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Threshold crossing time theory for quasicycles with application to brain rhythms
Mathias Roman1, Ivan L'Heureux1, Arthur Powanwe1
1Department of Physics, <a href="https://ror.org/03c4mmv16">University of Ottawa</a>, Ottawa, Ontario, Canada K1N 6N5.
Physical Review. E
|November 20, 2024
Summary
This study analyzes random amplitude fluctuations in physical and biological systems. We found that first return times (FRTs) to a threshold are governed by a single parameter and exhibit exponential behavior, offering insights into neural dynamics.
Area of Science:
- Physics
- Neuroscience
- Stochastic Processes
Background:
- The amplitude of a two-dimensional Ornstein-Uhlenbeck colored noise process follows a one-dimensional Rayleigh process, modeling random amplitude fluctuations in quasicycles.
- These quasicycles are noise-induced oscillations around an equilibrium with complex eigenvalues, relevant in physical and biological systems.
Purpose of the Study:
- To investigate the probability density of time intervals (first return times - FRTs) for amplitude staying below or above a fixed threshold.
- To characterize brain rhythm power excursions (bursts), avalanches, and branching processes using FRT statistics.
Main Methods:
- Utilizing a recently proposed technique involving Fokker-Planck eigenfunctions expansion and normalization correction to address non-normalizable FRT density.
- Comparing analytical expressions for FRT density with numerical realizations of the Rayleigh process across various parameters.
Main Results:
- Analytical FRT density expressions for the Rayleigh process show good agreement with numerical simulations for both above and below threshold trajectories.
- FRTs are governed by a single meta-parameter (Δ), the ratio of noise strength to linear stability coefficient.
- The mean FRT is invariant to the ratio of threshold to sqrt[Δ], and FRT density exhibits exponential behavior over time.
Conclusions:
- The study provides insights into threshold crossing time statistics in quasicycles and stochastic Wilson-Cowan neural equations.
- Established the absence of strict power-law scaling in these threshold-crossing statistics, revealing universal properties governed by a single meta-parameter.
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