Near-Pulse Solutions of the FitzHugh–Nagumo Equations on Cylindrical Surfaces

Abstract

We introduce a geometrical extension of the FitzHugh–Nagumo equations describing propagation of electrical impulses in nerve axons. In this extension, the axon is modeled as a warped cylinder, rather than a straight line, as is usually done. Nearly planar pulses propagate on its surface, along the cylindrical axis, as is the case with real axons. We prove the stability of electrical impulses for a straight (or standard) cylinder and existence and stability of pulse-like solutions for warped cylinders whose radii are small and vary slowly along their lengths and depend also on the azimuthal angle.

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Acknowledgements

It is a pleasure to thank Mary Pugh and Adam Stinchcombe for many stimulating discussions, and the third author is grateful to Daniel Sigal for very helpful discussions of the mechanism of propagation of pulses in axons. We are grateful to the anonymous referee for many useful remarks and suggestions. The research for this paper was supported in part by NSERC through Discovery Grant No. 311685 (A.B.) and Grant No. NA7901 (I.M.S.).

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Talidou, A., Burchard, A. & Sigal, I.M. Near-Pulse Solutions of the FitzHugh–Nagumo Equations on Cylindrical Surfaces. J Nonlinear Sci 31, 57 (2021). https://doi.org/10.1007/s00332-021-09710-8

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