On a Kinetic Fitzhugh–Nagumo Model of Neuronal Network

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

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LJLL - Laboratoire Jacques-Louis Lions
3 CIRB - Centre interdisciplinaire de recherche en biologie
4 Labex MemoLife
5 Mycenae - Multiscale dYnamiCs in neuroENdocrine AxEs

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 semi-group 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. Communications in Mathematical Physics, 2016, 342 (3), pp.1001-1042. ⟨10.1007/s00220-015-2556-9⟩. ⟨hal-01108872v4⟩