standard, non-connectionist AI in its focus on self-organization and emergence. Rather than specifying the update rules of the neurons and synapses, one is specifying the overall emergent structure of the autopoietic system.

    In particular, what I will present here is a theory of cortical structure. The brain is a highly complex system -- it is complex on many different levels, from overall architecture to neuronal wiring to biochemical dynamics. However, it is possible to single out certain aspects of neural complexity as being more important than others. What differentiates the human brain from the primate brain is, above all, our greatly enlarged neocortex. Elucidation of the workings of the cortex would thus seem to be a particularly important task.

    Thanks to recent advances in neurobiology, we now know a great deal about the structure and function of the cortex. What we do not know, however, is what cortex computes. Or, to put it in different terminology, we do not know the dynamics of the cortex. On a very crude level, it is clear that the cortex is largely responsible for such things as high-level pattern recognition, abstract thought, and creative inspiration. But how this particular configuration of brain cells accomplishes these fantastic things -- this is the sixty-four million dollar question. This is the question that I will attempt to answer here.

3.2 NEURONS AND NEURAL ASSEMBLIES

    In this section I will present a few basic facts about the structure and function of the brain. The goal is to give the minimum background information required to understand the model of cortex to be presented. For an adequate review of neuroanatomy and neurochemistry, the reader is referred elsewhere. An excellent, if dated, overview of the brain may be found in (Braitenberg, 1978); a more modern and thorough treatment may be found in any number of textbooks. Finally, the comprehensive reference on cognitive neuroscience is (Gazzaniga, 1995).

    First, the neuron is generally considered the basic unit of brain structure. Neurons are fibrous cells, but unlike the fibrous cells in other body tissues such as muscles or tendons, they have a tendency to ramify and branch out. The outer cell membrane of a neuron is shaped into extensive branches called dendrites, which receive electrical input from other neurons; and into a structure called an axon which, along its main stalk and its collateral ramifications, sends electrical output to other neurons. The gap between the dendrite of one neuron and the axon of another is called the synapse: signals are carried across synapses by a variety of chemicals called neurotransmitters. There is also a certain amount of diffusion of charge through the cellular matrix. The dynamics of the individual neuron are quite complex, but may be approximated by a mathematical "threshold law," whereby the neuron sums up its inputs and then gives an output which rapidly increases from minimal to maximal as its total input exceeds a certain threshold level.

    By passing signals from one neuron to another in complex circuits, the brain creates and stores information. Unlike nerve cells in the skin or the retina, which transmit information about the external world, neurons in the brain mostly trade information around among themselves. One may thus say with somejustification that the brain is a sense organ which senses itself.

    The neuron is in itself a complex dynamical system. However, many theorists have found it useful to take a simplified view of the neuron, to think of it as an odd sort of electrical switch, which takes charge in through certain "input wires" and puts charge out through certain "output wires." These wires are the biologist's synapses. Some of the wires give positive charge -- these are "excitatory" connections. Some turn positive charge into negative charge -- these are "inhibitory." Each wire has a certain conductance, which regulates the percentage of charge that gets through. But, as indicated above, the trick is that, until enough charge has built up in the neuron, it doesn't fire at all. When the magic "threshold" value of charge is reached, all of a sudden it shoots its load.

    This "electrical switch" view of the neuron is the basis of most computer models of the brain. One takes a bunch of these neuron-like switches and connects them up to each other, thus obtaining a vaguely brain-like system which displays remarkable learning and memory properties. But it is important to remember what is being left out in such models. First of all, in the brain, the passage of charge from one neuron to the other is mediated by chemicals called neurotransmitters. Which neurotransmitters a given neuron sends out or receives can make a big difference. Secondly, in the brain, there are many different types of neurons with different properties, and the arrangement of these types of neurons in particular large-scale patterns is of great importance. Each part of the brain has different concentrations of the different neurotransmitters, and a different characteristic structure. Here we will be concerned in particular with the cortex, which poses a serious problem for the brain theorist, as it has much less of an obvious architecture than such areas as the hindbrain or the cerebellum.

Cell Assemblies

    In the late 1940's, in his book The Organization of Behavior, Donald Hebb (Hebb, 1949) proposed a neuronal theory of high-level brain function. He hypothesized that learning takes place by the adaptive adjustment of the conductances of the connections between neurons. And he argued that thoughts, ideas and feelings arose in the brain as neural assemblies -- groups of neurons that mutually stimulate each other, and in this way maintain a collective dynamical behavior. While crude and clearly in need of biochemical, neuroanatomical and mathematical elaboration, Hebb's conceptual framework is still the best guide we have to understanding the emergence of mind from brain. It has inspired many current theorists, most notably Edelman (1987) and Palm (1982), and it underlies the ideas to be presented here.

