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Analyse et Comportement Asymptotique de Systèmes Dynamiques Complexes. Application en Neuroscience

Abstract : This thesis is devoted to mathematical modeling in neuroscience and mathematical analysis of coupled Hodgkin-Huxley (HH) systems. Two aspects were studied separately. The first focuses on the (HH) model when we take into account only the differential system (ODE), the second on the corresponding reaction-diffusion model (PDE). Therefore, firstly, a bifurcation analysis of the system is made, using the cur- rent of injection as a parameter. For this aim, we use a strong spectral method (called method of harmonic balance) to detect stable and unstable solutions. This help us in finding, in a more effective way, all the periodic solutions of the ODE (HH) system for various values of the parameter (i.e. current of injection). Secondly, we study the PDE-(HH) system as well as complex systems obtained by coupling many PDE-(HH). The existence and uniqueness of global solutions for initial functions from a Banach space are proved, and the proof of the existence of global attractor is also done. The last chapter gives a numerical study based on classical methods of discretization (finite differences and finite elements) coupled with a splitting method.
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Submitted on : Thursday, July 5, 2018 - 12:11:07 PM
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Aymen Balti, Spécialité Mathématiques. Analyse et Comportement Asymptotique de Systèmes Dynamiques Complexes. Application en Neuroscience. Systèmes dynamiques [math.DS]. Normandie Université, 2016. Français. ⟨tel-01830706⟩

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