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Analyzing critical propagation in a reaction-diffusion-advection model using unstable slow waves

  • Frederike KneerEmail author
  • Klaus Obermayer
  • Markus A. Dahlem
Regular Article

Abstract

The effect of advection on the propagation and in particular on the critical minimal speed of traveling waves in a reaction-diffusion model is studied. Previous theoretical studies estimated this effect on the velocity of stable fast waves and predicted the existence of a critical advection strength below which propagating waves are not supported anymore. In this paper, an analytical expression for the advection-velocity relation of the unstable slow wave is derived. In addition, the critical advection strength is calculated taking into account the unstable slow wave solution. We also analyze a two-variable reaction-diffusion-advection model numerically in a wide parameter range. Due to the new control parameter (advection) we can find stable wave propagation in the otherwise non-excitable parameter regime, if the advection strength exceeds a critical value. Comparing theoretical predictions to numerical results, we find that they are in good agreement. Theory provides an explanation for the observed behaviour.

Graphical abstract

Keywords

Flowing matter: Nonlinear Physics 

References

  1. 1.
    J.P. Keener, J.J. Tyson, Physica D 21, 307 (1986).CrossRefADSzbMATHMathSciNetGoogle Scholar
  2. 2.
    R. Kapral, K. Showalter, Chemical Waves and Patterns (Kluwer, Dordrecht, 1995).Google Scholar
  3. 3.
    A.L. Hodgkin, A.F. Huxley, J. Physiol. 117, 500 (1952).CrossRefGoogle Scholar
  4. 4.
    A.A.P. Leão, J. Neurophysiol. 7, 359 (1944).Google Scholar
  5. 5.
    J.P. Dreier, Nat. Med. 17, 439 (2011).CrossRefGoogle Scholar
  6. 6.
    P. Camacho, J.D. Lechleiter, Science 260, 226 (1993).CrossRefADSGoogle Scholar
  7. 7.
    M. Falcke, L. Tsimring, H. Levine, Phys. Rev. E 62, 2636 (2000).CrossRefADSGoogle Scholar
  8. 8.
    S. Jakubith, H.H. Rotermund, W. Engel, A. von Oertzen, G. Ertl, Phys. Rev. Lett. 65, 3013 (1990).CrossRefADSGoogle Scholar
  9. 9.
    C. Beta, M.G. Moula, A.S. Mikhailov, H.H. Rotermund, G. Ertl, Phys. Rev. Lett. 93, 188302 (2004).CrossRefADSGoogle Scholar
  10. 10.
    M. Bär, M. Falcke, M. Hildebrand, M. Neufeld, H. Engel, M. Eiswirth, Int. J. Bifur. Chaos 4, 499 (1994).Google Scholar
  11. 11.
    H. Sevcikova, M. Marek, Physica D 21, 61 (1986).CrossRefADSGoogle Scholar
  12. 12.
    M. Gómez-Gesteira, J. Mosquera, V.A. Davydov, V. Pérez-Muñuzuri, A.P. Muñuzuri, V.G. Morozov, V. Pérez-Villar, Phys. Lett. A 231, 389 (1997).CrossRefADSGoogle Scholar
  13. 13.
    O. Steinbock, J. Schütze, S.C. Müller, Phys. Rev. Lett. 68, 248 (1992).CrossRefADSGoogle Scholar
  14. 14.
    J.M. Chomaz, Phys. Rev. Lett. 69, 1931 (1992).CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. 15.
    B. Grafstein, J. Neurophysiol. 19, 308 (1956).Google Scholar
  16. 16.
    R.B. Lipton, D.W. Dodick, S.D. Silberstein, J.R. Saper, S.K. Aurora, S.H. Pearlman, R.E. Fischell, P.L. Ruppel, P.J. Goadsby, Lancet Neurol. 9, 373 (2010).CrossRefGoogle Scholar
  17. 17.
    J. Schoenen, B. Vandersmissen, S. Jeangette, L. Herroelen, M. Vandenheede, P. Gerard, D. Magis, Neurology 80, 697 (2013).CrossRefGoogle Scholar
  18. 18.
    Y. Mori, C. Liu, R.S. Eisenberg, Physica D 240, 1835 (2011).CrossRefADSzbMATHGoogle Scholar
