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Nonlocal control of pulse propagation in excitable media

  • Clemens Bachmair
  • Eckehard SchöllEmail author
Regular Article

Abstract

We study the effects of nonlocal control of pulse propagation in excitable media. As a generic example for an excitable medium the FitzHugh-Nagumo model with diffusion in the activator variable is considered. Nonlocal coupling in form of an integral term with a spatial kernel is added. We find that the nonlocal coupling modifies the propagating pulses of the reaction-diffusion system such that a variety of spatio-temporal patterns are generated including acceleration, deceleration, suppression, or generation of pulses, multiple pulses, and blinking pulse trains. It is shown that one can observe these effects for various choices of the integral kernel and the coupling scheme, provided that the control strength and spatial extension of the integral kernel is appropriate. In addition, an analytical procedure is developed to describe the stability borders of the spatially homogeneous steady state in control parameter space in dependence on the parameters of the nonlocal coupling.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    H. Haken, Synergetics, An Introduction, 3 edn. (Springer, Berlin, 1983) Google Scholar
  2. 2.
    Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, Berlin, 1984) Google Scholar
  3. 3.
    A.S. Mikhailov, in Foundations of Synergetics, 2nd edn. (Springer, Berlin, 1994), Vol. I Google Scholar
  4. 4.
    Chemical Waves and Patterns, edited by R. Kapral, K. Showalter (Kluwer, Dordrecht, 1995) Google Scholar
  5. 5.
    J.P. Keener, J. Sneyd, Mathematical Physiology (Springer, New York, Berlin, 1998) Google Scholar
  6. 6.
    H. Karatas, S.E. Erdener, Y. Gursoy-Ozdemir, S. Lule, E. Eren-Kocak, Z.D. Sen, T. Dalkara, Science 339, 1092 (2013) ADSCrossRefGoogle Scholar
  7. 7.
    J.P. Dreier, Nat. Med. 17, 439 (2011) CrossRefGoogle Scholar
  8. 8.
    M.A. Dahlem, R. Graf, A.J. Strong, J.P. Dreier, Y.A. Dahlem, M. Sieber, W. Hanke, K. Podoll, E. Schöll, Physica D 239, 889 (2010) ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    B.N. Belintsev, M.A. Livshits, M.V. Volkenstein, Z. Phys. B 44, 345 (1981) ADSCrossRefMathSciNetGoogle Scholar
  10. 10.
    N. Mazouz, G. Flätgen, K. Krischer, Phys. Rev. E 55, 2260 (1997) ADSCrossRefGoogle Scholar
  11. 11.
    M. Sheintuch, O. Nekhamkina, J. Chem. Phys. 107, 8165 (1997) ADSCrossRefGoogle Scholar
  12. 12.
    Y. Kuramoto, D. Battogtokh, H. Nakao, Phys. Rev. Lett. 81, 3543 (1998) ADSCrossRefGoogle Scholar
  13. 13.
    M. Hildebrand, H. Skødt, K. Showalter, Phys. Rev. Lett. 87, 088303 (2001) ADSCrossRefGoogle Scholar
  14. 14.
    Y.J. Li, J. Oslonovitch, N. Mazouz, F. Plenge, K. Krischer, G. Ertl, Science 291, 2395 (2001) ADSCrossRefGoogle Scholar
  15. 15.
    E.M. Nicola, M. Or-Guil, W. Wolf, M. Bär, Phys. Rev. E 65, 055101 (2002) ADSCrossRefGoogle Scholar
  16. 16.
    H. Varela, C. Beta, A. Bonnefont, K. Krischer, Phys. Rev. Lett. 94, 174104 (2005) ADSCrossRefGoogle Scholar
  17. 17.
    E.M. Nicola, M. Bär, H. Engel, Phys. Rev. E 73, 066225 (2006) ADSCrossRefGoogle Scholar
  18. 18.
    G. Bordyougov, H. Engel, Phys. Rev. E 74, 016205 (2006) ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    L. Gelens, G. Gomila, G. Van der Sande, M.A. Matías, Phys. Rev. Lett. 104, 154101 (2010) ADSCrossRefGoogle Scholar
  20. 20.
    P. Colet, M.A. Matías, L. Gelens, D. Gomila, Phys. Rev. E 89, 012914 (2014) ADSCrossRefGoogle Scholar
  21. 21.
