TY - JOUR
T1 - A variational method for analyzing limit cycle oscillations in stochastic hybrid systems
AU - Bressloff, Paul C.
AU - Maclaurin, James
N1 - Funding Information: P.C.B. and J.N.M. were supported by the National Science Foundation (Grant No. DMS-1613048). Publisher Copyright: © 2018 Author(s).
PY - 2018/6/1
Y1 - 2018/6/1
N2 - 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).
AB - 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).
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U2 - https://doi.org/10.1063/1.5027077
DO - https://doi.org/10.1063/1.5027077
M3 - Article
C2 - 29960393
SN - 1054-1500
VL - 28
JO - Chaos
JF - Chaos
IS - 6
M1 - 063105
ER -