On a kinetic FitzHugh-Nagumo model of neuronal network

Stéphane Mischler (1) , Cristóbal Quiñinao (2, 3) , Jonathan Touboul (4, 5, 3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LJLL - Laboratoire Jacques-Louis Lions
3 Mathematical Neuroscience Team
4 Mycenae - Multiscale dYnamiCs in neuroENdocrine AxEs
5 Labex MemoLife

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.

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Analyse fonctionnelle [math.FA]

Stéphane Mischler, Cristóbal Quiñinao, Jonathan Touboul. On a kinetic FitzHugh-Nagumo model of neuronal network. 2015. ⟨hal-01108872v2⟩