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Traveling pulse solutions to FitzHugh–Nagumo equations

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Abstract

Many patterns emerge from homogeneous media that are destabilized by a spatial modulation. Near such a bifurcation, Turing patterns are more or less uniformly distributed. Besides these regular patterns, particle-like structures are commonly observed in physical, chemical and biological systems. Depending on the system parameters and initial conditions, localized dissipative structures may stay at rest or propagate with a dynamically stabilized velocity. The system of FitzHugh–Nagumo equations has been extensively studied for diffusion-induced instability and emergence of patterns. By utilizing the inherited Hamiltonian structure, the existence of standing pulses has been established in a recent work. Our aim in this paper is to apply variational arguments to obtain a traveling pulse solution. With an investigation of the speed when the diffusivity of activator is small, we show that the profile of such a traveling pulse is in close proximity to a trivial background state except for one localized spatial region where change is substantial.

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References

  1. Acerbi, E., Fusco, N., Morini, M.: Minimality via second variation for a nonlocal isoperimetric problem (preprint)

  2. Akhmediev, N., Ankiewicz, A.: Three sources and three component parts of the concept of dissipative solitons. In: Dissipative Solitons: From Optics to Biology and Medicine. Lecture Notes in Physics, vol. 751, Springer, Berlin (2008)

  3. Alexander, J., Gardner, R., Jones, C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Proceedings of the Tulane Program in Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, vol. 446, p. 549. Springer, Berlin (1975)

  5. Berestycki, H., Larrouturou, B., Lions, P.-L.: Multidimensional traveling wave solutions of a flame propagation model. Arch. Ration. Mech. Anal. 111, 33–49 (1990)

    Article  MATH  Google Scholar 

  6. Berestycki, H., Nirenberg, L.: Traveling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linaire 9, 497–572 (1992)

  7. Bisgard, J.: Homoclinics for a Hamiltonian systems with wells at different levels. Calc. Var. 29, 1–30 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bode, M., Liehr, A.W., Schenk, C.P., Purwins, H.-G.: Interaction of dissipative solitons: particle-like behaviour of localized structures in a three-component reaction–diffusion system. Physica D 161, 45–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carpenter, G.: A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Differ. Equ. 23, 335–367 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, C.-N., Choi, Y.S.: Standing pulse solutions to FitzHugh–Nagumo equations. Arch. Ration. Mech. Anal. 206, 741–777 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, C.-N., Hu, X.: Stability criteria for reaction–diffusion systems with skew-gradient structure. Commun. PDE 33, 189–208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, C.-N., Hu, X.: Maslov index for homoclinic orbits of Hamiltonian systems. Ann. Inst. H. Poincare Anal. Non Linearie 24, 589–603 (2007)

  13. Chen, C.-N., Hu, X.: Stability analysis for standing pulse solutions to FitzHugh-Nagumo equations. Calc. Var. Partial Differ. Equ. 49, 827–845 (2014)

  14. Chen, C.-N., Kung, S.-Y., Morita, Y.: Planar standing wavefronts in the FitzHugh-Nagumo equations. SIAM J. Math. Anal. 46, 657–690 (2014)

  15. Chen, X., Oshita, Y.: An application of the modular function in nonlocal variational problems. Arch. Ration. Mech. Anal. 186, 109–132 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dancer, E.N., Yan, S.: A minimization problem associated with elliptic systems of FitzHugh–Nagumo type. Ann. Inst. H. Poincaré Anal. Nonlinéaire 21, 237–253 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. De Wit, A.: Spatial patterns and spatiotemporal dynamics in chemical systems. In: Advances in Chemical Physics, vol. 109, pp. 435–513. Wiley, New York (1999)

  18. Doelman, A., van Heijster, P., Kaper, T.: Pulse dynamics in a three-component system: existence analysis. J. Dyn. Differ. Equ. 21, 73–115 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Evans, J.W.: Nerve axon equations. IV: The stable and the unstable impulse. Indiana Univ. Math. J. 24, 1169–1190 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fife, P.C.: Mathematical Aspects of Reacting and Diffusing Systems. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  21. Fife, P.C.: Long time behavior of solutions of bistable nonlinear diffusion equations. Arch. Ration. Mech. Anal. 70, 31–46 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fife, P.C., Mcleod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  23. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Article  Google Scholar 

  24. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence (1988)

  25. Hastings, S.P.: On traveling wave solutions of the Hodgkin–Huxley equations. Arch. Ration. Mech. Anal. 60, 229–257 (1976)

    Article  MATH  Google Scholar 

  26. Heinze, S.: A variational approach to traveling waves. Max Planck Institute for Mathematical Sciences. Preprint 85 (2001)

  27. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)

