A scaling limit of the Glauber dynamics for a class of neural network models

Abstract

The time evolution of d mean field variables is considered for networks of N neurons whose connection matrices JN have d distinct rows. Certain assumptions are made about the large N behavior of JN, which guarantee the convergence of a free-energy density function. These assumptions are known to be satisfied, e.g., in the Hopfield model with p stored patterns, for d=2p. It is proved that in a scaling limit, where N tends to infinity and d stays fixed, the time evolution approaches that of a diffusion process in Rd. This process describes in detail, and for times up to ℴ(N3/2) iterations, the dynamics of the mean field fluctuations near a local minimum of the free-energy density.