A spherical Hopfield model

TL;DR

This paper introduces a spherical Hopfield neural network with continuous variables, analyzing its thermodynamics and dynamics, and demonstrating a retrieval phase through added quartic interactions, with exact solutions and numerical validation.

Contribution

It presents a novel spherical Hopfield model with continuous neurons and patterns, providing exact thermodynamic solutions and dynamic equations for the first time.

Findings

Thermodynamics are exactly solvable with replica symmetry.

A quartic term enables a retrieval phase.

Numerical results confirm theoretical predictions.

Abstract

We introduce a spherical Hopfield-type neural network involving neurons and patterns that are continuous variables. We study both the thermodynamics and dynamics of this model. In order to have a retrieval phase a quartic term is added to the Hamiltonian. The thermodynamics of the model is exactly solvable and the results are replica symmetric. A Langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks. The stationary limit corresponds to the thermodynamic results. Numerical calculations illustrate our findings.