Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay

In this paper, the dynamic behaviors of coupled reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay under the Neumann boundary conditions are investigated. By constructing a basis of phase space based on the eigenvectors of Laplace operator, the characteristic equation of this system is obtained. Then, the local stability of zero solution and the occurrence of Hopf bifurcation are established by regarding the time delay as the bifurcation parameter. In particular, by using the normal form theory and center manifold theorem of the partial differential equation, the normal forms are obtained, which determine the bifurcation direction and the stability of the periodic solutions. Finally, two examples are given to verify the theoretical results.