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

TL;DR

This work analyzes two-memory Hopfield networks with non-reciprocal couplings, uncovering a limit-cycle phase bounded by a Hopf bifurcation line and a fold bifurcation line. Using mean-field theory, complex normal-form reductions, Master Equation formalisms, Liouvillian diagonalization, and Glauber dynamics, the authors identify two distinct critical regions with dynamical exponents $\zeta=\tfrac{1}{2}$ (Hopf) and $\zeta=\tfrac{1}{3}$ (fold), and derive scaling laws for autocorrelations and response to external drive: resonant forcing yields $w/F \sim F^{-2/3}$ on the Hopf line, while on the fold line static driving cannot sustain limit cycles and switching exhibits $\theta$-scaling $\sim F^{1/2}$. Finite-size effects are quantified via the Liouvillian spectrum and Langevin analyses, with ensemble damping and noise-induced phase slips near the fold line. The framework provides a mechanistic description of cyclic memory and critical transitions relevant to biological sequential processes and suggests routes to generalize to more patterns and driven, out-of-equilibrium neural-like systems.

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.

Figures (13)

• Figure 1: Phase portraits (left) and the phase diagram (right) for the two-memory cyclic Hopfield model. Left: The panel shows the dynamical behavior for different values of $\lambda_+$ and $\lambda_-$. The six phase portraits ( a to f) show the trajectory dynamics, with green arrows indicating the vector fields of the derivatives. In e and f, empty circles represent saddle points, while solid circles denote stable points (sinks). Right: The phase diagram is divided into three regions of different dynamical behavior: Limit Cycles (diagonal lines with a purple background), Memory Retrieval (dotted dark green background), and Paramagnetic (vertical stripes with a light green background). These phases are bounded by bifurcation lines: fold bifurcation lines (dark green lines) and the Hopf bifurcation (purple vertical line). The positions of the six trajectory plots ( a - f) are indicated by corresponding labels on the phase diagram.
• Figure 2: Representative trajectories at the fold transition as a function of $t$ with $N = 1300$ and $D = 1$. One can notice several features that are absent (deep) in the limit cycle phase: First, there is a larger variation between different trajectories. This highlights the role of noise in inducing phase slips. Second, the period of jumps (or the frequency of the limit cycle rotation) is roughly of the order $T \sim 10$ while deep in the limit cycle phase it is of order 1. Again this is due to noise as the period should diverge when $N \to \infty$.
• Figure 3: Effective potential $V(\theta)$ for different values of $\epsilon$. Depending on its sign, $\epsilon$ alters the steepness in $V(\theta)$, thereby affecting system's stability. A negative $\epsilon$ initiates an uphill start in $V$, which poses a potential hill for $\theta$ as a metastable state till it overcomes the hill. For $\epsilon$ = 0, $V$ starts flat, then slips down at a faster rate than in the negative $\epsilon$ scenario. A positive $\epsilon$ triggers an immediate downhill movement in $V$, swiftly driving the system into the oscillatory phase at an even faster rate.
• Figure 4: Real and Imaginary part of the second smaller eigenvalue of the Liouvillian matrix, $\Lambda_2$, as a function of $\beta \lambda_+$ for a fixed value of $\beta \lambda_-=0.17$ and $N_S=N_D=N/2$.
• Figure 5: Exact $\langle m_2(t) \rangle$ solved from the master equation for different system sizes $N$ with $\beta\lambda_+ = 1.3$ and $\beta \lambda_- = 0.17$. As $N$ increases, the oscillations become slower and more pronounced.