Image evolution in Hopfield networks

We consider neural networks of the Hopfield type with couplings Jᵢⱼ 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_μ, 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.