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

Abstract

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.

Figures (16)

• Figure 1: Effective Diffusion in a Mixed-Mode Oscillatorkaramchandani_pulse-coupled_2018. (a): Sample voltage traces of a neural mixed-mode oscillator driven by noise. Only the large-amplitude excursions are events in which the oscillator produces “ output” ; the small-amplitude excursions are non-events. In the absence of noise, a fixed, periodic pattern of alternating large-amplitude and small-amplitude oscillations is produced. With noise $D_{\textit{in}}$, “ phase slips” perturb the pattern, in which events are added or omitted. (b): The phase diffusion $D_{\textit{phase}}$ from phase reduction theory and the effective diffusion $D_{\textit{eff}}$ (same data shown on two different scales), treating large-amplitude oscillations as events. As a function of the input diffusion strength, $D_{\textit{eff}}$ initially agrees with $D_{\textit{phase}}$. But it becomes greatly amplified for moderate noise strengths and then eventually decreases as the noise strength increases. We call this qualitative non-monotonic behavior “ unruly” . (c): A phase diagram for population states in globally coupled oscillators. Standard phase reduction with the Fokker-Planck theory, (\ref{['eq:Fokker-Planck']}), predicts that the boundary between stable coherent states and stable incoherence is linear. The theory only agrees with the full, coupled-oscillator simulation once $D_{\textit{phase}}$ is replaced with $D_{\textit{eff}}$.
• Figure 2: Events in a Noise-driven Limit-cycle Oscillator. (a): Noisy trajectory crossing a Poincare section $S$ and an event sub-section $E$. (b): The linearized, stochastic, discrete-time dynamics for $x_{k}$ on the Poincare section $S$. The shading gives the probability of $x_{k+1}$ conditioned on $x_{k}$. (c): The designation $1_{E}\left(x_{k}\right)$ of each $x_{k}$ as an event or non-event and the two associated point processes, $T_{k}^{S}$ and $T_{k}^{E}$.
• Figure 3: Temporal variance growth rate $\mathcal{V}_{E}^{\left(t\right)}$ for the 2-state toy model (\ref{['eq:toy_model']}). The “ Markov-only” and mixed components of the TVGR are non-monotonic and the temporal component is monotonic. In (a), the oscillatory case, their sum, the full TVGR, has the characteristic unruly quality: an initial linear growth, a strong nonlinear amplification, a maximum, and a subsequent decrease. The choice of an asymmetric interval $E_{\Delta T}$ about $\Delta T^{S}=1$ is only made so that $\overline{\Delta t}_{1}\ne\overline{\Delta t}_{0}$ and the mixed term is nonzero. It is not essential to the appearance of unruliness. In (b), the excitable case, the TVGR is missing the initial linear growth.
• Figure 4: The $D_{\textit{in}}$-dependent Elements of the TVGR $\mathcal{V}_{E}^{\left(t\right)}$. (a): A schematic depiction of the steady-state distribution on the Poincare section $S$ (black line) for $\delta>0$ and various $D_{\textit{in}}$. $\mathrm{\mathcal{E}}$ is the probability mass associated with $E$ (red shading above the red interval), and $x_{E}$ is its center of mass. (b): Plots of the TVGR elements as a function $D_{\textit{in}}$ for various values of $\delta$. The gray arrows indicate roughly how the graphs change with increasing $\delta$; all converge to the solid black curve as $\delta\rightarrow\frac{1}{2}$. The top row shows the “ raw” elements, $\mathcal{E}$ and $x_{E}$. Note that $x_{E}$ is identically $0$ for $\delta=0$. In the quasi-renewal limit (see Section \ref{['subsec:The-Quasi-Renewal-Case']}), the TVGR is a linear combination of the components plotted in (b.iii) and (b.v), $\mathcal{E}\left(1-\mathcal{E}\right)$ and $D_{\textit{in}}\mathcal{E}^{2}$.
• Figure 5: Unruliness requires $c$ to be sufficiently small. Sample TVGRs in the quasi-renewal limit that are (a) not unruly with $c$ above the unruliness threshold $c_{crit,r}$, (b) marginally unruly with $c$ at the threshold, and (c) unruly with $c$ below the threshold (\ref{['eq: c_crit quasi-renewal']}). The presence of a local maximum (red circle) does not distinguish these three cases. In the plots, we take $\delta=\frac{1}{4}$, but they are qualitatively representative for $\delta\in\left[0,\frac{1}{2}\right)$.