Hopfield model with quasi-diagonal connection matrix

TL;DR

This paper studies a Hopfield neural network with a quasi-diagonal connection matrix, providing a simple description of its fixed points and their dependence on matrix elements, including boundary condition variations.

Contribution

It introduces a detailed analysis of Hopfield networks with quasi-diagonal matrices, extending understanding of fixed points under different boundary conditions.

Findings

Fixed points depend on the quasi-diagonal matrix elements.

Analysis includes open and periodic boundary conditions.

Provides a framework for understanding neural dynamics with structured matrices.

Abstract

We analyze a Hopfield neural network with a quasi-diagonal connection matrix. We use the term "quasi-diagonal matrix" to denote a matrix with all elements equal zero except the elements on the first super- and sub-diagonals of the principle diagonal. The nonzero elements are arbitrary real numbers. Such matrix generalizes the well-known connection matrix of the one dimensional Ising model with open boundary conditions where all nonzero elements equal +1. We present a simple description of the fixed points of the Hopfield neural network and their dependence on the matrix elements. The obtained results also allow us to analyze the cases of a) the nonzero elements constitute arbitrary super- and sub-diagonals and b) periodic boundary conditions.