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A Lattice-Gas Cellular Automaton Model for Discrete Excitable Media

  • Simon SygaEmail author
  • Josué M. Nava-Sedeño
  • Lutz Brusch
  • Andreas Deutsch
Chapter
Part of the The Frontiers Collection book series (FRONTCOLL)

Abstract

How do ordered structures like spirals cope with stochastic events? Several phenomena in chemistry and biology provide examples of excitable media and spiral pattern formation and are intrinsically stochastic. Here, we present a novel lattice-gas cellular automaton model for discrete excitable media. In this stochastic model, two discrete interacting biological species determine each other’s birth and death probabilities. We show that this birth-death process, coupled to a random walk, is equivalent to a classical partial differential equation (PDE) model of excitable media in the macroscopic limit, and able to form spiral density waves. Importantly, our cellular automaton model includes a parameter which defines the maximum local number of individuals and influences the onset of spiral waves. We find that small values of this parameter allow spiral pattern formation even in situations where the corresponding deterministic PDE model predicts that no spirals are formed, reminiscent of stochastic resonance effects.

References

  1. 1.
    M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).  https://doi.org/10.1103/RevModPhys.65.851
  2. 2.
    E. Meron, Pattern formation in excitable media. Phys. Rep. 218, 1–66 (1992)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    J.C. Dallon et al., in Dynamics of Cell and Tissue Motion, ed. by W. Alt, A. Deutsch, G.A. Dunn (Birkhäuser, Basel, 1997), pp. 193–202.  https://doi.org/10.1007/978-3-0348-8916-2_23
  4. 4.
    J.D. Murray, E.A. Stanley, D.L. Brown, On the spatial spread of rabies among foxes. Proc. R. Soc. Lond. B 229, 111–150 (1986)ADSCrossRefGoogle Scholar
  5. 5.
    D. Barkley, A model for fast computer simulation of waves in excitable media. Phys. D 49, 61–70 (1991)CrossRefGoogle Scholar
  6. 6.
    E.M. Cherry, F.H. Fenton, Visualization of spiral and scroll waves in simulated and experimental cardiac tissue. New J. Phys. 10, 125016 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    H. Zhang, A.V. Holden, Chaotic meander of spiral waves in the FitzHugh-Nagumo system. Chaos Solitons Fractals 5, 661–670 (1995)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    W. Jahnke, W.E. Skaggs, A.T. Winfree, Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable Oregonator model. J. Phys. Chem. 93, 740–749 (1989)CrossRefGoogle Scholar
  9. 9.
    J.P. Keener, A geometrical theory for spiral waves in excitable media. SIAM J. Appl. Math. 46, 1039–1056 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S.A. Wolfram, A New Kind of Science (Wolfram Media, Inc., 2002)Google Scholar
  11. 11.
    J. Von Neumann, A.W. Burks, Theory of Self-reproducing Automata (University of Illinois Press, Urbana, 1996)Google Scholar
  12. 12.
    J.M. Greenberg, S.P. Hastings, Spatial patterns for discrete models of diffusion in excitable media. SIAM J. Appl. Math. 34, 515–523 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L.V. Reshodko, J. Bureš, Computer simulation of reverberating spreading depression in a network of cell automata. Biol. Cybern. 18, 181–189 (1975)CrossRefzbMATHGoogle Scholar
  14. 14.
    B.F. Madore, W.L. Freedman, Computer simulations of the Belousov-Zhabotinsky reaction. Science 222, 615–616 (1983)ADSCrossRefGoogle Scholar
  15. 15.
    A.T. Winfree, E.M. Winfree, H. Seifert, Organizing centers in a cellular excitable medium. Phys. D 17, 109–115 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    V.S. Zykov, A.S. Mikhailov, Rotating spiral waves in a simple model of an excitable medium. Sov. Phys. Dokl. 31, 51–52 (1986)ADSGoogle Scholar
  17. 17.
    A.S. Mikhailov, Foundations of Synergetics I: Distributed Active Systems (Springer, Berlin, 1990), pp. 32–80.  https://doi.org/10.1007/978-3-642-97269-0_3
  18. 18.
    M. Gerhardt, H. Schuster, J.J. Tyson, A cellular automation model of excitable media including curvature and dispersion. Science 247, 1563–1566 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    M. Markus, B. Hess, Isotropic cellular automaton for modelling excitable media. Nature 347, 56–58 (1990)ADSCrossRefGoogle Scholar
  20. 20.
    J.R. Weimar, J.J. Tyson, L.T. Watson, Third generation cellular automaton for modeling excitable media. Phys. D 55, 328–339 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    U. Frisch, B. Hasslacher, Y. Pomeau, Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Lett. 56, 1505–1508 (1986)ADSCrossRefGoogle Scholar
  22. 22.
    A. Deutsch, S. Dormann, Cellular Automaton Modeling of Biological Pattern Formation: Characterization, Applications, and Analysis, 2nd edn. (Birkhäauser, Boston, 2018)Google Scholar
  23. 23.
