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On a kinetic FitzHugh-Nagumo model of neuronal network. (English) Zbl 1338.35001
Summary: We investigate existence and uniqueness of solutions of a McKean-Vlasov evolution PDE representing the macroscopic behaviour of interacting FitzHugh-Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We prove existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we demonstrate uniqueness of the stationary solution in the weakly nonlinear regime. Eventually, using a semigroup factorisation method, we show exponential nonlinear stability in the small connectivity regime.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
35H10 Hypoelliptic equations
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