A multiscale study of stochastic spatially-extended conductance-based models for excitable systems

Alexandre Genadot 1
1 Probabilité, Statistique et Biologie
LPMA - Laboratoire de Probabilités et Modèles Aléatoires
Abstract : The purpose of the present thesis is the mathematical study of probabilistic models for the generation and propagation of an action potential in neurons and more generally of stochastic models for excitable cells. Indeed, we want to study the effect of noise on multiscale spatially extended excitable systems. We address the intrinsic as well as the extrinsic source of noise in such systems. Below, we first describe the mathematical content of the thesis. We then consider the physiological situation described by the considered models. To study the intrinsic or internal noise, we consider Hilbert-valued Piecewise Deterministic Markov Processes (PDMPs). We are interested in the multiscale and long time behavior of these processes. In a first part, we study the case where the fast component is a discrete component of the PDMP. We prove a limit theorem when the speed of the fast component is accelerated. In this way, we obtain the convergence of a class of Hilbert-valued PDMPs with multiple timescales toward so-called averaged processes which are, in some cases, still PDMPs. Then, we study the fluctuations of the multiscale model around the averaged one and show that the fluctuations are Gaussians through the proof of a Central Limit Theorem. In a second part, we consider the case where the fast component is itself a PDMP. This requires knowledge about the invariant measure of Hilbert-valued PDMPs. We show, under some conditions, the existence and uniqueness of an invariant measure and the exponential convergence of the process toward this measure. For a particular class of PDMPs that we call diagonals, the invariant measure is made explicit. This, in turn, allow us to obtain averaging results for "fast" PDMPs fully coupled to "slow" continuous time Markov chains. To study the extrinsic or external noise, we consider systems of Stochastic Partial Differential Equations (SPDEs) driven by colored noises. On bounded domains of $\mathbb{R}^2$ or $\mathbb{R}^3$, we analyze numerical schemes based on finite differences in time and finite elements in space. For a class of linear SPDEs, we obtain the strong error of convergence of such schemes. For simulations, we show the emergence of re-entrant patterns due to the presence of noise in spatial domains of dimension two for the Barkley and Mitchell-Schaeffer models.
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Alexandre Genadot. A multiscale study of stochastic spatially-extended conductance-based models for excitable systems. Probability [math.PR]. Université Pierre et Marie Curie - Paris VI, 2013. English. ⟨tel-00905886⟩

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