Dynamics and statistical mechanics of the Hopfield model

Abstract

The authors present a study of the Hopfield model of the memory characteristics of a network of interconnected two-state neuron variables. The fraction of nominated configurations which the model stores without error is calculated analytically as a function of the number, N, of neurons and the number, n, of the nominated configurations. The calculation is tested by computer simulation. The noise-free (zero-temperature) phase diagram of the model is determined within a replica-symmetric solution of the mean-field equations. The model exhibits a phase transition at alpha ( identical to n/N)= alpha _c approximately=0.069; at this point the thermodynamic states having macroscopic overlap with the nominated configurations disappear, implying a discontinuous change in the fraction of bits (of any nominated configuration) recalled correctly. Large scale Monte Carlo simulations using a distributed array processor provide some support for the existence of a phase transition close to the predicted value.