Phase synchronization of pulse-coupled excitable clocks

Consider a distributed network on a simple graph $G=(V,E)$ where each node has a phase oscillator revolving on $S^{1}=\mathbb{R}/\mathbb{Z}$ with unit speed. Pulse-coupling is a class of distributed time evolution rule for such networked clock system inspired by biological oscillators, where local interactions between neighboring nodes are event-triggered and do not depend on any kind of global information. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on any tree with maximum degree at most 3 and diameter $d$ synchronizes in at most $24d$ times. We extend this pulse-coupling by making each oscillator adjust its phase response curve depending on local events, and show that arbitrary initial configuration on any finite tree with diameter $d$ synchronizes in at most $51d$ times. As an application, we obtain a universal Las Vegas self-stabilizing distributed phase clock synchronization algorithm, using $O(\log \Delta)$ memory per node with sublinear $d^{O(1)}+O(\Delta^{2}\log |V|)$ expected time complexity, where $d$ and $\Delta$ denote the diameter and the maximum degree of $G$, respectively.