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Qualitative Properties of Solutions for the Noisy Integrate & Fire model in Computational Neuroscience

José Carrillo 1 Benoît Perthame 2, 3 Delphine Salort 4 Didier Smets 2
3 MAMBA - Modelling and Analysis for Medical and Biological Applications
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt
4 LCQB-MMB - Mathematical Modeling in Biology [LCQB]
LCQB - Biologie Computationnelle et Quantitative = Laboratory of Computational and Quantitative Biology
Abstract : The Noisy Integrate-and-Fire equation is a standard non-linear Fokker-Planck Equation used to describe the activity of a homogeneous neural network characterized by its connectivity b (each neuron connected to all others through synaptic weights); b > 0 describes excitatory networks and b < 0 inhibitory networks. In the excitatory case, it was proved that, once the proportion of neurons that are close to their action potential V F is too high, solutions cannot exist for all times. In this paper, we show a priori uniform bounds in time on the firing rate to discard the scenario of blow-up, and, for small connectivity, we prove qualitative properties on the long time behavior of solutions. The methods are based on the one hand on relative entropy and Poincaré inequalities leading to L 2 estimates and on the other hand, on the notion of 'universal super-solution' and parabolic regularizing effects to obtain L ∞ bounds.
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José Carrillo, Benoît Perthame, Delphine Salort, Didier Smets. Qualitative Properties of Solutions for the Noisy Integrate & Fire model in Computational Neuroscience. Nonlinearity, IOP Publishing, 2015, 28 (9), pp.3365 ⟨10.1088/0951-7715/28/9/3365⟩. ⟨hal-01079381⟩

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