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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10152_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(1+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> dimensional (‘radial’) system (the effective system) involving the surface radius function in its coefficients. This perhaps explains why solutions of the original <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10152_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(1+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> dimensional FitzHugh-Nagumo system agree so well with experimental data on electrical impulse propagation.</p>

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The FitzHugh-Nagumo System on Undulated Cylinders: Spontaneous Symmetrization and Effective System

  • 期刊论文

  • 发表日期：

    2025年4月17日

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  • G. Karali,
  • K. Tzirakis,
  • I. M. Sigal

摘要

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 \(1+1\) 1 + 1 dimensional (‘radial’) system (the effective system) involving the surface radius function in its coefficients. This perhaps explains why solutions of the original \(1+1\) 1 + 1 dimensional FitzHugh-Nagumo system agree so well with experimental data on electrical impulse propagation.