Storing Cycles in Hopfield-type Networks with Pseudoinverse Learning Rule - Retrievability and Bifurcation Analysis

TL;DR

This paper analyzes the retrievability and bifurcation behavior of cycles in Hopfield-type networks using the pseudoinverse learning rule, revealing how structural features influence network dynamics and stability.

Contribution

It provides a comprehensive stability and bifurcation analysis of admissible cycles in Hopfield networks with pseudoinverse learning, including numerical illustrations of various dynamic regimes.

Findings

Admissible cycles can be stored and retrieved as different attractors depending on network parameters.

The paper characterizes all local bifurcations of the trivial solution in these networks.

Transitions from fixed points to limit cycles involve multiple saddle-node bifurcations.

Abstract

In this paper, we study retrievability of admissible cycles and the dynamics of the networks constructed from admissible cycles with the pseudoinverse learning rule. Retrievability of admissible cycles in networks with $C_0>0$ and $\lambda$ sufficiently large are discussed. Based on the linear stability analysis we derive a complete description of all possible local bifurcations of the trivial solution for the networks constructed from admissible cycles. We illustrate numerically that, depending on the structural features, the admissible cycles are respectively stored and retrieved as attracting limit cycles, unstable periodic solutions and delay-induced long-lasting transient oscillations, and the transition from fixed points to the attracting limit cycle bifurcating from the trivial solution takes place through multiple saddle-nodes on limit cycle bifurcations.