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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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.

Article information

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

Dates
First available in Project Euclid: 27 August 2010

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

Subjects
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]

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

Citation

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. http://projecteuclid.org/euclid.aap/1282924062.


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