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Open Physics

formerly Central European Journal of Physics

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Volume 5, Issue 4


Volume 13 (2015)

Stochastic cellular automata modeling of excitable systems

Tamás Szakály / István Lagzi / Ferenc Izsák
  • Institute of Mathematics, Eötvös University, 1117, Budapest, Pázmány P. stny. 1/C, Hungary
  • Department of Applied Mathematics, University of Twente, 7500 AE, Enschede, The Netherlands
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/ László Roszol / András Volford
Published Online: 2007-12-01 | DOI: https://doi.org/10.2478/s11534-007-0032-7


A stochastic cellular automaton is developed for modeling waves in excitable media. A scale of key features of excitation waves can be reproduced in the presented framework such as the shape, the propagation velocity, the curvature effect and spontaneous appearance of target patterns. Some well-understood phenomena such as waves originating from a point source, double spiral waves and waves around some obstacles of various geometries are simulated. We point out that unlike the deterministic approaches, the present model captures the curvature effect and the presence of target patterns without permanent excitation. Spontaneous appearance of patterns, which have been observed in a new experimental system and a chemical lens effect, which has been reported recently can also be easily reproduced. In all cases, the presented model results in a fast computer simulation.

Keywords: Belousov-Zhabotinsky reaction; stochastic model; front propagation; cellular automata

PACS: 82.20.-w; 82.20.wt; 82.40.Ck; 82.40.Qt

  • [1] A.T. Winfree: “Varieties of spiral wave behavior: an experimentalist’s approach to the theory of excitable media”, Chaos, Vol. 1, (1991), pp. 303–334. http://dx.doi.org/10.1063/1.165844CrossrefGoogle Scholar

  • [2] A.S. Mikhailov: Foundations of Synergetics I. Distributed Active Systems 2nd ed., Springer, Berlin, 1994. Google Scholar

  • [3] D. Barkley: “A model for fast computer-simulation of waves in excitable meadia”, Physica D, Vol. 49, (1991), pp. 61–70. http://dx.doi.org/10.1016/0167-2789(91)90194-ECrossrefGoogle Scholar

  • [4] B. Chopard and M. Droz: Cellular Automata Modeling of Physical Systems, Cambridge University Press, Cambridge, 1998. Google Scholar

  • [5] S. Wolfram: Theory and Applications of Cellular Automata, World Scientific, Singapore, 1986. Google Scholar

  • [6] M. Gerhardt, H. Schuster and J. Tyson: “A cellular automaton model of excitable media. 2. Curvature, dispersion, rotating waves and meandering waves”, Physica D, Vol. 46, (1990), pp. 392–415. http://dx.doi.org/10.1016/0167-2789(90)90101-TCrossrefGoogle Scholar

  • [7] M. Gerhardt, H. Schuster and J. Tyson: “A cellular automaton model of excitable media. 3. Fitting the Belousov-Zhabotinskii reaction”, Physica D, Vol. 46 (1990), pp. 416–426. Google Scholar

  • [8] D. Chowdhury, L. Santen and A. Schadschneider: “Statistical physics of vehicular traffic and some related systems”, Phys. Rep., Vol. 329, (2000), pp. 199–329. http://dx.doi.org/10.1016/S0370-1573(99)00117-9CrossrefGoogle Scholar

  • [9] K. Nishinari, M. Fukui and A. Schadschneider: “A stochastic cellular automaton model for traffic flow with multiple metastable states”, J. Phys. A-Math. Gen., Vol. 37, (2004), pp. 3101–3110. http://dx.doi.org/10.1088/0305-4470/37/9/003CrossrefGoogle Scholar

  • [10] A. Kirchner and A. Schadschneider: “Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics”, Physica A, Vol. 312, (2002), pp. 260–276. http://dx.doi.org/10.1016/S0378-4371(02)00857-9Web of ScienceCrossrefGoogle Scholar

  • [11] P. Bak, K. Chen and C. Tang: “A forest fire model and some thoughts on turbulence”, Phys. Lett. A, Vol. 147, (1990), pp. 297–300. http://dx.doi.org/10.1016/0375-9601(90)90451-SCrossrefGoogle Scholar

  • [12] R.B. Schinazi: “On the spread of drug-resistant diseases”, J. Stat. Phys., Vol. 97, (1999), pp. 409–417. http://dx.doi.org/10.1023/A:1004635606196CrossrefGoogle Scholar

