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

Alan Dyson

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 $ε=0$ 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.

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

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 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.

Paper Structure

This paper contains 22 sections, 26 theorems, 134 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Hypotheses
  4. Singular Solution, Previous Results, and Outline
  5. Construction of Singular Homoclinic Orbit
  6. Comparison to Previous Results
  7. Outline of Technical Analysis
  8. Calculation of Unique $(a_\epsilon,c_\epsilon)$
  9. Uniqueness of the Front and Back Speed
  10. Preparation for the Implicit Function Theorem
  11. The Continuous Extension of $F$
  12. The Continuous Extension of $DF$
  13. Applying the Implicit Function Theorem
  14. Threshold Requirements and Closeness of $\mathcal{S}_\epsilon$ to $\mathcal{S}_0$
  15. Preliminaries and Proof Outline
  16. ...and 7 more sections

Key Result

Theorem 1.1

When $0<\epsilon \ll 1,$ there exists a unique $C^2$ (modulo translation) fast traveling pulse solution $(U_\epsilon,Q_\epsilon)$ to system eq: intro_system_1-eq: intro_system_2. In particular, with $z=x+ct$, there exists unique $C^1$ curves $c_\epsilon=c(\epsilon)$ and $a_\epsilon=a(\epsilon)$ such is the Hausdorff distance between sets in $(U,Q)$ space with respect to the standard Euclidean metr

Figures (4)

  • Figure 1: Generalized phase space of $\mathcal{S}_0$ (black) along with $\mathcal{S}_\epsilon$ (red) for $\epsilon\ll 1$
  • Figure 2: A traveling pulse solution when $\theta=0.25$, $\gamma=0.2$, $\epsilon=0.005$, and $K(x)= \IfNoValueTF{4a} {\frac{1}{1+a^2}} {\frac{1+a^2}{4a}}e^{-a|x|}(a\sin(|x|)+\cos(x))$ with $a=0.3$. The front and back are non-monotone, but cross the threshold only once
  • Figure 3: A schematic of the pulse solution (grey), broken down by region. The pulse is close to the front (red) and back (blue) overlaying regions $R_1$ and $R_3$, respectively
  • Figure 4: The pulse as a homoclinic orbit. The pivot points $s_\epsilon(z_0)$, $s_\epsilon(a_\epsilon-z_0)$, and $s_\epsilon(a_\epsilon+z_0)$ are used to define the inner regions $R_1$ and $R_3$

Theorems & Definitions (52)

  • Theorem 1.1: Existence and Uniqueness of Fast Pulses
  • Corollary 1.1: First-Order Approximations of $\tau(\epsilon)$ and $c(\epsilon)$
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 42 more