A maximum principle for an explicit finite difference scheme approximating the Hodgkin-Huxley model

BIT Numerical Mathematics

Volumn 45, Issue 4, 2005, Pages 725-741

  Hanslien, Monica ^a   Karlsen, Kenneth H ^b   Tveito, Aslak ^a  

^a SIMULA RESEARCH LABORATORY   (Norway)

^b UNIVERSITY OF OSLO   (Norway)

Author keywords

Compactness; Convergence; Explicit scheme; Maximum principle

Indexed keywords

EID: 29144440043     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-005-0023-2     Document Type: Conference Paper

References (11)

1. R. Y. Chegis, Investigation of difference schemes for a class of models of excitability, Comput. Math. Math. Phys., 32 (1992), pp. 757-767.
2. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, R.I., 1998.
3. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis VII, pp. 713-1020, North-Holland, Amsterdam, 2000.
4. A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve (reprinted from J. Phys., 117 (1952), pp. 500-544), Bull. Math. Biol., 52 (1990), pp. 25-71.
5. J. Keener and J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics, 8, Springer, 1980.
6. O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1967.
7. G. T. Lines, M. L. Buist, P. Grøttum, A. J. Pullan, J. Sundnes and A. Tveito, Mathematical Models and Numerical Methods for the Forward Problem in Cardiac Electrophysiology, Research Report, Simula, 2002.
8. M. Mascagni, An initial-boundary value problem of physical significance for equations of nerve conduction, Commun. Pure Appl. Math., 42 (1989), pp. 213-227.
9. M. Mascagni, The backward Euler method for numerical solution of the Hodgkin-Huxley equations of nerve conduction, SIAM J. Numer. Anal., 27 (1990), pp. 941-962.
10. S. Sugiyama, On the stability problems of difference equations, Bull. Sci. Eng. Res. Lab. Waseda Univ., 35 (1969), pp. 140-144.
11. A. Tveito and R. Winther, Introduction to Partial Differential Equations, Texts in Applied Mathematics, 29, Springer, New York, 1998.