Fast numerical integration of relaxation oscillator networks based on singular limit solutions

Relaxation oscillations exhibiting more than one time scale arise naturally from many physical systems. When relaxation oscillators are coupled in a way that resembles chemical synapses, we propose a fast method to numerically integrate such networks. The numerical technique, called the singular limit method, is derived from analysis of relaxation oscillations in the singular limit. In such a limit, system evolution gives rise to time instants at which fast dynamics take place and intervals between them during which slow dynamics take place. A full description of the method is given for a locally excitatory globally inhibitory oscillator network (LEGION), where fast dynamics, characterized by jumping which leads to dramatic phase shifts, is captured in this method by iterative operation and slow dynamics is entirely solved. The singular limit method is evaluated by computer experiments, and it produces remarkable speedup compared to other methods of integrating these systems. The speedup makes it possible to simulate large-scale oscillator networks.