Rapid Phase Locking in Systems of Pulse-Coupled Oscillators with Delays

The dynamical evolution of a system of integrate-and-fire units with delayed excitatory coupling is analyzed. The connectivity is arbitrary except for a normalization of the total input to each unit. It is shown that the system converges to a periodic solution where all units are phase locked but do not necessarily fire in unison. In the case of discrete and uniform delays, a periodic solution is reached after a finite time. For a delay distribution with finite support, an attractor is, in general, only reached asymptotically.