Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system

We consider a network of three identical neurons with multiple discrete signal transmission delays. The model for such a network is a system of nonlinear delay differential equations. After some consideration of the absolute synchronization of the system and the global attractivity of the zero solution, we present a detailed discussion about the boundaries of the stability region of the trivial solution. This allows us to determine the possible codimension one bifurcations which occur in the system. In particular, we show the existence of standard Hopf bifurcations giving rise to synchronized periodic solutions and of D3D3 equivariant Hopf bifurcations giving rise to three types of periodic solutions: phase-locked, mirror-reflecting, and standing waves. Hopf–Hopf and Hopf–steady state bifurcations interactions are shown to exist and give rise to coexistence of stable synchronized and desynchronized solutions. Perturbation techniques coupled with the Floquet theory are used to determine the stability of the phase-locked oscillations.