    In dynamical systems terms, we may recast Hebb's model by saying that mental entities are activation patterns of neural networks, which may arise in two ways: either as relatively ephemeral patterns of charge passing through networks, or as persistent attractors of subnetworks of the brain's neural network. The process of learning is then a process of adaptive modification of neuronal connections, so as to form networks withdesired attractors and transient activation patterns. The transient case corresponds to formal neural network models of perceptual and motor function, principally feedforward network models (Rumelhart et al, 1986). In these models the goal is to modify synaptic conductances so that the network will compute a desired function. The attractor case, on the other hand, corresponds to formal neural network models of associative memory (see Serra and Zanarini, 1991). In these models, the goal is to modify synaptic conductances so as to endow a network with a given array of attractors. The network may then remain constantly in a certain attractor state, or, alternately, it may possess a variety of different attractor states. A given attractor state may be elicited into by placing the network in another state which is in the basin of the desired state.

    This modernized Hebbian view of neural network learning is the basis of many formal neural network models, and in this sense it is known to be mathematically plausible. Biologically, it cannot be regarded as proven, but the evidence is nevertheless fairly convincing. On the one hand, researchers are beginning to document the existence of complex periodic and chaotic attractor states in the cortex -- the best example is Freeman's (1992) work on the olfactory cortex. And, on the other hand, the search for biochemical mechanisms of synaptic modification has turned up two main candidates: the number of vesicles on the presynaptic side of the synapse and the thickness of the spine on the postsynaptic side of those synapses involving dendritic spines (Braitenberg and Schuz, 1994 and references therein).

    The neural assembly model is quite general. What it does not give us is a clear picture of how simple assemblies build up to form more complex assemblies, and how the dynamics of simple assemblies relate to the dynamics of the more complex assemblies of which they are parts. To get at issues such as this, one needs to look at the architecture of particular regions of the brain, in this case the cortex.

3.3 THE STRUCTURE OF THE CORTEX

    The cortex is a thin, membrane-like tissue, about two millimeters thick. It is folded into the brain in a very complex way, and is generally understood to be structured in two orthogonal directions. First it has a laminar structure, a structure of layers upon layers upon layers. The consensus is that there are six fairly distinct layers, although in some areas these six may blend with each other, and in others some of these six may subdivide into distinct sublayers. Then, perpendicular to these six layers, there are large neurons called pyramidal neurons, which connect one layer with another. These pyramidal neurons are surrounded by smaller neurons, most notably the interneurons, and form the basis for cortical columns, which extend across layers.

    Pyramidal cells comprise about 85% of the neurons in the cortex, and tend to feed into each other with excitatory connections; there is good reason to consider the network of pyramidal cells as the "skeleton" of cortical organization. Pyramidal cells are distinguished by the possession of two setsof dendrites: basal dendrites close to the main body of the cell, and apical dendrites distant from the main cell body, connected by a narrow shaft-like membrane formation. Pyramidal cells in the cortex transmit signals mainly from the top down. They may receive input from thousands of other neurons -- less than the 10,000 inputs of a Purkinje cell, but far more than the few hundred inputs of the smallest neurons. Pyramidal neurons can transmit signals over centimeters, spanning different layers of the cortex. Lateral connections between pyramidal cells can also occur, with a maximum range of 2-3 millimeters, either directly through the collateral branches of the axons, or indirectly through small intervening interneurons. In many cases, there is a pattern of "on-center, off-surround," in which pyramidal neurons stimulate their near neighbors, but inhibit their medium-distance neighbors.

    The columnar structure imposed by pyramidal neurons is particularly vivid in the visual cortex, where it is well-established that all cells lying on a line perpendicular to the cortical layers will respond in a similar way. A column of, say, 100 microns in width might correspond to line segments of a certain orientation in the visual field. In general, it is clear that neurons of the same functional class, in the same cortical layer, and separated by several hundred microns or less, share almost the same potential synaptic inputs. The inputs become more and more similar as the cell bodies get closer together. What this suggests is that the brain uses redundancy to overcome inaccuracy. Each neuron is unreliable, but the average over 100 or 1000 neurons may yet be reliable. For instance, in the case of motion detection neurons, each individual neuron may display an error of up to 80% or 90% in estimating the direction of motion; yet the population average may be exquisitely accurate.

