MACROSCOPIC LYAPUNOV FUNCTIONS FOR SEPARABLE STOCHASTIC NEURAL NETWORKS WITH DETAILED BALANCE

We derive macroscopic Lyapunov functions for large, long-range, Ising-spin neural networks with separable symmetric interactions, which evolve in time according to local field alignment. We generalize existing constructions, which correspond to deterministic (zero-temperature) evolution and to specific choices of the interaction structure, to the case of stochastic evolution and arbitrary separable interaction matrices, for both parallel and sequential spin updating. We find a direct relation between the form of the Lyapunov functions (which describe dynamical processes) and the saddle-point integration that results from performing equilibrium statistical mechanical studies of the present type of model.