Mathematical Models of Action Potential



Mathematical models of action potential first appear at the beginning and the middle of twentieth century as both models of analogy (Ostwald [1], Van Der Pol [2]), FitzHugh [3], Nagumo[4]) and pure phenomenological models ( Wienner and Rosenblut [5], Moe et al [6]., Krinski [7]) including models based on finite automata representations. With significant developments of experimental technique and computer technology, and due to classical pioneering research accomplished by a group of scientists lead by Hodgkin and Huxley [8-10], the semi phenomenological ionic models have received recognition and wide applications for nerve AP models and were then modified to cardiac AP models in fundamental investigations accomplished by D. Noble and his group [11-13].

All ionic AP mathematical models are based on a balance of the electrical currents through a cell membrane. The existing ionic mathematical models reflect different knowledge of ionic currents flowing through the membrane and are based on experimental finding that ionic channel currents have stochastic character. There exist two approaches (see Chapter 3) in formulating the probability that an ionic channel, s, is in the open state. The first, introduced by Hodgkin-Huxley [8], is based on the assumption of mutual independence in time of channel gate variable processes, which describe different possible states of a channel. It is important to note that there were many concerns [14] about the validity of this assumption.


Purkinje Fiber Pace Rate Diastolic Interval Restitution Curve Gate Variable 


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  1. 1.
    Ostwald, W., Periodische Erscheinungen bei der Auflosung des Chrom in Sauren. Zeit. Phys. Chem., 1900. 35: p. 33-76 and 204-256.Google Scholar
  2. 2.
    Van Der Pol, B. andJ. Van Der Mark, The Heartbeat Considered as Relaxation Oscillations, and an Electrical model of the Heart. Archives Neerlandaises Physiologe De L'Homme et des Animaux, 1929. XIV: p. 418-443.Google Scholar
  3. 3.
    FitzHugh, R., Mathematical Models of Excitation and Propagation in Nerve, in Biological Engineering, H.P. Schwan, Editor. 1969, McGraw-Hill: New York. p. 1-85.Google Scholar
  4. 4.
    Nagumo, J., S. Arimoto, and S. Yoshizawa, An Active Pulse Transmission Line Simulating Nerve Axon. Proceedings of the IRE, 1962. 50(10): p. 2061-2070.CrossRefGoogle Scholar
  5. 5.
    Wiener, N. and A. Rosenblueth, The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. Mexico, 1946. 16(3): p. 205-265.MathSciNetGoogle Scholar
  6. 6.
    Moe, G.K., W.C. Rheinboldt, and J.A. Abildskov, A computer model of atrial fibrillation. Am Heart J, 1964. 67(2): p. 200-220.CrossRefGoogle Scholar
  7. 7.
    Krinsky, V.I., Fibrillation in excitable media. Problems in Cybernetics, 1968. 20: p. 59-80.Google Scholar
  8. 8.
    Hodgkin, A.L. and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J.Physiol., 1952. 117: p. 507-544.Google Scholar
  9. 9.
    Cole, K.S. and J.W. Moore, Potassium ion current in the squid giant axon: dynamic characteristic. Biophys J, 1960. 1: p. 1-14.CrossRefGoogle Scholar
  10. 10.
    Hodgkin, A.L., A.F. Huxley, and B. Katz, Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J Physiol, 1952. 116(4): p. 424-48.Google Scholar
  11. 11.
    Noble, D., Modification of Hodgkin-Huxley Equations Applicable to Purkinje Fibre Action and Pace-Maker Potentials. Journal of Physiology-London, 1962. 160(2): p. 317-&.Google Scholar
  12. 12.
    McAllister, R.E., Computed action potentials for Purkinje fiber membranes with resistance and capacitance in series. Biophys J, 1968. 8(8): p. 951-64.CrossRefGoogle Scholar
  13. 13.
    DiFrancesco, D. and D. Noble, A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos Trans R Soc Lond B Biol Sci, 1985. 307(1133): p. 353-98.CrossRefGoogle Scholar
  14. 14.
    Hoyt, R.C., The Squid Giant Axon. Mathematical Models. Biophys J, 1963. 3: p. 399-431.CrossRefGoogle Scholar
  15. 15.
    Hoyt, R.C., Sodium inactivation in nerve fibers. Biophys J, 1968. 8(10): p. 1074-97.CrossRefGoogle Scholar
  16. 16.
