One+Infinite Dimensional Attractor Neural Networks

Abstract

We solve a class of attractor neural network models with a mixture of 1D nearest-neighbour and infinite-range interactions, which are of a Hebbian-type form. Our solution is based on a combination of mean-field methods, transfer matrices and 1D random-field techniques, and is obtained for Boltzmann-type equilibrium (following sequential Glauber dynamics) and Peretto-type equilibrium (following parallel dynamics). Competition between the alignment forces mediated via short-range interactions, and those mediated via infinite-range ones, is found to generate novel phenomena, such as multiple locally stable `pure' states, first-order transitions between recall states, 2-cycles and non-recall states, and domain formation leading to extremely long relaxation times. We test our results against numerical simulations and simple benchmark cases and find excellent agreement.

24 pages and 23 postscript figures

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