Concentration Phenomena in Fitzhugh–Nagumo Equations: A Mesoscopic Approach

Large Coupling in a Fitzhugh-Nagumo Neural Network: Quantitative and Strong Convergence Results

“…A natural improvement of our result consists in removing the well-preparedness condition (2.4). The same difficulty was successfully dealt with in the context of concentration phenomena for neural networks [11,9,10]. Indeed, the well-preparedness condition was removed in [9] thanks to a time dependent re-scaling of the internal variable.…”

Wasserstein contraction for the stochastic Morris-Lecar neuron model

“…In order to investigate the large time behavior of solutions to (1), we use a mathematical approach based on the coupling of stochastic processes as those in (2). It has two advantages compared to other deterministic methods used to investigate the long time behavior of kinetic Fokker-Planck equations, such as hypocoercivity methods [29,16].…”

“…Main results. In the following, we state our result for the stochastic system (2). Denote by (P t ) t⩾0 the semigroup associated to (2), namely…”

A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions

“…A byproduct of our approach and energy estimates, is that they yield regularity estimates for the solution that depend on the regularity of the initial data and the source term. It is worth noting that a well-posedness result for the hyperbolic system was given in [21] and hypocoercivity estimates in [4]. However, a novelty in our approach is that we use a vanishing viscosity method to prove the well-posedness of the hyperbolic system and to obtain the corresponding energy estimates.…”

“…and we will consistently assume that e −U (x) ∈ L 1 (R d , dx). Our function spaces are defined with respect to the measures µ and η introduced in (4…”