Reaction-Diffusion Patterns and Waves: From Chemical Reactions to Cardiac Arrhythmias

  • Markus BärEmail author
Part of the The Frontiers Collection book series (FRONTCOLL)


Reaction-diffusion processes are behind many instances of pattern formation in chemical reactions and biological systems. Continuum reaction-diffusion equations have proved useful models for a wide variety of pattern dynamics starting with seminal work by Turing on the chemical basis of morphogenesis and by Hodgkin and Huxley on the propagation of electrical impulses along neurons in 1952. This article reviews basic concepts for and applications of reaction-diffusion models with an emphasis on spiral and vortex dynamics, related instabilities like spiral and scroll wave breakup and their potential role in cardiac arrhythmias.


  1. 1.
    M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851 (1993)Google Scholar
  2. 2.
    A.M. Turing, The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B 237, 37 (1952)Google Scholar
  3. 3.
    A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. Lond. 117, 500 (1952)Google Scholar
  4. 4.
    J. Keener, J. Sneyd, Mathematical Physiology (Springer, New York, 1998)Google Scholar
  5. 5.
    E. Meron, Pattern formation in excitable media. Phys. Rep. 218, 1 (1992)Google Scholar
  6. 6.
    R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445 (1961)Google Scholar
  7. 7.
    A. Gierer, H. Meinhardt, Theory for biological pattern formation. Kybernetik 12, 30 (1972)Google Scholar
  8. 8.
    D. Barkley, A model for fast computer simulation of waves in excitable media. Physica D 49, 61 (1991)Google Scholar
  9. 9.
    M. Bär, M. Eiswirth, Turbulence due to spiral breakup in a continuous excitable medium. Phys. Rev. E 48, 1635 (1993)Google Scholar
  10. 10.
    R. Kapral, K. Showalter (eds.), Chemical Waves and Patterns (Kluwer, Dordrecht, 1994)Google Scholar
  11. 11.
    A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behavior (Cambridge University, Cambridge, 1996)Google Scholar
  12. 12.
    A.N. Zaikin, A.M. Zhabotinsky, Concentration wave propagation in 2-dimensional liquid phase self-oscillating system. Nature 225, 535 (1970)Google Scholar
  13. 13.
    A.T. Winfree, Spiral waves of chemical activity. Science 175, 634 (1972)Google Scholar
  14. 14.
    R.J. Field, R.M. Noyes, E. Körös, Oscillations in chemical systems 2: Thorough analysis of temporal oscillations in Bromate-Cerium-Malonic acid system. J. Am. Chem. Soc. 94, 8649 (1972)Google Scholar
  15. 15.
    S.C. Müller, Th. Plesser, B. Hess, The structure of the core of the spiral wave in the Belousov-Zhabotinsky reaction. Science 230, 661 (1985)Google Scholar
  16. 16.
    V. Castets, E. Dulos, J. Boissonade, P. de Kepper, Experimental evidence of a sustained standing turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953 (1990)Google Scholar
  17. 17.
    Q. Ouyang, H.L. Swinney, Transition from a uniform state to hexagonal and striped turing patterns. Nature 352, 610 (1991)ADSCrossRefGoogle Scholar
  18. 18.
    K.J. Lee, W.D. McCormick, Q. Ouyang, H.L. Swinney, Pattern formation by interacting chemical fronts. Science 261, 183 (1993)ADSGoogle Scholar
  19. 19.
    K.J. Lee, W.D. McCormick, J. Pearson, H.L. Swinney, Experimental observation of self-replicating spots in a reaction-diffusion system. Nature 369, 215 (1994)ADSCrossRefGoogle Scholar
  20. 20.
    A. Hagberg, E. Meron, Complex patterns in reaction-diffusion systems: a tale of two front instabilities. Chaos 4, 477 (1994)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    S. Jakubith, A. von Oertzen, W. Engel, H.H. Rotermund, G. Ertl, Spatiotemporal concentration patterns in a surface reaction - propagating and standing waves, rotating spirals, and turbulence. Phys. Rev. Lett. 65, 3013 (1990)Google Scholar
  22. 22.
    R. Imbihl, G. Ertl, Oscillatory kinetics in heterogeneous catalysis. Chem. Rev. 95, 697 (1995)Google Scholar
  23. 23.
    Q. Ouyang, J.M. Flesselles, Transition from spirals to defect turbulence driven by a convective instability. Nature 379, 143 (1996)ADSCrossRefGoogle Scholar
  24. 24.
    Q. Ouyang, H.L. Swinney, G. Li, Transition from spirals to defect-mediated turbulence driven by a Doppler instability. Phys. Rev. Lett. 84, 1047 (2000)Google Scholar
  25. 25.
    L.Q. Zhou, Q. Ouyang, Experimental studies on long-wavelength instability and spiral breakup in a reaction-diffusion system. Phys. Rev. Lett. 85, 1650 (2000)Google Scholar
  26. 26.
    G. Gerisch, Periodic signals control pattern formation in cell aggregations. Naturwissenschaften 58, 430 (1971)ADSCrossRefGoogle Scholar
  27. 27.
