Patterns of oscillation in a Ring of Identical Cells with Delayed Coupling

We investigate the behaviour of a neural network model consisting of three neurons with delayed self and nearest-neighbour connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D3-equivariant, Hopf and pitchfork bifurcation points. We use numerical simulation and numerical bifurcation analysis to investigate the dynamics near the pitchfork-Hopf interaction points. Our numerical investigations reveal that multiple secondary Hopf bifurcations and pitchfork bifurcations of limit cycles may emanate from the pitchforkHopf points. Further, these secondary bifurcations give rise to 10 different types of periodic solutions. In addition, the secondary bifurcations can lead to multistability between equilibrium points and periodic solutions in some regions of parameter space. We conclude by generalizing our results into conjectures about the secondary bifurcations that emanate from codimension two pitchfork-Hopf bifurcation points in systems with Dn symmetry.