Fluid limit theorems for stochastic hybrid systems with application to neuron models
K. Pakdaman, M. Thieullen, and G. Wainrib
Source: Adv. in Appl. Probab. Volume 42, Number 3
(2010), 761-794.
Abstract
In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.
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Keywords: Stochastic hybrid system; piecewise-deterministic Markov process; fluid limit; Langevin approximation; kinetic model; neuron model; Hodgkin-Huxley; Morris-Lecar; stochastic ion channels
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Permanent link to this document: http://projecteuclid.org/euclid.aap/1282924062
Digital Object Identifier: doi:10.1239/aap/1282924062
Zentralblatt MATH identifier: 05820052
Mathematical Reviews number (MathSciNet): MR2779558
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