A variational method for analyzing limit cycle oscillations in stochastic hybrid systems

Paul C. Bressloff, James Maclaurin

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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).

Original languageEnglish (US)
Article number063105
Issue number6
StatePublished - Jun 1 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics


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