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Kinematics of Spiral Waves in Excitable Media

  • Vladimir S. ZykovEmail author
Chapter
Part of the The Frontiers Collection book series (FRONTCOLL)

Abstract

Spiral waves rigidly rotating in excitable media sometimes play a constructive role in self-organization, while in many cases they cause an undesirable and dangerous activity. An understanding of spiral wave kinematics can help to control or to prevent this self-sustained activity. A description of the spiral wave kinematics performed by use of a free-boundary approach, reveals the selection principle which determines the shape and the rotation frequency of spiral waves in an unbounded medium with a given excitability. It is shown that a rigidly rotating spiral in a medium with strongly reduced refractoriness is supported within an excitability range restricted by two universal limits. At the low excitability limit, the spiral core radius diverges, while it vanishes at the high excitability limit and the spiral wave resembles the Yin-Yang pattern.

References

  1. 1.
    G. Gerisch, Periodische Signale steuern die Musterbildung in Zellverbänden. Naturwissenschaften 58, 430–438 (1971)Google Scholar
  2. 2.
    A.T. Winfree, Spiral waves of chemical activity. Science 175, 634–636 (1972)ADSCrossRefGoogle Scholar
  3. 3.
    M.A. Allessie, F.I.M. Bonke, F.J.G. Schopman, Circus movement in rabbit atrial muscle as a mechanism of tachycardia. Circ. Res. 33, 54–62 (1973)CrossRefGoogle Scholar
  4. 4.
    N.A. Gorelova, J. Bures, Spiral waves of spreading depression in the isolated chicken retina. J. Neurobiol. 14, 353–363 (1983)CrossRefGoogle Scholar
  5. 5.
    S. Jakubith, H.H. Rotermund, W. Engel, A. von Oertzen, G. Ertl, Spatiotemporal concentration patterns in a surface reaction: propagating and standing waves, rotating spirals, and turbulence. Phys. Rev. Lett. 65, 3013–3016 (1990)ADSCrossRefGoogle Scholar
  6. 6.
    T. Mair, S.C. Müller, Traveling NADH and proton waves during oscillatory glycolysis in vitro. J. Biol. Chem. 271, 627–630 (1996)CrossRefGoogle Scholar
  7. 7.
    S.C. Müller, T. Plesser, B. Hess, The structure of the core of the spiral wave in the Belousov-Zhabotinskii reaction. Science 230, 661–663 (1985)ADSCrossRefGoogle Scholar
  8. 8.
    G.S. Skinner, H.L. Swinney, Periodic to quasiperiodic transition of chemical spiral rotation. Physica D 48, 1–16 (1991)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    K.I. Agladze, V.A. Davydov, A.S. Mikhailov, An observation of resonance of spiral waves in distributed excitable medium. JETP Lett. 45, 601–603 (1987)Google Scholar
  10. 10.
    O. Steinbock, V.S. Zykov, S.C. Müller, Control of spiral-wave dynamics in active media by periodic modulation of excitability. Nature 366, 322–324 (1993)ADSCrossRefGoogle Scholar
  11. 11.
    V.N. Biktashev, A. Holden, Design principles of a low voltage cardiac defibrillator based on the effect of feedback resonant drift. J. Theor. Biol. 169, 101–112 (1994)CrossRefGoogle Scholar
  12. 12.
    A.V. Panfilov, S.C. Müller, V.S. Zykov, J.P. Keener, Elimination of spiral waves in cardiac tissue by multiple electrical shocks. Phys. Rev. E 61, 4644–4647 (2000)ADSCrossRefGoogle Scholar
  13. 13.
    N. Wiener, 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. de Mex. 16, 205–265 (1946)MathSciNetzbMATHGoogle Scholar
  14. 14.
    W.K. Burton, N. Cabrera, F.C. Frank, The growth of crystals and the equilibrium structure of their surfaces. Philos. Trans. R. Soc. Lond. Ser. A 243, 299–358 (1951)Google Scholar
  15. 15.
    P. Pelcé, J. Sun, Wave front interaction in steadily rotating spirals. Physica D 48, 353–366 (1991)Google Scholar
  16. 16.
    F.B. Gul’ko, A.A. Petrov, Mechanism of formation of closed propagation pathways in excitable media. Biofizika 17, 261–270 (1972)Google Scholar
  17. 17.
    V.S. Zykov, Simulation of Wave Processes in Excitable Media (Manchester University Press, Manchester, 1987)zbMATHGoogle Scholar
  18. 18.
    V.S. Zykov, Kinematics of rigidly rotating spiral waves. Physica D 238, 931–940 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    I. Aranson, L. Kramer, The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    O. Rudzik, A.S. Mikhailov, Front reversals, wave traps, and twisted spirals in periodically forced oscillatory media. Phys. Rev. Lett. 96, 018302 (2006)ADSCrossRefGoogle Scholar
  21. 21.
    V. Hakim, A. Karma, Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications. Phys. Rev. E 60, 5073–5105 (1999)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    J.J. Tyson, J. Keener, Singular perturbation theory of traveling waves in excitable media (a review). Physica D 32, 327–361 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. Karma, Universal limit of spiral wave propagation in excitable media. Phys. Rev. Lett. 66, 2274–2277 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    V.S. Zykov, K. Showalter, Wave front interaction model of stabilized propagating wave segments. Phys. Rev. Lett. 94, 068302 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    A. Kothe, V.S. Zykov, H. Engel, Second universal limit of wave segment propagation in excitable media. Phys. Rev. Lett. 103, 154102 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    V.S. Zykov, E. Bodenschatz, Periodic sequence of stabilized wave segments in an excitable medium. Phys. Rev. E 97(3), 030201(R) (2018)ADSCrossRefGoogle Scholar
  27. 27.
    V.S. Zykov, N. Oikawa, E. Bodenschatz, Selection of spiral waves in excitable media with a phase wave at the wave back. Phys. Rev. Lett. 107, 254101 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    V.S. Zykov, E. Bodenschatz, Stabilized wave segments in an excitable medium with a phase wave at the wave back. New J. Phys. 16, 043030 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    N. Oikawa, E. Bodenschatz, V. Zykov, Unusual spiral wave dynamics in the Kessler-Levine model of an excitable medium. Chaos 25, 053115 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    V.S. Zykov, E. Bodenschatz, Continuous transition between two limits of spiral wave dynamics in an excitable medium. Phys. Rev. Lett. 112, 054101 (2014)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Max Planck Institute for Dynamics and Self-OrganizationGöttingenGermany

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