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Perturbation et excitabilité dans des modèles stochastiques de transmission de l’influx nerveux

Abstract : The stochastic FitzHugh-Nagumo equations is a qualitative model for the dynamics of neuronalaction potential. This slow-fast system is written εdxt = (xt - xt3 + yt) dt + √εσ1 dWt(1), dyt = (a - bxt - cyt) dt + σ2 dwt(2) where a, b and c are real numbers, ε is a small positive real number, σ1 et σ2 are two positivereal number representing the intensity of noise, Wt(1) et Wt(2) are two standard Brownian motion independent.In this thesis, we first study the associated deterministic system (σ1 = σ2 = 0) and we show this system is excitable. Then we are interested in the particular case b = 0. In this case, the behaviorin the neighborhood of the equilibrium is the same as the Morris-Lecar model. We study the law ofthe exit time of this neighborhood. In the general case, we show there are three main regimes. Westudy the distribution of the number of small oscillations N between two consecutive spikes using a substochastic Markov chain. Then we obtain results in the case of the weak-noise regime.
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Submitted on : Wednesday, November 14, 2012 - 5:52:11 PM
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Damien Landon. Perturbation et excitabilité dans des modèles stochastiques de transmission de l’influx nerveux. Mathématiques générales [math.GM]. Université d'Orléans, 2012. Français. ⟨NNT : 2012ORLE2021⟩. ⟨tel-00752088⟩

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