One+Infinite Dimensional Attractor Neural Networks

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.