    L. Gelens, M.A. Matías, D. Gomila, T. Dorissen, P. Colet, Phys. Rev. E 89, 012915 (2014) ADSCrossRefGoogle Scholar
  22. 22.
    J. Löber, R. Coles, J. Siebert, H. Engel, E. Schöll, in Engineering of Chemical Complexity II, edited by A.S. Mikhailov, G. Ertl (World Scientific, Singapore, 2014) Google Scholar
  23. 23.
    L.M. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer Series in Synergetics (Springer, Berlin, 2006) Google Scholar
  24. 24.
    V. Petrov, M.J. Crowley, K. Showalter, Phys. Rev. Lett. 72, 2955 (1994) ADSCrossRefGoogle Scholar
  25. 25.
    D. Tanaka, Y. Kuramoto, Phys. Rev. E 68, 026219 (2003) ADSCrossRefGoogle Scholar
  26. 26.
    S.-I. Shima, Y. Kuramoto, Phys. Rev. E 69, 036213 (2004) ADSCrossRefGoogle Scholar
  27. 27.
    J. Siebert, S. Alonso, M. Bär, E. Schöll, Phys. Rev. E 89, 052909 (2014) ADSCrossRefGoogle Scholar
  28. 28.
    U. Middya, M. Sheintuch, M.D. Graham, D. Luss, Physica D 63, 393 (1992) ADSCrossRefGoogle Scholar
  29. 29.
    A. Yochelis, M. Sheintuch, Phys. Rev. E 81, 025203 (2010) ADSCrossRefGoogle Scholar
  30. 30.
    A. Rovinsky, M. Menzinger, Phys. Rev. Lett. 70, 778 (1993) ADSCrossRefGoogle Scholar
  31. 31.
    Y. Khazan, L.M. Pismen, Phys. Rev. Lett. 75, 4318 (1995) ADSCrossRefGoogle Scholar
  32. 32.
    H. Malchow, J. Mar. Syst. 7, 193 (1996) ADSCrossRefGoogle Scholar
  33. 33.
    J. von Hardenberg, E. Meron, M. Shachak, Y. Zarmi, Phys. Rev. Lett. 87, 198101 (2001) ADSCrossRefGoogle Scholar
  34. 34.
    M. Bär, M. Falcke, M. Hildebrand, M. Neufeld, H. Engel, M. Eiswirth, Int. J. Bifurc. Chaos 4, 499 (1994) Google Scholar
  35. 35.
    C. Beta, M.G. Moula, A.S. Mikhailov, H.H. Rotermund, G. Ertl, Phys. Rev. Lett. 93, 188302 (2004) ADSCrossRefGoogle Scholar
  36. 36.
    F. Plenge, P. Rodin, E. Schöll, K. Krischer, Phys. Rev. E 64, 056229 (2001) ADSCrossRefGoogle Scholar
  37. 37.
    M. Meixner, P. Rodin, E. Schöll, A. Wacker, Eur. Phys. J. B 13, 157 (2000) ADSCrossRefGoogle Scholar
  38. 38.
    E. Schöll, in Nonlinear Spatio-temporal Dynamics and Chaos in Semiconductors, Nonlinear Science Series (Cambridge University Press, Cambridge, 2001), Vol. 10 Google Scholar
  39. 39.
    A.S. Mikhailov, K. Showalter, Phys. Rep. 425, 79 (2006) ADSCrossRefMathSciNetGoogle Scholar
  40. 40.
    J. Christoph, M. Eiswirth, Chaos 12, 215 (2002) ADSCrossRefGoogle Scholar
  41. 41.
    Y. Kuramoto, D. Battogtokh, Nonlin. Phen. Complex Syst. 5, 380 (2002) Google Scholar
  42. 42.
    D.M. Abrams, S.H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004) ADSCrossRefGoogle Scholar
  43. 43.
    A.M. Hagerstrom, T.E. Murphy, R. Roy, P. Hövel, I. Omelchenko, E. Schöll, Nat. Phys. 8, 658 (2012) CrossRefGoogle Scholar
  44. 44.
    M.R. Tinsley, S. Nkomo, K. Showalter, Nat. Phys. 8, 662 (2012) CrossRefGoogle Scholar
  45. 45.
    E.A. Martens, S. Thutupalli, A. Fourrière, O. Hallatschek, Proc. Natl. Acad. Sci. 110, 10563 (2013) ADSCrossRefGoogle Scholar
  46. 46.