    The visual cortex highlights a deep mystery of cortical function which has attracted a great deal of attention in the last few years, the so-called binding problem. The visual cortex contains a collection of two-dimensional maps of the scene in front of the eyes. Locations within these maps indicate the presence of certain features -- e.g. the presence at a certain location of a line at a certain orientation, or a certain color. The question is, how does the brain know which features correspond to the same object? The different features corresponding to, say, a cat in the visual field may be stored all over the cortex, and will generally be all mixed up with features of other objects in the visual field. How does the cortex "know" which features go with the cat? This is related to the problem of consciousness, in that one of the main functions of consciousness is thought to be the binding of disparate features into coherent perceived objects. The current speculation is that binding is a result of temporal synchronization -- that neurons corresponding to features of the same object will tend to fire at the same time (Singer, 1994). But this has not been conclusively proven; it is the subject of intense current research.

    A recent study by Braitenberg, Shuz and others at the Max Planck Institute for Biological Cybernetics sheds much light upon the statistics of cortical structure (Braitenberg and Shuz, 1994). They have done a detailed quantitative study of theneurons and synapses in the mouse cortex, with deeply intriguing results. They find that, in the system of pyramidal cell to pyramidal cell connections, the influence of any single neuron on any other one is very weak. Very few pairs of pyramidal cells are connected by more than one synapse. Instead, each pyramidal cell reaches out to nearly as many other pyramidal cells as it has synapses -- a number which they estimate at 4000. Furthermore, the cells to which a given pyramidal cell reaches can be spread over quite a large distance. The conclusion is that no neuron is more than a few synapses away from any other neuron in the cortex. The cortex "mixes up" information in a most remarkable way.

    Braitenberg and Shuz give a clever and convincing explanation for the emergence of columnar structure from this sprawling pyramidal network; they show how patches of lumped inhibitory interneurons, spaced throughout the cortex, could cause the pyramidal neurons inbetween them to behave as columnar feature receptors, in spite of having connections extending at least two or three columns out in any direction. This is fascinating, as it shows how the columnar structure fits in naturally with excitatory/inhibitory neuron dynamics.

    Finally, the manner in which the cortex deals with sensory input and motor output must be noted. Unlike the multilayered feedforward neural networks often studied in cognitive science, which take their inputs from the bottom layer and give their outputs from the top layer, the cortex takes both its input and its outputs from the bottom layers. The top layers help to process the input, but if time is short, their input may be overlooked and processing may proceed on the basis of lower-level neural assemblies.

3.4 A THEORY OF CORTICAL DYNAMICS

    Given the highly selective account of neural dynamics and cortical structure which I have presented here, the broad outlines of the relation between the psynet model and the cortex become almost obvious. However, there are still many details to be worked out. In particular, the emergence of abstract symbolic activity from underlying neural dynamics is a question to which I will devote special attention.

    Before going into details, it may be useful to cite the eight principles of brain function formulated by Michael Posner and his colleagues (Posner and Raichle, 1994), on the basis of their extensive work with PET brain scanning technology. Every one of these principles fits in neatly with the psynet view of brain/mind:

    Elementary mental operations are located in discrete

        neural areas...

    Cognitive tasks are performed by a network of widely

        distributed neural systems...

    Computations in a network interact by means of "re-

        entrant" processes...

    Hierarchical control is a property of network operation...

    Activating a computation from sensory input (bottom-

        up) and from attention (top-down) involves many of the same neurons...

    Activation of a computation produces a temporary

        reduction in the threshold for its reactivation...

    When a computation is repreated its reduced threshold

        is accompanied by reduced effort and less attention...

    Practice in the performance of any computation will

        decrease the neural networks necessary to perform it...

Posner's principles emphasize pattern recognition, hierarchical structure, distributed processing and self-organization (re-entrant processes) -- qualities which the psynet model ties up in a neat and synergetic bundle. What they do not give, however -- what does not come out of brain imaging studies at all, at the current level of technology -- is an explanation of how these processes and structures emerge from underlying neurodynamics. In order to probe this issue, one must delve deeper, and try to match up particular properties of the cortex with particular aspects of mental function.

