Critical Dynamics and Cyclic Memory Retrieval in Non-reciprocal Hopfield Networks

TL;DR

This paper explores the complex dynamics of non-reciprocal Hopfield networks, revealing phase transitions and critical behavior that could model biological cyclic processes.

Contribution

It introduces a detailed analysis of phase transitions and critical phenomena in non-reciprocal Hopfield networks, combining analytical and numerical methods.

Findings

Identification of dynamical phases including no memory, multiple memory, and limit-cycle states.

Characterization of bifurcation lines with distinct critical exponents.

Numerical verification of analytical predictions using a Master Equation approach.

Abstract

We study Hopfield networks with non-reciprocal coupling inducing switches between memory patterns. Dynamical phase transitions occur between phases of no memory retrieval, retrieval of multiple point-attractors, and limit-cycle attractors. The limit cycle phase is bounded by two critical regions: a Hopf bifurcation line and a fold bifurcation line, each with unique dynamical critical exponents and sensitivity to perturbations. A Master Equation approach numerically verifies the critical behavior predicted analytically. We discuss how these networks could model biological processes near a critical threshold of cyclic instability evolving through multi-step transitions.