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arXiv · 2511.17328

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

Alan Dyson
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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.

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Alan Dyson. 2025-11-21. Existence and Uniqueness of Fast Traveling Pulses in Singularly Perturbed Nonlocal Neural Fields With Heaviside Nonlinearities: a Complete Proof. https://arxiv.org/abs/2511.17328

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