    There are many different ways to map the psynet model onto the structure of the cortex. The course taken here is to look at the most straightforward and natural correspondence, which can be summarized in four principles:

    Proposed Psynet-Cortex Correspondence

    1. Neural assemblies may be viewed as "magicians" which

        transform each other

    3. What assemblies of cortical pyramidal neurons do is to

        recognize patterns in their inputs

    3. The multiple layers of the cortex correspond to the

        hierarchical network of the dual network

    4. The pyramidal cells based in each level of the cortex

        are organized into attractors that take the form of two-dimensional, heterarchical networks, in which cells represent emergent patterns among neighboring cells

    The first principle is essentially a reinterpretation of the cell assembly theory. If one accepts that cell assemblies have persistent attractors, and if one accepts that synapses are modified by patterns of use, then it follows that cell assemblies can, by interacting with each other, modify each other's synapses. Thus cell assemblies transform each other.

    The second principle, that neural processes recognize patterns, is also more programmatic than empirical, since virtually any process can, with a stretch of the imagination, be interpreted as recognizing a pattern. The real question is whether it is in any way useful to look at neural processes as pattern-recognizers.

    The pattern-recognition view is clearly useful in the visual cortex -- feature detectors are naturally understood as patternrecognizers. And I believe that it is also useful in a more general context. Perhaps the best way to make this point is to cite the last three of Posner's principles, given above. These state, in essence, that what the brain does is to recognize patterns. The two principles before the last state that components of the brain are more receptive to stimuli similar to those they have received in the recent past -- a fact which fact can be observed in PET scans as reduced blood flow and reduced activation of attention systems in the presence of habituated stimuli. And the final principle, in particular, provides a satisfying connection between neuroscience and algorithmic information theory. For what it says is that, once the brain has recognized something as a repeated pattern, it will use less energy to do that thing. Thus, where the brain is concerned, energy becomes approximately proportional to subjective complexity. Roughly speaking, one may gauge the neural complexity of a behavior by the amount of energy that the brain requires to do it.

    Turning to the third principle of the proposed psynet-cortex correspondence (that the multiple layers of the cortex are layers of more and more abstract patterns, ascending upwards), one may once again say that this is the story told by the visual cortex, in which higher levels correspond to more and more abstract features of a scene, composed hierarchically from the simpler features recognized on lower levels. Similar stories emerge for the olfactory and auditory regions of the cortex, and for the motor cortex. Numerous connections have been identified between perceptual and motor regions, on both lower and higher levels in the hierarchy (Churchland et al, 1995), thus bolstering the view that the lower levels of the cortex form a unified "perceptual-motor hierarchy." It would be an exaggeration to say that the layers of cortex have been conclusively proved to function as a processing hierarchy. However, there are many pieces of evidence in favor of this view, and, so far as I know, none contradicting it.

    The final principle, the correspondence between the heterarchical network and the organization of attractors in single layer of the cortex, is the least obvious of the four. In the case of the visual cortex, one can make the stronger hypothesis that the columnar organization corresponds to the heterarchical network. In this case the organization of the heterarchical network is based on the organization of the visual scene. Feature detectors reside near other feature detectors which are "related" to them in the sense of responding to the same type of feature at a location nearby in the visual field. This organization makes perfect sense as a network of emergence, in that each small region of a scene can be approximately determined by the small regions of the scene immediately surrounding it. The dynamic and inventive nature of this network representation of the visual world is hinted at by the abundance of perceptual illusions, which can often be generated by lateral inhibition effects in neural representations of scenes.

    In order for the fourth principle to hold, what is required is that other regions of the cortex contain maps like those of the visual cortex -- but not based on the structure of physical space, based rather on more general notions of relatedness. Thisis not an original idea; it has been explored in detail by Teuvo Kohonen (1988), who has shown that simple, biologically plausible two-dimensional formal neural networks can be used to create "self-organizing feature maps" of various conceptual spaces. All the formal neurons in one of his feature map networks receive common input from the same collection of formal neurons on an hypothetical "lower level"; each formal neuron also exchanges signals with the other formal neurons in the feature map network, within a certain radius.

    This is a crude approximation to the behavior of pyramidal cells within a cortical layer, but it is not outrageously unrealistic. What happens is that, after a number of iterations, the feature map network settles into a state where each formal neuron is maximally receptive to a certain type of input. The two-dimensional network then mirrors the topological structure of the high-dimensional state space of the collection of inputs, in the sense that nearby formal neurons correspond to similar types of input.

