Reduced Models for Noise-driven Limit-cycle Oscillators

Karamchandani, Avinash Jagdish

doi:https://doi.org/10.21985/n2-hpq8-3e02

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Reduced Models for Noise-driven Limit-cycle Oscillators

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Motivated by rhythms in the brain, we investigate the synchronization of noisy and all-to-all pulse-coupled oscillators. We consider a case where the oscillatory excursions are of varying amplitude and where only sufficiently large excursions result in the output pulses that drive the interactions between the oscillators. In the regime of weak noise and weak interaction strength, we use a standard reduction to a Kuramoto-type phase-description with continuous, non-sinusoidal coupling and the associated Fokker-Planck equation for the population density of oscillator phases. Its linear stability analysis identifies the appearance of various emergent cluster states. Direct simulations of the oscillator equations reveal that, in order to achieve quantitative agreement with the phase reduction theory, the coupling strength and the noise have to be extremely small. Even moderate noise leads to significant variation in the timing of the large oscillations, which can enhance the diffusion coefficient in the Fokker-Planck equation by orders of magnitude. Introducing an effective diffusion coefficient extends the range of agreement significantly. We find that the effective diffusion coefficient, which can be computed efficiently via simulations of a single, noise-driven oscillator, is highly nonlinear as of a function of the input noise strength. In a broader setting, for any limit-cycle oscillator that produces output conditional on the amplitude of its oscillations, we also treat the effective diffusion coefficient theoretically. In a novel framework, we model the outputs by “events” that correspond to distinguished crossings of a Poincare section. Using a linearization of the noisy Poincare map and its description under phase-isostable coordinates, we derive the effective 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 reduction analysis, and, strikingly, it also decreases when the input noise strength is increased further. The appearance of ``unruliness” thus reflects a break down of the standard phase reduction and, in the context of coupled oscillators, of the Fokker-Planck theory. We provide a thorough analysis in the case of planar oscillators, which exhibit unruliness in a finite region of the natural parameter space.

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