Many natural systems exhibit oscillations that show sizeable fluctuations in frequency and amplitude. This variability can arise from a wide variety of physical mechanisms. Phase descriptions that work for deterministic oscillators have a limited applicability for stochastic oscillators. In my talk I review attempts to generalize the phase concept to stochastic oscillations, specifically, the mean-return-time phase and the asymptotic phase.
For stochastic systems described by Fokker-Planck and Kolmogorov-backward equations, I introduce a mapping of the system’s variables to a complex pointer (instead of a real-valued phase) that is based on the eigenfunction of the Kolmogorov equation. Under the new (complex-valued) description, the statistics of the oscillator’s spontaneous activity, of its response to external perturbations, and of the coordinated activity of (weakly) coupled oscillators, is brought into a universal and greatly simplified form. The theory is tested for three theoretical models of noisy oscillators arising from fundamentally different mechanisms: a damped harmonic oscillator with dynamical noise, a fluctuation-perturbed limit-cycle system, and an excitable system in which oscillations require noise to occur.
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