Excitable medium

Toth A and Showalter K 1995 Logic gates in excitable media J. Chem. Rhys. 103 2058-66... 

Markin V S, Pastushenko V F and Chizmadzhev Y A 1987 Theory of Excitable Media (New York Wiley)... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Ca waves in systems [ike Xenopus laevis oocytes and pancreatic (3 cells fall into this category Electrochemical waves in cardiac and nerve tissue have this origin and the appearance and/or breakup of spiral wave patterns in excitable media are believed to be responsible for various types of arrhythmias in the heart [39, 40]. Figure C3.6.9 shows an excitable spiral wave in dog epicardial muscle [41]. 

There exist many different CA models exhibiting BZ-like spatial waves. One of the simplest, and earliest, described in the next section, is a model proposed by Greenberg and Hastings in 1978 [green78], and based on an earlier excitable media model by Weiner and Rosenbluth [weiner46]. One of the earliest and simplest mathematical models of the BZ reaction, called the Orcgonator, is due to Field and Noyes [field74]. 

Kadar, S., Wang, J. and Showalter, K. (1998) Noise-supported travelling waves in sub-excitable media. Nature, 391,... 

Reaction-diffusion systems can readily be modeled in thin layers using CA. Since the transition rules are simple, increases in computational power allow one to add another dimension and run simulations at a speed that should permit the simulation of meaningful behavior in three dimensions. The Zaikin-Zhabotinsky reaction is normally followed in the laboratory by studying thin films. It is difficult to determine experimentally the processes occurring in all regions of a three-dimensional segment of excitable media, but three-dimensional simulations will offer an interesting window into the behavior of such systems in the bulk. 

Tyson, J. J. and Keener, J. P. (1988). Singular perturbation theory of traveling waves in excitable media (a review). Physica, D 32, 327-61. (December)... 

The major characteristics of excitable media, such as oscillating chemical reactions, and some important concepts necessary for understanding their behaviour have been discussed. The capacity of Belousov-Zhabotinsky (BZ) reactions for spontaneous spatiotemporal auto-organization is described 289... 

V. S. Zykov, Simulation of Wave Processes in Excitable Media, Manchester University Press, Manchester, 1987. 

J. P. Keener, Knotted scroll wave filamentin excitable media , Physica, D34, 378 (1989). 

Thus the stirred excitable media under random perturbations can produce regular coherent oscillations whose frequency is controlled by the intensity of mixing. Note that this is in contrast with the... 

E. Meron. Pattern formation in excitable media. Phys. Rep., 218 1-66, 1992. 

Z. Neufeld. Excitable media in a chaotic flow. Phys. Rev. Lett., 87 108301, 2001. 

Z. Neufeld, C. Lopez, E. Hernandez-Garcfa, and O. Piro. Excitable media in open and closed chaotic flows. Phys. Rev. E, 66 066208, 2002c. 

J.E. Truscott and J. Brindley. Ocean plankton populations as excitable media. Bull. Math. Biol., 56 981-998, 1994. 

V. Beato, H. Engel, and L. Schimansky-Geier. Pulse trains propagating through excitable media subjected to external noise. 2006. submitted to PRE. 

J. Garcfa-Ojalvo and L. Schimansky-Geier. Noise-induced spiral dynamics in excitable media. Europhys. Lett, 47 298-303, 1999. 

S. Kadar, J. Wang, and K. Showalter. Noise-supported traveling waves in suh-excitable media. Nature, 391 770, 1998. 

In nature, spatio-temporal patterns in excitable media occur in seemingly unlimited variety. As early as 1946 Wiener and Rosenblueth [81] introduced the concept of excitable media to explain the propagation of electrical excitation fronts in the heart. Waves of electrical activity in the heart muscle assist its rhythmic contractions. The presence of spiral waves can indicate dangerous fibrillation. This is one of the motivations why the dynamics and control of spiral waves are studied. Furthermore spiral waves are typical, almost ubiquitous, patterns in excitable media see [88] in this volume. 

S. Alonso, F. Sagues, and A. S. Mikhailov. Periodic forcing of scroll rings and control of Winfree turbulence in excitable media. Chaos, 16 1-11, 2006. 

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