Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network Rings

Abstract

We study a Hopfield-type network that consists of a pair of one-way rings each with three neurons and two-way coupling between the rings. The rings have symmetric group Γ=Z3×Z2, which means the global symmetry Z2 and internal symmetry Z3. We discuss the spatiotemporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatiotemporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling; that is, different ranges of delays correspond to different patterns of neural network oscillators. The oscillations of corresponding neurons in the two loops can be in phase, antiphase, T/3, 2T/3, 4T/3, 5T/6, or 7T/6 periods out of phase depending on the delay. Some numerical simulations support our analysis results.