Zero-Hopf bifurcation and multistability coexistence on a four-neuron network model with multiple delays

In this paper, a four-neuron neural system with four delays is investigated to exhibit the effects of multiple delays and coupled weights on system dynamics. Zero-Hopf bifurcation is obtained, where the system characteristic equation has a simple zero and a simple pair of pure imaginary eigenvalues. The coupled weight and time delay are considered as bifurcation parameters to study dynamical behaviors derived from zero-Hopf bifurcation. Various dynamical behaviors are analyzed near the bifurcation singularity qualitatively and quantitatively in detail by using perturbation- incremental method, and bifurcation diagrams are obtained. Numerical simulations and theoretical results are given to display a stable resting state, multistability coexistence of two resting states and a pair of periodic activities in the neighbor of the zero-Hopf bifurcation point.