Skip to main content
SpringerLink
Log in

Mathematical Modeling of the Electrical Activity of Cardiac Cells

  • Chapter
Book cover Theory of Heart

Part of the book series: Institute for Nonlinear Science ((INLS))

  • 512 Accesses

Abstract

We introduce the Hodgkin-Huxley (HH) formulation describing the flow of ionic currents across the membrane of a cardiac cell, paying particular attention to the central concepts of activation and inactivation. We indicate a few situations in which HH-type modeling of cardiac cells has been useful, and show that continuous models of the HH-type break down when one observes phenomena in which single-channel behavior becomes important. Finally, we show that there are some intriguing parallels between the behavior of single ionic channels, which are currently thought to be governed by stochastic processes, and the behavior of chaotic systems, which are governed not by stochastic, but rather by deterministic rules.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F.T. Arrechi and A. Califano. Noise-induced trapping at the boundary between two attractors: A source of 1/f spectrum and nonlinear dynamics. Europhys. Lett., 3, 5–10, 1987.

    Article  ADS  Google Scholar 

  2. G.W. Beeler and H. Reuter. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol (Lond.), 268: 177–210, 1977.

    Google Scholar 

  3. W.K. Bleeker, A.J.C. Mackaay, M. Masson-Pévet, L.N. Bouman, and A.E. Becker. Functional and morphological organization of the rabbit sinus node. Circ. Res., 46: 11–22, 1980.

    Google Scholar 

  4. A. Cavalié, D. Pelzer, and W. Trautwein. Fast and slow gating behaviour of single calcium channels in cardiac cells. Pflüg. Arch., 406: 241–258, 1986.

    Article  Google Scholar 

  5. D.R. Chialvo and J. Jalife. Non-linear dynamics of cardiac excitation and impulse propagation. Nature, 330: 749–752, 1988.

    Article  ADS  Google Scholar 

  6. J.R. Clay and L.J. DeFelice. The relationship between membrane excitability and single channel open-close kinetics. Biophys. J., 42: 151–157, 1983.

    Article  Google Scholar 

  7. D. Colquhoun and A.G. Hawkes. The principles of the stochastic interpretation of ion channel mechanisms. In B. Sakmann and E. Neher, editors, Single-Channel Recording, pages 135–175. Plenum, New York, 1983.

    Google Scholar 

  8. F. Conti. Noise analysis and single-channel recordings. Curr. Top. Memb. Transport, 22: 371–405, 1984.

    Google Scholar 

  9. D. DiFrancesco and D. Noble. A model of electrical activity incorporating ionic pumps and concentration changes. Phil. Trans. Roy. Soc. Lond., B 307: 353–398, 1985.

    Article  ADS  Google Scholar 

  10. M. Feingold, D.L. Gonzalez, O. Piro, and H. Viturro. Phase locking, period doubling, and chaotic phenomena in externally driven excitable systems. Physical Rev., A 37: 4060–4063, 1988.

    ADS  Google Scholar 

  11. R. Fischmeister and G. Vassort. The electrogenic Na-Ca exchange and the cardiac electrical activity. I-Simulation of Purkinje fiber action potentials. J. Physiol. (Paris), 77: 705–709, 1981.

    Google Scholar 

  12. R. FitzHugh. Impulses and physiological states in theoretical models of nerve membrane. Biophys J., 1: 445–466, 1961.

    Article  ADS  Google Scholar 

  13. R. FitzHugh. Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys., 17: 257–278, 1955.

    Article  Google Scholar 

  14. C. Grebogi, E. Ott, F. Romieras, and J.A. Yorke. Critical exponents for crisis-induced intermittency. Physical Rev., 36A: 5365–5380, 1987.

    ADS  Google Scholar 

  15. M.R. Guevara. Chaotic Cardiac Dynamics. Ph.D. thesis, McGill University, Montreal, Quebec, 1984.

    Google Scholar 

  16. M.R. Guevara. Spatiotemporal patterns of block in an ionic model of cardiac Purkinje fibre, In M. Markus, S.C. Müller, and G. Niclois, editors, From Chemical to Biological Organization, pages 273–281. Springer, Berlin, 1988.

