Synchrony, stability, and firing patterns in pulse-coupled oscillators

We study non-trivial firing patterns in small assemblies of pulse-coupled oscillatory maps. We find conditions for the existence of waves in rings of coupled maps that are coupled bi-directionally. We also find conditions for stable synchrony in general all-to-all coupled oscillators. Surprisingly, we find that for maps that are derived from physiological data, the stability of synchrony depends on the number of oscillators. We describe rotating waves in two-dimensional lattices of maps and reduce their existence to a reduced system of algebraic equations which are solved numerically.