Analysis of the “Unruly” Breakdown of Phase Reduction Theory: an Application of Markov Renewal Processes to Noise-driven Oscillators
Abstract: We investigate the noise response properties of nonlinear, dissipative oscillators, i.e. systems of ODEs that have a stable limit cycle in the absence of noise. We find that the response of such oscillators to noise is often quite “unruly”: measures of the output diffusion are non-monotonic in the input noise strength and greatly enhanced as compared to a linear prediction from the standard phase reduction analysis.
The dynamics of the principal neurons of the mammalian olfactory bulb serves as a motivating example. The cell's membrane potential exhibits mixed-mode oscillations (MMOs) with both the large amplitude, action potential “spikes” typical of neurons as well as small amplitude, sub-threshold oscillations (STOs). Noisy input from upstream neurons can strongly perturb the patterns of spikes and STOs displayed, as is reflected in an unruly diffusion coefficient for spike times. The noise-driven perturbations can greatly affect the spike-driven interactions between cells, and we note the effect that the enhanced diffusion has on coherent, population-level behavior.
Finally, we propose a technique for the analysis of noise-driven oscillators that accounts for unruly corrections to the standard phase reduction. We center our analysis on linearized, random Poincare map dynamics. Taking the segregation of MMOs into spikes and STOs as motivation, we label each crossing of the Poincare section as either an event of interest (e.g. a spike) or a non-event. Using Markov renewal theory, we compute the diffusion coefficient of the resulting point process for the occurrence and timing of events. We argue that for many oscillator models, the corresponding point process exhibits unruly diffusion. In particular, we offer a thorough analysis in the case of planar oscillators, which exhibit unruliness in a finite region of a natural parameter space.
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