The FitzHugh-Nagumo system on undulated cylinders: spontaneous symmetrization and effective system

TL;DR

This work analyzes the FitzHugh-Nagumo system on undulated cylindrical surfaces to model electrical impulses along axons. By decomposing the dynamics into radial and angular components and employing Lyapunov-type functionals, the authors prove an angular contraction: for small radii and near-radial initial data, solutions converge to their radial averages. They then show these radial averages are well approximated by a 1D radial FHN system with coefficients determined by the radius profile rho(x). The findings provide a theoretical justification for the empirical success of 1D FHN models in describing axial nerve pulse propagation and offer a rigorous pathway to effective reduced models on curved neuronal geometries.

Abstract

We consider the FitzHugh-Nagumo system on undulated cylindrical surfaces modeling nerve axons. We show that for sufficiently small radii and for initial conditions close to radially symmetrical ones, (i) the solutions converge to their radial averages, and (ii) the latter averages can be approximated by solutions of a 1+1 dimensional ('radial') system (the effective system) involving the surface radius function in its coefficients. This perhaps explains why solutions of the original 1+1 dimensional FitzHugh-Nagumo system agree so well with experimental data on electrical impulse propagation.

Figures (1)

• Figure 1: The graph of the polynomial $p(Z).$

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• proof : Proof of Theorem \ref{['thm:rad-conv']}