  • [13] M. Small and C.K. Tsea: “Clustering model for transmission of the SARS virus: application to epidemic control and risk assessment”, J. Phys. A-Math. Gen., Vol. 351, (2005), pp. 499–511. Google Scholar

  • [14] E. Domany and W. Kinzel: “Equivalence of cellular automata to Ising-models and directed percolation”, Phys. Rev. Lett., Vol. 53, (1984), pp. 311–314. http://dx.doi.org/10.1103/PhysRevLett.53.311CrossrefGoogle Scholar

  • [15] H. Fukś: “Probabilistic cellular automata with conserved quantities”, Nonlinearity, Vol. 17, (2004), pp. 159–173. http://dx.doi.org/10.1088/0951-7715/17/1/010CrossrefGoogle Scholar

  • [16] Y.C. Lee and S. Quian: “Adaptive stochastic cellular automata-Theory”, Physica D, Vol. 45, (1990), pp. 159–180. http://dx.doi.org/10.1016/0167-2789(90)90180-WCrossrefGoogle Scholar

  • [17] http://www.getfreesofts.com/download/66/2971/Five_Cellular_Automata.html Google Scholar

  • [18] http://ccl.northwestern.edu/netlogo/models/B-ZReaction Google Scholar

  • [19] M. Gerhardt and H. Schuster: “A cellular automaton describing the formation of spatially ordered structures in chemical systems”, Physica D, Vol. 36, (1989), pp. 209–221. http://dx.doi.org/10.1016/0167-2789(89)90081-XCrossrefGoogle Scholar

  • [20] A.K. Dewdney: “Computer recreations: The hodgepodge machine makes waves”, Scientific American, Vol. 43, (1988), pp. 104–107. http://dx.doi.org/10.1038/scientificamerican0888-104CrossrefGoogle Scholar

  • [21] J. S. Kiraldy: “Spontaneous evolution of spatiotemporal patterns in materials”, Report and Progress in Physics, Vol. 55, (1992), pp. 723–795. http://dx.doi.org/10.1088/0034-4885/55/6/002CrossrefGoogle Scholar

  • [22] J. Weimar and J-P. Boon: “Class of cellular automata for reaction-diffusion systems”, Phys. Rev. E, Vol. 49, (1994), pp. 1749–1752. http://dx.doi.org/10.1103/PhysRevE.49.1749CrossrefGoogle Scholar

  • [23] A. Adamatzky and O. Holland: “Phenomenology of excitation in 2-D cellular automata and swarm systems”, Chaos. Soliton. Fract., Vol. 9, (1998), pp. 1233–1265. http://dx.doi.org/10.1016/S0960-0779(97)00123-9CrossrefGoogle Scholar

  • [24] C. Beauchemin, J. Samuel and J. Tuszynski: “A simple cellular automaton model for influenza A viral infections”, J. Theor. Biol., Vol. 232, (2005), pp. 223–234. http://dx.doi.org/10.1016/j.jtbi.2004.08.001CrossrefGoogle Scholar

  • [25] B. Drossel and F. Schwabl: “Formation of space-times structure in a forest-fire model”, Physica A, Vol. 204, (1994), pp. 212–229. http://dx.doi.org/10.1016/0378-4371(94)90426-XCrossrefGoogle Scholar

  • [26] A.N. Zaikin and A.M. Zhabotinsky: “Concentration wave propagation in 2-dimensional liquid-phase self-oscillating system”, Nature, Vol. 225, (1970) pp. 535–537. http://dx.doi.org/10.1038/225535b0CrossrefGoogle Scholar

  • [27] F. Falo, A.R. Bishop, P.S. Lomdahl and B. Horowitz: “Langevin molecular dynamics of interfaces: Nucleation versus spiral growth”, Phys. Rev. B., Vol. 43, (1991), pp. 8081–8088. http://dx.doi.org/10.1103/PhysRevB.43.8081CrossrefGoogle Scholar

  • [28] P. Grassberger and H. Kantz: “On a forest fire model with supposed self-organized criticality”, J. Stat. Phys., Vol. 63, (1991), pp. 685–700. http://dx.doi.org/10.1007/BF01029205CrossrefGoogle Scholar

  • [29] J.P. Keener: “A geometrical theory for spiral waves in excitable media”, SIAM J. Appl. Math., Vol. 46, (1986), pp. 1039–1056. http://dx.doi.org/10.1137/0146062CrossrefGoogle Scholar

