September  2019, 39(9): 5017-5083. doi: 10.3934/dcds.2019205

Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions

Mathematisch Instituut - Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands

* Corresponding author: w.m.schouten@math.leidenuniv.nl

Received  May 2018 Revised  February 2019 Published  May 2019

Fund Project: Both authors acknowledge support from the Netherlands Organization for Scientific Research (NWO) (grant 639.032.612)

We establish the existence and nonlinear stability of travelling pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions close to the continuum limit. For the verification of the spectral properties, we need to study a functional differential equation of mixed type (MFDE) with unbounded shifts. We avoid the use of exponential dichotomies and phase spaces, by building on a technique developed by Bates, Chen and Chmaj for the discrete Nagumo equation. This allows us to transfer several crucial Fredholm properties from the PDE setting to our discrete setting.

Citation: Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205
References:
[1]

P. W. BatesX. Chen and A. Chmaj, Traveling Waves of Bistable Dynamics on a Lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002. Google Scholar

[2]

M. BeckH. J. HupkesB. Sandstede and K. Zumbrun, Nonlinear Stability of Semidiscrete Shocks for Two-Sided Schemes, SIAM J. Math. Anal., 42 (2010), 857-903. doi: 10.1137/090775634. Google Scholar

[3]

M. BeckB. Sandstede and K. Zumbrun, Nonlinear stability of time-periodic viscous shocks, Archive for Rational Mechanics and Analysis, 196 (2010), 1011-1076. doi: 10.1007/s00205-009-0274-1. Google Scholar

[4]

M. Beck, G. Cox, C. Jones, Y. Latushkin, K. McQuighan and A. Sukhtayev, Instability of pulses in gradient reaction–diffusion systems: A symplectic approach, Phil. Trans. R. Soc. A, 376 (2018), 20170187, 20pp. doi: 10.1098/rsta.2017.0187. Google Scholar

[5]

S. Benzoni-GavageP. Huot and F. Rousset, Nonlinear Stability of Semidiscrete Shock Waves, SIAM J. Math. Anal., 35 (2003), 639-707. doi: 10.1137/S0036141002418054. Google Scholar

[6]

J. Bos, Fredholm Eigenschappen van Systemen met Interactie Over een Oneindig Bereik, Bachelor Thesis, Leiden University, 2015.Google Scholar

[7]

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, Journal of Physics A: Mathematical and Theoretical, 45 (2012), 033001,109 pp. doi: 10.1088/1751-8113/45/3/033001. Google Scholar

[8]

P. C. Bressloff, Waves in Neural Media: From single Neurons to Neural Fields, Lecture notes on mathematical modeling in the life sciences., Springer, 2014. doi: 10.1007/978-1-4614-8866-8. Google Scholar

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J. W. Cahn, Theory of Crystal Growth and Interface Motion in Crystalline Materials, Acta Met., 8 (1960), 554-562. Google Scholar

[10]

G. Carpenter, A Geometric Approach to Singular Perturbation Problems with Applications to Nerve Impulse Equations, J. Diff. Eq., 23 (1977), 335-367. doi: 10.1016/0022-0396(77)90116-4. Google Scholar

[11]

P. CarterB. de Rijk and B. Sandstede, Stability of traveling pulses with oscillatory tails in the FitzHugh–Nagumo system, Journal of Nonlinear Science, 26 (2016), 1369-1444. doi: 10.1007/s00332-016-9308-7. Google Scholar

[12]

P. Carter and B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh–Nagumo system, SIAM Journal on Mathematical Analysis, 47 (2015), 3393-3441. doi: 10.1137/140999177. Google Scholar

[13]

C.-N. Chen and X. Hu, Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 827-845. doi: 10.1007/s00526-013-0601-0. Google Scholar

[14]

X. Chen, Existence, Uniqueness and Asymptotic Stability of Traveling Waves in Nonlocal Evolution Equations, Adv. Diff. Eq., 2 (1997), 125-160. Google Scholar

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S. N. ChowJ. Mallet-Paret and W. Shen, Traveling Waves in Lattice Dynamical Systems, J. Diff. Eq., 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478. Google Scholar