    Article  Google Scholar 

  28. Jones, C.: Stability of the travelling wave solution of the FitzHugh–Nagumo system. Trans. Am. Math. Soc. 286, 431–469 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  29. Keener, J., Sneyd, J.: Mathematical Physiology I: Cellular Physiology, 2nd edn. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  30. Kondo, S., Asai, R.: A reaction–diffusion wave on the skin of the marine angelfish pomacanthus. Nature 376–31, 765–768 (1995)

    Article  Google Scholar 

  31. Langer, R.: Existence of homoclinic travelling wave solutions to the FitzHugh-Nagumo equations. Ph.D. Thesis, Northeastern University (1980)

  32. Liehr, A.W.: Dissipative Solitons in Reaction–Diffusion Systems. Springer Series in Synergetics, vol. 70. Springer, Berlin (2013)

  33. Liehr, A.W., Bödeker, H.U., Röttger, M.C., Frank, T.D., Friedrich, R., Purwins, H.-G.: Drift bifurcation detection for dissipative solitons. New J. Phys. 5, 1–9 (2003)

    Article  MathSciNet  Google Scholar 

  34. Lucia, M., Muratov, C., Novaga, M.: Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction–diffusion equations invading an unstable equilibrium. Commun. Pure Appl. Math. 57, 616–636 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lucia, M., Muratov, C.B., Novaga, M.: Existence of traveling waves of invasion for Ginzburg–Landau-type problems in infinite cylinders. Arch. Ration. Mech. Anal. 188(3), 475–508 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mather, J.N.: Variational construction of orbits of twist diffeomorphisms. J. Am. Math. Soc. 4, 207–263 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  37. Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142 (1986)

    MathSciNet  MATH  Google Scholar 

  38. Muratov, C.B.: A global variational structure and propagation of disturbances in reaction–diffusion systems of gradient type. Discrete Contin. Dyn. Syst. Ser. B 4(4), 867–892 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)

    Article  Google Scholar 

  40. Nishiura, Y., Teramoto, T., Yuan, X., Udea, K.-I.: Dynamics of traveling pulses in heterogeneous media. CHAOS 17(3), 037104 (2007)

  41. Oshita, Y.: On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh–Nagumo equations in higher dimensions. J. Differ. Equ. 188, 110–134 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  43. Rabinowitz, P.H., Stredulinsky, E.W.: Extensions of Moser-Bangert Theory: Local Minimal Solutions. Progress in Nonlinear Differential Equations and Their Applications, vol. 81. Birkhauser/Springer, New York (2011)

  44. Reineck, J.F.: Traveling wave solutions to a gradient system. Trans. Am. Math. Soc. 307, 535–544 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  45. Reinecke, C., Sweers, G.: A positive solution on \({\mathbb{R}}^n\) to a equations of FitzHugh–Nagumo type. J. Differ. Equ. 153, 292–312 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ren, X., Wei, J.: Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology. SIAM J. Math. Anal. 39, 1497–1535 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Roquejoffre, J.M., Terman, D., Volpert, V.A.: Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems. SIAM J. Math. Anal. 27, 1261–1269 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  48. Smoller, J.: Shock Waves and Reaction Diffusion Equations. Springer, New York (1994)

    Book  MATH  Google Scholar 

  49. Terman, D.: Infinitely many traveling wave solutions of a gradient system. Trans. Am. Math. Soc. 301, 537–556 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  50. Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)

    Article  MathSciNet  Google Scholar 

  51. Vega, J.M.: The asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains. J. Differ. Equ. 102, 119–152 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  52. Volpert, A., Volpert, V.: Existence of multidimensional traveling waves and systems of waves. Commun. Partial Differ. Equ. 26, 421–459 (2001)

    Article  MATH  Google Scholar 

  53. Volpert, A., Volpert, V., Volpert, V.: Traveling Wave Solutions of Parabolic Systems. American Mathematical Society, Providence (1994)

    MATH  Google Scholar 

  54. Wei, J., Winter, M.: Clustered spots in the FitzHugh–Nagumo system. J. Differ. Equ. 213, 121–145 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  55. Yanagida, E.: Standing pulse solutions in reaction–diffusion systems with skew-gradient structure. J. Dyn. Differ. Equ. 14, 189–205 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yanagida, E.: Stability of fast travelling pulse solutions of the FitzHugh–Nagumo equations. J. Math. Biol. 22, 81–104 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Part of the work was done at National Changhua University of Education. Research is supported in part by the Mathematics Research Promotion Center, National Changhua University of Education, and Ministry of Science and Technology, Taiwan, ROC.

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Authors and Affiliations

  1. Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan

    Chao-Nien Chen

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Y. S. Choi

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  1. Chao-Nien Chen
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  2. Y. S. Choi
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Correspondence to Chao-Nien Chen.

Additional information

Communicated by P. Rabinowitz.

Dedicated to Jen-Chung Chuan on the occasion of his 65th birthday.

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Chen, CN., Choi, Y.S. Traveling pulse solutions to FitzHugh–Nagumo equations. Calc. Var. 54, 1–45 (2015). https://doi.org/10.1007/s00526-014-0776-z

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