Existence of pulses in excitable media with nonlocal coupling

Abstract

We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a PDE oriented approach, relying on exponential weights, Fredholm theory, and commutator estimates.

Rigorous Derivation of the Nonlocal Reaction-Diffusion Fitzhugh--Nagumo System

“…where I ext stands for the input current the neuron receives, N (v) is a bistable nonlinearity which models the cell excitability, and τ ≥ 0, a ∈ R and b ≥ 0 are some given constants. Without loss of generality [2,20,34,35], we assume that N (v) is given by the following cubic nonlinearity 1…”

“…where L ρ 0 (V ) can be interpreted as a nonlocal diffusion operator in x. In the limiting case ρ 0 ≡ 1 2 , such a system has already been well studied especially regarding the formation of propagating waves (traveling fronts and pulses) in both cases τ = 0 and 0 < τ ≪ 1 [3,20]. We also mention the classical works regarding the local FitzHugh-Nagumo system, that is when L ρ 0 (V ) is replaced by the standard diffusion operator [11,24,28], and the more recent advances for the discrete case [26,27].…”

Pinning and Unpinning in Nonlocal Systems

“…This would certainly require making the formal asymptotics in [14] rigorous, without using the dynamical systems techniques referenced here [11]. Some ideas on how to approach such questions in nonlocal problems can be found in [7].…”

Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions

“…e −k 2 [u j+k + u j−k − 2u j ] + g(u j ; r 0 ) − w j w j = ρ[u j − γw j ], (1.9) in which κ > 0 is a normalisation constant. In [22], Faye and Scheel studied equations such as (1.9) for discretisations with infinite-range interactions featuring exponential decay in the coupling strength. They circumvented the need to use a state space as in [35], which enabled them to construct pulses to (1.9) for arbitrary discretisation distance h. Very recently [23], they developed a center manifold approach that allows bifurcation results to be obtained for neural field equations.…”