Oscillatory Instability and Criticality in Non-Reciprocal Hopfield Networks

Traditional Hopfield models rely on symmetric spin interactions to describe memory retrieval dynamics. However, incorporating non-reciprocal interactions makes the dynamics of these systems more complex. We propose a Hopfield model with two memory patterns in a system of N spins with non-reciprocal couplings. We show that non-reciprocal interactions can induce cyclic switching between the two predefined memory patterns, driving the system toward an oscillatory instability threshold. Non-reciprocity breaks time-reversal symmetry and enriches the phase diagram, revealing a limit cycle phase bounded by Hopf bifurcation and Fold bifurcation critical lines. We analytically examine the criticality in the auto-correlation function, C(τ), near and on these critical lines. The C(τ) scales as C ~ N^α C̃(τ/N^ζ), where C̃ and ζ represent universal scale-invariant functions and a dynamical critical exponent, respectively. Notably, we find ζ to be 1/2 along the Hopf line and 1/3 along the Fold line, suggesting distinct critical behaviors in the two regions. In addition, we derive an exact form of the Master Equation and explore the large N limit through Glauber Monte Carlo simulations. Our numerical results validate these critical behaviors and critical exponents. This work provides a framework for investigating out-of-equilibrium state switching in complex systems, such as cyclic biological processes occurring in cell division.