High-order synchronization in identical neurons with asymmetric pulse coupling.

The phenomenon of high-order (p/q) synchronization, induced by two different frequencies in the system, is well known and studied extensively in forced oscillators including neurons and to a lesser extent in coupled oscillators. Their frequencies are locked such that for every p cycles of one oscillator there are q cycles of the other. We demonstrate this phenomenon in a pair of coupled neurons having identical frequencies but asymmetric coupling. Specifically, we focus on an excitatory-inhibitory (E-I) neuron pair where such an asymmetry is naturally present even with equal reciprocal synaptic strengths (g) and inverse time constant (α). We thoroughly investigate the asymmetric coupling-induced p/q frequency-locking structure in (g,α) parameter space through simulations and analysis. Simulations display quasiperiodicity, devil staircase, a Farey arrangement of spike sequences, and presence of reducible and irreducible p/q regions. We introduce an analytical method, based on event-driven maps, to determine the existence and stability of any spike sequence of the two neurons in a p/q frequency-locked state. Specifically, this method successfully deals with nonsmooth bifurcations and we could utilize it to obtain solutions for the case of identical E-I neuron pair under arbitrary coupling strength. In contrast to the so-called Arnold tongues, the p/q regions obtained here are not structureless. Instead they have their own internal bifurcation structure with varying levels of complexity. Intrasequence and intersequence multistability, involving spike sequences of same p/q state, are found. Additionally, multistability also arises by overlap of p/q with p^{'}/q^{'}. The boundaries of both reducible and irreducible p/q regions are defined by saddle node and nonsmooth grazing bifurcations of various types.