Advances in Applied Probability

Fluid limit theorems for stochastic hybrid systems with application to neuron models

K. Pakdaman, M. Thieullen, and G. Wainrib

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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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Adv. in Appl. Probab. Volume 42, Number 3 (2010), 761-794.

First available in Project Euclid: 27 August 2010

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Mathematical Reviews number (MathSciNet)

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60J75: Jump processes
Secondary: 92C20: Neural biology 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) [See also 80A30]

Stochastic hybrid system piecewise-deterministic Markov process fluid limit Langevin approximation kinetic model neuron model Hodgkin-Huxley Morris-Lecar stochastic ion channels


Pakdaman, K.; Thieullen, M.; Wainrib, G. Fluid limit theorems for stochastic hybrid systems with application to neuron models. Advances in Applied Probability 42 (2010), no. 3, 761--794. doi:10.1239/aap/1282924062.

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