References
- A.A. Aderogba, M. Chapwanya, J. Djoko Kamdem, and J.M.-S. Lubuma, Coupling finite volume and nonstandard finite difference schemes for a singularly perturbed Schrödinger equation, Int. J. Comput. Math., 93 (2016), pp. 1833–1844. doi: 10.1080/00207160.2015.1076569
- A.A. Aderogba, M. Chapwanya, and O.A. Jejeniwa, Finite difference discretisation of a model for biological nerve conduction, in International Conference of Numerical Analysis and Applied Mathematics 2015 (ICNAAM 2015), Rhodes, Greece, Vol. 1738, AIP Publishing, p. 030009.
- A.A. Aderogba and M. Chapwanya, An explicit nonstandard finite difference scheme for the Allen–Cahn equation, J. Differ. Equ. Appl. 21 (2015), pp. 875–886. doi: 10.1080/10236198.2015.1055737
- J.G. Alford and G. Auchmuty, Rotating wave solutions of the FitzHugh–Nagumo equations, J. Math. Biol. 53 (2006), pp. 797–819. doi: 10.1007/s00285-006-0022-1
- R. Anguelov, P. Kama, and J.M-S. Lubuma, On non-standard finite difference models of reaction–diffusion equations, J. Comput. Appl. Math. 175 (2005), pp. 11–29. doi: 10.1016/j.cam.2004.06.002
- R. Anguelov and J.M-S. Lubuma, Contributions to the mathematics of the nonstandard finite difference method and applications, Numer. Methods Partial Differ. Equ. 17 (2001), pp. 518–543. doi: 10.1002/num.1025
- R. Anguelov and J.M.-S. Lubuma, Nonstandard finite difference method by nonlocal approximation, Math. Comput. Simul. 61 (2003), pp. 465–475. doi: 10.1016/S0378-4754(02)00106-4
- G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh–Nagumo equation, Nonlinear Anal. 113 (2015), pp. 51–70. doi: 10.1016/j.na.2014.09.023
- G.A. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations, J. Differ. Equ. 23 (1977), pp. 335–367. doi: 10.1016/0022-0396(77)90116-4
- A. Carpio and L.L. Bonilla, Pulse propagation in discrete systems of coupled excitable cells, SIAM J. Appl. Math. 63 (2003), pp. 619–635. doi: 10.1137/S0036139901391732
- M. Chapwanya, J.M.-S. Lubuma, and R.E. Mickens, From enzyme kinetics to epidemiological models with Michaelis–Menten contact rate: Design of nonstandard finite difference schemes, Comput. Math. Appl. 64 (2012), pp. 201–213. doi: 10.1016/j.camwa.2011.12.058
- M. Chapwanya, J.M.-S. Lubuma, and R.E. Mickens, Nonstandard finite difference schemes for Michaelis–Menten type reaction-diffusion equations, Numer. Methods Partial Differ. Equ. 29 (2013), pp. 337–360. doi: 10.1002/num.21733
- Z. Chen, A.B. Gumel, and R.E. Mickens, Nonstandard discretizations of the generalized Nagumo reaction-diffusion equation, Numer. Methods Partial Differ. Equ. 19 (2003), pp. 363–379. doi: 10.1002/num.10048
- R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961), pp. 445–466. doi: 10.1016/S0006-3495(61)86902-6
- P. Gordon, Nonsymmetric difference equations, J. Soc. Indust. Appl. Math. 13 (1965), pp. 667–673. doi: 10.1137/0113044
- A.R. Gourlay, Hopscotch: A fast second-order partial differential equation solver, IMA J. Appl. Math. 6 (1970), pp. 375–390. doi: 10.1093/imamat/6.4.375
- J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst. 9 (2010), pp. 138–153. doi: 10.1137/090758404
- J.P. Keener and J. Sneyd, Mathematical Physiology, Vol. 1, Springer, New York, 1998.
- T. Kostova, R. Ravindran, and M. Schonbek, Fitzhugh–Nagumo revisited: Types of bifurcations, periodical forcing and stability regions by a Lyapunov functional, Int. J. Bifurc. Chaos 14 (2004), pp. 913–925. doi: 10.1142/S0218127404009685
- Y.N. Kyrychko, M.V. Bartuccelli, and K.B. Blyuss, Persistence of travelling wave solutions of a fourth order diffusion system, J. Comput. Appl. Math. 176 (2005), pp. 433–443. doi: 10.1016/j.cam.2004.07.028
- R.E. Mickens, Nonstandard finite difference models of differential equations, World Scientific, Singapore, 1994.
- R.E. Mickens, Nonstandard finite difference schemes for reaction-diffusion equations, Numer. Methods Partial Differ. Equ. 15 (1999), pp. 201–214. doi: 10.1002/(SICI)1098-2426(199903)15:2<201::AID-NUM5>3.0.CO;2-H
- S. Mischler, C. Quininao, and J. Touboul, On a kinetic FitzHugh–Nagumo model of neuronal network, Comm. Math. Phys. 342 (2016), pp. 1001–1042. doi: 10.1007/s00220-015-2556-9
- J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, Springer, New York, 2011.
- J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 50 (1962), pp. 2061–2070. doi: 10.1109/JRPROC.1962.288235
- D. Olmos and B.D. Shizgal, Pseudospectral method of solution of the FitzHugh–Nagumo equation, Math. Comput. Simulation 79 (2009), pp. 2258–2278. doi: 10.1016/j.matcom.2009.01.001
- J. Rauch and J. Smoller, Qualitative theory of the FitzHugh-Nagumo equations, Adv. Math. 27 (1978), pp. 12–44. doi: 10.1016/0001-8708(78)90075-0
- C. Reinecke and G. Sweers, Existence and uniqueness of solutions on bounded domains to a FitzHugh–Nagumo type elliptic system, Pacific J. Math. 197 (2001), pp. 183–211. doi: 10.2140/pjm.2001.197.183
- C. Reinecke and G. Sweers, Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type, Math. Nachr. 251 (2003), pp. 64–87. doi: 10.1002/mana.200310031
- C. Rocsoreanu, A. Georgescu, and N. Giurgiteanu, The FitzHugh-Nagumo Model: Bifurcation and Dynamics, Mathematical Modelling: Theory and Applications, Vol. 10, Springer-Science + Business Media, B.V., Kluwer Academic, New York, 2000.
- L-I.W. Roeger, Nonstandard finite difference schemes for differential equations with n+1 distinct fixed-points, J. Differ. Equ. Appl. 15 (2009), pp. 133–151. doi: 10.1080/10236190801987912
- L-I.W. Roeger and R.E Mickens, Exact finite-difference schemes for first order differential equations having three distinct fixed-points, J. Differ. Equ. Appl. 13 (2007), pp. 1179–1185. doi: 10.1080/10236190701466439
- A.A. Soliman, Numerical simulation of the FitzHugh-Nagumo equations, Abstr. Appl. Anal. 2012 (2012), ID 762516. Available at https://www.hindawi.com/journals/aaa/2012/762516/.
- E.H. Twizell, A.B. Gumel and Q. Cao, A second-order scheme for the Brusselator reaction–diffusion system, J. Math. Chem. 26 (1999), pp. 297–316. doi: 10.1023/A:1019158500612
- B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differ. Equ. 96 (1992), pp. 1–27. doi: 10.1016/0022-0396(92)90142-A