Robust Nonlinear Dynamics Conditions That Arise in Self-oscillatory Networks

Abstract

Self-oscillation is an emergent behavior naturally occurring in biological neural circuits, facilitating the coordination of complex locomotion and cognitive functions. Recently, numerous discrete dynamical system models, termed Self-Oscillatory Networks (SONs), have been proposed to model the functional behavior of such neural circuits. In brief, SONs are recurrent neural networks that generate spontaneous, self-sustaining rhythmic patterns without any input. However, the internal dynamics of SONs, especially in systems of high dimensionality, remain unexplored due to their complexity. This paper analyzes the robust nonlinear dynamics that arise within SONs. Through numerical analyses, we examine the influence of spectral radius on the emergence of dynamic attractors, particularly limit cycles. Following that, we identify the critical value of the spectral radius that induces a supercritical Hopf bifurcation in the system of SONs. We also perform stability analysis using Lyapunov exponents and phase shift to demonstrate that SONs exhibit robust behavior against perturbations. Therefore, we conclude that SONs contain cyclic attractors that maintain stable limit cycles, even under perturbations.