  19. 19.
    B.I. Henry, T.A.M. Langlands, S.L. Wearne, Phys. Rev. Lett. 100, 128103 (2008).CrossRefADSGoogle Scholar
  20. 20.
    A. Yochelis, M. Sheintuch, Phys. Rev. E 80, 056201 (2009).CrossRefADSGoogle Scholar
  21. 21.
    A. Yochelis, M. Sheintuch, Phys. Chem. Chem. Phys. 11, 9210 (2009).CrossRefGoogle Scholar
  22. 22.
    A. Yochelis, M. Sheintuch, Phys. Rev. E 81, 025203 (2010).CrossRefADSGoogle Scholar
  23. 23.
    A. Yochelis, M. Sheintuch, Phys. Chem. Chem. Phys. 12, 3957 (2010).CrossRefGoogle Scholar
  24. 24.
    G. Zhao, S. Ruan, J. Diff. Eq. 257, 1078 (2014).CrossRefADSzbMATHMathSciNetGoogle Scholar
  25. 25.
    R. Friedrichs, A. Engel, Europhys. Lett. 63, 8614 (2003).CrossRefGoogle Scholar
  26. 26.
    G. Bordyugov, H. Engel, Phys. Rev. E 74, 016205 (2006).CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    P. Colet, M.A. Matías, L. Gelens, D. Gomila, Phys. Rev. E 89, 012914 (2014).CrossRefADSGoogle Scholar
  28. 28.
    J. Siebert, S. Alonso, M. Bär, E. Schöll, Phys. Rev. E 89, 052909 (2014).CrossRefADSGoogle Scholar
  29. 29.
    V.S. Zykov, Simulation of Wave Processes in Excitable Media (John Wiley & Sons Ltd (english translation from 1992) Moscow, 1984).Google Scholar
  30. 30.
    V.A. Davydov, N. Manz, O. Steinbock, S.C. Müller, Europhys. Lett. 59, 344 (2002).CrossRefADSGoogle Scholar
  31. 31.
    J.J. Tyson, J.P. Keener, Physica D 32, 327 (1988).CrossRefADSzbMATHMathSciNetGoogle Scholar
  32. 32.
    Petra Foerster, Stefan C. Müller, Benno Hess, Science 241, 685 (1988).CrossRefADSGoogle Scholar
  33. 33.
    O. Steinbock, V. Zykov, S.C. Müller, Nature 366, 322 (1993).CrossRefADSGoogle Scholar
  34. 34.
    K.F. Bonhoeffer, J. Gen. Physiol. 32, 69 (1948).CrossRefGoogle Scholar
  35. 35.
    R. FitzHugh, Biophys. J. 1, 445 (1961).CrossRefADSGoogle Scholar
  36. 36.
    J. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IRE 50, 2061 (1962).CrossRefGoogle Scholar
  37. 37.
    G. Bordyugov, H. Engel, Physica D 228, 49 (2007).CrossRefADSzbMATHMathSciNetGoogle Scholar
  38. 38.
    V.A. Davydov, V.G. Morozov, N.V. Davydov, Phys. Lett. A 307, 265 (2003).CrossRefADSzbMATHMathSciNetGoogle Scholar
  39. 39.
    F. Kneer, E. Schöll, M.A. Dahlem, New J. Phys. 16, 053010 (2014).CrossRefADSGoogle Scholar
  40. 40.
    J. Schütze, O. Steinbock, S.C. Müller, Nature 356, 45 (1992).CrossRefADSGoogle Scholar
  41. 41.
    B. Schmidt, P. De Kepper, S.C. Müller, Phys. Rev. Lett. 90, 118302 (2003).CrossRefADSGoogle Scholar
  42. 42.
    M. Krupa, B. Sandstede, P. Szmolyan, J. Diff. Eq. 133, 49 (1997).CrossRefADSzbMATHMathSciNetGoogle Scholar
  43. 43.
    E.J. Doedel, B.E. Oldeman, Auto-07P: Continuation and bifurcation software for ordinary differential equations, Concordia University, Montreal, Canada (2009).Google Scholar
  44. 44.
    R.G. Casten, H. Cohen, P.A. Lagerstrom, Q. Appl. Math. 32, 365 (1975).zbMATHMathSciNetGoogle Scholar
  45. 45.
    A.S. Mikhailov, Foundations of Synergetics, Vol. I, 2nd edition (Springer, Berlin, 1994).Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Frederike Kneer
    • 1
    Email author
  • Klaus Obermayer
    • 1
  • Markus A. Dahlem
    • 2
  1. 1.Department of Software Engineering and Theoretical Computer ScienceTechnische Universität BerlinBerlinGermany
  2. 2.Department of PhysicsHumboldt Universität zu BerlinBerlinGermany

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