Dynamical phases and computation in nonlinear networks with correlated couplings

Recurrent networks with random couplings can serve as minimal models of diverse systems including neural networks, large ecosystems, and annealers for combinatorial optimization. The statistics of fixed points and chaos in different parameter regimes can be studied using random matrix methods and dynamic mean-field theory. To date, such studies have usually assumed network units without intrinsic dynamics beyond linear relaxation, and treatments with nonlinear self-couplings have only considered uncorrelated cross-couplings. We extend these methods to networks with both nonlinear self-couplings and nonzero coupling covariances, and analyze the disintegration of the energy landscape and transition to chaos as the network is tuned away from symmetric couplings. We study the computational relevance of different dynamical phases for the operation of modern neuromorphic annealing hardware, quantifying both the implications of unwanted asymmetric coupling noise and the potential for advantageous chaotic annealing via a tunable coupling asymmetry. Along the way, we derive self-consistent equations for the spectrum of the most general class of large Gaussian random matrices.