Solutions of Probabilistic Equations for Neural Networks

Abstract

A discussion of spontaneous behaviour in neural networks has recently been made by Taylor.¹ We investigate some solution of eq. 12) of Reference 1) for particular networks which will be specified later. The most important result is the existence of steady state solutions where the probability vector remains fixed in time. The system will usually tend towards one of these steady state solutions which correspond to a statistical reverbertation of period one in the general case. The absence of longer reverbertation periods is in direct contrast to the results of Caian-ello ², where deterministic neural networks were considered. The steady states of the systems are entirely due to, and eventually dominated by, the spontaneous firing of the individual neurons. Any information injected into such a system will thus eventually be lost in a time determined by the physical parameters describing the neurons and their couplings.

There also exist special cases of these equations where a critical choice of coupling coefficients leads to a degenerate family of final state solutions with an oscillatory behaviour. In this case the final state behaviour depends upon the initial state of the system.

We consider briefly the number of these “uniform chaos” final states, and find that these are expected to increase rapidly with the total number of neurons in the network and with the degree of connectivity of the net. We conclude that such networks may be very useful in pattern recognition, and allow the construction of a new class of computer.