Stability switching and Hopf bifurcation in a multiple-delayed neural network with distributed delay

We consider a network of three identical neurons incorporating distributed and discrete signal transmission delays. The model for such a network is a system of coupled nonlinear delay differential equations. It is established that two cases of a single Hopf bifurcation may occur at the trivial equilibrium of the system, as a consequence of the D3D3 symmetry of the network. These single Hopf bifurcations are the simple and the double root. The present paper looks at the simple root case, and addresses the issue of absolute stability of the trivial equilibrium and stability switching, leading up to calculation of the critical delay and formulation of a Hopf bifurcation theorem.