    K. Böttger, H. Hatzikirou, A. Voss-Böhme, E.A. Cavalcanti-Adam, M.A. Herrero, A. Deutsch, An emerging Allee effect is critical for tumor initiation and persistence. PLOS Comput. Biol. 11, 1–14 (2015).  https://doi.org/10.1371/journal.pcbi.1004366
  24. 24.
    H. Hatzikirou, K. Böttger, A. Deutsch, Model-based comparison of cell density-dependent cell migration strategies. Math. Model. Nat. Phenom. 10, 94–107 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    K. Böttger, H. Hatzikirou, A. Chauviere, A. Deutsch, Investigation of the migration/proliferation dichotomy and its impact on avascular glioma invasion. Math. Model. Nat. Phenom. 7, 105–135 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    C. Mente, I. Prade, L. Brusch, G. Breier, A. Deutsch, A lattice-gas cellular automaton model for in vitro sprouting angiogenesis. Acta Phys. Pol. B 5, 99–115 (2012)zbMATHGoogle Scholar
  27. 27.
    S. De Franciscis, H. Hatzikirou, A. Deutsch, Analysis of lattice-gas cellular automaton models for tumor growth by means of fractal scaling. Acta Phys. Pol. B Proc. Suppl. 4, 167 (2011)CrossRefGoogle Scholar
  28. 28.
    M. Tektonidis et al., Identification of intrinsic in vitro cellular mechanisms for glioma invasion. J. Theor. Biol. 287, 131–147 (2011)CrossRefzbMATHGoogle Scholar
  29. 29.
    B. Chopard, R. Ouared, A. Deutsch, H. Hatzikirou, D. Wolf-Gladrow, Lattice-gas cellular automaton models for biology: from fluids to cells. Acta Biotheor. 58, 329–340 (2010)CrossRefGoogle Scholar
  30. 30.
    H. Hatzikirou, A. Deutsch, Cellular automata as microscopic models of cell migration in heterogeneous environments. Curr. Top. Dev. Biol. 81, 401–434 (2008)CrossRefGoogle Scholar
  31. 31.
    S. Dormann, A. Deutsch, Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. Silico Biol. 2, 393–406 (2002)Google Scholar
  32. 32.
    S. Dormann, A. Deutsch, A.T. Lawniczak, Fourier analysis of Turing-like pattern formation in cellular automaton models. Futur. Gener. Comput. Syst. 17, 901–909 (2001).  https://doi.org/10.1016/S0167-739X(00)00068-6
  33. 33.
    A. Deutsch, A new mechanism of aggregation in a lattice-gas cellular automaton model. Math. Comput. Model. 31, 35–40 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    H.J. Bussemaker, A. Deutsch, E. Geigant, Mean-field analysis of a dynamical phase transition in a cellular automaton model for collective motion. Phys. Rev. Lett. 78, 5018–5021 (1997).  https://doi.org/10.1103/PhysRevLett.78.5018ADSCrossRefGoogle Scholar
  35. 35.
    A. Deutsch, Towards analyzing complex swarming patterns in biological systems with the help of lattice-gas cellular automata. J. Biol. Syst. 3, 947–955 (1995)CrossRefGoogle Scholar
  36. 36.
    J.M. Nava-Sedeño, H. Hatzikirou, R. Klages, A. Deutsch, Cellular automaton models for time-correlated random walks: derivation and analysis. Sci. Rep. 7, 16952 (2017)Google Scholar
  37. 37.
    J.M. Nava-Sedeño, H. Hatzikirou, F. Peruani, A. Deutsch, Extracting cellular automaton rules from physical Langevin equation models for single and collective cell migration. J. Math. Biol. 75, 1075–1100 (2017)Google Scholar
  38. 38.
    H. Hatzikirou, L. Brusch, A. Deutsch, From cellular automaton rules to a macroscopic mean-field description. Acta Phys. Pol. B Proc. Suppl. 3, 399–416 (2010)Google Scholar
  39. 39.
    C. Mente, I. Prade, L. Brusch, G. Breier, A. Deutsch, Parameter estimation with a novel gradient-based optimization method for biological lattice-gas cellular automaton models. J. Math. Biol. 63, 173–200 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    A. Deutsch, A.T. Lawniczak, Probabilistic lattice models of collective motion and aggregation: from individual to collective dynamics. Math. Biosci. 156, 255–269 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    B. Lindner, J. García-Ojalvo, A. Neiman, L. Schimansky-Geier, Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)ADSCrossRefGoogle Scholar
  42. 42.
    D. Barkley, EZ-spiral: a code for simulating spiral waves Version 3.2 (2007), http://homepages.warwick.ac.uk/staff/D.Barkley/Software/ez_software.html
  43. 43.
    D. Barkley, Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett. 72, 164–167 (1994).  https://doi.org/10.1103/PhysRevLett.72.164
  44. 44.
    S. Kadar, J.C. Wang, K. Showalter, Noise-supported travelling waves in sub-excitable media. Nature 391, 770–772 (1998)ADSCrossRefGoogle Scholar
  45. 45.
    M. Perc, Spatial coherence resonance in excitable media. Phys. Rev. E 72, 016207 (2005).  https://doi.org/10.1103/PhysRevE.72.016207

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Simon Syga
    • 1
    Email author
  • Josué M. Nava-Sedeño
    • 1
  • Lutz Brusch
    • 1
  • Andreas Deutsch
    • 1
  1. 1.Centre for Information Services and High Performance Computing, Technische Universität DresdenDresdenGermany

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