Existence and Uniqueness of Fast Traveling Pulses in Singularly Perturbed Nonlocal Neural Fields With Heaviside Nonlinearities: a Complete Proof

TL;DR

This paper rigorously proves the existence and uniqueness of fast traveling pulse solutions in singularly perturbed nonlocal neural field models with Heaviside nonlinearities, overcoming longstanding mathematical challenges.

Contribution

It provides the first complete proof of pulse existence and uniqueness, including new approximations and geometric singular perturbation analysis for such neural systems.

Findings

Established existence and uniqueness of pulses.

Derived first-order approximations of pulse speed and width.

Showed the formal pulse closely approximates the true solution.

Abstract

We rigorously prove the existence and uniqueness of fast traveling pulse solutions to the singularly perturbed neural field system with linear feedback and Heaviside nonlinearity structure within a spatial convolution. Although a long-standing open problem, the pulse is well-accepted to often exist based on its original singular construction, closed form when it exists, and follow-ups, but prior to this study, there has not been a proof that overcomes the difficulties of (i) solving for the fast speed and width functions using the implicit function theorem at $\epsilon=0$ and (ii) tracking the resultant formal homoclinic orbit near its singular orbit during fast, slow, and mixed time scales. First, we provide new, first-order approximations of the pulse speed and width. We then show that the formal pulse is close to the front and back at threshold crossing points and that the Hausdorff distance between the formal pulse and the singular homoclinic orbit converges to zero, demonstrating that the formal pulse is a true solution to the original system. Broadly, our methods provide insight into nonlocal geometric singular perturbation theory.