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Part of the book series: Lecture Notes in Biomathematics ((LNBM,volume 66))

Abstract

The beautiful spiral waves of oxidation in the Belousov-Zhabotinskii reaction [20] are the source of many interesting questions about periodic structures in excitable media. Because they are easy to produce and photograph, pictures of these spirals have appeared in a number of popular science oriented magazines. The study of spirals takes on a more personal interest when one realizes that fibrillation and sudden cardiac arrest from heart attacks may also be due to the appearance of rotating spiral waves of electrical activity on the ventricular myocardium [1], [14]. Immediately questions like “How do spirals form?” and “Can Spirals be prevented?” or “Can one predict if a heart attack will be fatal?” spring to mind. Going beyond questions of self preservation, we may also ask about the properties of spirals, such as their wavelength and frequency, or the conditions necessary to. sustain spiral activity in a given medium.

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References

  1. M. A. Allessie, F. I. M. Bonke, and F. J. G. Schopman, Circus movement in rabbit atrial muscle as a mechanism of tachycardia, Circ. Res. 41 9–18(1977).

    Google Scholar 

  2. R. G. Casten, H. Cohen, P. A. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory, Quart. Appl. Math., 32, 365–402(1975).

    MATH  MathSciNet  Google Scholar 

  3. N. El-Sherif, R. Mehra, W. B. Gough, and R. H. Zeiler, Ventricular Activation Patterns of Spontaneous and Induced Ventricular Rhythms in Canine One-Day-Old Myocardial Infraction, Circ. Res. 51, pp. 152–166(1982).

    Google Scholar 

  4. P. C. Fife, Singular perturbation and wave front techniques in reaction-diffusion problems, Proc. AMS-SIAM Symp. on Asymptotic Methods and Singular Perturbation, New York, 1976.

    Google Scholar 

  5. P. C. Fife, Propagator — Controller Systems and Chemical Patterns, in Nonequilibrium Dynamics in Chemical Systems, C. Vidal and A. Pacault, eds., Springer-Verlag, 1984.

    Google Scholar 

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

    Article  Google Scholar 

  7. R. FitzHugh, Mathematical Models of Excitation and Propagation in Nerve, in Biological Engineering (ed. H. P. Schwan) pp. 1–85, McGraw-Hill, 1969.

    Google Scholar 

  8. J. P. Keener, Geometrical theory for spirals in excitable media, to appear.

    Google Scholar 

  9. J. P. Keener, Waves in Excitable Media, SIAM J. Appl. Math. 39, pp. 528–548(1980).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. P. Keener and J. J. Tyson, Spiral Waves in the Belousov-Zhabotinskii Reaction, to appear.

    Google Scholar 

  11. J. B. Keller, Geometrical Acoustics 1, The Theory of Weak Shocks, J. Appl. Phys. 25, 1938–947(1954).

    Google Scholar 

  12. J. B. Keller, A Geometrical Theory of Diffraction, Proc. Symp. Appl. Math., 8 27–52(1958).

    Google Scholar 

  13. H. P. McKean, Nagumo’s equation, Adv. in Math. 4, 209–223(1970).

    Article  MATH  MathSciNet  Google Scholar 

  14. G. R. Mines, On Circulating Excitation on Heart Muscles and their possible relation to tachycardia and fibrillation, Trans. Roy. Soc. Can. 4, 43–53(1914).

    Google Scholar 

  15. J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation, Biophysics J. 13, 1313–1337(1973).

    Article  Google Scholar 

  16. J. J. Tyson, On scaling the Oregonator equations, In: Nonlinear Phenomena in Chemical Dynamics, eds. C. Vidal and A. Pacault, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  17. J. J. Tyson and P. C. Fife, Target Patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys. 73, (5) 2224–2237(1980).

    Article  MathSciNet  Google Scholar 

  18. J. J. Tyson and V. Manoranjan, The speed of propagation of oxidizing and reducing wave fronts in the Belousov-Zhabotinskii reaction, In: Nonequilibrium Dynamics in Chemical Systems, eds. C. Vidal and A. Pacault, Springer-Verlag, Berlin, 1984.

    Google Scholar 

  19. A. T. Winfree, The Geometry of Biological Time, Springer-Verlag, Berlin, 1980, p. 245.

    MATH  Google Scholar 

  20. A. T. Winfree, Rotating chemical reactions, Scientific American, June 1974.

    Google Scholar 

  21. A. T. Winfree, Spiral waves of chemical activity, Science 175 (1972) 634–636.

    Article  Google Scholar 

  22. A. T. Winfree, Two kinds of waves in an oscillating chemical solution, Faraday Symp. Chem. Soc. 9 (1974) 38–46.

    Google Scholar 

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© 1986 Springer-Verlag Berlin Heidelberg

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Keener, J.P. (1986). Spiral Waves in Excitable Media. In: Othmer, H.G. (eds) Nonlinear Oscillations in Biology and Chemistry. Lecture Notes in Biomathematics, vol 66. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93318-9_7

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  • DOI: https://doi.org/10.1007/978-3-642-93318-9_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16481-4

  • Online ISBN: 978-3-642-93318-9

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