Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of Traveling Fronts of the FitzHugh-Nagumo Equations on Cylindrical Surfaces

Afroditi Talidou

TL;DR

The paper addresses the stability of traveling fronts in the FitzHugh-Nagumo system on cylindrical surfaces, motivated by axonal geometry. It develops a spectral-semiclassical stability framework on standard cylinders, proving nonlinear orbital stability with exponential decay, and extends the approach to warped cylinders by treating radius variation as a small perturbation, yielding persistence of front-like solutions. The main contributions are rigorous exponential decay for perturbations off the front manifold on $ S_R$, a perturbative persistence result on $ S_ ho$ with explicit δ-bounds, and supporting numerical simulations on both geometries. The work advances the understanding of how surface geometry influences wave propagation in reaction-diffusion systems, with potential implications for modeling neural signal transmission in morphologically realistic axons.

Abstract

The FitzHugh-Nagumo equations are known to admit traveling front solutions in one spatial dimension that are nonlinearly stable. This paper concerns the stability of traveling front solutions propagating on cylindrical surfaces. It is shown that such traveling fronts are nonlinearly stable on the surface of standard cylinders of constant radius. The analysis is extended to warped cylinders with slowly varying radius, where persistence of front-like solutions is established. Numerical simulations support the theoretical findings.

Stability of Traveling Fronts of the FitzHugh-Nagumo Equations on Cylindrical Surfaces

TL;DR

The paper addresses the stability of traveling fronts in the FitzHugh-Nagumo system on cylindrical surfaces, motivated by axonal geometry. It develops a spectral-semiclassical stability framework on standard cylinders, proving nonlinear orbital stability with exponential decay, and extends the approach to warped cylinders by treating radius variation as a small perturbation, yielding persistence of front-like solutions. The main contributions are rigorous exponential decay for perturbations off the front manifold on , a perturbative persistence result on with explicit δ-bounds, and supporting numerical simulations on both geometries. The work advances the understanding of how surface geometry influences wave propagation in reaction-diffusion systems, with potential implications for modeling neural signal transmission in morphologically realistic axons.

Abstract

The FitzHugh-Nagumo equations are known to admit traveling front solutions in one spatial dimension that are nonlinearly stable. This paper concerns the stability of traveling front solutions propagating on cylindrical surfaces. It is shown that such traveling fronts are nonlinearly stable on the surface of standard cylinders of constant radius. The analysis is extended to warped cylinders with slowly varying radius, where persistence of front-like solutions is established. Numerical simulations support the theoretical findings.

Paper Structure

This paper contains 28 sections, 18 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results.
  3. Outline.
  4. Front solutions on standard cylinders
  5. The semigroup generated by $L$
  6. Spectrum of the linearization $L$
  7. The spectral projection
  8. Resolvent estimate for the zero Fourier mode
  9. Estimate on $S_1$:
  10. Estimate on $S_2$:
  11. Estimate on $S_3$:
  12. Exponential decay of the linear semigroup
  13. Proof of Theorem \ref{['thm:nonli-stab']}
  14. Front-like solutions on warped cylinders
  15. Linear dynamics
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\mathcal{S}_R$ be a standard cylinder of constant radius $R\leq 1$ and consider Eq. eq:FHN on that geometry. Assume $0< \varepsilon \ll 1$, $\gamma>0$ and $\alpha \in \left( 0, \frac{1}{2}\right)$ are such that Eq. eq:FHN has a front solution $\Phi(x-ct)$, and that $C_\mathrm{m}, r_{int} >0$. T for $K_1 > 0$ independent of $\varepsilon$, $\mu \gtrsim\varepsilon$, and $\tilde{h}\in{\mathbb R}$

Figures (2)

  • Figure 1: Traveling fronts on the surface of a constant cylinder.a-c.Top: Heat maps on three-dimensional cylindrical surfaces illustrating the geometries considered. Bottom: Snapshots of traveling front solutions at three time instances, $t = 60, 100, 180$. Note that the front preserves its shape as it propagates along the cylinder. The parameter values are $\alpha = 0.01$, $\varepsilon = 0.0001$, $\gamma = 7$, $C_m = 1$, $r_{\mathrm{int}} = 0.1$, and $L = 1000$.
  • Figure 2: Traveling fronts on warped cylinders.a-c.The cylinder radius is given by $\rho(x) = 0.8 + 0.1 e^{sin(\frac{6 \pi x}{L})}$, illustrating an example of a pearl-on-a-string morphology of neuronal axons. d-f. The cylinder radius is given by $\rho(x) = 0.8 + s\frac{x}{L} + A e^{-\frac{(x-x_0)^2}{2\sigma^2}}$, representing a localized axonal swelling. The parameter values are the same as in Fig. \ref{['fig:constant']}. The near-front solutions remain stable but propagate with different speeds, highlighting the influence of axonal morphology on signal transmission.

Theorems & Definitions (32)

  • Theorem 1.1: Stability of fronts on $\mathcal{S}_R$
  • Theorem 1.2: Persistance of front-like solutions on $\mathcal{S}_{\rho}$
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Proposition \ref{['prop:lin_semigroup']}
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 22 more