Bifurcation, Synchronization, and Multistability of Two Interacting Networks with Multiple Time Delays

This paper reveals the dynamical properties of two interacting neural networks with multiple couplings. Different time delays are introduced into the nearest-neighbor links and long-range connections in each layer and the couplings between different substructures. The delay-dependent and delay-independent stability and the oscillations bifurcated from the trivial equilibrium of the network are analyzed. The conditions of the existence of nontrivial equilibria and pitchfork bifurcation are discussed. Numerical simulations are performed to validate the theoretical results and interesting neuronal activities are observed, such as completely synchronous oscillations, three types of asynchronous oscillations, multiple switches between the rest states and different oscillations, coexistence of different oscillations, and coexistence of nontrivial equilibria and oscillations.