91-2 Koch H.

A Scaling Limit of the Glauber Dynamics for a Class of Neural Network Models. (227K, ps) Jul 10, 91

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Abstract. We consider the time evolution of d mean field variables for networks of N neurons whose connection matrices J_N have d distinct rows. Certain assumptions are made about the large N behavior of J_N, 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=2^p. 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 R^d. This process describes in detail, and for times up to o(N^(3/2)) iterations, the dynamics of the mean field fluctuations near a local minimum of the free energy density.

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