Rapid Phase Locking in Systems of Pulse-Coupled Oscillators with Delays
Physical Review Letters
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
... al, only reached asymptotically. PACS numbers: 87.10.+e, 05.20.-y, 64.60.Ht Networks of pulse-coupled oscillators have attracted an increasing amount of interest [1-6]. Theoretical results on synchronization and phase locking have been applied to a wide range of phenomena including synchronously flashing fireflies , biological clocks , oscillating neuronal activity , and earthquake cycles . Most of the analytical studies [1,2] have focused on fully connected networks where mean-field methods can be applied. In a different line of research aiming at an understanding of self-organized criticality, networks with local connections have been studied    . Recently, it has been shown that homogeneous networks of integrate-and-fire units with arbitrary, local or long-ranged, connectivity and no leakage are amenable to mathematical analysis whenever the total input to each unit is normalized . In the model network presented below we use this general class of connectivity. In systems of identical integrate-and-fire units without leakage, many degenerate cyclic solutions with the same period can coexist    . The convergence time to the set of periodic solutions is short. More precisely, a periodic solution is reached as soon as every unit has fired once . This result, however, is limited to networks with delayless interaction and instantaneous reset of the state variable after each firing. Naturally, the question arises whether fast phase locking is specific for the delayless situation or generic in the sense that it holds for a broader class of oscillator models. In this paper, oscillator networks with delayed excitatory interaction and partially delayed reset are studied. Such systems can be considered as extremely simplified models of neural networks or earthquake faults  . There are several questions concerning the dynamics of such systems. Are there periodic solutions? If so, what is their period? What is the asymptotic system behavior? How fast is an attractor reached? These questions are addressed below. It is shown that all attractors are periodic. As in the delayless case, units are phase locked but not necessarily in synchrony. Furthermore, the attractors are reached after a finite time, if delays are discrete and shorter than the period of a cyclic solution.