Image evolution in Hopfield networks

Abstract

We consider neural networks of the Hopfield type with couplings $J_{\mathrm{ij}}$ which need not be symmetric. From the master equation for microscopic states we derive an evolution equation for the probability density of the macroscopic parameters $q_{\mu }$, which measure the overlap of the instantaneous microscopic state (or image) with one of the built-in patterns. No restrictions are imposed on the choice of the patterns. For three different temperatures this equation is used to illustrate retrieval in the standard Hopfield network and limit-cycle behavior in nonsymmetric models.