    F. Siegert, C. Weijer, Three dimensional scroll waves organize dictyostelium slugs. Proc. Natl. Acad. Sci. USA 89, 6433 (1992)Google Scholar
  28. 28.
    E. Ben-Jacob, I. Cohen, H. Levine, Cooperative self-organization of microorganism. Adv. Phys. 49, 395 (2000)Google Scholar
  29. 29.
    J. Lechleiter, S. Girard, E. Peralta, D. Clapham, Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes. Science 252, 123 (1992)Google Scholar
  30. 30.
    M. Falcke, Reading the patterns in living cells: the physics of Ca\(^{2+}\) signaling. Adv. Phys. 53, 255 (2004)Google Scholar
  31. 31.
    D.M. Raskin, P.A. De Boer, Rapid pole-to-pole oscillation of a protein required for directing division to the middle of Escherichia Coli. Proc. Natl. Acad. Sci. USA 96, 4971 (1999)Google Scholar
  32. 32.
    M. Loose, E. Fischer-Friedrich, J. Ries, K. Kruse, P. Schwille, Spatial regulators for bacterial cell division self-organize into surface waves in vitro. Science 320, 789–792 (2008)Google Scholar
  33. 33.
    C. Beta, K. Kruse, Intracellular oscillations and waves. Ann. Rev. Cond. Math. Phys. 8, 239–264 (2017)Google Scholar
  34. 34.
    J. Halatek, F. Brauns, E. Frey, Self-organization principles of intracellular pattern formation. Phil. Trans. R. Soc. B 373, 20170107 (2018)CrossRefGoogle Scholar
  35. 35.
    V.K. Vanag, I.R. Epstein, Pattern formation in a tunable medium: the Belousov-Zhabotinsky reaction in an aerosol OT microemulsion. Phys. Rev. Lett. 87, 228301 (2001)ADSCrossRefGoogle Scholar
  36. 36.
    S. Alonso, F. Sagués, A.S. Mikhailov, Taming winfree turbulence of scroll waves in excitable media. Science 299, 1722 (2003)ADSCrossRefGoogle Scholar
  37. 37.
    A.S. Mikhailov, K. Showalter, Control of waves, patterns and turbulence in chemical systems. Phys. Rep. 425, 79–194 (2006)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    T. Bánsági Jr., O. Steinbock, Nucleation and collapse of scroll rings in excitable media. Phys. Rev. Lett. 97, 198301 (2006)ADSCrossRefGoogle Scholar
  39. 39.
    C. Luengviriya, U. Storb, G. Lindner, S.C. Müller, M. Bär, M.J.B. Hauser, Scroll wave instabilities in an excitable chemical medium. Phys. Rev. Lett. 100, 148302 (2008)Google Scholar
  40. 40.
    T. Bánsági, V.K. Vanag, I.R. Epstein, Tomography of reaction-diffusion microemulsions reveals three-dimensional turing patterns. Science 331, 1309–1312 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    A.T. Winfree, When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias (Princeton University, Princeton, 1987)Google Scholar
  42. 42.
    S.V. Pandit, J. Jalife, Rotors and the dynamics of cardiac fibrillation. Circ. Res. 112, 849 (2013)CrossRefGoogle Scholar
  43. 43.
    A. Karma, Physics of cardiac arrhythmogenesis. Ann. Rev. Cond. Math. Phys. 4, 313 (2013)Google Scholar
  44. 44.
    Z. Qu, G. Hu, A. Garfinkel, J.N. Weiss, Nonlinear and stochastic dynamics in the heart. Phys. Rep. 543, 61 (2014)Google Scholar
  45. 45.
    S. Alonso, M. Bär, B. Echebarria, Nonlinear physics of electrical wave propagation in the heart: a review. Rep. Prog. Phys. 79, 096601 (2016)ADSCrossRefGoogle Scholar
  46. 46.
    A.T. Winfree, Electrical turbulence in three-dimensional heart muscle. Science 266, 1003 (1994)ADSCrossRefGoogle Scholar
  47. 47.
    M. Bär, L. Brusch, Breakup of spiral waves caused by radial dynamics: Eckhaus and finite wavenumber instabilities. New J. Phys. 6, 5 (2004)ADSCrossRefGoogle Scholar
  48. 48.
    F.H. Fenton, E.M. Cherry, H.M. Hastings, S.J. Evans, Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 12, 852 (2002)ADSCrossRefGoogle Scholar
  49. 49.
    P. Wheeler, D. Barkley, Computation of spiral spectra. SIAM J. Appl. Dyn. Syst. 5, 15777 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    I.S. Aranson, L. Kramer, The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 92 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    S. Alonso, M. Bär, Reentry near the percolation threshold in a heterogeneous discrete model for cardiac tissue. Phys. Rev. Lett. 110, 158101 (2013)ADSCrossRefGoogle Scholar
  52. 52.
    A.S. Mikhailov, G. Ertl, Chemical Complexity - Self-Organization Processes in Molecular Systems (Springer International Publishing Company, Cham, 2017)Google Scholar
  53. 53.
    J. Christoph, M. Chebbok, C. Richter, J. Schröder-Schetelig, P. Bittihn, S. Stein, I. Uzelac, F.H. Fenton, G. Hasenfuß, R.F. Gilmour Jr., S. Luther, Electro-mechanical vortex filaments during cardiac fibrillation. Nature 555, 667 (2018)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Physikalisch-Technische BundesanstaltBerlinGermany

Personalised recommendations