Local and Global Existences of Synchronized Periodic Solutions in a Tri-neuron Network Model with Delay

The coupled identical FHN models with synaptic connection are considered in this paper. The synchronization conditions of the tri-neuron network model are first given and the existence of local Hopf bifurcations of synchronized solutions is investigated, then the normal form method and center manifold theory are employed to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. A global Hopf bifurcations theorem due to Wu and Bendixson´s criterion are used to obtain the existence conditions of synchronized periodic solutions when the delay is sufficiently large. Finally, numerical simulations are carried out to support the analytic results.