Full-text resources of PSJD and other databases are now available in the new Library of Science.
Visit https://bibliotekanauki.pl


Preferences help
enabled [disable] Abstract
Number of results


2007 | 5 | 4 | 471-486

Article title

Stochastic cellular automata modeling of excitable systems


Title variants

Languages of publication



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.


  • Institute of Chemistry, Eötvös University, 1117, Budapest, Pázmány P. stny. 1/A, Hungary
  • Institute of Chemistry, Eötvös University, 1117, Budapest, Pázmány P. stny. 1/A, Hungary
  • Department of Chemical Physics, University of Technology and Economics, 1521, Budapest, Hungary
  • Department of Chemical Physics, University of Technology and Economics, 1521, Budapest, Hungary


  • [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/10.1063/1.165844[Crossref]
  • [2] A.S. Mikhailov: Foundations of Synergetics I. Distributed Active Systems 2nd ed., Springer, Berlin, 1994.
  • [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-E[Crossref]
  • [4] B. Chopard and M. Droz: Cellular Automata Modeling of Physical Systems, Cambridge University Press, Cambridge, 1998.
  • [5] S. Wolfram: Theory and Applications of Cellular Automata, World Scientific, Singapore, 1986.
  • [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-T[Crossref]
  • [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.
  • [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)00117-9[Crossref]
  • [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/10.1088/0305-4470/37/9/003[Crossref]
  • [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)00857-9[WoS][Crossref]
  • [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-S[Crossref]
  • [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:1004635606196[Crossref]
  • [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.
  • [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.53.311[Crossref]
  • [15] H. Fukś: “Probabilistic cellular automata with conserved quantities”, Nonlinearity, Vol. 17, (2004), pp. 159–173. http://dx.doi.org/10.1088/0951-7715/17/1/010[Crossref]
  • [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-W[Crossref]
  • [17] http://www.getfreesofts.com/download/66/2971/Five_Cellular_Automata.html
  • [18] http://ccl.northwestern.edu/netlogo/models/B-ZReaction
  • [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-X[Crossref]
  • [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/scientificamerican0888-104[Crossref]
  • [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/10.1088/0034-4885/55/6/002[Crossref]
  • [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.49.1749[Crossref]
  • [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)00123-9[Crossref]
  • [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.2004.08.001[Crossref]
  • [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-X[Crossref]
  • [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/10.1038/225535b0[Crossref]
  • [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.43.8081[Crossref]
  • [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/BF01029205[Crossref]
  • [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/10.1137/0146062[Crossref]
  • [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/BF01166721[Crossref]
  • [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/10.1063/1.166115[Crossref]
  • [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/j100053a029[Crossref]
  • [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/ft9969202903[Crossref]
  • [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/10.1063/1.166270[Crossref]
  • [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.
  • [36] S.K. Scott: Oscillations, Waves and Chaos in Chemical Kinetics, Oxford University Press, Oxford, 1995.
  • [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/jp9824609[Crossref]
  • [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)00331-3[Crossref]
  • [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.0000004066.71858.06[Crossref]
  • [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/10.1063/1.440418[Crossref]
  • [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/la0623432[WoS][Crossref]
  • [42] M. Fialkowski, A. Bitner and B.A. Grzybowski: “Wave Optics of Liesegang Rings”, Phys. Rev. Lett., Vol. 94, (2005), art. 018303.
  • [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.2005.08.082[Crossref]

Document Type

Publication order reference


YADDA identifier

JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.