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Periodicity induced by noise and interaction in the kinetic mean-field FitzHugh-Nagumo model. (English) Zbl 1479.60198

Summary: We consider the long-time behavior of a population of mean-field oscillators modeling the activity of interacting excitable neurons in a large population. Each neuron is represented by its voltage and recovery variables, which are the solution to a FitzHugh-Nagumo system, and interacts with the rest of the population through a mean-field linear coupling, in the presence of noise. The aim of the paper is to study the emergence of collective oscillatory behaviors induced by noise and interaction on such a system. The main difficulty of the present analysis is that we consider the kinetic case, where interaction and noise are only imposed on the voltage variable. We prove the existence of a stable cycle for the infinite population system, in a regime where the local dynamics is small.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
35K55 Nonlinear parabolic equations
35Q84 Fokker-Planck equations
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
92C20 Neural biology
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