Skip to main content
Log in

On the formation of circulating patterns of excitation in anisotropic excitable media

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

We present a model of excitable media with the feature that it has a vulnerable phase during which a premature current stimulus will result in the formation of a reentrant selfsustained wave of excitation. The model exploits anisotropic coupling of identical cells, and is therefore useful as a model for the myocardium. We give rigorous verification that there is a vulnerable phase, and demonstrate numerically that permanently rotating waves are formed. Finally, it is shown that the direction of fastest propagation in myocardium is not necessarily the direction of highest safety factor, contrary to commonly accepted opinion.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Canada)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Chi, H., Bell, J., Hassard, B.: Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory, J. Math. Biol. 4, 583–607 (1986)

    Google Scholar 

  2. Clerc, L.: Directional differences of impulse spread in trabecular muscle from mammalian heart. J. Physiol. 255, 335–346 (1976)

    Google Scholar 

  3. Cranefield, P. F.: The conduction of the cardiac impulse. Mt. Kisco, N.Y.: Futura 1975

    Google Scholar 

  4. Diaz, P. J., Rudy, Y., Plonsey, R.: Intercalated discs as a cause for discontinuous propagation in cardiac muscle: a theoretical simulation. Ann. Biomed. Eng. 11, 177–189 (1983)

    Google Scholar 

  5. Dillon, S., Ursell, P. C., Wit, A. L.: Pseudo-block caused by anisotropic conduction: a new mechanism for sustained reentry. Circ. Suppl. 72, III-279 (1985)

    Google Scholar 

  6. FitzHugh, R.: Impulse and physiological states in models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Google Scholar 

  7. Han, J., Garcia de Jalon, P. D., Moe, G. K.: Fibrillation threshold of premature ventricular responses. Circ. Res. 43, 18–25 (1966)

    Google Scholar 

  8. Joyner, R. W.: Effects of the discrete pattern of electrical coupling on propagation through an electrical synctium. Circ. Res. 50, 192–200 (1982)

    Google Scholar 

  9. Keener, J. P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572 (1987)

    Google Scholar 

  10. Keener, J. P.: Propagation and its failure in the discrete Nagumo equation. In: Sleeman, B. D., Jarvis, R. J. (eds.) Ordinary and partial differential equations. Pitman Res. Notes Math., vol. 157, New York: Longman 1987.

    Google Scholar 

  11. Krinskii, V. I.: Spread of excitation in an inhomogeneous medium (state similar to cardiac fibrillation). Biofizika 11, 676–683 (1966)

    Google Scholar 

  12. Miller, R., Rinzel, J.: The dependence of impulse propagation speed on firing frequency, dispersion for the Hodgkin-Huxley models. Biophys. J. 34, 227–259 (1981)

    Google Scholar 

  13. Moe, G. K., Rheinholdt, W. C., Abildshov, J. A.: A computer model of atrial fibrillation. Am. Heart J. 67, 200–220 (1964)

    Google Scholar 

  14. Rinzel, J., Keller, J. B.: Traveling wave solutions of a nerve conduction equation. Biophys. J. 13, 1313–1337 (1973)

    Google Scholar 

  15. Smith, J. M., Cohen, R. J.: Simple finite-element model accounts for wide range of cardiac dysrhythmias. Proc. Natl. Acad. Sci. USA 81, 233–237 (1984)

    Google Scholar 

  16. Spach, M. S., Kootsey, J. M.: The nature of electrical propagation in cardiac muscle. Am. J. Physiol. 244, H3-H22, (1983)

    Google Scholar 

  17. Spach, M. S., Miller, W. T., Geselowitz, D. B., Barr, R. C., Kootsey, J. M., Johnson, E. A.: The discontinuous nature of propagation in normal canine cardiac muscle. Circ. Res. 48, 39–54 (1981)

    Google Scholar 

  18. Van Cappelle, F. J. L., Durrer, D.: Computer simulation of arrhythmias in a network of coupled excitable elements. Circ. Res. 47, 454–466 (1980)

    Google Scholar 

  19. Winfree, A. T.: Sudden cardiac death, a problem in topology. Sci. Am. 248, 114–161 (1983)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Keener, J.P. On the formation of circulating patterns of excitation in anisotropic excitable media. J. Math. Biology 26, 41–56 (1988). https://doi.org/10.1007/BF00280171

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00280171

Key words

Navigation

-