Stochastic analyses of the dynamics of generalized Little-Hopfield-Hemmen type neural networks

Stochastic analyses are conducted of model neural networks of the generalized Little-Hopfield-Hemmen type, in which the synaptic connections with linearly embeddedp sets of patterns are free of symmetric ones, and a Glauber dynamics of a Markovian type is assumed. Two kinds of approaches are taken to study the stochastic dynamical behavior of the network system. First, by developing the method of the nonlinear master equation in the thermodynamic limitN, an exact self-consistent equation is derived for the time evolultion of the pattern overlaps which play the role of the order parameters of the system. The self-consistent equation is shown to describe almost completely the macroscopic dynamical behavior of the network system. Second, conducting the system-size expansion of the master equation for theN-body probability distribution of the Glauber dynamics makes it possible to analyze the fluctuations. In the course of the analysis, the self-consistent equation for the pattern overlaps is derived again. The main result of the rigorous fluctuation analysis is that as far as the fluctuations are concerned, the time course of the pattern overlap fluctuations behaves independently of the fluctuations in the remaining modes of the system's macrovariables, in accordance with the self-determining property of the macroscopic motion of the pattern overlaps for neural networks with linear synaptic couplings.