Skip to main content

Part of the book series: NATO ASI Series ((NSSB,volume 270))

  • 258 Accesses

Abstract

Excitable systems play an important role in the field of spatio-temporal self-organisation under conditions far from thermodynamic equilibrium [1]. A particular property of these systems is that they can be assumed to exist in one of three different states: as long as a stimulus is absent, they remain in a quiescent state which is excitable; by application of a stimulus, they are excited to an active state; after excitation there follows a refractory period during which the system is not yet excitable, but relaxes to the previous quiescent and newly excitable situation. It is well known that such systems support the formation of traveling waves of excitation with different front geometries, the complexity of which depends on the dimensionality of the system and the influence of internal and external perturbations [2].

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. G. Nicolis and I. Prigogine, “Self-Organization in Nonequilibrium Systems”, Wiley, New York (1977).

    Google Scholar 

  2. J.J. Tyson and J.P. Keener, Singular Perturbation Theory of Traveling Waves in Excitable Media (A Review), Physica D32;327 (1988).

    Google Scholar 

  3. R. Fitzhugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J. 1:445 (1961).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  4. G. Matsumoto, H. Shimizu, J. Phvs. Soc. Jpn 44:1399 (1978).

    Article  Google Scholar 

  5. A. Gierer and H. Meinhardt, “A Theory of Biological Pattern Formation”, Kybernetik 12:30 (1972).

    Article  CAS  PubMed  Google Scholar 

  6. M.A. Allessie, F.I.M. Bonke and F.J.G. Schopman, Circus Movement in Rabbit Atrical Muscle as a Mechanism of Tachycardia, Circ. Res. 41:9 (1977).

    Article  PubMed  CAS  Google Scholar 

  7. A.T. Winfree, “When Time Breaks Down”, Princeton Univ. Press, Princeton (1987).

    Google Scholar 

  8. R.J. Field and M. Burger (eds.), “Oscillations and Travelling Waves in Chemical Systems”, John Wiley, New York (1985).

    Google Scholar 

  9. G. Gerisch, Chemotaxis in Dictyostelium, A. Rev. Physiol. 44:535 (1982)

    Article  CAS  Google Scholar 

  10. S.C. Müller, Th. Plesser, and B. Hess, The Structure of the Core of the Spiral Wave in the Belousov-Zhabotinskii Reaction, Science 230:661 (1985).

    Article  PubMed  Google Scholar 

  11. K.J. Tomchik and P.N. Devreotes, Adenosine 3′,5′ Monophosphate Waves in Dictyostelium discoideum: a Demonstration by Isotope Dilution-Fluorography, Science 212:443 (1981).

    Article  PubMed  CAS  Google Scholar 

  12. P. Foerster, S.C. Müller, and B. Hess, Curvature and Spiral Geometry in Aggregation Patterns of Dictyostelium discoideum, Development 109: 11 (1990).

    Google Scholar 

  13. E. Meron and P. Pelcé, Model for Spiral Wave Formation in Excitable Media, Phys. Rev. Lett. 60:1880 (1988).

    Article  CAS  Google Scholar 

  14. D. Barkley, M. Kness, and L.S. Tuckerman, Spiral-Wave Dynamics in a Simple Model of Excitable Media, Phys. Rev. A 42:2489 (1990).

    Article  PubMed  CAS  Google Scholar 

  15. M. Markus and B. Hess, Isotropic Cellular Automata for Modelling Excitable Media, Nature 347:56 (1990).

    Article  CAS  Google Scholar 

  16. J.P. Keener and J.J. Tyson, Spiral Waves in the Belousov-Zhabotinskii Reaction, Physica D21:307 (1986).

    Google Scholar 

  17. J.-L. Martiel and A. Goldbeter, A Model Based on Receptor Desensitization for Cyclic AMP Signaling in Dictyostelium Cells, Biophvs. J. 52:807 (1987).

    Article  CAS  Google Scholar 

  18. J.J. Tyson, K.A. Alexander, V.S. Manoranjan, and J.D. Murray, Spiral Waves of Cyclic AMP in a Model of Slime Mould Aggregation, Physica D34:193 (1989)

    Google Scholar 

  19. S.C. Müller, Th. Plesser, and B. Hess, Two-Dimensional Spectrophotometry of Spiral Wave Propagation in the Belousov-Zhabotinskii Reaction. Part II, Physica D24:87 (1987).

    Google Scholar 

  20. P. Foerster, S.C. Müller, and B. Hess, Critical Size and Curvature of Wave Formation in an Excitable Chemical Medium, Proc. Natl. Acad. Sci. USA 86:6831 (1989).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  21. V.A. Davydov, V.S. Zykov, and A.S. Mikhailov, Kinematical Theory of Autowave Patterns in Excitable Media, in: “Nonlinear Waves in Active Media”, J. Engelbrecht, ed., Springer, Berlin (1989).

    Google Scholar 

  22. J. Ross, S.C. Müller, and C. Vidal, Chemical Waves, Science 240:460 (1988).

    Article  PubMed  CAS  Google Scholar 

  23. H. Miike, Y. Kurihara, H. Hashimoto, and K. Koga, Velocity-Field Measurements by Pixel-Based Temporal Mutual-Correlation Analysis of Dynamic Image. Trans. IEICE Japan E 69:877 (1986).

    Google Scholar 

  24. O. Steinbock, H. Hashimoto, and S.C. Müller, Quantitative Analysis of Periodic Chemotaxis in Aggregation Patterns of Dictyostelium discoideum, Physica D, in press.

    Google Scholar 

  25. F. Siegert and C. Weijer, Analysis of Optical Density Wave Propagation and Cell Movement in the Cellular Slime Mould, Physica D, in press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Plenum Press, New York

About this chapter

Cite this chapter

Müller, S.C. (1991). Vortex Formation in Excitable Media. In: Mosekilde, E., Mosekilde, L. (eds) Complexity, Chaos, and Biological Evolution. NATO ASI Series, vol 270. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-7847-1_24

Download citation

  • DOI: https://doi.org/10.1007/978-1-4684-7847-1_24

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4684-7849-5

  • Online ISBN: 978-1-4684-7847-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics