Dynamically selected steady states and criticality in non-reciprocal networks

Abstract

Diverse equilibrium systems with heterogeneous interactions lie at the edge of stability. Such marginally stable states are dynamically selected as the most abundant ones or as those with the largest basins of attraction. On the other hand, systems with non-reciprocal (or asymmetric) interactions are inherently out of equilibrium, and exhibit a rich variety of steady states, including fixed points, limit cycles and chaotic trajectories. How are steady states dynamically selected away from equilibrium? We address this question in a simple neural network model, with a tunable level of non-reciprocity. Our study reveals different types of ordered phases and it shows how non-equilibrium steady states are selected in each phase. In the spin-glass region, the system exhibits marginally stable behaviour for reciprocal (or symmetric) interactions and it smoothly transitions to chaotic dynamics, as the non-reciprocity (or asymmetry) in the couplings increases. Such region, on the other hand, shrinks and eventually disappears when couplings become anti-symmetric. Our results are relevant to advance the knowledge of disordered systems beyond the paradigm of reciprocal couplings, and to develop an interface between statistical physics of equilibrium spin-glasses and dynamical systems theory.

References in corpus (11)

• Landau-Ginzburg theory of cortex dynamics: Scale-free avalanches emerge at the edge of synchronization
• Marginal Stability in Structural, Spin and Electron Glasses
• Topological and Dynamical Complexity of Random Neural Networks
• Generalized Lotka-Volterra equations with random, non-reciprocal interactions: the typical number of equilibria
• The window at the edge of chaos in a simple model of gene interaction networks
• Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates
• Extended Anderson Criticality in Heavy-Tailed Neural Networks
• Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems
• Competitive exclusion and Hebbian couplings in random generalised Lotka-Volterra systems
• Double-replica theory for evolution of genotype-phenotype interrelationship
• Large-scale fluctuations of the largest Lyapunov exponent in diffusive systems

Cited by in corpus (4)

• Dynamical Mean-Field Theory of Complex Systems on Sparse Directed Networks
• Dynamical Theory for Adaptive Systems
• Irreversibility in Non-reciprocal Chaotic Systems
• Zero-temperature dynamics of the spherical model with non-reciprocal interactions