Threshold crossing time theory for quasicycles with application to brain rhythms

The amplitude of a two-dimensional Ornstein-Uhlenbeck colored noise process evolves according to the one-dimensional Rayleigh process. This is a general model for the random amplitude fluctuations of a quasicycle, i.e., of a noise-induced oscillation around an equilibrium with complex eigenvalues in physical and biological systems. We consider the probability density of time intervals during which the amplitude is either below or above a fixed threshold. The statistics of such first return times (FRTs) are of particular interest in neuroscience to characterize brain rhythm power excursions known as bursts, as well as avalanches and other branching processes. In contrast with the density of first passage times computed using Fokker-Planck theory between a start point and a different endpoint, the density of FRTs is non-normalizable. A recently proposed technique reframes the problem using an expansion of the Fokker-Planck eigenfunctions along with a correction to the normalization. Analytical expressions for the FRT density for the Rayleigh process are shown to be in good agreement with those computed from numerical realizations over a wide range of parameters, both for trajectories above and below threshold. Special care is required to evaluate the theory above threshold due to the crowded roots of the Tricomi confluent hypergeometric function. The results provide insight into the statistics of threshold crossing times in quasicycles generally, and in the stochastic Wilson-Cowan neural equations in particular. Surprisingly, FRTs are governed by a single meta-parameter Δ given by the ratio of the noise strength and the linear stability coefficient. We find the universal property that the mean FRT is invariant to the ratio of threshold to sqrt[Δ]. The FRT density further exhibits exponential behavior over medium to long timescales, and mixtures of exponentials at shorter FRTs, thereby establishing the absence of strict power-law scaling in these threshold-crossing statistics.