Turing Patterns and Excitable Media

  • Andreas Deutsch
  • Sabine Dormann
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


In this chapter we demonstrate how cellular automata can be used to analyze Turing-type interactions and excitable media. In particular, we show that mean-field analysis of the cellular automaton models allows to deduce the important pattern characteristics observed in simulations.


  1. Bär, M., and L. Brusch. 2004. Breakup of spiral waves caused by radial dynamics: Eckhaus and finite wave number instabilities. New Journal of Physics 6: 5.CrossRefGoogle Scholar
  2. Bär, M., and M. Eiswirth. 1993. Turbulence due to spiral breakup in a continuous excitable media. Physics Review E 48: 1635–1637.CrossRefGoogle Scholar
  3. Barkley, D. 1991. A model for fast computer simulation of waves in excitable media. Physica D 49: 61–70.CrossRefGoogle Scholar
  4. Bussemaker, H. 1995. Pattern Formation and Correlations in Lattice Gas Automata. Ph.D. thesis, Instituut voor Theoretische Fysica, Universiteit Utrecht, The Netherlands.Google Scholar
  5. Bussemaker, H., A. Deutsch, and E. Geigant. 1997. Mean-field analysis of a dynamical phase transition in a cellular automaton model for collective motion. Physical Review Letters 78: 5018–5021.CrossRefGoogle Scholar
  6. Bussemaker, H. J. 1996. Analysis of a pattern forming lattice-gas automaton: mean-field theory and beyond. Physical Review E 53(2): 1644–1661.CrossRefGoogle Scholar
  7. Castets, V., E. Dulos, J. Boissonade, and P. de Kepper. 1990. Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Physical Review Letters 64: 2953.CrossRefGoogle Scholar
  8. Comins, H. N., M. P. Hassell, and R. M. May. 1992. The spatial dynamics of host-parasitoid systems. The Journal of Animal Ecology 61: 735–748.CrossRefGoogle Scholar
  9. Cross, M., and P. Hohenberg. 1993. Pattern formation outside of equilibrium. Reviews of Modern Physics 65: 851.CrossRefzbMATHGoogle Scholar
  10. Deutsch, A. 1996. Orientation-induced pattern formation: swarm dynamics in a lattice-gas automaton model. International Journal of Bifurcation and Chaos 6: 1735–1752.CrossRefzbMATHGoogle Scholar
  11. Deutsch, A. 1999b. Principles of biological pattern formation: swarming and aggregation viewed as self-organization phenomena. Journal of Biosciences 24: 115–120.CrossRefGoogle Scholar
  12. Dormann, S. 2000. Pattern Formation in Cellular Automaton Models – Characterisation, Examples and Analysis. Ph.D. thesis, University of Osnabrück, Department of Mathematics/Computer Science, Applied Systems Science.Google Scholar
  13. Engelhardt, R. 1994. Modelling pattern formation in reaction diffusion systems. Master’s thesis, Department of Chemistry, University of Copenhagen, Denmark.Google Scholar
  14. Ermentrout, G. B., and L. Edelstein-Keshet. 1993. Cellular automata approaches to biological modeling. Journal of Theoretical Biology 160: 97–133.CrossRefGoogle Scholar
  15. Fast, V. G., and I. R. Efimov. 1991. Stability of vortex rotation in an excitable cellular medium. Physica D 49: 75–81.CrossRefGoogle Scholar
  16. Gerhardt, M., H. Schuster, and J. Tyson. 1990a. A cellular automaton model of excitable media. II. curvature, dispersion, rotating waves and meandering waves. Physica D 46: 392–415.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Gierer, A., and H. Meinhardt. 1972. A theory of biological pattern formation. Kybernetik 12: 30–39.CrossRefzbMATHGoogle Scholar
  18. Greenberg, J. M., B. D. Hassard, and S. P. Hastings. 1978. Pattern formation and periodic structures in systems modeled by reaction-diffusion equations. Bulletin of the American Mathematical Society 84: 1296–1327.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Haken, H. 1978b. Synergetics. An Introduction. Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry and Biology. Berlin: Springer.zbMATHGoogle Scholar
  20. Haken, H., and H. Olbrich. 1978. Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis. Journal of Mathematical Biology 6: 317–331.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Hasslacher, B., R. Kapral, and A. Lawniczak. 1993. Molecular Turing structures in the biochemistry of the cell. Chaos 3(1): 7–13.CrossRefGoogle Scholar
  22. Kapral, R., A. T. Lawniczak, and P. Masiar. 1991. Oscillations and waves in a reactive lattice-gas automaton. Physical Review Letters 66(19):2539–2542.CrossRefGoogle Scholar
  23. Kruse, K. 2002. A dynamic model for determining the middle of Escherichia coli. Biophysical Journal 82: 618.CrossRefGoogle Scholar
  24. Landau, L. D., and E. M. Lifshitz, 1979. Fluid Mechanics. Oxford: Pergamon Press.zbMATHGoogle Scholar
