Skip to main content

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 51))

Abstract

An element of an excitable medium returns to its initial state of rest after a burst of activity initiated by a supercritical external perturbation. The role of such a perturbation can be played by a diffusional flow from the neighboring elements of the medium. This results in propagation of a travelling excitation pulse.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. B.P. Belousov: “Periodically acting reaction and its mechanisms”, in Sbornik Referatov po Radiatsionnoy Meditsine (Medgiz, Moscow 1959) p. 145

    Google Scholar 

  2. A.I. Zhabotinskii: Dokl. Akad. Nauk SSSR 157, 392–395 (1964)

    Google Scholar 

  3. A.I. Zhabotinskii: Concentration Self-Oscillations (Nauka, Moscow 1974)

    Google Scholar 

  4. J.J. Tyson: The Belousov-Zhabotinsky Reaction, Lect. Notes in Biomath., Vol. 10 (Springer, Berlin, Heidelberg 1976)

    Google Scholar 

  5. J.J. Tyson, P.C. Fife: J. Chem. Phys. 73, 2224–2236 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  6. R.J. Field, E. Köros, R. Noyes: J. Am. Chem. Soc. 94, 8649–8664 (1972)

    Article  Google Scholar 

  7. A.C. Scott: Rev. Mod. Phys. 47, 47–92 (1975)

    Article  Google Scholar 

  8. V.A. Vasilev, Yu.M. Romanovskii, D.S. Chernavskii, V.G. Yakhno: Autowave Processes in Kinetic Systems (Reidel, Dordrecht 1986)

    Google Scholar 

  9. P. Ortoleva, J. Ross: J. Chem. Phys. 63, 3398–3431 (1975)

    Article  ADS  Google Scholar 

  10. L.A. Ostrovskii, V.G. Yakhno: Biofizika 20, 489–496 (1975)

    Google Scholar 

  11. R.C. Casten, H. Cohen, P.A. Lagerstrom: Quart. Appl. Math. 32, 365–402 (1975)

    MATH  MathSciNet  Google Scholar 

  12. J. Murray: Lectures on Nonlinear Differential Equation Models in Biology (Clarendon Press, Oxford 1977)

    Google Scholar 

  13. A.S. Mikhailov, V.I. Krinsky: Physica 9D, 346–376 (1983)

    ADS  MathSciNet  Google Scholar 

  14. J.A. Feroe: Biophys. J. 21, 103–110 (1978)

    Article  ADS  Google Scholar 

  15. K. Maginu: J. Math. Biol. 6, 49–57 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Rinzel, J.B. Keller: Biophys. J. 13, 1313–1337 (1973)

    Article  ADS  Google Scholar 

  17. L.S. Polak, A.S. Mikhailov: Self-Organization in Nonequilibrium Physicochemical Systems (Nauka, Moscow 1983)

    Google Scholar 

  18. V.S. Zykov: Biofizika 25, 888–892 (1980)

    Google Scholar 

  19. V.S. Zykov: Simulation of Wave Processes in Excitable Media (Nauka, Moscow, 1984) [English transl.: Manchester Univ. Press, 1987]

