Global dynamics in neural networks. III

Abstract

A transform is introduced that maps discrete neural network dynamics to almost everywhere topologically conjugate dynamical systems on the unit interval. In many cases this correspondence gives rise to continuous conjugates, in which case the tranform preserves entropy. The transform also allows transfer of many dynamical properties of continuous systems to a large class of infinite discrete neural networks. For instance, it is proved that the network dynamics of very simple classes of neural networks, even with highly symmetric weights and architectures, have chaotic regions of evolution (in the sense of existence of scrambled sets -Li & Yorke-and configurations of arbitrarily large periods -Sarkovskii). These results raise the possibility of fully modeling parallel computability on real-valued dynamical systems by discrete neural nets.