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Stability of pulse solutions for the discrete FitzHugh-Nagumo system

Authors: H. J. Hupkes and B. Sandstede
Journal: Trans. Amer. Math. Soc. 365 (2013), 251-301
MSC (2010): Primary 34A33, 34K26, 34D35, 34K08
Published electronically: July 11, 2012
MathSciNet review: 2984059
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Abstract: We show that the fast travelling pulses of the discrete FitzHugh-Nagumo system in the weak-recovery regime are nonlinearly stable. The spectral conditions that need to be verified involve linear operators that are associated to functional differential equations of mixed type. Such equations are ill-posed and do not admit a semi-flow, which precludes the use of standard Evans-function techniques. Instead, we construct the potential eigenfunctions directly by using exponential dichotomies, Fredholm techniques and an infinite-dimensional version of the Exchange Lemma.

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  • [BC99] P. W. Bates and A. Chmaj, A Discrete Convolution Model for Phase Transitions, Arch. Rational Mech. Anal. 150 (1999), 281-305. MR 1741258 (2001c:82026)
  • [BG02] S. Benzoni-Gavage, Stability of Semi-Discrete Shock Profiles by Means of an Evans Function in Infinite Dimensions, J. Dynam. Diff. Equations 14 (2002), 613-674. MR 1917653 (2003j:35204)
  • [BGHR03] S. Benzoni-Gavage, P. Huot, and F. Rousset, Nonlinear Stability of Semidiscrete Shock Waves, SIAM J. Math. Anal. 35 (2003), 639-707. MR 2048404 (2005j:35147)
  • [BHSZ10] M. Beck, H. J. Hupkes, B. Sandstede, and K. Zumbrun, Nonlinear Stability of Semidiscrete Shocks for Two-Sided Schemes, SIAM J. on Mathematical Analysis 42 (2010), 857-903. MR 2644362 (2011d:35301)
  • [Cah60] J. W. Cahn, Theory of Crystal Growth and Interface Motion in Crystalline Materials, Acta Met. 8 (1960), 554-562.
  • [Car77] G. Carpenter, A Geometric Approach to Singular Perturbation Problems with Applications to Nerve Impulse Equations, J. Diff. Eq. 23 (1977), 335-367. MR 0442379 (56:762a)
  • [Car05a] A. Carpio, Asymptotic Construction of Pulses in the Discrete Hodgkin-Huxley Model for Myelinated Nerves, Physical Review E 72 (2005), 011905. MR 2178366 (2006e:92008)
  • [Car05b] -, Wave Trains, Self-Oscillations and Synchronization in Discrete Media, Physica D 207 (2005), 117-136. MR 2166978 (2006d:37165)
  • [CB02] A. Carpio and L. L. Bonilla, Pulse Propagation in Discrete Systems of Coupled Excitable Cells, SIAM J. Appl. Math. 63 (2002), 619-635. MR 1951953 (2003m:92027)
  • [CH99] X. Chen and S. P. Hastings, Pulse Waves for a Semi-Discrete Morris-Lecar Type Model, J. Math. Bio. 38 (1999), 1-20. MR 1674025 (2000b:92006)
  • [CMPS98] S. N. Chow, J. Mallet-Paret, and W. Shen, Traveling Waves in Lattice Dynamical Systems, J. Diff. Eq. 149 (1998), 248-291. MR 1646240 (2000b:37092)
  • [CR93] L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circ. Syst. 40 (1993), 147-156.
  • [EN93] T. Erneux and G. Nicolis, Propagating Waves in Discrete Bistable Reaction-Diffusion Systems, Physica D 67 (1993), 237-244. MR 1234443 (94e:92009)
  • [Eva75] J. W. Evans, Nerve Axon Equations, IV: The Stable and Unstable Impulse, Indiana Univ. Math. J. 24 (1975), 1169-1190. MR 0393891 (52:14698)
  • [EVV05] C. E. Elmer and E. S. Van Vleck, Spatially Discrete FitzHugh-Nagumo Equations, SIAM J. Appl. Math. 65 (2005), 1153-1174. MR 2147323 (2006a:35167)
  • [GS07] A. Ghazaryan and B. Sandstede, Nonlinear Convective Instability of Turing-Unstable Fronts Near Onset: A Case Study, SIAM J. Appl. Dyn. Sys. 6 (2007), 319-347. MR 2318657 (2008d:37137)
  • [Has76] S. Hastings, On Travelling Wave Solutions of the Hodgkin-Huxley Equations, Arch. Rat. Mech. Anal. 60 (1976), 229-257. MR 0402302 (53:6123)
  • [Hen81] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981. MR 610244 (83j:35084)
  • [HHK52] A. L. Hodgkin, A. F. Huxley, and B. Katz, Measurement of Current-Voltage Relations in the Membrane of the Giant Axon of Loligo, J. Physiology 116 (1952), 424-448.
  • [HS49] A. F. Huxley and R. Stampfli, Evidence for Saltatory Conduction in Peripheral Meylinated Nerve Fibres, J. Physiology 108 (1949), 315-339.
  • [HS10] H. J. Hupkes and B. Sandstede, Travelling Pulse Solutions for the Discrete FitzHugh-Nagumo System, SIAM J. Appl. Dyn. Sys. 9 (2010), 827-882. MR 2665452
  • [HVL93] J. K. Hale and S. M. Verduyn-Lunel, Introduction to Functional-Differential Equations, Springer-Verlag, New York, 1993. MR 1243878 (94m:34169)
  • [HVL05] H. J. Hupkes and S. M. Verduyn-Lunel, Analysis of Newton's Method to Compute Travelling Waves in Discrete Media, J. Dyn. Diff. Eq. 17 (2005), 523-572. MR 2165558 (2006e:65129)
  • [HVL07] -, Center Manifold Theory for Functional Differential Equations of Mixed Type, J. Dyn. Diff. Eq. 19 (2007), 497-560. MR 2333418 (2008c:34156)
  • [HVL09] -, Lin's Method and Homoclinic Bifurcations for Functional Differential Equations of Mixed Type, Indiana Univ. Math. J. 58 (2009), 2433-2487. MR 2603755 (2011d:34147)
  • [HZ93] D. Hankerson and B. Zinner, Wavefronts for a Cooperative Tridiagonal System of Differential Equations, J. Dyn. Diff. Eq. 5 (1993), 359-373. MR 1223452 (94c:34120)
  • [JKL91] C. K. R. T. Jones, N. Kopell, and R. Langer, Construction of the FitzHugh-Nagumo Pulse Using Differential Forms, Patterns and Dynamics in Reactive Media (New York) (H. Swinney, G. Aris, and D. G. Aronson, eds.), IMA Volumes in Mathematics and its Applications, vol. 37, Springer, 1991, pp. 101-116. MR 1228919 (94e:92015)
  • [Jon84] C. K. R. T. Jones, Stability of the Travelling Wave Solutions of the FitzHugh-Nagumo System, Trans. AMS 286 (1984), 431-469. MR 760971 (86b:35011)
  • [Kee87] J. P. Keener, Propagation and its Failure in Coupled Systems of Discrete Excitable Cells, SIAM J. Appl. Math. 47 (1987), 556-572. MR 889639 (88k:92028)
  • [KS98] J. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, 1998. MR 1673204 (2000c:92010)
  • [LE92] J. P. Laplante and T. Erneux, Propagation Failure in Arrays of Coupled Bistable Chemical Reactors, J. Phys. Chem. 96 (1992), 4931-4934.
  • [Lil25] R. S. Lillie, Factors Affecting Transmission and Recovery in the Passive Iron Nerve Model, J. of General Physiology 7 (1925), 473-507.
  • [MP99a] J. Mallet-Paret, The Fredholm Alternative for Functional Differential Equations of Mixed Type, J. Dyn. Diff. Eq. 11 (1999), 1-48. MR 1680463 (2000j:34107)
  • [MP99b] J. Mallet-Paret, The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems, J. Dyn. Diff. Eq. 11 (1999), 49-128. MR 1680459 (2000k:37125)
  • [MPVL] J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations, J. Diff. Eq., to appear.
  • [Ran78] L. A. Ranvier, Lećons sur l'histologie du système nerveux, par m. l. ranvier, recueillies par m. ed. weber, F. Savy, Paris, 1878.
  • [Rus89] A. Rustichini, Functional Differential Equations of Mixed Type: The Linear Autonomous Case, J. Dyn. Diff. Eq. 11 (1989), 121-143. MR 1010963 (91b:34108)
  • [Ton03] A. Tonnelier, McKean Caricature of the FitzHugh-Nagumo Model: Traveling Pulses in a Discrete Diffusive Medium, Physical Review E 67 (2003), 036105. MR 1976825 (2004c:35228)
  • [Yan85] E. Yanagida, Stability of Fast Travelling Wave Solutions of the FitzHugh-Nagumo Equations, J. Math. Biol. 22 (1985), 81-104. MR 802737 (87a:92019)
  • [Zin91] B. Zinner, Stability of Traveling Wavefronts for the Discrete Nagumo Equation, SIAM J. Math. Anal. 22 (1991), 1016-1020. MR 1112063 (92d:34146)

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Additional Information

H. J. Hupkes
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
Address at time of publication: Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands

B. Sandstede
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
Email: bjorn{\textunderscore}

Keywords: Discrete FitzHugh–Nagumo system, travelling pulses, stability, singular perturbation theory, lattice differential equations, functional differential equations.
Received by editor(s): May 6, 2010
Received by editor(s) in revised form: February 1, 2011
Published electronically: July 11, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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