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O. Ciaurri, L. Roncal, P. Stinga, J. Torrea and J. Varona, Fractional discrete Laplacian versus discretized fractional Laplacian, preprint, arXiv: 1507.04986.Google Scholar

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F. CiuchiA. MazzullaN. ScaramuzzaE. Lenzi and L. Evangelista, Fractional diffusion equation and the electrical impedance: Experimental evidence in liquid-crystalline cells, The Journal of Physical Chemistry C, 116 (2012), 8773-8777. doi: 10.1021/jp211097m. Google Scholar

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P. Cornwell, Opening the maslov box for traveling waves in skew-gradient systems, preprint, arXiv: 1709.01908.Google Scholar

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P. Cornwell and C. K. Jones, On the existence and stability of fast traveling waves in a doubly-diffusive FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst., 17 (2018), 754–787, arXiv: 1709.09132. doi: 10.1137/17M1149432. Google Scholar

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J. Evans, Nerve axon equations: Ⅲ. stability of the nerve impulse, Indiana Univ. Math. J., 22 (1972), 577-593. doi: 10.1512/iumj.1973.22.22048. Google Scholar

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G. Faye and A. Scheel, Fredholm properties of nonlocal differential operators via spectral flow, Indiana University Mathematics Journal, 63 (2014), 1311-1348. doi: 10.1512/iumj.2014.63.5383. Google Scholar

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G. Faye and A. Scheel, Existence of pulses in excitable media with nonlocal coupling, Advances in Mathematics, 270 (2015), 400-456. doi: 10.1016/j.aim.2014.11.005. Google Scholar

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G. Faye and A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370 (2018), 5843-5885. doi: 10.1090/tran/7190. Google Scholar

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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), 335-361. doi: 10.1007/BF00250432. Google Scholar

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R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J., 1 (1966), 445-466. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar

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R. FitzHugh, Mathematical Models of Excitation and Propagation in Nerve, Publisher Unknown, 1966.Google Scholar

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R. Fitzhugh, Motion picture of nerve impulse propagation using computer animation, Journal of Applied Physiology, 25 (1968), 628-630. doi: 10.1152/jappl.1968.25.5.628. Google Scholar

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T. Gallay and E. Risler, A variational proof of global stability for bistable travelling waves, Differential and Integral Equations, 20 (2007), 901-926. Google Scholar

[29]

Q. Gu, E. Schiff, S. Grebner, F. Wang and R. Schwarz, Non-Gaussian transport measurements and the Einstein relation in amorphous silicon, Physical Review Letters, 76 (1996), 3196. doi: 10.1103/PhysRevLett.76.3196. Google Scholar

[30]

C. H. S. Hamster and H. J. Hupkes, Stability of travelling waves for reaction-diffusion equations with multiplicative noise, SIAM J. Appl. Dyn. Syst., 18 (2019), 205-278. doi: 10.1137/17M1159518. Google Scholar

[31]

S. Hastings, On Travelling Wave Solutions of the Hodgkin-Huxley Equations, Arch. Rat. Mech. Anal., 60 (1976), 229-257. doi: 10.1007/BF01789258. Google Scholar

[32]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiology, 117.Google Scholar

[33]

P. Howard and A. Sukhtayev, The Maslov and Morse indices for Schrödinger operators on [0, 1], Journal of Differential Equations, 260 (2016), 4499-4549. doi: 10.1016/j.jde.2015.11.020. Google Scholar

[34]

H. J. Hupkes and E. Augeraud-Véron, Well-posed of initial value problems on Hilbert spaces, In preparation.Google Scholar

[35]

H. J. Hupkes and B. Sandstede, Travelling Pulse Solutions for the Discrete FitzHugh-Nagumo System, SIAM J. Appl. Dyn. Sys., 9 (2010), 827-882. doi: 10.1137/090771740. Google Scholar

[36]

H. J. Hupkes and B. Sandstede, Stability of Pulse Solutions for the Discrete FitzHugh-Nagumo System, Transactions of the AMS, 365 (2013), 251-301. doi: 10.1090/S0002-9947-2012-05567-X. Google Scholar

[37]

H. J. Hupkes and E. S. Van Vleck, Negative diffusion and traveling waves in high dimensional lattice systems, SIAM J. Math. Anal., 45 (2013), 1068-1135. doi: 10.1137/120880628. Google Scholar

