Abstract
Stochastic oscillations can be characterized by a corresponding point process; this is a common practice in computational neuroscience, where oscillations of the membrane voltage under the influence of noise are often analyzed in terms of the interspike interval statistics, specifically the distribution and correlation of intervals between subsequent threshold-crossing times. More generally, crossing times and the corresponding interval sequences can be introduced for different kinds of stochastic oscillators that have been used to model variability of rhythmic activity in biological systems. In this paper we show that if we use the so-called mean-return-time (MRT) phase isochrons (introduced by Schwabedal and Pikovsky) to count the cycles of a stochastic oscillator with Markovian dynamics, the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process. We first outline the general mathematical argument for this finding and illustrate it numerically for three models of increasing complexity: (i) the isotropic Guckenheimer–Schwabedal–Pikovsky oscillator that displays positive interspike interval (ISI) correlations if rotations are counted by passing the spoke of a wheel; (ii) the adaptive leaky integrate-and-fire model with white Gaussian noise that shows negative interspike interval correlations when spikes are counted in the usual way by the passage of a voltage threshold; (iii) a Hodgkin–Huxley model with channel noise (in the diffusion approximation represented by Gaussian noise) that exhibits weak but statistically significant interspike interval correlations, again for spikes counted when passing a voltage threshold. For all these models, linear correlations between intervals vanish when we count rotations by the passage of an MRT isochron. We finally discuss that the removal of interval correlations does not change the long-term variability and its effect on information transmission, especially in the neural context.
Highlights
A number of biological systems of rather different nature display stochastic oscillations
We report the remarkable observation that counting rotations in terms of the MRT phase in planar whitenoise-driven oscillators leads to a sequence of interphase intervals (IPIs), for which linear correlations vanish
We count rotations by the crossings of, or, put differently, the return to this curve after the completion of a rotation. The latter condition is important: Crossings of a curve are a subtle issue in dynamical systems driven by white noise because even if we restrict the crossings to be counted only when occurring into the direction of rotation, there will be infinitely many of them in a finite time if we count them in a naive way without the condition of the completed rotation (for the general problem of the number of crossings for a stochastic process, see Stratonovich (1967))
Summary
A number of biological systems of rather different nature display stochastic oscillations. Cao et al (2020) proposed an analytical definition for a special class of planar white-noise-driven oscillators, which is based on the well-known partial differential equation for the mean-firstpassage time with an unusual jump condition Another simplifying approach to oscillatory systems is to associate a point process with the repetitive features of the system: In neurons, for instance, upcrossings of a voltage threshold have been used to define a spike train or, equivalently, an ordered sequence of interspike intervals (ISIs); in heart dynamics, the intervals between heartbeats have been analyzed in a similar way. We report the remarkable observation that counting rotations in terms of the MRT phase in planar whitenoise-driven oscillators leads to a sequence of interphase intervals (IPIs), for which linear correlations vanish. We conclude the paper with a brief discussion of the implications of our result for modeling stochastic oscillations
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