On a kinetic FitzHugh-Nagumo model of neuronal network

Abstract

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.

Cited by in corpus (13)

• Power-law statistics and universal scaling in the absence of criticality
• Complex oscillations in the delayed Fitzhugh-Nagumo equation
• Noise-induced synchronization and anti-resonance in excitable systems; Implications for information processing in Parkinson's Disease and Deep Brain Stimulation
• Emergence of oscillatory behaviors for excitable systems with noise and mean-field interaction, a slow-fast dynamics approach
• The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states
• Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels
• On quantitative hypocoercivity estimates based on Harris-type theorems
• Rigorous derivation of the nonlocal reaction-diffusion FitzHugh-Nagumo system
• General time elapsed neuron network model: well-posedness and strong connectivity regime
• Relaxation in time elapsed neuron network models in the weak connectivity regime
• The kinetic Fokker-Planck equation with weak confinement force
• On the Krein-Rutman theorem and beyond
• The Kinetic Fokker-Planck Equation with General Force