A statistical-physics analysis of two-memory Hopfield networks whose non-reciprocal (asymmetric) couplings break time-reversal symmetry and drive cyclic switching between stored memory patterns. The dynamical phase diagram contains no-retrieval, multiple point-attractor, and limit-cycle phases; the limit-cycle phase is bounded by a Hopf bifurcation line with dynamical critical exponent zeta=1/2 and a fold bifurcation line with zeta=1/3, each showing distinct autocorrelation scaling and distinct perturbation-response times (|F|^{-2/3} on the Hopf line, |F|^{-1/2} on the fold line). Mean-field predictions are verified via exact Master-Equation (Liouvillian) diagonalization and large-N Glauber Monte Carlo, modeling finite-size biological programs near a threshold of cyclic instability.
This paper studies two-memory Hopfield networks with non-reciprocal (asymmetric) couplings, which break time-reversal symmetry and drive the system toward oscillatory instability. It maps the dynamic…