Part 15

such that

dim(X′) = dim(X″), (9)

where

X′ = ⨂Xᵢ (10) I′

X″ = ⨂Xᵢ. I″

(If dim (X) is odd, then we have to cheat a little by putting in an extra random dummy dimension.) And then the decomposition of the space

X = ⨂Xᵢ (11) I

has to be carried out so that this partitioning is preserved. Since this partitioning is arbitrary (as far as the mathematics is concerned), it is obvious that a space which is not partitioned will have many (equivalent) decompositions. On the other hand, if the partitioning is into more than two parts, then the existence of a decomposition is not guaranteed.

A slight penalty has to be paid for the use of this partitioning, namely: instead of eventually obtaining a random cartesian product of one-dimensional spaces, we obtain an extended channel (with random input) of single-dimensional channels. It is obvious that if we were to drop the partitioning temporarily, each such single-dimensional channel would be further decomposed into two random components. This decomposition is not unique. But one of these equivalent decompositions is particularly convenient; namely, that decomposition where we take the component out of the original X′ and that which is random to it, say V. This V (as well as the cartesian product of all such V’s, which of necessity are random) is called the linearly additive noise. The name “linearly additive” is justified because it is just the statistical concept isomorphic to the linear addition of vectors in orthogonal Euclidean geometry. (The proof of this last statement is not completed as yet.)

Denumerable Space

The procedure for this decomposition was worded to de-emphasize the possible presence of a denumerable (component of the) space. Such a component may be given outright; otherwise, it results if the space was not simply connected. Any denumerable space is zero dimensional, as may be verified easily from the full information theoretic definition of dimensionality.

The obvious way of disposing of a denumerable space is to use the conventional mapping that converts a Stieltjes to a Lebesque integral, using fixed length segments. (It can be shown that H is invariant under such a mapping.) Unfortunately, while this mapping followed by a repetition of the preceding procedure will always solve a given problem (no new denumerable component need be generated on the second pass), little insight is provided into the structure of the resulting space. On the other hand, because channels under cascading constitute a group, any such denumerable space is a representation of a denumerable group.

Only non-cyclic irreducible (wrt direct product) denumerable group components of the old denumerable space will remain.

SUMMARY

In summary, the original metrizable topological space was decomposed into an orthogonal Euclidean space times a denumerable random cartesian product of irreducible (wrt direct product) denumerable groups. Thus, since any individual component of a random cartesian product may be studied independently of the others, all that one needs to study is: (1) a Gaussian distribution on a single real axis and (2) the irreducible denumerable groups.

Random cartesian product.

Finally, it should be emphasized that there are only these two ways of decomposing a metrizable topology; (1) if a (statistical) basis is given, use the diagonalization of a symmetric matrix algorithm described earlier (and given in detail in the three channels in cascade problem), and (2) otherwise use a suitable network of the NPO’s with n₀=1. Of course, any hybrid of these two methods may be employed as well.

On Functional Neuron Modeling

C. E. HENDRIX

Space-General Corporation El Monte, California

There are two very compelling reasons why mathematical and physical models of the neuron should be built. Model building, while widely used in the physical sciences, has been largely neglected in biology. However, there can be little doubt that building neuron models will increase our understanding of the function of real neurons, if experience in the physical sciences is any guide. Secondly, neuron models are extremely interesting in their own right as new technological devices. Hence, the interest in, and the reason for symposia on self-organizing systems.

We should turn our attention to the properties of real neurons, and see which of them are the most important ones for us to imitate. Obviously, we cannot hope to imitate all the properties of a living neuron, since that would require a complete simulation of a living, metabolizing cell, and a highly specialized one at that; but we can select those functional properties which we feel are the most important, and then try to simulate those.

The most dramatic aspect of neuron function is, of course, the axon discharge. It is this which gives the neuron its “all-or-nothing” character, and it is this which provides it with a means for propagating its output pulses over a distance. Hodgkin and Huxley (1) have developed a very complete description of this action. Their model is certainly without peer in describing the nature of the real neuron.

On the technological side, Cranes’ “neuristors” (2) represent a class of devices which imitate the axonal discharge in a gross sort of way, without all the subtle nuances of the Hodgkin-Huxley model. Crane has shown that neuristors can be combined to yield the various Boolean functions needed in a computer.

However, interesting as such models of the axon are, there is some question as to their importance in the development of self-organizing systems. The pulse generation, “all-or-nothing” part of the axon behavior could just as well be simulated by a “one-shot” trigger circuit. The transmission characteristic of the axon is, after all, only Nature’s way of sending a signal from here to there. It is an admirable solution to the problem, when one considers that it evolved, and still works, in a bath of salt water. There seems little point, however, in a hardware designer limiting himself in this way, especially if he has an adequate supply of insulated copper wire.

If the transmission characteristic of the axon is deleted, the properties of the neuron which seem to be the most important in the synthesis of self-organizing systems are:

a. The neuron responds to a stimulus with an electrical pulse of standard size and shape. If the stimulus continues, the pulses occur at regular intervals with the rate of occurrence dependent on the intensity of stimulation.

b. There is a threshold of stimulation. If the intensity of the stimulus is below this threshold, the neuron does not fire.