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].
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References
G. Nicolis and I. Prigogine, “Self-Organization in Nonequilibrium Systems”, Wiley, New York (1977).
J.J. Tyson and J.P. Keener, Singular Perturbation Theory of Traveling Waves in Excitable Media (A Review), Physica D32;327 (1988).
R. Fitzhugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J. 1:445 (1961).
G. Matsumoto, H. Shimizu, J. Phvs. Soc. Jpn 44:1399 (1978).
A. Gierer and H. Meinhardt, “A Theory of Biological Pattern Formation”, Kybernetik 12:30 (1972).
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).
A.T. Winfree, “When Time Breaks Down”, Princeton Univ. Press, Princeton (1987).
R.J. Field and M. Burger (eds.), “Oscillations and Travelling Waves in Chemical Systems”, John Wiley, New York (1985).
G. Gerisch, Chemotaxis in Dictyostelium, A. Rev. Physiol. 44:535 (1982)
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).
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).
P. Foerster, S.C. Müller, and B. Hess, Curvature and Spiral Geometry in Aggregation Patterns of Dictyostelium discoideum, Development 109: 11 (1990).
E. Meron and P. Pelcé, Model for Spiral Wave Formation in Excitable Media, Phys. Rev. Lett. 60:1880 (1988).
D. Barkley, M. Kness, and L.S. Tuckerman, Spiral-Wave Dynamics in a Simple Model of Excitable Media, Phys. Rev. A 42:2489 (1990).
M. Markus and B. Hess, Isotropic Cellular Automata for Modelling Excitable Media, Nature 347:56 (1990).
J.P. Keener and J.J. Tyson, Spiral Waves in the Belousov-Zhabotinskii Reaction, Physica D21:307 (1986).
J.-L. Martiel and A. Goldbeter, A Model Based on Receptor Desensitization for Cyclic AMP Signaling in Dictyostelium Cells, Biophvs. J. 52:807 (1987).
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)
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).
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).
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).
J. Ross, S.C. Müller, and C. Vidal, Chemical Waves, Science 240:460 (1988).
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).
O. Steinbock, H. Hashimoto, and S.C. Müller, Quantitative Analysis of Periodic Chemotaxis in Aggregation Patterns of Dictyostelium discoideum, Physica D, in press.
F. Siegert and C. Weijer, Analysis of Optical Density Wave Propagation and Cell Movement in the Cellular Slime Mould, Physica D, in press.
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© 1991 Plenum Press, New York
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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
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DOI: https://doi.org/10.1007/978-1-4684-7847-1_24
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