The emergence of the deterministic Hodgkin???Huxley equations as a limit from the underlying stochastic ion-channel mechanism, The Annals of Applied Probability, vol.18, issue.4, pp.1279-1325, 2008. ,
DOI : 10.1214/07-AAP494
An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution, Journal of Mathematical Biology, vol.2, issue.3, 2011. ,
DOI : 10.1007/s00285-010-0395-z
Spontaneous action potentials due to channel fluctuations, Biophysical Journal, vol.71, issue.6, pp.3013-3021, 2000. ,
DOI : 10.1016/S0006-3495(96)79494-8
URL : http://doi.org/10.1016/s0006-3495(96)79494-8
Stability and Ergodicity of Piecewise Deterministic Markov Processes, SIAM Journal on Control and Optimization, vol.47, issue.2, pp.1053-1077, 2008. ,
DOI : 10.1137/060670109
URL : https://hal.archives-ouvertes.fr/hal-00268162
Singular Perturbation for the Discounted Continuous Control of Piecewise Deterministic Markov Processes, Applied Mathematics & Optimization, vol.216, issue.1, pp.357-384, 2011. ,
DOI : 10.1007/s00245-010-9124-7
URL : https://hal.archives-ouvertes.fr/hal-00545824
Convergence of stochastic gene networks to hybrid piecewise deterministic processes, The Annals of Applied Probability, vol.22, issue.5, 2011. ,
DOI : 10.1214/11-AAP814
URL : https://hal.archives-ouvertes.fr/hal-00553482
Piecewise-deterministic Markov processes, J. R. Statist. Soc, vol.46, pp.353-388, 1984. ,
DOI : 10.1007/978-1-4899-4483-2_2
Markov models and optimization, Monograph on Stat. and Appl. Prob, vol.49, 1993. ,
DOI : 10.1007/978-1-4899-4483-2
Chaotic states in a random world: Relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels, Journal of Statistical Physics, vol.28, issue.1-2, pp.339-354, 1993. ,
DOI : 10.1007/BF01053972
Markov Processes: Characterization and Convergence, 1986. ,
DOI : 10.1002/9780470316658
Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes, To appear in Markov Proc. Rel. Fields 137, pp.259-304, 2009. ,
Ion-Channel Noise Places Limits on the Miniaturization of the Brain???s Wiring, Current Biology, vol.15, issue.12, pp.1143-1149, 2005. ,
DOI : 10.1016/j.cub.2005.05.056
Thresholds and Plateaus in the Hodgkin-Huxley Nerve Equations, The Journal of General Physiology, vol.43, issue.5, pp.867-896, 1960. ,
DOI : 10.1085/jgp.43.5.867
Contributions of sustained INa and IKv43 to transmural heterogeneity of early repolarization and arrhythmogenesis in canine left ventricular myocytes, AJP: Heart and Circulatory Physiology, vol.291, issue.6, pp.2617-2629, 2006. ,
DOI : 10.1152/ajpheart.00350.2006
Quantitative Estimates for the Long Time Behavior of a PDMP Describing the Movement of Bacteria ,
Stochastic versions of the Hodgkin-Huxley equations, Biophysical Journal, vol.72, issue.5, p.20682074, 1997. ,
DOI : 10.1016/S0006-3495(97)78850-7
Mechanisms of Excitation-Contraction Coupling in an Integrative Model of the Cardiac Ventricular Myocyte, Biophysical Journal, vol.90, issue.1, pp.77-91, 2006. ,
DOI : 10.1529/biophysj.105.065169
Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol.840, 1981. ,
DOI : 10.1007/BFb0089647
Ionic Channels of Excitable Membranes, 2001. ,
A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, vol.117, issue.4, pp.500-544, 1952. ,
DOI : 10.1113/jphysiol.1952.sp004764
Limit Theorems for Stochastic Processes, 1987. ,
DOI : 10.1007/978-3-662-02514-7
Convergence faible et principe d'invariance pour des martingalesàmartingalesà valeurs dans des espaces de Sobolev, Ann. I.H.P, vol.20, pp.329-348, 1984. ,
Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic Markov processes, Proc. Appl ,
DOI : 10.1016/j.spa.2012.03.005
URL : https://hal.archives-ouvertes.fr/hal-00714354
Multiscale Methods, Averaging and Homogenization ,
An Introduction to Partial Differential Equations ,
Almost sure convergence of numerical approximations for Piecewise Deterministic Markov Processes, Journal of Computational and Applied Mathematics, vol.239, pp.10-34, 2010. ,
DOI : 10.1016/j.cam.2012.09.021
Limit Theorems for Infinite Dimensional Piecewise Deterministic Processes and Applications to Stochastic Neuron Models, 2011. ,
Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, 1997. ,
Nonlinear Functional Analysis and its Applications II/B, Nonlinear Monotone Operators, 1990. ,