The neurons of the cortex are organized in clusters, each containing 50 to 10,000 neurons. The

neurons of each cluster are connected primarily to other neurons in the same cluster. Edelman

(1988) has proposed that it makes sense to think of connections between clusters, not just

individual neurons, as being reinforced or inhibited; and he has backed this up with a detailed

mathematical model of neural behavior. Following this line of thought, it is the nature of the

interaction between neural clusters which interests us here. We shall show that, according to a

simple model inspired by Edelman’s ideas, the interaction of neural clusters gives rise to a simple

form of structural analogy.

Edelman’s theory of "Neural Darwinism" divides the evolution of the brain into two phases.

The first, which takes place during fetal development, is the phase of cluster formation. And the

second, occurring throughout the remainder of life, is the phase of repermutation: certain

arrangements of clusters are selected from the set of possible arrangements. This is not an

outlandish hypothesis; Changeux (1985), among others, has made a similar suggestion. Edelman,

however, has formulated the theory as a sequence of specific biochemical hypotheses, each of

which is supported by experimental results.

Roughly speaking, a set of clusters which is habitually activated in a certain order is called a

map. Mental process, according to Edelman’s theory, consists of the selection of maps, and the

actual mapping of input — each map receiving input from sensory sources and/or other maps,

and mapping its output to motor control centers or other maps. Mathematically speaking, a map

is not necessarily a function, since on different occasions it may conceivably give different

outputs for the same input. A map is, rather, a dynamical system in which the output yt at time t

and the internal state St at time t are determined by yt=f(xt-1,St-1), St=g(xt-1,Mt-1), where xt is

the input at time t. And it is a dynamical system with a special structure: each map may be

approximately represented as a serial/parallel composition of simple maps which are composedof

single clusters. One key axiom of Edelman’s theory is the existence of numerous "degenerate"

clusters, clusters which are different in internal structure but, in many situations, act the same

(i.e. often produce the same output given the same input). This implies that each map is in fact a

serial/parallel combination of simple component maps which are drawn from a fairly small set,

or at least extremely similar to elements of a fairly small set.

There is much more to Neural Darwinism than what I have outlined here — for instance, I have

not even mentioned Edelman’s intriguing hypotheses as to the role of adhesion molecules in the

development of mentality. But there are nevertheless a number of mysteries at the core of the

theory. Most importantly, it is not known exactly how the behavior of a cluster varies with its

strengths of interconnection and its internal state.

Despite this limitation, however, Neural Darwinism is in my opinion the only existing

effective theory of low-to-intermediate-level brain structure and function. Philosophically, it

concords well with our theory of mind: it conceives the brain as a network of "maps", or

"functions". In previous chapters we spoke of a network of "programs," but the terms "program"

and "map" are for all practical purposes interchangeable. The only difference is that a program is

discrete whereas the brain is considered "continuous," but since any continuous system may be

approximated arbitrarily closely by some discrete system, this is inessential.

So, consider a set of n processors Pi, each one possessing an internal state Sit at each time t.

Assume, as above, that the output yit at time t and the internal state Sit at time t are determined

by yit=fi(xit-1,Sit-1), Sit=gi(xit-1,Sit-1), where xit is the input at time t. Define the "design" of

each processor as the set of all its possible states. Assume the processors are connected in such a

way that the input of each processor is composed of subsets of the output of some set of k

processors (where k is small compared to n, say O(logn)). This is a basic "network of

processors".

Note that we have defined f and g to vary with i. Strictly speaking, this means that the

different processors are different dynamical systems. In this case, what it really means,

intuitively, is that fi and gi may vary with the pattern of flow of the dynamical system. In fact,

from here on we shall assume gi=g for all i; we shall not consider the possibility of varying the

gi. However, we shall be concerned with minor variations in the fi; in particular, with

"strengthening" and "weakening" the connections between one processor and another. For this

purpose, we may as well assume that the space of outputs yit admits is Rn or a discrete

approximation of some subset thereof. Let si,j denote the scalar strength of connection from Pi to

Pj. For each Pi, let jfit denote the portion of the graph of fi which is connected to Pj at time t.

Assume that if j and l are unequal, then jfir and lfit are disjoint for any t and r; and that jfir= si,j jfit

for all t and r. According to all this, then, the only possible variations of fi over time are merely

variations of strength of connectivity, or "conductance".

As observed above, our model of mind is expressed as a network of processors, and Edelman’s

theory expresses brain function as a result of thedynamics of a network of neuronal clusters,

which are specialized processors. In the context of neuronal clusters, the "design" as defined

above is naturally associated with the graph of interconnection of the neurons in the cluster; a

particular state is then an assignation of charge levels to the neurons in the cluster, and a

specification of the levels of various chemicals, most importantly those concerned with the

modification of synaptic strength. According to Edelman’s theory, the designs of the million or

so processors in the brain’s network are highly repetitive; they do not vary much from a much

smaller set of fundamental designs. And it is clear that all the functions fi regulating connection

between neuronal clusters are essentially the same, except for variations in conductance.

THE NOISY HEBB RULE

D.O. Hebb, in his classic Organization of Behavior (1949), sought to explain mental process in

terms of one very simple neural rule: when a synaptic connection between two neurons is used,

its "conductance" is temporarily increased. Thus a connection which has proved somehow useful

in the past will be reinforced. This provides an elegant answer to the question: where does the

learning take place? There is now physiological evidence that Hebbian learning does indeed

occur. Two types of changes in synaptic strength have been observed (Bliss, 1979): "post-tetanic

potentiation", which lasts for at most a few minutes, and "enhancement", which can last for hours

or days.

Hebb proceeded from this simple rule to the crucial concepts of the cell-assembly and the

phase-sequence:

Any frequently repeated, particular stimulation will lead to the slow development of a "cell-

assembly", a diffuse structure comprising cells in the cortex and diencephalon… capable of

acting briefly as a closed system, delivering facilitation to other such systems and usually having

a specific motor facilitation. A series of such events constitutes a "phase sequence" — the thought

process. Each assembly action may be aroused by a preceding assembly, by a sensory event, or —

normally — by both.

This theory has been criticized on the physiological level; but this is really irrelevant. As Hebb

himself said, "it is… on a class of theory that I recommend you to put your money, rather than

any specific formulation that now exists" (1963, p.16). The more serious criticism is that Hebb’s

ideas do not really explain much about the most interesting aspects of mental process. Simple

stimulus-response learning is a long way from analogy, associative memory, deduction, and the

other aspects of thought which Hebb hypothesizes to be special types of "phase sequences".

Edelman’s ideas mirror Hebb’s on the level of neural clusters rather than neurons. In the

notation given above, Edelman has proposed that if the connection from P1 to P2 is used often

over a certain interval of time, then its"conductance" s1,2 is temporarily increased.

Physiologically, this is a direct consequence of Hebb’s neuron-level principle; it is simply more

specific. It provides a basis for the formation of maps: sets of clusters through which information

very often flows according to a certain set of paths. Without this Hebbian assumption, lasting

maps would occur only by chance; with it, their emergence from the chaos of neural flow is

virtually guaranteed. At bottom, what the assumption amounts to is a neural version of the

principle of induction. It says: if a pathway has been useful in the past, we shall assume it will be

useful in the future, and hence make it more effective.

Unfortunately, there is no reason to believe that the cluster-level interpretation of Hebb’s

theory is sufficient for the explanation of higher mental processes. By constructing a simulation

machine, Edelman has shown that a network of entities much like neural clusters, interacting

according to the Hebbian rule, can learn to perceive certain visual phenomena with reasonable

accuracy. But this work — like most perceptual biology — has not proceeded past the lower levels

of the perceptual hierarchy.

In order to make a bridge between these neural considerations and the theory of mind, I would

like to propose a substantially more general hypothesis: that if the connection between P1 and P3

is used often over a certain interval of time, and the network is structured so that P2 can

potentially output into P3, then the conductance s2,3 is likely to be temporarily increased.

As we shall see, this "noisy Hebb rule" leads immediately to a simple explanation of the

emergence of analogy from a network of neural clusters. Although I know of no evidence either

supporting or falsifying the noisy Hebb rule, it is certainly not biologically unreasonable. One

way of fulfilling it would be a spatial imprecision in the execution of Hebbian neural induction.

That is, if the inductive increase in conductance following repeated use of a connection were by

some means spread liberally around the vicinity of the connection, this would account for the

rule to within a high degree of approximation. This is yet another case in which imprecision may

lead to positive results.

THE NOISY HEBB RULE AND STRUCTURAL ANALOGY

Consider the situation in which two maps, A and B, share a common set of clusters. This

should not be thought an uncommon occurrence; on the contrary, it is probably very rare for a

cluster to belong to one map only. Let B-A (not necessarily a map) denote the set of clusters in B

but not A. The activation of map A will cause the activation of some of those clusters in map B-

A which are adjacent to clusters in A. And the activation of these clusters may cause the

activation of some of the clusters in B-A which are adjacent to them — and so on. Depending on

what is going on in the rest of B, this process might peter out with little effect, or it might result

in the activation of B. In the latter case, what has occurred is the most primitive form of

structural analogy.

Structural analogy, as defined earlier, may be very roughly described asreasoning of the form:

A and B share a common pattern, so if A is useful, B may also be useful. The noisy Hebb rule

involves only the simplest kind of common pattern: the common subgraph. But it is worth

remembering that analogy based on common subgraphs also came up in the context of

Poetszche’s approach to analogical robot learning. Analogy by common subgraphs works. There

is indeed a connection between neural analogy and conceptual analogy. And — looking ahead

to chapter 7 — it is also worth noting that, in this simple case, analogy and structurally

associative memory are inextricably intertwined: A and B have a common pattern and are

consequently stored near each other (in fact, interpenetrating each other); and it is this

associative storage which permits neural analogy to lead to conceptual analogy.

Kaynak: A New Mathematical Model of Mind

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