Probabilistic Convergence Guarantees for Type II Pulse Coupled Oscillators

TL;DR

This paper proves that many pulse coupled oscillators reliably synchronize with high probability on various graphs with delays, using probabilistic analysis and phase response curve classification.

Contribution

It introduces a new classification scheme for Type II phase response curves and provides rigorous probabilistic convergence bounds for pulse coupled oscillators.

Findings

High probability convergence on large graphs with delays

Development of a computational technique for convergence analysis

Insights into biological oscillators and sensor network synchronization

Abstract

We show that a large class of pulse coupled oscillators converge with high probability from random initial conditions on a large class of graphs with time delays. Our analysis combines previous local convergence results, probabilistic network analysis, and a new classification scheme for Type II phase response curves to produce rigorous lower bounds for convergence probabilities based on network density. These bounds are then used to develop a simple, fast and rigorous computational analytic technique. These results suggest new methods for the analysis of pulse coupled oscillators, and provide new insights into the operation of biological Type II phase response curves and also the design of decentralized and minimal clock synchronization schemes in sensor nets.