Hopf bifurcating periodic orbits in a ring of neurons with delays

In this paper, we consider a ring of neurons with self-feedback and delays. The linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Based on the normal form approach and the center manifold theory, we derive the formula for determining the properties of Hopf bifurcating slowly oscillating periodic orbits for a ring of neurons with delays, including the direction of Hopf bifurcation, stability of the Hopf bifurcating slowly oscillating periodic orbits, and so on. Moreover, by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delay of signal transmission on the pattern formation, but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns.