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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Arioli, G., Koch, H.: Existence and stability of traveling pulse solutions of the FitzHugh–Nagumo equation. Nonlinear Anal. 113, 51–70 (2015)
Beck, M., Jones, ChKRT, Schaeffer, D., Wechselberger, M.: Electrical waves in a one-dimensional model of cardiac tissue. SIAM J. Appl. Dyn. Syst. 7(4), 1558–1581 (2008)
Carpenter, G.: A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Differ. Equ. 23, 152–173 (1977)
Carter, P., Sandstede, B.: Fast pulses with oscillatory tails in the FitzHugh–Nagumo system. SIAM J. Math. Anal. 47(5), 3393–3441 (2015)
Carter, P., de Rijk, B., Sandstede, B.: Stability of traveling pulses with oscillatory tails in the FitzHugh–Nagumo system. J. Nonlinear Sci. 26, 1369–1444 (2016)
Chen, C., Choi, Y.S.: Traveling pulse solutions to FitzHugh–Nagumo equations. Calc. Var. 54, 1–45 (2015)
Chen, C., Hu, X.: Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations. Calc. Var. 49, 827–845 (2014)
Cornwell, P., Jones, ChKRT: On the existence and stability of fast traveling waves in a doubly diffusive FitzHugh–Nagumo System. SIAM J. Appl. Dyn. Syst. 17(1), 754–787 (2018)
Doelman, A.A., Kaper, T., Promislow, K.: Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer–Meinhardt model. SIAM J. Math. Anal. 38(6), 1760–1787 (2007)
Engel, K.-J., Nagel, R.: A Short Course on Operator Semigroups. Springer, New York (2006)
Evans, J.W.: Nerve axon equations, IV: the stable and the unstable impulse. Indiana Univ. Math. J. 24, 1169–1190 (1975)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Flores, G.: Stability analysis for the slow traveling pulse of the FitzHugh–Nagumo system. SIAM J. Math. Anal. 22(2), 392–399 (1991)
Hastings, S.P.: On the existence of homoclinic and periodic orbits for the FitzHugh–Nagumo equations. Q. J. Math. Oxf. Ser. 2(27), 123–134 (1976)
Hastings, S.P.: Single and multiple pulse waves for the FitzHugh–Nagumo equations. SIAM J. Appl. Math. 42(2), 247–260 (1982)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Hupkes, H.J., Sandstede, B.: Stability of pulse solutions for the discrete FitzHugh–Nagumo system. Trans. Am. Math. Soc. 365, 251–301 (2013)
Hupkes, H.J., Morelli, L., Schouten-Straatman, W.M., Van Vleck, E.S.: Traveling waves and pattern formation for spatially discrete bistable reaction-diffusion equations. In: Bohner, M., Siegmund, S., Šimon, Hilscher, R., Stehlík, P. (eds.) Difference Equations and Discrete Dynamical Systems with Applications. ICDEA 2018. Springer Proceedings in Mathematics and Statistics, vol. 312. Springer (2020)
Ikeda, H., Mimura, M., Tsujikawa, T.: Slow traveling wave solutions to the Hodgkin–Huxley equations. Lect. Notes Numer. Appl. Anal. 9, 1–73 (1987)
Ito, M.: Traveling train solutions of FitzHugh–Nagumo systems. Lect. Notes Numer. Appl. Anal. 9, 75–104 (1987)
Jones, ChKRT: Stability of the traveling wave solution of the FitzHugh–Nagumo system. Trans. AMS 286(2), 431–469 (1984)
Jones, ChKRT, Kopell, N., Langer, R.: Construction of the FitzHugh–Nagumo pulse using differential forms. In: Swinney, H., Aris, G., Aronson, D.G. (eds.) Patterns and Dynamics in Reactive Media, IMA Volumes in Mathematics and Its Applications, vol. 37, pp. 101–116. Springer, New York (1991)
Kapitula, T., Promislow, K.: Spectral and Dynamical Stability of Nonlinear Waves. Applied Mathematical Sciences, vol. 185. Springer, New York (2013)
Keener, J.P.: A geometrical theory for spiral waves in excitable media. SIAM J. Appl. Math. 46, 1039–1056 (1986)
Krupa, M., Sandstede, B., Szmolyan, P.: Fast and slow waves in the FitzHugh–Nagumo equation. J. Differ. Equ. 133, 49–97 (1997)
Langer, R.: Existence and uniqueness of pulse solutions to the FitzHugh–Nagumo equations. PhD thesis, Northeastern Univ. (1980)
Maginu, K.: Existence and stability of periodic travelling wave solutions to Nagumo’s nerve equation. J. Math. Biol. 10, 133–153 (1980)
Mikhailov, A.S., Krinskii, V.I.: Rotating spiral waves in excitable media: the analytical results. Physics 9D, 346–371 (1983)
Moore, B.R., Promislow, K.: The semi-strong limit of multipulse interaction in a thermally driven optical system. J. Differ. Equ. 245(6), 1616–1655 (2008)
Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1964)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Prüss, J.: On the spectrum of \(C_0\)-semigroups. Trans. Am. Math. Soc. 284, 847–857 (1984)
Sandstede, B.: Stability of traveling waves, chapter 18. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2. Gulf, pp. 983–1055 (2002)
Schouten-Straatman, W.M., Hupkes, H.J.: Traveling waves for spatially discrete systems of FitzHugh–Nagumo type with periodic coefficients. SIAM J. Math. Anal. 51(4), 3492–3532 (2019)
Tsujikawa, T., Nagai, T., Mimura, M., Kobayashi, R., Ikeda, H.: Stability properties of traveling pulse solutions of the higher dimensional FitzHugh–Nagumo equations. Jpn. J. Appl. Math. 6, 341–366 (1989)
Yanagida, E.: Stability of the fast traveling pulse solutions of the FitzHugh–Nagumo equations. J. Math. Biol. 22, 81–104 (1985)
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.).
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Dejan Slepcev.
About this article
Cite this article
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