Pattern Formation of the FitzHugh-Nagumo Model: Cellular Automata Approach

Document Type : Research Article

Authors

1 Laboratory of Systems Biology and Bioinformatics (LBB), Institute of Biochemistry and Biophysics, University of Tehran, P.O. Box 13145-1384 Tehran, I.R. IRAN

2 Department of Chemistry, Tarbiat Moallem University, P.O. Box 15719-14911 Tehran, I.R. IRAN

3 Department of Chemistry, K.N. Toosi University of Technology, P.O. Box 4416-15875 Tehran, I.R. IRAN

10.30492/ijcce.2011.6365

Abstract

FitzHugh-Nagumo (FHN) model is a famous Reaction-Diffusion System which first introduced for the conduction of electrical impulses along a nerve fiber. This model is also considered as an abstract model for pattern formation. Here, we have used the Cellular Automata method to simulate the pattern formation of the FHN model. It is shown that the pattern of this model is very similar to those of a kind of a rabbitfish which implies natural patterns could be based on reaction-diffusion systems. We have also considered the effects of different parameters of the FHN model on changing the initial pattern.

Keywords

Main Subjects


[1] Brown  K.J.,  Lacey  A.A.,  "Reaction-Diffusion Equations",OxfordUniversityPress, (1990)
[2] Zhen Mei, “Numerical Bifurcation Analysis for Reaction-Diffusion Equations”, Springer, (2000).
[3] Cross M.C., Hohenberg P.C., Pattern Formation Outside of Wquilibrium, Rev. Mod. Phys., 65(3),
p. 851 (1993).
[4] Meinhardt H., Pattern Formation in Biology: a Comparison of Models and Experiments, Rep. Prog. Phys., 55, p. 797 (1992).
[5] Koch A.J., Meinhardt H., Biological Pattern Formation: from Basic Mechanisms to Complex Structures, Rev. Mod. Phys. 66(4), 1481-1507, (1994).
[6] Von-Neumann J., “Theory of Self-Reproducing Automata”,Urbana:UniversityofIllinoisPress, (1966).
[7] Wolfram S., “A New Kind of Science”, Wolfram Media Inc., (2002).
[8] Wolfram S., “Theory and Applications of Cellular Automata, Including Selected Papers (1983-1986), Vol. 1 of Advanced Series on Complex Systems”,Singapore, World Scientific, (1986).
[9] FitzHugh R., Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J., 1, p. 445 (1961).
[10] Nagumo J., Arimoto S., Yoshizawa S., An Active Pulse Transmission Line Simulating Nerve Axon, Proc. IRE, 50, p. 2061 (1962).
[11] Scott A.C., The Electrophysics of a Nerve Fiber, Rev. Mod. Phys., 47(2), p. 487 (1975).
[12] Hodgkin  A.L.,  Huxley  A.F.,  A  Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve, J. Physiol., 117, p. 500 (1952).
[13] Winfree A.T., Varieties of Spiral Wave Behavior: An Experimentalist’s Approach to the Theory of Excitable Media, Chaos, 1(3), p. 303 (1991).
[14] Weimar J.R., Boon J.P., Class of Cellular Automata for Reaction-Diffusion Systems, Phys. Rev. E, 49(2), p. 1749 (1994).
[15] Weimar  J.R.,  "Cellular Automata  for  Reactive Systems" Ph.D Thesis,UniversitéLibre de Bruxelles,Belgium, (1995).
[16] Vanag  V.K.,  Reviews  of  Topological  Problems: Study of Spatially Extended Dynamical Systems Using Probabilistic Cellular Automata, Physics-Uspekhi, 42(5), p. 413 (1999).
[17] Glendinning P., Stability, “Instability and Chaos”,CambridgeUniversityPress, (1994).
[18] Kapral R., Malevanets A., Microscopic Model for FitzHugh-Nagumo Dynamics, Phys. Rev. E., 55, p. 5657 (1997).