Spatially discrete Fitzhugh-Nagumo equations

SIAM Journal on Applied Mathematics

Volumn 65, Issue 4, 2005, Pages 1153-1174

  Elmer, Christopher E ^a   Van Vleck, Erik S ^b  

^a NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF KANSAS   (United States)

Author keywords

Discrete diffusion; Traveling fronts and pulses

Indexed keywords

DEPOLARIZATION; DISCRETE DIFFUSION; HYPERPOLARIZATION; TRAVELING FRONTS AND PULSES;

DIFFUSION; POLARIZATION;

MATHEMATICAL TECHNIQUES;

EID: 25144464603     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S003613990343687X     Document Type: Article

References (45)

1. A. R. A. ANDERSON AND B. D. SLEEMAN, Wave front propagation and its failure in coupled systems of discrete bistable cells modelled by FitzHugh-Nagumo dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), pp. 63-74.
2. G.G. ARONSON AND H. F. WEINBERGER, Multidimensional nonlinear diffusion arising from population genetics, Adv. Math., 30 (1978), pp. 33-76.
3. P. W. BATES, X. CHEN, AND A. J. J. CHMAJ, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), pp. 520-546.
4. J. BELL AND C. COSNER, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42 (1984), pp. 1-114.
5. S. N. CHOW, J. MALLET-PARET, AND W. SHEN, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), pp. 248-291.
6. J. W. CAHN, Theory of crystal growth and interface motion in crystalline materials, Acta Met., 6 (1960), pp. 554-561.
7. J. W. CAHN, J. MALLET-PARET, AND E. S. VAN VLECK, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1998), pp. 455-493.
8. A. CARPIO AND L. L. BONILLA, Pulse propagation in discrete systems of coupled excitable cells, SIAM J. Appl. Math., 63 (2002), pp. 619-635.
9. S. BINCZAK, J. C. EILBECK, AND A. C. SCOTT, Ephaptic coupling of myelinated nerve fibers, Phys. D, 148 (2001), pp. 159-174.
10. B. DENG, The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations, SIAM J. Math. Anal., 22 (1991), pp. 1631-1650.
11. C. E. ELMER, Multiple Planar Interfaces for Crystalline Materials with Spatially Discrete Gradient Energy, preprint, 2005.
12. C. E. ELMER AND E. S. VAN VLECK, Analysis and computation of traveling wave solutions of bistable differential-difference equations, Nonlinearity, 12 (1999), pp. 771-798.
13. C. E. ELMER AND E. S. VAN VLECK, Traveling wave solutions for bistable differential-difference equations with periodic diffusion, SIAM J. Appl. Math., 61 (2001), pp. 1648-1679.
14. T. ERNEUX AND G. NICOLIS, Propagating waves in discrete bistable reaction-diffusion systems, Phys. D, 67 (1993), pp. 237-244.
15. J. EVANS, Nerve axon equations I: Linear approximations, Indiana Univ. Math. J., 21 (1972), pp. 877-955.
16. J. EVANS, Nerve axon equations II: Stability at rest, Indiana Univ. Math. J., 22 (1972), pp. 75-90.
17. J. EVANS, Nerve axon equations III: Stability of the nerve impulse, Indiana Univ. Math. J., 22 (1972), pp. 577-594.
18. J. EVANS, Nerve axon equations IV: The stable and unstable impulse, Indiana Univ. Math. J., 24 (1975), pp. 1169-1190.
19. G. FATH, Propagation failure of traveling waves in discrete bistable medium, Phys. D, 116 (1998), pp. 176-190.
20. J. A. FEROE, Existence and stability of multiple impulse solutions of a nerve equation, SIAM J. Appl. Math., 42 (1982), pp. 235-246.
21. P. C. FIFE AND J. B. MCLEOD, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), pp. 335-361.
22. R. FITZHUGH, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), pp. 445-466.
23. W.-Z. GAO, Threshold behavior and propagation for a differential- difference system, SIAM J. Math. Anal., 24 (1993), pp. 89-115.
24. S. HEINZE, G. PAPANICOLAOU, AND A. STEVENS, Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math., 62 (2001), pp. 129-148.
25. C. K. R. T. JONES, Stability of the traveling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc., 286 (1984), pp. 431-469.
26. J. P. KEENER, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), pp. 556-572.
27. J. P. KEENER, The effects of discrete gap junction coupling on propagation in myocardium, J. Theoret. Biol., 148 (1991), pp. 49-82.
28. J. KEENER AND J. SNEYD, Mathematical Physiology, Springer-Verlag, New York, 1998.
29. M. KRUPA, B. SANDSTEDE, AND P. SZMOLYAN, Fast and slow waves in the FitzHugh-Nagumo equation, J. Differential Equations, 133 (1997), pp. 49-97.
30. K. MAGINU, Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction-diffusion systems, SIAM J. Appl. Math., 45 (1985), pp. 750-774.
31. J. MALLET-PARET, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), pp. 1-48.
32. J. MALLET-PARET, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), pp. 49-128.
33. H. MCKEAN, Nagumo's equation, Adv. Math., 4 (1970), pp. 209-223.
34. J. NAGUMO, S. ARIMOTO, AND S. YOSHIZAWA, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Engrg., 50 (1964), pp. 2061-2070.
35. J. RINZEL AND J. B. KELLER, Traveling wave solutions of a nerve conduction equation, Biophys. J., 13 (1973), pp. 1313-1337.
36. A. SCOTT, Neuroscience, A Mathematical Primer, Springer-Verlag, New York, 2002.
37. L. F. SHAMPINE, R. C. ALLEN, AND S. PRUESS, Fundamentals of Numerical Computing, Wiley, New York, 1997.
38. A. TONNELIER, Wave propagation in discrete media, J. Math. Biol., (2001), pp. 1-19.
39. A. TONNELIER, The McKean's caricature of the FitzHugh-Nagumo model I. The space-clamped system, SIAM J. Appl. Math., 63 (2002), pp. 459-484.
40. A. TONNELIER, The McKean's caricature of the FitzHugh-Nagumo model: Traveling pulses in discrete diffusive medium, Phys. Rev. E (3), 67 (2003), 036105.
41. W.-P. WANG, Multiple impulse solutions to McKean's caricature of the nerve equation. I. Existence, Comm. Pure Appl. Math., 41 (1988), pp. 71-103.
42. W.-P. WANG, Multiple impulse solutions to McKean's caricature of the nerve equation. II. Stability, Comm. Pure Appl. Math., 41 (1988), pp. 997-1025.
43. E. YANAGIDA, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equation, J. Math. Biol., 22 (1985), pp. 81-104.
44. B. ZINNER, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), pp. 1016-1020.
45. B. ZINNER, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), pp. 1-27.