    Kohonen's feature map networks are not, intrinsically, networks of emergence; the notion of "relatedness" which they embody is simply proximity in the high-dimensional space of lower-level inputs. However, these feature maps do provide a vivid illustration of the spontaneous formation of two-dimensional neural maps of conceptual space. What Kohonen's work suggests is a restatement of Principle 4 of the hypothesized psynet/cortex correspondence:

    4'. Each cortical layer consists of a network of pyramidal neurons organized into Kohonen-style feature maps, whose topology is based on a structural notion of relatedness between nearby pyramidal neurons.

    In order for this restated principle to hold true, it is sufficient that two properties should hold. These criteria lie at the borderline of mathematics and biology. It is possible that they could be proved true mathematically, in such a general sense that they would have to hold true in the cortex. On the other hand, it seems more likely that their mathematical validity relies on a number of conditions, the applicability of which to the cortex is a matter of empirical fact.

    The first property is that pyramidal neurons and neuronal groups which are close to each other, and thus have similar inputs from lower level neurons, should, on average, recognize similar patterns. Of course, there will be cases in which this does not hold: one can well have two neurons with almost the same inputs but entirely different synaptic conductances, or with different intervening interneurons turning excitatory connections to inhibitory ones. But this is not generally the case in the visual cortex -- there what we see is quite consistent with the idea of a continuity of level k+1 feature detectors corresponding to the continuity of their level k input.

    The second property is that higher-level patterns formed from patterns involved in a network of emergence should themselves naturally form into a network of emergence. We have seen that the associative-memory "network of emergence" structureis an attractor for networks of pattern recognition processes; what the fulfillment of this criterion hinges on is the basin of the network of emergence structure being sufficiently large that patterns recognized among level k attractors will gradually organize themselves into a level k+1 network of emergence.

3.5 EVOLUTION AND AUTOPOIESIS IN THE BRAIN

    I have delineated the basic structural correspondence between the psynet model and the cortex. In essence, the idea is that the two orthogonal structures of the cortex correspond to the two principal subnetworks of the dual network. The next natural question is: what about dynamics? The psynet model comes equipped with its own dynamics; how do these correspond to the dynamics of brain function?

    Recall that, in the previous chapter, a distinction was drawn between two types of dynamics in a dual network: evolution and autopoiesis. This is to some extent an artificial distinction, but it nevertheless useful in a neurobiological context. It is essentially the same as the distinction, in biology, between evolution and ecology. On a more basic, philosophical level, it is a distinction between a force of change and a force of preservation.

Evolution

    First, what about neural evolution? On the simplest level, one may say that the reinforcement of useful pathways between neural assemblies is a form of evolution. Edelman (1987) has called this view "neuronal group selection," or "Neural Darwinism." Essentially, in Neural Darwinism, one has survival of the fittest connections. Chaos and randomness in the neural circuits provide mutation, and long-term potentiation provides differential selection based on fitness. As in the dual network model, the progressive modification of synapses affects both associative memory (within a layer) and hierarchical perception/control (across layers).

    In this simplest model of neural evolution, there is no reproduction -- and also no crossover. Edelman argues that the lack of reproduction is compensated for by the vast redundancy of the cortex. For, in a sense, one doesn't need to reproduce connections, because almost every connection one might wish for is already there. There may not be many multiple connections between the same pair of pyramidal neurons, but pyramidal neurons tend to have similar connections to their neighbors, so there will be plenty of multiple connections from one cluster of similar pyramidal neurons to another.

    Edelman does not even mention the lack of crossover. From his perspective, mutation alone is a perfectly valid evolution strategy. From another point of view, however, one might argue for the necessity of neural crossover. As will be argued in Chapter Six, crossover is demonstrably a more powerful learning technique than mere mutation. Furthermore, if one considers the two methods as learning algorithms, crossover gives the power-law learning curve so familiar from psychology, while mutation givesa straight-line learning curve. Finally, intuition and introspection indicate that human creativity involves some form of combination or crossing-over of ideas.

    As I have argued in EM, synaptic modification, in the context of an hierarchical processing network, can provide a kind of reproduction by crossover. By appropriate strengthening and weakening of synapses, one can take two trees of neural assemblies and swap subtrees between them. This is very close to the kind of crossover studied by John Koza (1992) in his "genetic programming paradigm." The difference is that, instead of trees of neural assemblies, he has trees of LISP functions. There is therefore a sense in which the Neural Darwinist model of neural evolution can provide for crossover.