    Sakmann, B. and E. Neher, eds. Single Channel Recording. 1983, Plenum Press: New York.Google Scholar
  17. 17.
    Luo, C.H. and Y. Rudy, A Model of the Ventricular Cardiac Action-Potential - Depolarization, Repolarization, and Their Interaction. Circulation Research, 1991. 68(6): p. 1501-1526.Google Scholar
  18. 18.
    Gulko, F.B. and A.A. Petrov, On a mathematical model of excitation processes in Purkinje fiber. Biofizika (Russian), 1970. 15(3): p. 513.Google Scholar
  19. 19.
    Beeler, G.W. and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres. J.Physiol.(Lond), 1977. 268: p. 177-210.Google Scholar
  20. 20.
    van Capelle, F.J. and D. Durrer, Computer simulation of arrhythmias in a network of coupled excitable elements. Circ Res, 1980. 47(3): p. 454-66.Google Scholar
  21. 21.
    Karma, A., Spiral Breakup in Model-Equations of Action-Potential Propagation in Cardiac Tissue. Physical Review Letters, 1993. 71(7): p. 1103-1106.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Karma, A., Electrical alternans and spiral wave breakup in cardiac tissue. Chaos, 1994. 4(3): p. 461-472.CrossRefGoogle Scholar
  23. 23.
    Kogan, B.Y., W.J. Karplus, and M.G. Karpoukhin, The third-order action potential model for computer simulation of electrical wave propagation in cardiac tissue., in Computer Simulations in Biomedicine, H. Power and R.T. Hart, Editors. 1995, Computational Mechanics Publishers: Boston.Google Scholar
  24. 24.
    Bers, D., Excitation-Contraction Coupling and Cardiac Contractile Force. 2nd ed. 2001, Boston: Kluwer Academic Publishers. 294-300.Google Scholar
  25. 25.
    Lopez-Lopez, J.R., P.S. Shacklock, C.W. Balke, and W.G. Wier, Local calcium transients triggered by single L-type calcium channel currents in cardiac cells. Science, 1995. 268(5213): p. 1042-5.CrossRefGoogle Scholar
  26. 26.
    Bassani, J.W., W. Yuan, and D.M. Bers, Fractional SR Ca release is regulated by trigger Ca and SR Ca content in cardiac myocytes. Am J Physiol Cell Physiol, 1995. 268(5): p. C1313-1319.Google Scholar
  27. 27.
    Stern, M.D., M.C. Capogrossi, and E.G. Lakatta, Spontaneous calcium release from the sarcoplasmic reticulum in myocardial cells: mechanisms and consequences. Cell Calcium, 1988. 9(5-6): p. 247-56.CrossRefGoogle Scholar
  28. 28.
    Mahajan, A., Y. Shiferaw, X. Lai-Hua, R. Olcese, A. Baher, M.-J. Yang, A. Karma, P.-S. Chen, A. Garfinkel, Z. Qu, and J. Weiss, A rabbit ventricular action potential model replicating cardiac dynamics at rapid heart rates. preprint, 2006.Google Scholar
  29. 29.
    Fabiato, A., Time and calcium dependence of activation and inactivation of calcium-induced release of calcium from the sarcoplasmic reticulum of a skinned canine cardiac Purkinje cell. J Gen Physiol, 1985. 85(2): p. 247-89.CrossRefGoogle Scholar
  30. 30.
    Hilgemann, D.W. and D. Noble, Excitation-contraction coupling and intracellular calcium transients in rabbit atrium: reconstruction of basic cellular mechanisms. Proc.R.Soc.Lond., 1987. 230: p. 163-205.CrossRefGoogle Scholar
  31. 31.
    Earm, Y.E. and D. Noble, A model of the single atrial cell: relation between calcium current and calcium release. Proc R Soc Lond B Biol Sci, 1990. 240(1297): p. 83-96.CrossRefGoogle Scholar
  32. 32.
    Lindblad, D.S., C.R. Murphey, J.W. Clark, and W.R. Giles, A model of the action potential and underlying membrane currents in a atrial cell. Am.J.Physiol., 1996. 271: p. H1666-H1696.Google Scholar
  33. 33.
    Nygren, A., C. Fiset, L. Firek, J.W. Clark, D.S. Lindblad, R.B. Clark, and W.R. Giles, Mathematical model of an adult human atrial cell: The role of K+ currents in repolariztion. Circ.Res., 1998. 82: p. 63-81.Google Scholar
  34. 34.
    Courtemanche, M., R.J. Ramirez, and S. Nattel, Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model. Am J Physiol, 1998. 275(1 Pt 2): p. H301-21.Google Scholar
  35. 35.
    Nordin, C., Computer Model of Membrane Current and Intracellular Ca2+ Flux in the Isolated Guinea Pig Ventricular Myocyte. Am.J.Physiol., 1993. 265: p. H2117-H2136.Google Scholar
  36. 36.
