A variational method for analyzing limit cycle oscillations in stochastic hybrid systems
File(s)
Published version
Author(s)
Bressloff, Paul C
MacLaurin, James
Type
Journal Article
Abstract
Many systems in biology can be modeled through ordinary differential equations, which are piece-wise continuous, and switch between different states according to a Markov jump process known as a stochastic hybrid system or piecewise deterministic Markov process (PDMP). In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we develop a phase reduction method for stochastic hybrid systems that support a stable limit cycle in the deterministic limit. A classic example is the Morris-Lecar model of a neuron, where the switching Markov process is the number of open ion channels and the continuous process is the membrane voltage. We outline a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the switching rate ϵ−1
. That is, we show that for a constant C, the probability that the expected time to leave an O(a) neighborhood of the limit cycle is less than T scales as T exp (−Ca/ϵ)
.
. That is, we show that for a constant C, the probability that the expected time to leave an O(a) neighborhood of the limit cycle is less than T scales as T exp (−Ca/ϵ)
.
Date Issued
2018-06
Date Acceptance
2018-05-18
Citation
Chaos: an interdisciplinary journal of nonlinear science, 2018, 28 (6)
ISSN
1054-1500
Publisher
American Institute of Physics
Journal / Book Title
Chaos: an interdisciplinary journal of nonlinear science
Volume
28
Issue
6
Copyright Statement
This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Paul C. Bressloff, James MacLaurin; A variational method for analyzing limit cycle oscillations in stochastic hybrid systems. Chaos 1 June 2018; 28 (6): 063105. and may be found at https://doi.org/10.1063/1.5027077
Identifier
http://dx.doi.org/10.1063/1.5027077
Publication Status
Published
Article Number
063105
Date Publish Online
2018-06-05