Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-rtmr9 Total loading time: 0.223 Render date: 2021-06-17T04:52:40.200Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions

Published online by Cambridge University Press:  04 January 2016

Alexandre Genadot
Affiliation:
Université Pierre et Marie Curie
Michèle Thieullen
Affiliation:
Université Pierre et Marie Curie
Corresponding
Rights & Permissions[Opens in a new window]

Abstract

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.

Type
General Applied Probability
Copyright
© Applied Probability Trust 

Footnotes

This work was supported by the Agence Nationale de la Recherche through the project MANDy, Mathematical Analysis of Neuronal Dynamics, ANR-09-BLAN-0008-01.

References

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, 12791325.CrossRefGoogle Scholar
Buckwar, R. and Riedler, M. G. (2011). An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J. Math. Biol. 63, 10511093.CrossRefGoogle ScholarPubMed
Chow, C. C. and White, J. A. (1996). Spontaneous action potentials due to channel fluctuations. Biophys. J. 71, 30133021.CrossRefGoogle ScholarPubMed
Costa, O. L. V. and Dufour, F. (2008). Stability and ergodicity of piecewise deterministic Markov processes. SIAM J. Control Optimization 47, 10531077.CrossRefGoogle Scholar
Costa, O. L. V. and Dufour, F. (2011). Singular perturbation for the discounted continuous control of piecewise deterministic Markov processes. Appl. Math. Optimization 63, 357384.CrossRefGoogle Scholar
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. CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353388.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman and Hall, London.CrossRefGoogle Scholar
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, 339354.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
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, 497548.Google Scholar
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, 11431149.CrossRefGoogle ScholarPubMed
Fitzhugh, R. (1960). Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J. Gen. Physiol. 43, 867896.CrossRefGoogle ScholarPubMed
Flaim, S. N., Gilles, W. R. and McCulloch, A. D. (2006). Contributions of sustained I Na and I Kv43 to transmural heterogeneity of early repolarization and arrhythmogenesis in canine left ventricular myocytes. Am. J. Physiol. Heart Circ. Physiol. 291, 26172629.CrossRefGoogle ScholarPubMed
Fonbona, J., Guérin, H. and Malrieu, F. (2012). Quantitative estimates for the long time behavior of a PDMP describing the movement of bacteria. Submitted.Google Scholar
Fox, R. F. (1997). Stochastic versions of the Hodgkin-Huxley equations. Biophys. J. 72, 20682074.CrossRefGoogle ScholarPubMed
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, 7791.CrossRefGoogle Scholar
Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations (Lecture Notes Math. 840). Springer, Berlin.CrossRefGoogle Scholar
Hille, B. (1992). Ionic Channels of Excitable Membranes, 2nd edn. Sinauer, Sunderland, MA.Google ScholarPubMed
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, 500544.CrossRefGoogle ScholarPubMed
Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.CrossRefGoogle Scholar
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, 329348.Google Scholar
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.CrossRefGoogle Scholar
Pavliotis, G. A. and Stuart, A. M. (2008). Multiscale Methods (Texts Appl. Math. 53). Springer, New York.Google Scholar
Riedler, M. G. (2011). Almost sure convergence of numerical approximations for piecewise deterministic Markov processes. Preprint. Available at http://arxiv.org/abs/1112.1190v1.Google Scholar
Riedler, M. G., Thieullen, M. and Wainrib, G. (2012). Limit theorems for infinite-dimensional piecewise deterministic processes and applications to stochastic neuron models. Submitted.Google Scholar
Yin, G. G. and Zhang, Q. (1997). Continuous-Time Markov Chains and Applications. Springer, Berlin.Google Scholar
You have Access
9
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *