Shuyue Xue, Mohammad Maghrebi, George I. Mias, Carlo Piermarocchi
SciPost Phys. 19, 100 (2025) ·
published 16 October 2025
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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-cycles. The limit cycle phase is bounded by a Hopf bifurcation line and a fold bifurcation line. Autocorrelation scales as $\tilde{C}(\tau/N^\zeta)$, with $\zeta = 1/2$ on the Hopf line and $\zeta = 1/3$ on the fold line. Perturbations of strength $F$ on the Hopf line exhibit response times scaling as $|F|^{-2/3}$, while they induce switches in a controlled way within times scaling as $|F|^{-1/2}$ in the fold line. 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.
