Advances in Applied Probability

Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions

Alexandre Genadot and Michèle Thieullen

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In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this `two-time-scale' model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

Article information

Adv. in Appl. Probab., Volume 44, Number 3 (2012), 749-773.

First available in Project Euclid: 6 September 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60J75: Jump processes 35K57: Reaction-diffusion equations
Secondary: 92C20: Neural biology 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) [See also 80A30]

Piecewise-deterministic Markov process fully coupled system averaging principle reaction diffusion equation slow-fast system Markov chain neuron model Hodgkin-Huxley model


Genadot, Alexandre; Thieullen, Michèle. Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions. Adv. in Appl. Probab. 44 (2012), no. 3, 749--773. doi:10.1239/aap/1346955263.

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  • Austin, T. D. (2008). The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism. Ann. Appl. Prob. 18, 1279–1325.
  • Buckwar, R. and Riedler, M. G. (2011). An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J. Math. Biol. 63, 1051–1093.
  • Chow, C. C. and White, J. A. (1996). Spontaneous action potentials due to channel fluctuations. Biophys. J. 71, 3013–3021.
  • Costa, O. L. V. and Dufour, F. (2008). Stability and ergodicity of piecewise deterministic Markov processes. SIAM J. Control Optimization 47, 1053–1077.
  • Costa, O. L. V. and Dufour, F. (2011). Singular perturbation for the discounted continuous control of piecewise deterministic Markov processes. Appl. Math. Optimization 63, 357–384.
  • Crudu, A., Debussche, A., Muller, A. and Radulescu, O. (2012). Convergence of stochastic gene networks to hybrid piecewise deterministic processes. To appear in Ann. Appl. Prob.
  • Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.
  • Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman and Hall, London.
  • Defelice, L. J. and Isaac, A. (1993). Chaotic states in a random world: relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels. J. Statist. Phys. 70, 339–354.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.
  • Faggionato, A., Gabrielli, D. and Crivellari, M. R. (2010). Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes and applications to molecular motors. Markov Process. Relat. Fields 16, 497–548.
  • Faisal, A. A., White, J. A. and Laughlin, S. B. (2005). Ion-channel noise places limits on the miniaturization of the brain's wiring. Current Biol. 15, 1143–1149.
  • Fitzhugh, R. (1960). Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J. Gen. Physiol. 43, 867–896.
  • Flaim, S. N., Gilles, W. R. and McCulloch, A. D. (2006). Contributions of sustained $I_{\text{Na}}$ and $I_\text{Kv43}$ to transmural heterogeneity of early repolarization and arrhythmogenesis in canine left ventricular myocytes. Am. J. Physiol. Heart Circ. Physiol. 291, 2617–2629.
  • Fonbona, J., Guérin, H. and Malrieu, F. (2012). Quantitative estimates for the long time behavior of a PDMP describing the movement of bacteria. Submitted.
  • Fox, R. F. (1997). Stochastic versions of the Hodgkin-Huxley equations. Biophys. J. 72, 2068–2074.
  • Greenstein, J. L., Hinch, R. and Winslow, R. L. (2006). Mechanisms of excitation-contraction coupling in an integrative model of the cardiac ventricular myocyte. Biophys. J. 90, 77–91.
  • Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations (Lecture Notes Math. 840). Springer, Berlin.
  • Hille, B. (1992). Ionic Channels of Excitable Membranes, 2nd edn. Sinauer, Sunderland, MA.
  • Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • Métivier, M. (1984). Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev. Ann. Inst. H. Poincaré Prob. Statist. 20, 329–348.
  • Pakdaman, K., Thieullen, M. and Wainrib, G. (2012). Asymptotic expansion and central limit theorem for multiscale piecewise deterministic Markov processes. To appear in Stoch. Process Appl.
  • Pavliotis, G. A. and Stuart, A. M. (2008). Multiscale Methods (Texts Appl. Math. 53). Springer, New York.
  • Riedler, M. G. (2011). Almost sure convergence of numerical approximations for piecewise deterministic Markov processes. Preprint. Available at
  • Riedler, M. G., Thieullen, M. and Wainrib, G. (2012). Limit theorems for infinite-dimensional piecewise deterministic processes and applications to stochastic neuron models. Submitted.
  • Yin, G. G. and Zhang, Q. (1997). Continuous-Time Markov Chains and Applications. Springer, Berlin.