Pattern formation and continuation in a trineuron ring with delays

In this paper, we consider a single-directional ring of three neurons with delays. First, linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Next, we studied the local Hopf bifurcations and the spatio-temporal patterns of Hopf bifurcating periodic orbits. Basing on the normal form approach and the center manifold theory, we derive the formula for determining the properties of Hopf bifurcating periodic orbit, such as the direction of Hopf bifurcation. Finally, global existence conditions for Hopf bifurcating periodic orbits are derived by using degree theory methods.