Order Parameter Flow of Auto-Associative Memory in an XY Spin System

Abstract

A phase coupled oscillator model for auto-associative memory is studied. For the auto-associative memory of an Ising spin system, Coolen and Sherrington proposed the dynamical replica theory. In that theory, a non-Gaussian noise distribution is calculated when macroscopic order parameters are given. In this paper, we extend their theory to an XY spin system and derive the dynamical equations governing the macroscopic order parameters. In order to avoid technical difficulties, the noise distribution function is expanded with respect to a small parameter. Then, the noise distribution is decomposed into noise components. The zeroth-order of the noise component is a rotationally symmetric Gaussian, and all other components corresponding to the correction terms are asymmetric with respect to the origin. Comparison with Monte Carlo simulations suggests that, for a successful retrieval process, the correction terms must be much smaller than the zeroth-order component, so that the noise distribution can be regarded as Gaussian. For an unsuccessful retrieval process, the correction terms are larger, so that the noise distribution function is deformed to be non-Gaussian. For such a process, there are large discrepancies between the results for the temporal evolution of the overlap obtained from simulations and the theory. However, the trajectories of the overlap and the variance of the noise are qualitatively very similar for both successful and unsuccessful retrieval processes. The origin of this discrepancy and the validity of the theory are discussed.