    Zeng, J., K.R. Laurita, D.S. Rosenbaum, and Y. Rudy, Two components of the delayed rectifier K+ current in ventricular myocytes of the guinea pig type. Theoretical formulation and their role in repolarization. Circ Res, 1995. 77(1): p. 140-52.Google Scholar
  37. 37.
    Priebe, L. and D.J. Beuckelmann, Simulation study of cellular electric properties in heart failure. Circ Res, 1998. 82(11): p. 1206-23.Google Scholar
  38. 38.
    Jafri, M.S., J.J. Rice, and R.L. Winslow, Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load. Biophys J, 1998. 74(3): p. 1149–68.CrossRefGoogle Scholar
  39. 39.
    Noble, D., A. Varghese, P. Kohl, and P. Noble, Improved guinea-pig ventricular cell model incorporating a diadic space, IKr and IKs, and length- and tension-dependent processes. Can J Cardiol, 1998. 14(1): p. 123–34.Google Scholar
  40. 40.
    Chudin, E., J. Goldhaber, A. Garfinkel, J. Weiss, and B. Kogan, Intracellular Ca(2+) dynamics and the stability of ventricular tachycardia. Biophys J, 1999. 77(6): p. 2930–41.CrossRefGoogle Scholar
  41. 41.
    Shiferaw, Y., M.A. Watanabe, A. Garfinkel, J.N. Weiss, and A. Karma, Model of intracellular calcium cycling in ventricular myocytes. Biophys J, 2003. 85(6): p. 3666–86.CrossRefGoogle Scholar
  42. 42.
    Shiferaw, Y., D. Sato, and A. Karma, Coupled dynamics of voltage and calcium in paced cardiac cells. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 2005. 71(2): p. 021903.Google Scholar
  43. 43.
    Puglisi, J.L. and D.M. Bers, LabHEART: an interactive computer model of rabbit ventricular myocyte ion channels and Ca transport. Am J Physiol Cell Physiol, 2001. 281(6): p. C2049–60.Google Scholar
  44. 44.
    ten Tusscher, K.H., D. Noble, P.J. Noble, and A.V. Panfilov, A model for human ventricular tissue. Am J Physiol Heart Circ Physiol, 2004. 286(4): p. H1573–89.CrossRefGoogle Scholar
  45. 45.
    Luo, C.H. and Y. Rudy, A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res, 1994. 74(6): p. 1071–96.Google Scholar
  46. 46.
    Shaw, R.M., Theoretical studies in cardiac electrophysiology: role of membrane and gapjunctions in excitability and conduction, in Dept. of Biomed. Eng. 1966, Case Western Research University.Google Scholar
  47. 47.
    Faber, G.M. and Y. Rudy, Action potential and contractility changes in [Na(+)](i) overloaded cardiac myocytes: a simulation study. Biophys J, 2000. 78(5): p. 2392–404.CrossRefGoogle Scholar
  48. 48.
    Chudin, E., A. Garfinkel, J. Weiss, W. Karplus, and B. Kogan, Wave propagation in cardiac tissue and effects of intracellular calcium dynamics (computer simulation study). Prog Biophys Mol Biol, 1998. 69(2–3): p. 225–36.CrossRefGoogle Scholar
  49. 49.
    Keizer, J. and L. Levine, Ryanodine receptor adaptation and Ca2+(−)induced Ca2+ release-dependent Ca2+ oscillations. Biophys J, 1996. 71(6): p. 3477–87.CrossRefGoogle Scholar
  50. 50.
    Gyorke, S. and M. Fill, Ryanodine receptor adaptation – control mechanism of Ca-induced Ca release in heart. Science, 1993. 260: p. 807–809.CrossRefGoogle Scholar
  51. 51.
    Stern, M.D., Theory of excitation-contraction coupling in cardiac muscle. Biophys.J., 1992. 63: p. 497–517.CrossRefGoogle Scholar
  52. 52.
    Chernyavskiy, S., Computer simulation of the Jafri-Winslow action potential model. 1998, UCLA.Google Scholar
  53. 53.
    Samade, R. and B. Kogan. Calcium alternans in cardiac cell mathematical models. in International Conference on Bioinformatics and Computational Biology. 2007. Las Vegas, NV: CSREA Press.Google Scholar
  54. 54.
    Shannon, T.R., F. Wang, J. Puglisi, C. Weber, and D.M. Bers, A Mathematical Treatment of Integrated Ca Dynamics within the Ventricular Myocyte. Biophys. J., 2004. 87(5): p. 3351–3371.CrossRefGoogle Scholar
  55. 55.
    Koller, M.L., M.L. Riccio, and R.F. Gilmour, Jr., Dynamic restitution of action potential duration during electrical alternans and ventricular fibrillation. Am J Physiol, 1998. 275(5 Pt 2): p. H1635–42.Google Scholar
  56. 56.
    Carmeliet, E., Repolarisation and frequency in cardiac cells. J Physiol (Paris), 1977. 73(7): p. 903–23.Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of California, Los AngelesLos AngelesUSA

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