    I. Omelchenko, O.E. Omel’chenko, P. Hövel, E. Schöll, Phys. Rev. Lett. 110, 224101 (2013) ADSCrossRefGoogle Scholar
  47. 47.
    A. Zakharova, M. Kapeller, E. Schöll, Phys. Rev. Lett. 112, 154101 (2014) ADSCrossRefGoogle Scholar
  48. 48.
    M.J. Panaggio, D.M. Abrams, arXiv:1403.6204 (2014) Google Scholar
  49. 49.
    K. Kang, M. Shelley, H. Sompolinsky, Proc. Natl. Acad. Sci. 100, 2848 (2002) ADSCrossRefGoogle Scholar
  50. 50.
    X. Xu, W. Bosking, G. Sáry, J. Stefansic, D. Shima, V. Casagrande, J. Neurosci. 24, 6237 (2004) CrossRefGoogle Scholar
  51. 51.
    M.A. Dahlem, F.M. Schneider, E. Schöll, Chaos 18, 026110 (2008) ADSCrossRefMathSciNetGoogle Scholar
  52. 52.
    F.M. Schneider, E. Schöll, M.A. Dahlem, Chaos 19, 015110 (2009) ADSCrossRefMathSciNetGoogle Scholar
  53. 53.
    M.A. Dahlem, T.M. Isele, J. Math. Neurosci. 3, 7 (2013) CrossRefMathSciNetGoogle Scholar
  54. 54.
    F. Kneer, E. Schöll, M.A. Dahlem, New J. Phys. 16, 053010 (2014) ADSCrossRefGoogle Scholar
  55. 55.
    A.T. Winfree, When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias (Princeton University Press, Princeton, 1987) Google Scholar
  56. 56.
    S. Alonso, F. Sagués, A.S. Mikhailov, Science 299, 1722 (2003) ADSCrossRefGoogle Scholar
  57. 57.
    P. Dähmlow, S. Alonso, M. Bär, M.J.B. Hauser, Phys. Rev. Lett. 110, 234102 (2013) ADSCrossRefGoogle Scholar
  58. 58.
    A. Ahlborn, U. Parlitz, Phys. Rev. E 75, 65202 (2007) ADSCrossRefGoogle Scholar
  59. 59.
    Y.N. Kyrychko, K.B. Blyuss, S.J. Hogan, E. Schöll, Chaos 19, 043126 (2009) ADSCrossRefGoogle Scholar
  60. 60.
    S.V. Gurevich, R. Friedrich, Phys. Rev. Lett. 110, 014101 (2013) ADSCrossRefGoogle Scholar
  61. 61.
    M. Stich, C. Beta, Phys. Rev. E 88, 042910 (2013) ADSCrossRefGoogle Scholar
  62. 62.
    A. Hutt, M. Bestehorn, T. Wennekers, Netw. Comput. Neural Syst. 14, 351 (2003) CrossRefGoogle Scholar
  63. 63.
    L. Zhang, A. Hutt, J. Appl. Anal. Comput. 4, 1 (2014) zbMATHMathSciNetGoogle Scholar
  64. 64.
    R. FitzHugh, Biophys. J. 1, 445 (1961) ADSCrossRefGoogle Scholar
  65. 65.
    J. Nagumo, S. Arimoto, S. Yoshizawa., Proc. IRE 50, 2061 (1962) CrossRefGoogle Scholar
  66. 66.
    B. Lindner, J. García-Ojalvo, A.B. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004) ADSCrossRefGoogle Scholar
  67. 67.
    S. Kondo, T. Miura, Science 329, 1616 (2010) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  68. 68.
    M. Meixner, A. De Wit, S. Bose, E. Schöll, Phys. Rev. E 55, 6690 (1997) ADSCrossRefMathSciNetGoogle Scholar
  69. 69.
    C.A. Bachmair, Nonlocal control of pulse propagation in excitable media, Master’s thesis, TU Berlin, 2013 Google Scholar
  70. 70.
    R.V. Craster, R. Sassi, Technical Report 99, 1 (2006) Google Scholar
  71. 71.
    M. Stich, A.S. Mikhailov, Y. Kuramoto, Phys. Rev. E 79, 026110 (2009) ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für Theoretische Physik, Technische Universität BerlinBerlinGermany

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