    It is not clear, however, whether this kind of mutation-based crossover is enough. I have proposed as a speculative hypothesis that the brain, in its creative evolution, routinely carries out a more flexible kind of crossover -- that its neural networks are easily able to move assemblies from place to place. Memory reorganization would be more effective, it would seem, if memories were actually able to move from one part of the brain to another, rather than merely having the connections between them modified. And, from the hierarchical point of view, actual moving of neural assemblies would provide for a much more flexible crossover operation between trees and other systems of assemblies. This hypothesis finds some support both in neuroscience and in formal neural network theory. On the one hand, evidence is emerging that the brain is, in certain circumstances, able to move whole systems of assemblies from one place to another; even from one hemisphere to another (Blakeslee, 1991). And, on the other hand, one may show that simple Kohonen-style neural networks, under appropriate conditions, can give rise to spontaneously mobile activation bubbles (Goertzel, 1996a). It is not possible to draw any definite conclusions as yet, but the concept of "sexually" reproducing neural assemblies is looking more and more plausible.

Autopoiesis

    Autopoiesis has an obvious correlate in neural networks. If neural assemblies are magicians, then structural conspiracies are assemblies of neural assemblies -- neural meta-assemblies. Hebb, in the original statement of cell assembly theory, foresaw that neural assemblies would themselves group into self-organizing systems. Of course, self-organizing systems of neural assemblies need not be static, but may be in a continual process of mutual growth and change.

    Autopoiesis, in the psynet model, is asked to carry out a wide variety of functions. Essentially, anything involving the preservation of structure over time must be accomplished by pattern/process autopoiesis. This leads to a variety of intriguing hypotheses. For instance, consider the vexing question of symbolic versus connectionist processing. How do the messy, analogue statistical learning algorithms of the brain give rise to the precise symbolic manipulations needed for language and logic. According to the psynet model, this must come out of pattern/process autopoiesis. Thus, in the current brain theory,it must come out of autopoietic systems of neural assemblies.

    But how can symbol processing come out of autopoiesis? Intriguingly, mathematics provides a ready answer. The technique of symbolic dynamics, to be discussed in Chapter Five, deals precisely with the emergence of symbol systems and formal languages out of complex dynamical systems. To study a dynamical system using symbolic dynamics, one partitions the state space of the system into N+1 regions, and assigns each region a distinct code number drawn from {0,...,N}. The system's evolution over any fixed period of time may then be represented as a finite series of code numbers, the code number for time t representing the region of state space occupied by the system state S(t). This series of code numbers is called a "symbolic trajectory"; it may be treated as a corpus of text from an unknown language, and grammatical rules may be inferred from it. In particular, systems with complex chaotic dynamics will tend to give rise to interesting languages. Chaos, which involves dynamical unpredictability, does not rule out the presence of significant dynamic patterns. These patterns reveal themselves visually as the structure of the chaotic system's strange attractor, and they reveal themselves numerically as languages emergent from symbolic dynamics.

    Cohen and Eichenbaum (1995) have demonstrated that cortical-hippocampal feedback loops play a fundamental role in helping the neocortex to store and access symbolic, declarative information. The hypothesis to which the psynet model of brain leads us is that the cortical-hippocampal feedback loops in fact serve to encode and decode symbolic memories in the structures of the attractors of cortical neural assemblies. In fact, one may show that these encoding and decoding operations can be carried out by biologically plausible methods. This is an intriguing and falsifiable hypothesis which ensues directly from applying the simplicity of the psynet model to the complexity of the brain.

3.6 CONCLUSION

    It is important not to fall into the trap of believing neural network models, in particular, to exhaust the applicability of complex systems science to the study of brain function. The brain is a very complex system, and complex systems ideas can be applied to it on many different levels -- from the microtubular level stressed by Stuart Hameroff in Ultimate Computing and Roger Penrose in Shadows of Mind, up to the abstract mental-process level emphasized here.

    I am not a neurobiologist, and the cortical model presented here is plainly not a neurobiologist's model. It has an abstract structure which doubtless reflects my background as a mathematician. But, on the other hand -- and unlike many neural network models -- it is not a mere exercise in mathematical formalism. It is, rather, a conceptual model, an intuitive framework for understanding.

    The value of such models lies in their ability to guide thought. In particular, this model was developed not only to guide my own thinking about the brain, but to guide my ownthinking about the learning behavior of human, animals and artificial intelligence systems. My hope is that it may help others to guide their thoughts as well. For, after all, the project of understanding the brain/mind is just barely getting under way -- we need all the ideas we can muster.