Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay

In this paper, we study the effect of synaptic delay of signal transmission on the pattern formation and global continuation of nonlinear waves in a single-directed excitatory ring of four identical neurons. Firstly, under some suitable conditions, solutions of most initial conditions tend to stable equilibria, trajectories on the boundary separating the basin of attraction of these stable equilibria tend either to the unstable equilibrium, or are asymptotically periodic, but there are no orbits homoclinic to the unstable equilibrium. Secondly, linear stability of the model is investigated by analysing the associated characteristic transcendental equation. Thirdly, by using the symmetric bifurcation theory of delay differential equations coupled with representation theory of Lie groups, we discuss the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Finally, global existence conditions for Hopf bifurcating periodic orbits are derived using the equivariant degree theory developed by Geba et al (1994 Bull. Lond. Math. Soc. 69 377–98) and independently by Ize and Vignoli (2003 Equivariant Degree Theory (Berlin: de Gruyter & Co)). These theoretical results are important to complement the experimental and numerical observations made in living neuron systems and artificial neural networks, in order to better understand the mechanisms underlying the system's dynamics.