Exact results for the average dynamic behavior of some nonlinear neural networks (Q1095064)

A general equation of motion for homogeneous, synchronous, highly interconnected nets of binary neurons (state variables taking values \(\pm 1)\) with fixed, uniformly distributed excitatory and inhibitory weights, is derived. Two special cases are considered in detail. The first assumes that there is on average no correlation between states of individual neurons (and the topology of the net is irrelevant), which leads to an equation of the form \(<s(t+1)>=Tanh(V+W<s(t)>)\), where V is the threshold value, W specifies balance between excitatory and inhibitory weights and the brackets \(<\) \(>\) denote averaging. This gives a good characterization of (one to three) stable solutions. In the second case, the topology of Cayley trees is assumed, so that the only pathways through which mutual influence between neurons is manifested, is direct communication between neighbours. The formula for \(<s_ ks_ 1>\) enables then to estimate average correlation, which is shown either to disappear quickly or to be maintained for a long time before doing so. The average activity behaves similarly as in the preceding case. The interesting point is that a sort of stability can be achieved even in loop-free nets.