    Google Scholar 

  17. M.R. Guevara. Afterpotentials and pacemaker oscillations in an ionic model of cardiac Purkinje fibres, L. Rensing, U. an der Heiden and M.C. Mackey, editors, Temporal Disorder in Human Oscillatory Systems, pages 126–133. Springer, Berlin, 1987.

    Google Scholar 

  18. M.R. Guevara, F. Alonso, D. Jeandupeux, and A.C.G. van Ginneken. Alternans in periodically stimulated isolated ventricular myocytes: Experiment and model, In A. Goldbeter, editor, Cell-to-Cell Signalling: From Experiments to Theoretical Models, pages 551–563. Academic Press, London, 1989.

    Google Scholar 

  19. M.R. Guevara, L. Glass, M.C. Mackey, and S. Shrier. Chaos in neurobiology. IEEE Trans. Syst. Man. Cybern., SMC-13: 790–798, 1983.

    MATH  Google Scholar 

  20. M.R. Guevara, L. Glass, and A. Shrier. Phase locking, period doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. Science, 214: 1350–1353, 1981.

    Article  ADS  Google Scholar 

  21. M.R. Guevara, D. Jeandupeux, F. Alonso, and N. Morissette. Wenckebach rhythms in isolated ventricular heart cells, In St. Pnevmatikos, T. Bountis and Sp. Pnevmatikos, editors, Singular Behaviour and Nonlinear Dynamics, pages 629–642. World Scientific, Singapore, 1989.

    Google Scholar 

  22. M.R. Guevara and A. Shrier. Phase resetting in a model of cardiac Purkinje fibre. Biophys. J., 52: 165–175, 1987.

    Article  ADS  Google Scholar 

  23. M.R. Guevara, A. Shrier, and L. Glass. Phase resetting of spontaneously beating embryonic ventricular heart cell aggregates. Am. J. Physiol., 251: H1298-H1305, 1986.

    Google Scholar 

  24. M.R. Guevara, A.C.G. van Ginneken, and H.J. Jongsma. Patterns of activity in a reduced ionic model of a cell from the rabbit sinoatrial node, In H. Degn, A.V. Holden and L.F. Olsen, editors, Chaos in Biological Systems, pages 5–12. Plenum, London, 1987.

    Google Scholar 

  25. D.W. Hilgemann and D. Noble. Excitation-contraction coupling and extracellular calcium transients in rabbit atrium: Reconstruction of basic cellular mechanisms. Proc. Roy. Soc. Lond., B 230: 163–205, 1987.

    Article  ADS  Google Scholar 

  26. B. Hille. Ionic Channels of Excitable Membranes. Sinauer, Sunderland, 1984.

    Google Scholar 

  27. A.L. Hodgkin and A.F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.), 117: 500–544, 1952.

    Google Scholar 

  28. P. Holmes. “Strange” phenomena in dynamical systems and their physical implications. Appl. Math. Modelling, 1: 362–366, 1977.

    Article  MATH  Google Scholar 

  29. R. Horn and S.J. Korn. Graphical discrimination of Markov and fractal models of single channel gating. Comments Theor. Biol., 1: 39–46, 1988.

    Google Scholar 

  30. H. Irisawa and A. Noma. Pacemaker mechanisms of rabbit sinoatrial node cells, In L.N. Bouman and H.J. Jongsma, editors, Cardiac Rate and Rhythm, pages 35–51. Nijhoff, Amsterdam, 1982.

    Google Scholar 

  31. H. Ishii, H. Fujisawa, and M. Inoue. Breakdown of chaos symmetry and intermittency in the double-well potential system. Phys. Lett., 116A: 257–263, 1986.

