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On asymptotic behaviour of stochastic processes on neuroscience

Abstract : In this thesis, we focus on two stochastic models which can be applied to neuroscience : Hawkes model and FitzHugh-Nagumo model. We study their long-time behavior. The first chapter deals with cumulative processes, which are a larger processes class than renewal processes. These processes accumulate independent random variables over time. These random variables are added on time intervals given by a renewal process. Inspired by the work of Lefevere, Mariani and Zambotti (2011), we prove a Large Deviations Principle for these processes, and large deviations inequalities in a more general framework. The second chapter is dedicated to Hawkes processes, in a non-linear context, with a signed reproduction function. They model self-excitation and self-inhibition. We prove a law of large numbers, a central limit theorem and large deviations results for a unique Hawkes process. These results lie on a renewal structure for these processes introduced by Costa, Graham, Marsalle and Tran (2020), which leads to a comparison with cumulative processes. Thus, we use known results for cumulative processes and results obtained in Chapter 1. We also exhibit two examples with explicit computations. The last chapter is a joint work with Pierre Le Bris and is devoted to the study of stochastic FitzHugh-Nagumo processes in interaction. The specificity of this model, described by Stochastic Differential Equations, is its cubic term in the drift which is non-Lipschitz. We focus on mean-field interactions and we prove a propagation of chaos, non-uniform in time first, and then a uniform in time one. To do so, we use a combined coupling method, i.e. a synchronous coupling on a specific subspace and an antithetic coupling on the complementary subspace. We also exhibit explicit bounds for these results.
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Submitted on : Thursday, August 25, 2022 - 2:31:11 PM
Last modification on : Tuesday, September 6, 2022 - 9:58:50 AM

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Laetitia Colombani. On asymptotic behaviour of stochastic processes on neuroscience. Probability [math.PR]. Université Paul Sabatier - Toulouse III, 2022. English. ⟨NNT : 2022TOU30092⟩. ⟨tel-03760659⟩

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