Simplified Action Potential Models

  • Boris Ja. KoganEmail author


The basic motivations for simplifying the AP mathematical models are: To make computer simulation of excitation wave propagation in 3D-tissue model with complex configuration feasible. To find a qualitative relationship between normal AP generation and propagation.

There are at least three known approaches used to simplify AP mathematical models: Based on singular perturbation theory Based on clamp-experiment data Based on the Van der Pole relaxation generator

In some cases, the sensitivity analysis [1] allows the introduction of some simplifications to modern sophisticated mathematical models.


Action Potential Duration Outward Current Singular Perturbation Theory Repolarization Phase Versus Rest 
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. 1.
    Paulsen, R.A., Jr., J.W. Clark, Jr., P.H. Murphy, and J.A. Burdine, Sensitivity analysis and improved identification of a systemic arterial model. IEEE Trans Biomed Eng, 1982. 29(3): p. 164–77.CrossRefGoogle Scholar
  2. 2.
    Tikhonov, A.N., Sets of differential equations containing small parameters on derivatives. Math. Collection, 1952. 31(73): p. 575–586.Google Scholar
  3. 3.
    Krinsky, V.I. and Y.M. Kokos, Analysis of the equations of excitable membranes. III. Membrane of the Purkinje fibre. Reduction of the noble equations to a second order system. Analysis of automation by the graphs of the zero-isoclines. Biofizika, 1973. 18(6): p. 1067–1073.Google Scholar
  4. 4.
    Van Der Pol, B. and J. 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
  5. 5.
    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
  6. 6.
    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
  7. 7.
    Ivanitsky, G.R., V.I. Krinsky, and E.E. Selkov, Mathematical Biophysics of Cell. 1978, Moscow: Nauka Press.Google Scholar
  8. 8.
    Karpoukhin, M.G., B.Y. Kogan, and W.J. Karplus, The Application of a Massively Parallel Computer to the Simulation of Electrical Wave Propagation Phenomena in the Heart Muscle Using Simplified Models. Proceedings of the 28th Annual Hawaii International Conference on System Sciences, 1995: p. 112–122.Google Scholar
  9. 9.
    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
  10. 10.
    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
  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.
    Karma, A., Spiral Breakup in Model-Equations of Action-Potential Propagation in Cardiac Tissue. Physical Review Letters, 1993. 71(7): p. 1103–1106.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ermakova, E.A., A.V. Pertsov, and E.E. Shnol, On the interaction of vortices in two- dimensional active media. Physica D, 1989. 40: p. 185–195.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nadapurkar, P.I. and A.T. Winfree, A computation study of twisted linked scroll waves in excitable media. Physica D, 1987. 29: p. 69–83.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Pertsov, A.V., R.N. Chramov, and A.V. Panfilov, Sharp increase in refractory period induced by oxidation suppression in FitzHughNagumo model. New mechanism of anti-arrhythmic drug action. Biofizika, 1981. 6.Google Scholar
  16. 16.
    Pertsov, A.V. and A.V. Panfilov, Spiral waves in active media. The reverberator in the FitzHugh-Agumo Model, in Autowave Processes in Systems with Diffusion. 1981: Gor'kii. p. 77–91.Google Scholar
  17. 17.
    Winfree, A.T., When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias. 1987: Princeton Univ Press.Google Scholar
  18. 18.
    Winfree, A.T., Electrical instability in cardiac muscle: phase singularities and rotors. J Theor Biol, 1989. 138(3): p. 353–405.CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kogan, B.Y., W.J. Karplus, B.S. Billet, A.T. Pang, H.S. Karagueuzian, and S. Khan, The simplified Fitzhugh-Nagumo model with action potential duration restitution: effects on 2D-wave propagation. Physica D, 1991. 50: p. 327–340.zbMATHCrossRefGoogle 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

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

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

Personalised recommendations