Dynamically selected steady states and criticality in non-reciprocal networks

We consider a simple neural network model, evolving via non-linear coupled stochastic differential equations, where neural couplings are random Gaussian variables with non-zero mean and arbitrary degree of reciprocity. Using a path-integral approach, we analyse the dynamics, averaged over the network ensemble, in the thermodynamic limit. Our results show that for any degree of reciprocity in the couplings, two types of criticality emerge, corresponding to ferromagnetic and spin-glass order, respectively. The critical lines separating the disordered from the ordered phases is consistent with spectral properties of the coupling matrix, as derived from random matrix theory. As the non-reciprocity (or asymmetry) in the couplings increases, both ordered phases diminish in size, ultimately resulting in the disappearance of the spin-glass phase when the couplings become anti-symmetric. We investigate non-fixed point steady-state solutions for uncorrelated interactions. For such solutions the time-lagged correlation function evolves according to a gradient-descent dynamics on a potential, which depends on the stationary variance. Our analysis shows that in the spin-glass region, the variance dynamically selected by the system leads the correlation function to evolve on the separatrix curve, limiting different realizable steady states, whereas in the ferromagnetic region, a fixed point solution is selected as the only realizable steady state. In the spin-glass region, stationary solutions are unstable against perturbations that break time-translation invariance, indicating chaotic behaviour in large single network instances. Numerical analysis of Lyapunov exponents confirms that chaotic behaviour emerges throughout the spin-glass region, for any value of the coupling correlations. While negative correlations increase the strength of chaos, positive ones reduce it, with chaos disappearing for reciprocal (i.e. symmetric) couplings, where marginal stability is attained. On the other hand, in finite size non-reciprocal networks, fixed points and limit cycles can arise in the spin-glass region, especially close to the critical line. Finally, we show that chaos is suppressed when the strength of external noise exceeds a certain threshold. Intriguing analogies between chaotic phases in non-equilibrium systems and spin-glass phases in equilibrium are put forward.