    Google Scholar 

  20. N. Wiener, A. Rosenblueth: Arch. Inst. Cardiol. 16, 205–265 (1946)

    MathSciNet  Google Scholar 

  21. A. Rosenblueth: Amer. J. Physiol. 194, 491 (1958)

    Google Scholar 

  22. Selfridge: Arch. Inst. Cardiol. 53, 113 (1948)

    Google Scholar 

  23. I.S. Balakhovskii: Biofizika 10, 1063–1069 (1965)

    Google Scholar 

  24. V.I. Krinsky: “Fibrillation in excitable media” in Problemy Kibernetiki, Vol. 20 (Nauka, Moscow 1968) pp. 59–73.

    Google Scholar 

  25. L.V. Reshodko, J. Bureŝ: Biol. Cybern. 18, 181–189 (1975)

    Article  MATH  Google Scholar 

  26. B.F. Madore, W.L. Freedman: Science 222, 615–617 (1983)

    Article  ADS  Google Scholar 

  27. A.T. Winfree, E.M. Winfree, H. Seifert: Physica 17D, 109–121 (1985)

    ADS  MathSciNet  Google Scholar 

  28. A.S. Mikhailov, V.S. Zykov: Dokl. Akad. Nauk SSSR 286, 341–344 (1986)

    Google Scholar 

  29. B. Dorjsurengiyn: Diploma Thesis (Dept. Physics, Moscow State Univ., 1988)

    Google Scholar 

  30. A.S. Mikhailov: Priroda 3, 15–26 (1987)

    Google Scholar 

  31. L. Kuhnert: Nature 319, 393 (1986)

    Article  ADS  Google Scholar 

  32. A.M. Zhabotinskii, A.N. Zaikin: Nature 225, 535–538 (1970)

    ADS  Google Scholar 

  33. K.I. Agladze, V.I. Rrinsky: “On the mechanism of target pattern formation in the distributed Belousov-Zhabotinsky system” in Self-Organization: Autowaves and Structures far from Equilibrium, ed. by V.I. Krinsky (Springer, Berlin, Heidelberg 1984) pp. 147-149

    Google Scholar 

  34. A.M. Zhabotinskii, A.N. Zaikin: “Spatial effects in a self-oscillatory chemical system” in Kolebatelnye Protsessy v Biologicheskikh i Khimicheskikh Sistemakh [Oscillatory Processes in Biological and Chemical Systems], Vol. 2 (Biological Research Center Publications, Pushchino 1971) pp. 314-317

    Google Scholar 

  35. A.M. Zhabotinskii, A.N. Zaikin: J. Theor. Biol. 40, 45–61 (1973)

    Article  Google Scholar 

  36. A.T. Winfree: Science 175, 634–640 (1972)

    Article  ADS  Google Scholar 

  37. G.R. Ivanitskii, V.I. Krinsky, A.N. Zaikin, A.M. Zhabotinskii: Sov. Sci. Rev. 2D, 280–350 (1980)

    Google Scholar 

  38. K.I. Agladze, V.I. Krinsky: Nature 296, 424–426 (1982)

    Article  ADS  Google Scholar 

  39. S.C. Müller, T. Plesser, Hess: Science 230, 661 (1985)

    Article  ADS  Google Scholar 

  40. S.L. Müller, T. Plesser, Hess: Naturwissenschaften 73, 165–179 (1986)

    Article  ADS  Google Scholar 

  41. S.L. Müller, T. Plesser, Hess: Physica 24D, 71–86, 87-96 (1987)

    Article  ADS  Google Scholar 

  42. G.K. Korn, T.M. Korn: Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York 1961)

    MATH  Google Scholar 

  43. A.M. Pertsov, E.A. Ermakova, A.V. Panfilov: Physica 14D, 117–124 (1984)

    ADS  MathSciNet  Google Scholar 

  44. A.S. Mikhailov: “A theory of spiral waves” in Self-Organization: Autowaves and Structures far from Equilibrium ed. by V.I. Krinsky (Springer, Berlin, Heidelberg 1984) pp. 92–95

    Chapter  Google Scholar 

  45. P.K. Brazhnik, V.A. Davydov, V.S. Zykov, A.S. Mikhailov: Zh. Eksp. Teor. Fiz. 93, 1725–1736 (1987)

    ADS  Google Scholar 

  46. V.A. Davydov, A.S. Mikhailov, V.S. Zykov: “Kinematical theory of autowave patterns in excitable media”, in Nonlinear Waves in Active Media, ed. by Yu. Engelbrecht (Springer, Berlin, Heidelberg 1989) pp. 38-51

