Asymptotic properties of mathematical models of excitability

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 364, Issue 1842, 2006, Pages 1283-1298

  Biktasheva, I V ^a   Simitev, R D ^a   Suckley, R ^a   Biktashev, V N ^a  

^a UNIVERSITY OF LIVERPOOL   (United Kingdom)

Author keywords

Action potential; Front dissipation; Singular perturbations

Indexed keywords

EID: 33646515574     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2006.1770     Document Type: Article

References (25)

1. Arnold, V. I., Afrajmovich, V. S., Il'yashenko, Y. S. & Shil'nikov, L. P. 1994 Dynamical systems V. Bifurcation theory and catastrophe theory. Berlin: Springer.
2. Benson, A. P., Clayton, R. H., Holden, A. V., Kharche, S. & Tong, W. C. 2006 Endogenous driving and synchronization in cardiac and uterine virtual tissues: bifurcations and local coupling. Phil. Trans. R. Soc. A 364, 1313-1327. (doi:10.1098/rsta.2006.1772)
3. Biktashev, V. N. 2002 Dissipation of the excitation wavefronts. Phys. Rev. Lett. 89, 168 102. (doi:10.1103/PhysRevLett.89.168102)
4. Biktashev, V. N. 2003 A simplified model of propagation and dissipation of excitation fronts. Int. J. Bif. Chaos 13, 3605-3620. (doi:10.1142/ S0218127403008958)
5. Biktashev, V. N. & Biktasheva, I. V. 2005 Dissipation of excitation fronts as a mechanism of conduction block in re-entrant waves. Led. Notes Comput. Sci. 3504, 283-292.
6. Biktashev, V. N. & Suckley, R. 2004 Non-Tikhonov asymptotic properties of cardiac excitability. Phys. Rev. Lett. 93, 168 103. (doi:10.1103/PhysRevLett.93.168103)
7. Biktasheva, I. V., Biktashev, V. N., Dawes, W. N., Holden, A. V., Saumarez, R. C. & Savill, A. M. 2003 Dissipation of the excitation front as a mechanism of self-terminating arrhythmias. Int. J. Bif. Chaos 13, 3645-3656. (doi:10.1142/S0218127403008909)
8. Cherry, E. M. & Fenton, F. H. 2004 Suppression of alternans and conduction blocks despite steep APD restitution: electrotonic, memory, and conduction velocity restitution effects. Am. J. Physiol. - Heart C 286, H2332-H2341. (doi:10.1152/ajpheart.00747.2003)
9. Courtemanche, M., Ramirez, R. J. & Nattel, S. 1998 Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model. Am. J. Physiol. 275, H301-H321.
10. FitzHugh, R. 1961 Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445-456.
11. Hinh, R. 2004 Stability of cardiac waves. Bull. Math. Biol. 66, 1887-1908. (doi:10.1016/j.bulm. 2004.05.004)
12. Hodgkin, A. L. & Huxley, A. F. 1952 A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500-544.
13. Karma, A. 1994 Electrical alternans and spiral wave breakup in cardiac tissue. Chaos 4, 461-472. (doi: 10.1063/1.166024)
14. Karma, A., Levine, H. & Zou, X. Q. 1994 Theory of pulse instabilities in electrophysiological models of excitable tissues. Physica D 73, 113-127. (doi: 10.1016/0167-2789(94)90228-3)
15. Nagumo, J., Arimoto, S. & Yoshizawa, S. 1962 An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061-2070.
16. Noble, D. 1962 A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials. J. Physiol. 160, 317-352.
17. Nolasco, J. B. & Dahlen, R. W. 1968 A graphic method for the study of alternation in cardiac action potentials. J. Appl. Physiol. 25, 191-196.
18. Pontryagin, L. S. 1957 The asymptotic behaviour of systems of differential equations with a small parameter multiplying the highest derivatives. Izv. Akad. Nauk SSSR, Ser. Mat. 21, 107-155.
19. Steinberg, B. E., Glass, L., Shrier, A. & Bub, G. 2006 The role of heterogeneities and intercellular coupling in wave propagation in cardiac tissue. Phil. Trans. R. Soc. A 364, 1299-1311. (doi: 10.1098/rsta.2006.1771)
20. Suckley, R. 2004 The asymptotic structure of excitable systems of equations. Ph.D. thesis, University of Liverpool, UK.
21. Suckley, R. & Biktashev, V. N. 2003 Comparison of asymptotics of heart and nerve excitability. Phys. Rev. E 68, 011 902. (doi:10.1103/PhysRevE. 68.011902)
22. Tikhonov, A. N. 1952 Systems of differential equations, containing small parameters at the derivatives. Mat. Sbornik 31, 575-586.
23. Tyson, J. J. & Keener, J. P. 1988 Singular perturbation theory of traveling waves in excitable media (a review). Physica D 32, 327-361. (doi:10.1016/0167-2789(88)90062-0)
24. Weiss, J. N., Chen, P. S., Qu, Z., Karagueuzian, H. S. & Garfinkel, A. 2000 Ventricular fibrillation: how do we stop the waves from breaking? Circ. Res. 87, 1103-1107.
25. Wiener, N. & Rosenblueth, A. 1946 The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. Mex. 16, 205-265.