Non-linear waves in a ring of neurons

^** Corresponding author. Email: shangjguo{at}etang.com In this paper, we study the effect of synaptic delay of signaltransmission on the pattern formation and some properties ofnon-linear waves in a ring of identical neurons. First, linearstability of the model is investigated by analyzing the associatedcharacteristic transcendental equation. Regarding the delayas a bifurcation parameter, we obtained the spontaneous bifurcationof multiple branches of periodic solutions and their spatio-temporalpatterns. Second, global continuation conditions for Hopf bifurcatingperiodic orbits are derived by using the equivariant degreetheory developed by Geba et al. and independently by Ize &Vignoli. Third, we show that the coincidence of these periodicsolutions is completely determined either by a scalar delaydifferential equation if the number of neurons is odd, or bya system of two coupled delay differential equations if thenumber of neurons is even. Fourth, we summarize some importantresults about the properties of Hopf bifurcating periodic orbits,including the direction of Hopf bifurcation, stability of theHopf bifurcating periodic orbits, and so on. Fifth, in an excitatoryring network, solutions of most initial conditions tend to stableequilibria, the boundary separating the basin of attractionof these stable equilibria contains all of periodic orbits andhomoclinic orbits. Finally, we discuss a trineuron network toillustrate the theoretical results obtained in this paper andconclude that these theoretical results are important to complementthe experimental and numerical observations made in living neuronssystems and artificial neural networks, in order to understandthe mechanisms underlying the system dynamics better.