Bulletin of Mathematical Biology

, Volume 65, Issue 5, pp 767–793 | Cite as

A two-current model for the dynamics of cardiac membrane

  • Colleen C. Mitchell
  • David G. Schaeffer


In this paper we introduce and study a model for electrical activity of cardiac membrane which incorporates only an inward and an outward current. This model is useful for three reasons: (1) Its simplicity, comparable to the FitzHugh-Nagumo model, makes it useful in numerical simulations, especially in two or three spatial dimensions where numerical efficiency is so important. (2) It can be understood analytically without recourse to numerical simulations. This allows us to determine rather completely how the parameters in the model affect its behavior which in turn provides insight into the effects of the many parameters in more realistic models. (3) It naturally gives rise to a one-dimensional map which specifies the action potential duration as a function of the previous diastolic interval. For certain parameter values, this map exhibits a new phenomenon—subcritical alternans—that does not occur for the commonly used exponential map.


Bifurcation Diagram Action Potential Duration Solution Branch Balt Cardiac Action Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Banville, I. and R. Gray (2002). Effect of action potential duration and conduction velocity restitution and their spatial dispersion on alternans and the stability of arrhythmias. J. Cardiovasc. Electrophysiol. 13, 1141–1149.CrossRefGoogle Scholar
  2. Beeler, G. and H. Reuter (1977). Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. (Lond.) 268, 177–210.Google Scholar
  3. Bender, C. and S. Orszag (1978). Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill.Google Scholar
  4. Boyett, M. and D. Fedida (1984). Changes in the electrical activity of dog cardiac Purkinje fibres at high heart rates. J. Physiol. 350, 361–391.Google Scholar
  5. Euler, D. (1999). Cardiac Alternans: Mechanisms and pathophysiological significance. Cardiovasc. Res. 42, 583–590.CrossRefGoogle Scholar
  6. Fenton, F. and A. Karma (1998). Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos 8, 20–47.CrossRefzbMATHGoogle Scholar
  7. FitzHugh, R. (1960). Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J. Gen. Physiol. 43, 867–896.CrossRefGoogle Scholar
  8. FitzHugh, R. (1961). Impulse and physiological states in models of nerve membrane. Biophys. J. 1, 445–466.CrossRefGoogle Scholar
  9. Gilmour, R., N. Otani and M. Watanabe (1997). Memory and complex dynamics in cardiac Purkinje fibers. Am. J. Physiol. 272, 1826–1832.Google Scholar
  10. Glass, L. and M. Mackey (1988). From Clocks to Chaos: The Rhythms of Life, Princeton University Press.Google Scholar
  11. Golubitsky, M. and D. Schaeffer (1985). Singularities and Groups in Bifurcation Theory, Vol. I, Springer.Google Scholar
  12. Guevara, M., M. Ward, A. Shrier and L. Glass (1984). Electrical alternans and period-doubling bifurcations. Comput. Cardiology 167–170.Google Scholar
  13. Hall, G. M., S. Bahar and D. Gauthier (1999). Prevalence of rate-dependent behaviors in cardiac muscle. Phys. Rev. Lett. 82, 2995–2998.CrossRefGoogle Scholar
  14. Karma, A. (1993). Spiral breakup in model equations of action potential propagation in cardiac tissue. Phys. Rev. Lett. 71, 1103–1106.zbMATHMathSciNetCrossRefGoogle Scholar
  15. Karma, A. (1994). Electrical alternans and sprial wave breakup in cardiac tissue. Chaos 4, 461–472.CrossRefGoogle Scholar
  16. Kevorkian, J. and J. Cole (1981). Perturbation Methods in Applied Mathematics, Springer.Google Scholar
  17. Luo, C. and Y. Rudy (1991). A model of the ventricular cardiac action potential. Circ. Res. 68, 1501–1526.Google Scholar
  18. Luo, C. and Y. Rudy (1994). A dynamic model of the cardiac ventricular action potential. Circ. Res. 74, 1071–1096.Google Scholar
  19. Noble, D. (1960). Cardiac action and pacemaker potentials based on the Hodgkin-Huxley equations. Nature 188, 495–497.Google Scholar
  20. Noble, D. (1962). A modification of the Hodgkin-Huxley equations applicable to purkinje fibre action potentials. J. Physiol. 160, 317–352.Google Scholar
  21. Nolasco, J. and R. Dahlen (1968). A graphic method for the study of alternation in cardiac action potentials. J. Appl. Physiol. 25, 191–196.Google Scholar
  22. Strogatz, S. (1994). Nonlinear Dynamics and Chaos, Addison Wesley.Google Scholar
  23. Tolkacheva, E., D. Schaeffer, D. Gauthier and C. Mitchell (2002). Analysis of the Fenton-Karma model through approximation by a one dimensional map. Chaos 12, 1034–1042.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Winfree, A. (1987). When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias, Princeton University Press.Google Scholar

Copyright information

© Society for Mathematical Biology 2003

Authors and Affiliations

  • Colleen C. Mitchell
    • 1
  • David G. Schaeffer
    • 1
  1. 1.Department of MathematicsDuke University and Center for Nonlinear and Complex SystemsDurhamUSA

Personalised recommendations