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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[43]

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

[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.

[45]

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

[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.

[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.

[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.

[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.

[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.

[51]

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

[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.

[53]

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

[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.

[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.

[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.

[57]

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.

[58]

P. van Heijster and B. Sandstede, Bifurcations to Travelling Planar Spots in a Three-Component FitzHugh–Nagumo system, Physica D, 275 (2014), 19-34. doi: 10.1016/j.physd.2014.02.001.

[59]

E. Yanagida, Stability of Fast Travelling Wave Solutions of the FitzHugh-Nagumo Equations, J. Math. Biol., 22 (1985), 81-104. doi: 10.1007/BF00276548.

[60]

K. Zumbrun, Instantaneous Shock Location and One-Dimensional Nonlinear Stability of Viscous Shock Waves, Quarterly of applied mathematics, 69 (2011), 177-202. doi: 10.1090/S0033-569X-2011-01221-6.

[61]

K. Zumbrun and P. Howard, Pointwise Semigroup Methods and Stability of Viscous Shock Waves, Indiana Univ. Math. J., 47 (1998), 741-871. doi: 10.1512/iumj.1998.47.1604.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[43]

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

[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.

[45]

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

[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.

[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.

[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.

[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.

[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.

[51]

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

[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.

[53]

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

[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.

[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.

[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.

[57]

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.

[58]

P. van Heijster and B. Sandstede, Bifurcations to Travelling Planar Spots in a Three-Component FitzHugh–Nagumo system, Physica D, 275 (2014), 19-34. doi: 10.1016/j.physd.2014.02.001.

[59]

E. Yanagida, Stability of Fast Travelling Wave Solutions of the FitzHugh-Nagumo Equations, J. Math. Biol., 22 (1985), 81-104. doi: 10.1007/BF00276548.

[60]

K. Zumbrun, Instantaneous Shock Location and One-Dimensional Nonlinear Stability of Viscous Shock Waves, Quarterly of applied mathematics, 69 (2011), 177-202. doi: 10.1090/S0033-569X-2011-01221-6.

[61]

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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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