[38]

H. J. Hupkes and E. S. Van Vleck, Travelling Waves for Complete Discretizations of Reaction Diffusion Systems, J. Dyn. Diff. Eqns, 28 (2016), 955-1006. doi: 10.1007/s10884-014-9423-9. Google Scholar

[39]

H. J. Hupkes and S. M. Verduyn-Lunel, Center Manifold Theory for Functional Differential Equations of Mixed Type, J. Dyn. Diff. Eq., 19 (2007), 497-560. doi: 10.1007/s10884-006-9055-9. Google Scholar

[40]

C. K. R. T. Jones, Stability of the Travelling Wave Solutions of the FitzHugh-Nagumo System, Trans. AMS, 286 (1984), 431-469. doi: 10.1090/S0002-9947-1984-0760971-6. Google Scholar

[41]

C. K. R. T. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo Pulse using Differential Forms, in Patterns and Dynamics in Reactive Media (eds. H. Swinney, G. Aris and D. G. Aronson), vol. 37 of IMA Volumes in Mathematics and its Applications, Springer, New York, 1991,101–115. doi: 10.1007/978-1-4612-3206-3_7. Google Scholar

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C. Jones, Geometric singular perturbation theory, Dynamical Systems (Montecatini Terme, 1994), 44–118, Lecture Notes in Math., 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095239. Google Scholar

[43]

A. KaminagaV. K. Vanag and I. R. Epstein, A Reaction–Diffusion Memory Device, Angewandte Chemie International Edition, 45 (2006), 3087-3089. Google Scholar

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T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, vol. 457, Springer, 2013. doi: 10.1007/978-1-4614-6995-7. Google Scholar

[45]

J. Keener and J. Sneed, Mathematical Physiology, Springer–Verlag, New York, 1998. Google Scholar

[46]

M. KrupaB. Sandstede and P. Szmolyan, Fast and Slow Waves in the FitzHugh-Nagumo Equation, J. Diff. Eq., 133 (1997), 49-97. doi: 10.1006/jdeq.1996.3198. Google Scholar

[47]

R. S. Lillie, Factors Affecting Transmission and Recovery in the Passive Iron Nerve Model, J. of General Physiology, 7 (1925), 473-507. doi: 10.1085/jgp.7.4.473. Google Scholar

[48]

J. Mallet-Paret, The Fredholm Alternative for Functional Differential Equations of Mixed Type, J. Dyn. Diff. Eq., 11 (1999), 1-47. doi: 10.1023/A:1021889401235. Google Scholar

[49]

M. Or-Guil, M. Bode, C. P. Schenk and H. G. Purwins, Spot Bifurcations in Three-Component Reaction-Diffusion Systems: The Onset of Propagation, Physical Review E, 57 (1998), 6432.Google Scholar

[50]

D. Pinto and G. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: 1. traveling fronts and pulses, SIAM J. of Appl. Math., 62 (2001), 206-225. doi: 10.1137/S0036139900346453. Google Scholar

[51]

L. A. Ranvier, Lećons sur l'Histologie du Système Nerveux, par M. L. Ranvier, Recueillies par M. Ed. Weber, F. Savy, Paris, 1878.Google Scholar

[52]

A. Rustichini, Functional Differential Equations of Mixed Type: the Linear Autonomous Case, J. Dyn. Diff. Eq., 1 (1989), 121-143. doi: 10.1007/BF01047828. Google Scholar

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N. Sabourova, Real and Complex Operator Norms, Licentiate Thesis, Luleå University of Technology, 2007.Google Scholar

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C. P. Schenk, M. Or-Guil, M. Bode and H. G. Purwins, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Physical Review Letters, 78 (1997), 3781. doi: 10.1103/PhysRevLett.78.3781. Google Scholar

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W. M. Schouten-Straatman and H. J. Hupkes, Travelling waves for spatially discrete systems of FitzHugh-Nagumo type with periodic coefficients, preprint, arXiv: 1808.00761.Google Scholar

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J. Sneyd, Tutorials in Mathematical Biosciences II., vol. 187 of Lecture Notes in Mathematics, chapter Mathematical Modeling of Calcium Dynamics and Signal Transduction., New York: Springer, 2005. doi: 10.1007/b107088. Google Scholar

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A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B, 79 (2009), 144123. doi: 10.1103/PhysRevB.79.144123. Google Scholar

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show all references

References:
[1]

P. W. BatesX. Chen and A. Chmaj, Traveling Waves of Bistable Dynamics on a Lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002. Google Scholar

[2]

M. BeckH. J. HupkesB. Sandstede and K. Zumbrun, Nonlinear Stability of Semidiscrete Shocks for Two-Sided Schemes, SIAM J. Math. Anal., 42 (2010), 857-903. doi: 10.1137/090775634. Google Scholar

[3]

M. BeckB. Sandstede and K. Zumbrun, Nonlinear stability of time-periodic viscous shocks, Archive for Rational Mechanics and Analysis, 196 (2010), 1011-1076. doi: 10.1007/s00205-009-0274-1. Google Scholar

[4]

M. Beck, G. Cox, C. Jones, Y. Latushkin, K. McQuighan and A. Sukhtayev, Instability of pulses in gradient reaction–diffusion systems: A symplectic approach, Phil. Trans. R. Soc. A, 376 (2018), 20170187, 20pp. doi: 10.1098/rsta.2017.0187. Google Scholar

[5]

S. Benzoni-GavageP. Huot and F. Rousset, Nonlinear Stability of Semidiscrete Shock Waves, SIAM J. Math. Anal., 35 (2003), 639-707. doi: 10.1137/S0036141002418054. Google Scholar

[6]

J. Bos, Fredholm Eigenschappen van Systemen met Interactie Over een Oneindig Bereik, Bachelor Thesis, Leiden University, 2015.Google Scholar

[7]

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, Journal of Physics A: Mathematical and Theoretical, 45 (2012), 033001,109 pp. doi: 10.1088/1751-8113/45/3/033001. Google Scholar

[8]

P. C. Bressloff, Waves in Neural Media: From single Neurons to Neural Fields, Lecture notes on mathematical modeling in the life sciences., Springer, 2014. doi: 10.1007/978-1-4614-8866-8. Google Scholar

[9]

J. W. Cahn, Theory of Crystal Growth and Interface Motion in Crystalline Materials, Acta Met., 8 (1960), 554-562. Google Scholar

[10]

G. Carpenter, A Geometric Approach to Singular Perturbation Problems with Applications to Nerve Impulse Equations, J. Diff. Eq., 23 (1977), 335-367. doi: 10.1016/0022-0396(77)90116-4. Google Scholar

[11]

P. CarterB. de Rijk and B. Sandstede, Stability of traveling pulses with oscillatory tails in the FitzHugh–Nagumo system, Journal of Nonlinear Science, 26 (2016), 1369-1444. doi: 10.1007/s00332-016-9308-7. Google Scholar

[12]

P. Carter and B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh–Nagumo system, SIAM Journal on Mathematical Analysis, 47 (2015), 3393-3441. doi: 10.1137/140999177. Google Scholar

[13]

C.-N. Chen and X. Hu, Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 827-845. doi: 10.1007/s00526-013-0601-0. Google Scholar

[14]

X. Chen, Existence, Uniqueness and Asymptotic Stability of Traveling Waves in Nonlocal Evolution Equations, Adv. Diff. Eq., 2 (1997), 125-160. Google Scholar

[15]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling Waves in Lattice Dynamical Systems, J. Diff. Eq., 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478. Google Scholar

[16]

O. Ciaurri, L. Roncal, P. Stinga, J. Torrea and J. Varona, Fractional discrete Laplacian versus discretized fractional Laplacian, preprint, arXiv: 1507.04986.Google Scholar

[17]

F. CiuchiA. MazzullaN. ScaramuzzaE. Lenzi and L. Evangelista, Fractional diffusion equation and the electrical impedance: Experimental evidence in liquid-crystalline cells, The Journal of Physical Chemistry C, 116 (2012), 8773-8777. doi: 10.1021/jp211097m. Google Scholar

[18]

P. Cornwell, Opening the maslov box for traveling waves in skew-gradient systems, preprint, arXiv: 1709.01908.Google Scholar

[19]

P. Cornwell and C. K. Jones, On the existence and stability of fast traveling waves in a doubly-diffusive FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst., 17 (2018), 754–787, arXiv: 1709.09132. doi: 10.1137/17M1149432. Google Scholar

[20]

J. Evans, Nerve axon equations: Ⅲ. stability of the nerve impulse, Indiana Univ. Math. J., 22 (1972), 577-593. doi: 10.1512/iumj.1973.22.22048. Google Scholar

[21]

G. Faye and A. Scheel, Fredholm properties of nonlocal differential operators via spectral flow, Indiana University Mathematics Journal, 63 (2014), 1311-1348. doi: 10.1512/iumj.2014.63.5383. Google Scholar

[22]

G. Faye and A. Scheel, Existence of pulses in excitable media with nonlocal coupling, Advances in Mathematics, 270 (2015), 400-456. doi: 10.1016/j.aim.2014.11.005. Google Scholar

[23]

G. Faye and A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370 (2018), 5843-5885. doi: 10.1090/tran/7190. Google Scholar

[24]

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), 335-361. doi: 10.1007/BF00250432. Google Scholar

[25]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J., 1 (1966), 445-466. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar

[26]

R. FitzHugh, Mathematical Models of Excitation and Propagation in Nerve, Publisher Unknown, 1966.Google Scholar

[27]

R. Fitzhugh, Motion picture of nerve impulse propagation using computer animation, Journal of Applied Physiology, 25 (1968), 628-630. doi: 10.1152/jappl.1968.25.5.628. Google Scholar

[28]

T. Gallay and E. Risler, A variational proof of global stability for bistable travelling waves, Differential and Integral Equations, 20 (2007), 901-926. Google Scholar

[29]

Q. Gu, E. Schiff, S. Grebner, F. Wang and R. Schwarz, Non-Gaussian transport measurements and the Einstein relation in amorphous silicon, Physical Review Letters, 76 (1996), 3196. doi: 10.1103/PhysRevLett.76.3196. Google Scholar

[30]

C. H. S. Hamster and H. J. Hupkes, Stability of travelling waves for reaction-diffusion equations with multiplicative noise, SIAM J. Appl. Dyn. Syst., 18 (2019), 205-278. doi: 10.1137/17M1159518. Google Scholar

[31]

S. Hastings, On Travelling Wave Solutions of the Hodgkin-Huxley Equations, Arch. Rat. Mech. Anal., 60 (1976), 229-257. doi: 10.1007/BF01789258. Google Scholar

[32]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiology, 117.Google Scholar

[33]

P. Howard and A. Sukhtayev, The Maslov and Morse indices for Schrödinger operators on [0, 1], Journal of Differential Equations, 260 (2016), 4499-4549. doi: 10.1016/j.jde.2015.11.020. Google Scholar

[34]

H. J. Hupkes and E. Augeraud-Véron, Well-posed of initial value problems on Hilbert spaces, In preparation.Google Scholar

[35]

H. J. Hupkes and B. Sandstede, Travelling Pulse Solutions for the Discrete FitzHugh-Nagumo System, SIAM J. Appl. Dyn. Sys., 9 (2010), 827-882. doi: 10.1137/090771740. Google Scholar

[36]

H. J. Hupkes and B. Sandstede, Stability of Pulse Solutions for the Discrete FitzHugh-Nagumo System, Transactions of the AMS, 365 (2013), 251-301. doi: 10.1090/S0002-9947-2012-05567-X. Google Scholar

[37]

H. J. Hupkes and E. S. Van Vleck, Negative diffusion and traveling waves in high dimensional lattice systems, SIAM J. Math. Anal., 45 (2013), 1068-1135. doi: 10.1137/120880628. Google Scholar

[38]

H. J. Hupkes and E. S. Van Vleck, Travelling Waves for Complete Discretizations of Reaction Diffusion Systems, J. Dyn. Diff. Eqns, 28 (2016), 955-1006. doi: 10.1007/s10884-014-9423-9. Google Scholar

[39]

H. J. Hupkes and S. M. Verduyn-Lunel, Center Manifold Theory for Functional Differential Equations of Mixed Type, J. Dyn. Diff. Eq., 19 (2007), 497-560. doi: 10.1007/s10884-006-9055-9. Google Scholar

[40]

C. K. R. T. Jones, Stability of the Travelling Wave Solutions of the FitzHugh-Nagumo System, Trans. AMS, 286 (1984), 431-469. doi: 10.1090/S0002-9947-1984-0760971-6. Google Scholar

[41]

C. K. R. T. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo Pulse using Differential Forms, in Patterns and Dynamics in Reactive Media (eds. H. Swinney, G. Aris and D. G. Aronson), vol. 37 of IMA Volumes in Mathematics and its Applications, Springer, New York, 1991,101–115. doi: 10.1007/978-1-4612-3206-3_7. Google Scholar

[42]

C. Jones, Geometric singular perturbation theory, Dynamical Systems (Montecatini Terme, 1994), 44–118, Lecture Notes in Math., 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095239. Google Scholar

[43]

A. KaminagaV. K. Vanag and I. R. Epstein, A Reaction–Diffusion Memory Device, Angewandte Chemie International Edition, 45 (2006), 3087-3089. Google Scholar

[44]

T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, vol. 457, Springer, 2013. doi: 10.1007/978-1-4614-6995-7. Google Scholar

[45]

J. Keener and J. Sneed, Mathematical Physiology, Springer–Verlag, New York, 1998. Google Scholar

[46]

M. KrupaB. Sandstede and P. Szmolyan, Fast and Slow Waves in the FitzHugh-Nagumo Equation, J. Diff. Eq., 133 (1997), 49-97. doi: 10.1006/jdeq.1996.3198. Google Scholar

[47]

R. S. Lillie, Factors Affecting Transmission and Recovery in the Passive Iron Nerve Model, J. of General Physiology, 7 (1925), 473-507. doi: 10.1085/jgp.7.4.473. Google Scholar

[48]

J. Mallet-Paret, The Fredholm Alternative for Functional Differential Equations of Mixed Type, J. Dyn. Diff. Eq., 11 (1999), 1-47. doi: 10.1023/A:1021889401235. Google Scholar

[49]

M. Or-Guil, M. Bode, C. P. Schenk and H. G. Purwins, Spot Bifurcations in Three-Component Reaction-Diffusion Systems: The Onset of Propagation, Physical Review E, 57 (1998), 6432.Google Scholar

[50]

D. Pinto and G. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: 1. traveling fronts and pulses, SIAM J. of Appl. Math., 62 (2001), 206-225. doi: 10.1137/S0036139900346453. Google Scholar

[51]

L. A. Ranvier, Lećons sur l'Histologie du Système Nerveux, par M. L. Ranvier, Recueillies par M. Ed. Weber, F. Savy, Paris, 1878.Google Scholar

[52]

A. Rustichini, Functional Differential Equations of Mixed Type: the Linear Autonomous Case, J. Dyn. Diff. Eq., 1 (1989), 121-143. doi: 10.1007/BF01047828. Google Scholar

[53]

N. Sabourova, Real and Complex Operator Norms, Licentiate Thesis, Luleå University of Technology, 2007.Google Scholar

[54]

C. P. Schenk, M. Or-Guil, M. Bode and H. G. Purwins, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Physical Review Letters, 78 (1997), 3781. doi: 10.1103/PhysRevLett.78.3781. Google Scholar

[55]

W. M. Schouten-Straatman and H. J. Hupkes, Travelling waves for spatially discrete systems of FitzHugh-Nagumo type with periodic coefficients, preprint, arXiv: 1808.00761.Google Scholar

[56]

J. Sneyd, Tutorials in Mathematical Biosciences II., vol. 187 of Lecture Notes in Mathematics, chapter Mathematical Modeling of Calcium Dynamics and Signal Transduction., New York: Springer, 2005. doi: 10.1007/b107088. Google Scholar

[57]

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Figure 1.  Illustration of the regions $ R_1,R_2,R_3 $ and $ R_4 $. Note that the regions $ R_2 $ and $ R_3 $ grow when $ h $ decreases, while the regions $ R_1 $ and $ R_4 $ are independent of $ h $
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