  25. Lawniczak, A. T., D. Dab, R. Kapral, and J. Boon. 1991. Reactive lattice-gas automata. Physica D 47: 132–158.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Maini, P. K. 1999. Mathematical models in morphogenesis. In Mathematics Inspired by Biology. eds. V. Capasso and O. Dieckmann, 151–189. Berlin: Springer.Google Scholar
  27. Markus, M., and B. Hess. 1990. Isotropic cellular automaton for modeling excitable media. Nature 347(6288): 56–58.CrossRefGoogle Scholar
  28. Markus, M., and H. Schepers. 1993. Turing structures in a semi-random cellular automaton. In Mathematics Applied to Biology and Medicine, eds. J. Demongeot, and V. Capasso, 473–481. Winnipeg: Wuerz Publishing. Proceeding of the 1st European Conference on Mathematics applied to Biology and Medicine.Google Scholar
  29. Medvinsky, A. B., D. A. Tikhonov, J. Enderlein, and H. Malchow. 2000. Fish and plankton interplay determines both plankton spatio-temporal pattern formation and fish school walks. A theoretical study. Nonlinear Dynamics Psychol. Life Sci.Nonlinear Dynamics, Psychology, and Life Sciences 4(2): 135–152.Google Scholar
  30. Meinhardt, H., and P. A. J. de Boer. 2001. Pattern formation in Escherichia coli: A model for the pole-to-pole oscillations of Min proteins and the localization of the division site. Proceedings of the National Academy of Sciences of the United States of America 98(25): 14202–14207.CrossRefGoogle Scholar
  31. Mikhailov, A. S. 1994. Foundations of Synergetics I. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  32. Müller, P., K. W. Rogers, B. M. Jordan, J. S. Lee, D. Robson, S. Ramanathan, and A. F. Schier. 2012. Differential diffusivity of Nodal and Lefty underlies a reaction-diffusion patterning system. Science 336(6082): 721–724.CrossRefGoogle Scholar
  33. Murray, J. D. 2002. Mathematical Biology, 3rd ed. New York: Springer.zbMATHGoogle Scholar
  34. Newell, P. C. 1983. Attraction and adhesion in the slime mold Dictyostelium. In Fungal Differentiation: A Contemporary Synthesis. Mycology Series 43, ed. J. E. Smith, 43–71. New York: Marcel Decker.Google Scholar
  35. Nicola, E., M. Or-Guil, W. Wolf, and M. Bär. 2002. Drifting pattern domains in reaction-diffusion systems with nonlocal coupling. Physical Review E 65: 055101.CrossRefGoogle Scholar
  36. Ouyang, Q., and H. L. Swinney. 1991. Transitions from a uniform state to hexagonal and striped Turing patterns. Nature 352: 610–611.CrossRefGoogle Scholar
  37. Schenk, C. P., M. Or-Guil, M. Bode, and H. G. Purwins. 1997. Interacting pulses in three-component reaction-diffusion-systems on two-dimensional domains. Physical Review Letters 78: 3781–3783.CrossRefGoogle Scholar
  38. Schönfisch, B. 1993. Zelluläre Automaten und Modelle für Epidemien. Ph.D. thesis, Fakultät für Biologie, Universität Tübingen.Google Scholar
  39. Segel, L. A. 1984. Modeling Dynamic Phenomena in Molecular and Cellular Biology. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  40. Sepulchre, J. A., and A. Babloyantz. 1995. Spiral and target waves in finite and discontinuous media. In Chemical Waves and Patterns, eds. R. Kapral and K. Showalter. Dordrecht: Kluwer Academic Publishers.Google Scholar
  41. Turing, A. 1952. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society B 237: 37–72.MathSciNetCrossRefGoogle Scholar
  42. Weimar, J. R., J. J. Tyson, and L. T. Watson. 1992. Third generation cellular automaton for modeling excitable media. Physica D 55: 328–339.MathSciNetCrossRefzbMATHGoogle Scholar
  43. Wiener, N., and A. Rosenbluth. 1946. The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Archivos de Cardiología de México 16: 205–265.MathSciNetGoogle Scholar
  44. Winfree, A. T. 1972. Spiral waves of chemical activity. Science 175: 634–636.CrossRefGoogle Scholar
  45. Winfree, A. T. 1987. When Time Breaks Down. Princeton: Princeton University Press.Google Scholar
  46. Wyatt, T. 1973. The biology of Oikopleura dioica and Fritillaria borealis in the Southern Bight. Marine Biology 22: 137–158.CrossRefGoogle Scholar
  47. Young, D. A. 1984. A local activator-inhibitor model of vertebrate skin patterns. Mathematical Biosciences 72: 51–58.MathSciNetCrossRefGoogle Scholar
  48. Zhabotinskii, A. M., and A. N. Zaikin. 1970. Concentration wave propagation in a two-dimensional liquid-phase self-oscillating system. Nature 225:535–537.Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Andreas Deutsch
    • 1
  • Sabine Dormann
    • 2
  1. 1.Centre for Information Services and High Performance ComputingTechnische Universität DresdenDresdenGermany
  2. 2.FB Mathematik/InformatikUniversität OsnabrückOsnabrückGermany

Personalised recommendations