    ADS  Google Scholar 

  32. J.H. Jensen, P.L. Christiansen, and A.C. Scott. Chaos in the Beeler-Reuter system for the action potential of ventricular myocardial fibers. Physica, 13D: 269–277, 1984.

    MathSciNet  ADS  Google Scholar 

  33. R.W. Joyner and F.J.L. van Capelle. Propagation through electrically coupled cells: How a small SA node drives a large atrium. Biophys. J., 50: 1157–1164, 1986.

    Article  ADS  Google Scholar 

  34. R.W. Joyner, R. Veenstra, D. Rawling, and A. Chorro. Propagation through electrically coupled cells: Effects of a resistive barrier. Biophys. J., 45: 1017–1025, 1984.

    Article  Google Scholar 

  35. J.P. Keener. A mathematical model for the initiation of ventricular tachycardia in myocardium, In A. Goldbeter, editor, Cell to Cell Signalling: From Experiments to Theoretical Models, pages 589–608. Academic Press, London, 1989.

    Google Scholar 

  36. L.S. Liebovitch and J.M. Sullivan. Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons. Biophys. J., 52: 979–988, 1987.

    Article  Google Scholar 

  37. L.S. Liebovitch and T.I. Tóth. Fractal activity in cell membrane ion channels. Ann. NY Acad. Sci., 591: 375–391, 1990.

    Article  ADS  Google Scholar 

  38. P. Manneville. Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems. J. Phys.(Paris), 41: 1235–1243, 1980.

    Article  MathSciNet  Google Scholar 

  39. M.A. Masson-Pévet, W.K. Bleeker, E. Besselsen, B.W. Treytel, H.J. Jongsma, and L.N. Bouman. Pacemaker cell types in the rabbit sinus node: A correlative ultrastructural and electrophysiological study. J. Molec. Cell. Cardiol, 16: 53–63, 1984.

    Article  Google Scholar 

  40. R.E. McAllister, D. Noble, and R.W. Tsien. Reconstruction of the electrical activity of cardiac Purkinje fibres. J. Physiol. (Lond.), 251: 1–59, 1975.

    Google Scholar 

  41. D.C. Michaels, E.P. Matyas, and J. Jalife. A mathematical model of the effects of acetylcholine pulses on sinoatrial pacemaker activity. Circ. Res., 55: 89–101, 1984.

    Google Scholar 

  42. D.C. Michaels, E.P. Matyas, and J. Jalife. Mechanisms of sinoatrial pacemaker synchronization: A new hypothesis. Circ. Res., 61: 704–714, 1987.

    Google Scholar 

  43. T. Musha, H. Takeuchi, and T. Inoue. 1/f fluctuations in the spontaneous intervals of a giant snail neuron. IEEE Trans. Biomed. Eng., BME-30: 194–197, 1983.

    Article  Google Scholar 

  44. R.A. Nadeau, F.A. Roberge, and P. Bhéreur. The mechanism of the Wenckebach phenomenon. Isr. J. Med. Sci., 5: 814–818, 1969.

    Google Scholar 

  45. D. Noble. A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials. J. Physiol. (Lond.), 160: 317–352, 1962.

    Google Scholar 

  46. D. Noble and S.J. Noble. A model of the sino-atrial node electrical acitivity based on a modification of the DiFrancesco-Noble (1984) equations. Proc. Roy. Soc. Lond., B 222: 295–304, 1984.

    Article  ADS  Google Scholar 

  47. H. Othmer. The dynamics of forced excitable systems. In A.V. Holden, M. Markus, H. Othmer, editors, Nonlinear Wave Processes in Excitable Media, Plenum, London, 1990.

    Google Scholar 

  48. A.V. Panfilov and A.N. Rudenko. Two regimes of the scroll ring drift in the three-dimensional active media. Physica, 28D: 215–218, 1987.

    MathSciNet  ADS  Google Scholar 

  49. A.A. Petrov and B.N. F’eld. Analysis of the possible mechanism of origin of the extrasystole in local ischaemia of the myocardium using a mathematical model. Biophysics, 18: 1145–1150, 1973.

    Google Scholar 

  50. T. Post, H.W. Capel, and J.P. van de Weele. New results in intermittent switching, In St. Pnevmatikos, T. Bountis, Sp. Pnevmatikos, editors, Singular Behaviour and Nonlinear Dynamics, pages 195–200. World Scientific, Singapore, 1989.

    Google Scholar 

  51. I. Proccacia and H. Schuster. Functional renormalization group theory of universal 1/f noise in dynamical systems. Physical Rev., 28A: 1210–1212, 1983.

    ADS  Google Scholar 

  52. O.E. Rössler, R. Rössler, and H.D. Landahl. Arrhythmia in a periodically forced excitable system (abstract). In Proc. Sixth Int. Biophys. Congress, Kyoto, 1978.

    Google Scholar 

  53. P. Selepova. Single Ion Channel Dynamics. Master’s thesis, McGill University, Montreal, Quebec, 1986.

    Google Scholar 

  54. A.I. Shcherbunov, N.I. Kukushkin, and M.Y. Sakson. Reverberator in a system of interrelated fibers described by the Noble equation. Biophysics, 18: 547–554, 1973.

    Google Scholar 

  55. A. Shrier, H. Dubarsky, M. Rosengarten, M.R. Guevara, S. Nattel, and L. Glass. Prediction of complex atrioventricular conduction rhythms in human beings with use of the atrioventricular nodal recovery curve. Circulation, 76: 1196–1205, 1987.

    Article  Google Scholar 

  56. E. Skaugen. Firing behaviour in stochastic nerve membrane models with different pore densities. Acta. Physiol. Scand., 108: 49–60, 1980.

    Article  Google Scholar 

  57. J.M.T. Thompson and H.B. Stewart. Nonlinear Dynamics and Chaos. Wiley, Chichester, 1986.

    MATH  Google Scholar 

  58. F.J.L. van Capelle and D. Durrer. Computer simulation of arrythmias in a network of coupled excitable elements. Circ. Res., 47: 454–466, 1980.

    Google Scholar 

  59. B. van der Pol and J. van der Mark. The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Phil. Mag. (Series 7), 6: 763–775, 1928.

    Google Scholar 

  60. T. Watanabe, P.M. Rautaharju, and T.F. McDonald. Ventricular action potentials, ventricular extracellular potentials, and the ECG of guinea pig. Circ. Res., 57: 362–373, 1985.

    Google Scholar 

  61. A.T. Winfree. Electrical instability in cardiac muscle: phase singularities and rotors. J. Theor. Biol., 138: 353–405, 1989.

    Article  MathSciNet  Google Scholar 

  62. J.A. Yorke and E.D. Yorke. Metastable chaos: The transition to sustained chaotic behaviour in the Lorenz model. J. Stat. Phys., 21: 263–277, 1979.

    Article  MathSciNet  ADS  Google Scholar 

  63. V.S. Zykov. Simulation of Wave Processes in Excitable Media. Manchester University Press, Manchester, 1987.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physiology, McGill University, Montréal, Québec, H3G 1Y6, Canada

    Michael R. Guevara

Authors
  1. Michael R. Guevara
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Physiology, McGill University, Montreal, Quebec, H3G 1Y6, Canada

    Leon Glass

  2. Department of Engineering Science, University of Auckland, Auckland, New Zealand

    Peter Hunter

  3. Department of Applied Mechanics and Engineering Science—Bioengineering, University of California, San Diego, La Jolla, CA, 92093, USA

    Andrew McCulloch

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Guevara, M.R. (1991). Mathematical Modeling of the Electrical Activity of Cardiac Cells. In: Glass, L., Hunter, P., McCulloch, A. (eds) Theory of Heart. Institute for Nonlinear Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3118-9_10

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-3118-9_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7803-0

  • Online ISBN: 978-1-4612-3118-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation