Events in Noise-Driven Oscillators: Markov Renewal Processes and the "Unruly" Breakdown of Phase-Reduction Theory

We introduce an extension to the standard reduction of oscillatory systems to a single phase variable. The standard reduction is often insufficient, particularly when the oscillations have variable amplitude and the magnitude of each oscillatory excursion plays a defining role in the impact of that oscillator on other systems, i.e. on its output. For instance, large excursions in bursting or mixed-mode neural oscillators may constitute events like action potentials, which trigger output to other neurons, while smaller, sub-threshold oscillations generate no output and therefore induce no coupling between neurons. Noise induces diffusion-like dynamics of the oscillator phase on top of its otherwise constant rate-of-change, resulting in the irregular occurrence of these output events. We model the events as corresponding to distinguished crossings of a Poincare section. Using a linearization of the noisy Poincare map and its description under phase-isostable coordinates, we determine the diffusion coefficient for the occurrence and timing of the events using Markov renewal theory. We show that for many oscillator models the corresponding point process can exhibit "unruly" diffusion: with increasing input noise strength the diffusion coefficient vastly increases compared to the standard phase reduction analysis, and, strikingly, it also decreases when the input noise strength is increased further. We provide a thorough analysis in the case of planar oscillators, which exhibit unruliness in a finite region of the natural parameter space. Our results in part explain the surprising synchronization behavior obtained in pulse-coupled, mixed-mode oscillators as they arise, e.g., in neural systems.