  • [30] P.L. Simon and H. Farkas: “Geometric theory of trigger waves — A dynamical system approach” J. Math. Chem., Vol. 19, (1996), pp. 301–315. http://dx.doi.org/10.1007/BF01166721CrossrefGoogle Scholar

  • [31] A. Lázár, Z. Noszticzius and H. Farkas: “Involutes — The geometry of chemical waves rotating in annular membranes”, Chaos, Vol. 5, (1995), pp. 443–447. http://dx.doi.org/10.1063/1.166115CrossrefGoogle Scholar

  • [32] Á. Tóth, V. Gáspár and K. Showalter: “Signal transmission in chemical systems — Propagation of chemical waves through capillary tubes”, J. Phys. Chem., Vol. 98, (1994), pp. 522–531. http://dx.doi.org/10.1021/j100053a029CrossrefGoogle Scholar

  • [33] A. Lázár, H-D. Försterling, A. Volford and Z. Noszticzius: “Refraction of chemical waves propagating in modified membranes”, J. Chem. Soc., Faraday Trans., Vol. 92, (1996), pp. 2903–2909. http://dx.doi.org/10.1039/ft9969202903CrossrefGoogle Scholar

  • [34] A. Lázár, H-D. Försterling and H. Farkas: “Waves of excitation on nonuniform membrane rings, caustics, and reverse involutes”, Chaos, Vol. 7, (1997), pp. 731–737. http://dx.doi.org/10.1063/1.166270CrossrefGoogle Scholar

  • [35] O. Rudzick and A.S. Mikhailov: “Front Reversals, Wave Traps, and Twisted Spirals in Periodically Forced Oscillatory Media”, Phys. Rev. Lett., Vol. 96, (2006), art. 018302. Google Scholar

  • [36] S.K. Scott: Oscillations, Waves and Chaos in Chemical Kinetics, Oxford University Press, Oxford, 1995. Google Scholar

  • [37] A. Volford, Z. Noszticzius and V. Krinsky: “Amplitude control of chemical waves in catalytic membranes. Asymmetric wave propagation between zones loaded with different catalyst concentrations”, J. Phys. Chem. A, Vol. 102, (1998), pp. 8355–8361. http://dx.doi.org/10.1021/jp9824609CrossrefGoogle Scholar

  • [38] A. Volford, P. Simon, H. Farkas and Z. Noszticzius: “Rotating chemical waves: theory and experiments”, Physica A, Vol. 274, (1999), pp. 30–49. http://dx.doi.org/10.1016/S0378-4371(99)00331-3CrossrefGoogle Scholar

  • [39] K.A. Kály-Kullai: “A fast method to simulate travelling waves in nonhomogeneous chemical or biological media”, J. Math. Chem., Vol. 34, (2003), pp. 163–176. http://dx.doi.org/10.1023/B:JOMC.0000004066.71858.06CrossrefGoogle Scholar

  • [40] J. Tyson and P. Fife: “Target patterns in a realistic model of Belousov-Zhabotinsky reaction”, J. Chem. Phys., Vol. 73, (1980), pp. 2224–2237. http://dx.doi.org/10.1063/1.440418CrossrefGoogle Scholar

  • [41] A. Volford, F. Izsák, M. Ripszám and I. Lagzi: “Pattern Formation and Self-Organization in a Simple Precipitation System”, Langmuir, Vol. 23, (2007), pp. 961–964. http://dx.doi.org/10.1021/la0623432Web of ScienceCrossrefGoogle Scholar

  • [42] M. Fialkowski, A. Bitner and B.A. Grzybowski: “Wave Optics of Liesegang Rings”, Phys. Rev. Lett., Vol. 94, (2005), art. 018303. Google Scholar

  • [43] K. Kály-Kullai, L. Roszol and A. Volford: “Chemical lens”, Chem. Phys. Lett., Vol. 414, (2005), pp. 326–330. http://dx.doi.org/10.1016/j.cplett.2005.08.082CrossrefGoogle Scholar

About the article

Published Online: 2007-12-01

Published in Print: 2007-12-01

Citation Information: Open Physics, Volume 5, Issue 4, Pages 471–486, ISSN (Online) 2391-5471, DOI: https://doi.org/10.2478/s11534-007-0032-7.

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