    Google Scholar 

  47. W.K. Burton, N. Cabrera, F.C. Frank: Phil. Trans. Roy. Soc. 243, 299–312 (1951)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  48. V.S. Zykov: Biofizika 25, 319–322 (1980)

    Google Scholar 

  49. V.S. Zykov, O.L. Morozova: Biofizika 25, 1071–1076 (1980)

    Google Scholar 

  50. P.K. Brazhnik, V.A. Davydov, A.S. Mikhailov: Teor. Mat. Fiz. 74, 440–447 (1987)

    MathSciNet  Google Scholar 

  51. V.A. Davydov, A. S. Mikhailov: “Spiral waves in distributed active media” in Nonlinear Waves. Patterns and Bifurcations, ed. by A.V. Gaponov-Grekhov, M.I. Rabinovich (Nauka, Moscow, 1987) pp. 261–279

    Google Scholar 

  52. V.S. Zykov: “Investigations of some properties of self-sustained activity in excitable medium” in Controlling Complex Systems, ed. By Ya.A. Tsypkin (Nauka, Moscow, 1975) pp. 59–66

    Google Scholar 

  53. R.C. Brower, D.A. Kessler, J. Koplik, H. Levine: Phys. Rev. Lett. 29, 1335–1342 (1984)

    ADS  Google Scholar 

  54. E. Meron, P. Pelcé: Phys. Rev. Lett. 60, 1880–1883 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  55. P.K. Brazhnik, V.A. Davydov, A.S. Mikhailov: “Spiral waves and vortex rings in combustion with subsequent recovering of the initial properties of the medium” in Kinetics and Combustion, Proc. VIII Soviet Symp. on Combustion and Explosion, Tashkent, 1986 (Inst. Chem. Phys., Chernogolovka 1986) pp. 39–43

    Google Scholar 

  56. J.P. Keener, J.J. Tyson: Physica 21D, 300–324 (1986)

    ADS  MathSciNet  Google Scholar 

  57. J.P. Keener: SIAM J. Appl. Math. 46, 1039–1056 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  58. J.J. Tyson, J.P. Keener: Physica 29D, 215–222 (1987)

    ADS  Google Scholar 

  59. J.J. Tyson, J.P. Keener: Physica 32D, 327–361 (1988)

    ADS  MathSciNet  Google Scholar 

  60. P.K. Brazhnik, V.A. Davydov, V.S. Zykov, A.S. Mikhailov: Izv. VUZ. Radiofizika 31, 574–584 (1988)

    Google Scholar 

  61. K.I. Agladze, V.A. Davydov, A.S. Mikhailov: Pisma Zh. Eksp. Teor. Fiz. 45, 601–603 (1987)

    Google Scholar 

  62. V.A. Davydov, V.S. Zykov: Zh. Eksp. Teor. Fiz. 95, 139–148 (1989)

    Google Scholar 

  63. A.S. Mikhailov: “Kinematics of wave patterns in excitable media” in Nonlinear Wave Processes in Excitable Media, ed. by A.V. Holden, M. Markus, H.G. Othmer (Plenum Press, New York 1990) in press

    Google Scholar 

  64. W. Jahnke, W.E. Skeggs, A.T. Winfree: J. Phys. Chem. 95, 740–749 (1989)

    Article  Google Scholar 

  65. S.C. Müller, Th. Plesser: “Dynamics of Spiral Centers in the Ferroin-Catalyzed Belousov-Zhabotinskii Reaction” in Nonlinear Wave Processes in Excitable Media, ed. by A.V. Holden, M. Markus, H.G. Othmer (Plenum Press, New York 1990) in press

    Google Scholar 

  66. V.S. Zykov: Biofizika 32, 337–430 (1987)

    Google Scholar 

  67. V.S. Zykov, O.L. Morozova: “Kinematic method of investigation of stability of spiral autowaves” (Preprint, Inst. of Control Science, Moscow 1988); J. Nonlinear Biol. 1 (1990), in press

