Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays

This paper reveals the dynamical behaviors of a coupled network consisting of four neural sub-networks and multiple two-way couplings of neurons between the individual sub-networks. Different types of time delays are introduced into the internal connections within the individual sub-networks and the couplings between the sub-networks. Stability switches of the network equilibrium and Hopf bifurcation are analyzed by discussing the associated characteristic equation. The conditions for the existence of different patterns of oscillations are discussed. By using the Lyapunov's second method, the global stability of the network equilibrium is studied. Numerical simulations are performed to validate the theoretical results, and rich dynamical behaviors are observed, such as synchronous oscillations, multiple stability switches between the rest state and oscillations, phase-locked oscillations, asynchronous oscillations, and the coexistence of different patterns of bifurcated oscillations.