    Google Scholar 

  68. A.T. Winfree, S.H. Strogatz: Physica 9D, 35–49, 65-80, 333-345 (1983)

    ADS  Google Scholar 

  69. A.T. Winfree, S.H. Strogatz: Physica 13D, 221–233 (1984)

    ADS  MathSciNet  Google Scholar 

  70. Ya.B. Zeldovich, B.A. Malomed: Dokl. Akad. Nauk SSSR 254, 92–94 (1980)

    MathSciNet  Google Scholar 

  71. B.J. Welsh, J. Gomatam, A.E. Burgess: Nature 304, 611–614 (1983)

    Article  ADS  Google Scholar 

  72. A.B. Medvinsky, A.V. Panfilov, A.M. Pertsov: “Properties of rotating waves in three dimensions” in Self-Organization: Autowaves and Structures far from Equilibrium, ed. by V.I. Krinsky (Springer, Berlin, Heidelberg 1984) pp. 195-199

    Google Scholar 

  73. A.V. Panfilov, A.N. Rudenko: Physica 28D, 215–218 (1987)

    ADS  MathSciNet  Google Scholar 

  74. A.V. Panfilov, A.T. Winfree: Physica 17D, 323–330 (1985)

    ADS  MathSciNet  Google Scholar 

  75. A.V. Panfilov, R.R. Aliev, A.V. Mushinsky: Physica 36D, 181–186 (1989)

    ADS  MathSciNet  Google Scholar 

  76. P.K. Brazhnik: Geometric Methods in the Theory of Autowave Patterns (Ph.D. Thesis, Dept. Physics, Moscow State Univ. 1988)

    Google Scholar 

  77. N.V. Khrastova: Diploma Thesis (Dept. Physics, Moscow State Univ. 1989)

    Google Scholar 

  78. J.P. Keener: Physica 31D, 269–276 (1988)

    ADS  MathSciNet  Google Scholar 

  79. A.Yu. Abramychev, V.A. Davydov, A.S. Mikhailov: Biofizika 35, 100–105 (1989)

    Google Scholar 

  80. P.K. Brazhnik, V.A. Davydov, V.S. Zykov, A.S. Mikhailov: “Dynamics of three-dimensional autowave patterns” in Proc. II Soviet Conf. on Mathematical and Computational Methods in Biology (Biological Research Center Publications, Pushchino 1987) pp. 118-119

    Google Scholar 

  81. P.K. Brazhnik, V.A. Davydov, A.S. Mikhailov: Izv. VUZ. Radiofizika 32, 289–293 (1989)

    MathSciNet  Google Scholar 

  82. A.V. Panfilov, A.N. Rudenko, A.M. Pertsov: “Twisted scroll waves in three-dimensional active media” in Self-Organization: Autowaves and Structures far from Equilibrium, ed. by V.I. Krinsky (Springer, Berlin, Heidelberg 1984) pp. 103-105

    Google Scholar 

  83. A.S. Mikhailov, A.V. Panfilov, A.N. Rudenko: Phys. Lett. 109A, 246–250 (1985)

    ADS  MathSciNet  Google Scholar 

  84. V.N. Biktashev: Physica 36D, 167–172 (1989)

    ADS  MathSciNet  Google Scholar 

  85. P.J. Nandapurkar, A.T. Winfree: Physica 29D, 69–83 (1987)

    ADS  MathSciNet  Google Scholar 

  86. M. Gerhardt, H. Schuster, J.J. Tyson: Science (1990), in press

    Google Scholar 

  87. M. Markus, Hess: Nature (1990), in press

    Google Scholar 

  88. G.S. Skinner, H.L. Swinney: Physica D (1990), in press

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Mikhailov, A.S. (1990). Excitable Media. In: Foundations of Synergetics I. Springer Series in Synergetics, vol 51. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78556-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-78556-6_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-78558-0

  • Online ISBN: 978-3-642-78556-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics