Advertisement

Delay-induced chimeras in neural networks with fractal topology

  • Jakub SawickiEmail author
  • Iryna Omelchenko
  • Anna Zakharova
  • Eckehard Schöll
Regular Article
  • 41 Downloads

Abstract

We study chimera states, which are partial synchronization patterns consisting of spatially coexisting domains of coherent (synchronized) and incoherent (desynchronized) dynamics, in ring networks of FitzHugh-Nagumo oscillators with fractal connectivities. In particular, we focus on the interplay of time delay in the coupling term and the network topology. In the parameter plane of coupling strength and delay time we find tongue-like regions of existence of chimera states alternating with regions of coherent dynamics. We show analytically and numerically that the period of the synchronized dynamics as a function of delay is characterized by a sequence of piecewise linear branches. In between these branches various chimera states and other partial synchronization patterns are induced by the time delay. By varying the time delay one can deliberately choose and stabilize desired spatio-temporal patterns.

Graphical abstract

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    A. Pikovsky, M.G. Rosenblum, J. Kurths, Synchronization: a universal concept in nonlinear sciences (Cambridge University Press, Cambridge, 2001) Google Scholar
  2. 2.
    S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.U. Hwang, Phys. Rep. 424, 175 (2006) ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Stein, S. Luther, U. Parlitz, New J. Phys. 19, 063040 (2017) ADSCrossRefGoogle Scholar
  4. 4.
    M.J. Panaggio, D.M. Abrams, Nonlinearity 28, R67 (2015) ADSCrossRefGoogle Scholar
  5. 5.
    E. Schöll, Eur. Phys. J. Special Topics 225, 891 (2016) ADSCrossRefGoogle Scholar
  6. 6.
    Y. Kuramoto, D. Battogtokh, Nonlin. Phen. Complex Syst. 5, 380 (2002) Google Scholar
  7. 7.
    D.M. Abrams, S.H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004) ADSCrossRefGoogle Scholar
  8. 8.
    I. Omelchenko, Y. Maistrenko, P. Hövel, E. Schöll, Phys. Rev. Lett. 106, 234102 (2011) ADSCrossRefGoogle Scholar
  9. 9.
    N. Semenova, A. Zakharova, E. Schöll, V.S. Anishchenko, Europhys. Lett. 112, 40002 (2015) ADSCrossRefGoogle Scholar
  10. 10.
    T.E. Vadivasova, G. Strelkova, S.A. Bogomolov, V.S. Anishchenko, Chaos 26, 093108 (2016) ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    S. Bogomolov, A. Slepnev, G. Strelkova, E. Schöll, V.S. Anishchenko, Commun. Nonlinear Sci. Numer. Simul. 43, 25 (2017) ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Bukh, E. Rybalova, N. Semenova, G. Strelkova, V. Anishchenko, Chaos 27, 111102 (2017) ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    N. Semenova, G. Strelkova, V.S. Anishchenko, A. Zakharova, Chaos 27, 061102 (2017) ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    I. Omelchenko, B. Riemenschneider, P. Hövel, Y. Maistrenko, E. Schöll, Phys. Rev. E 85, 026212 (2012) ADSCrossRefGoogle Scholar
  15. 15.
    I. Omelchenko, O.E. Omel’chenko, P. Hövel, E. Schöll, Phys. Rev. Lett. 110, 224101 (2013) ADSCrossRefGoogle Scholar
  16. 16.
    J. Hizanidis, V. Kanas, A. Bezerianos, T. Bountis, Int. J. Bifurcat. Chaos 24, 1450030 (2014) CrossRefGoogle Scholar
  17. 17.
    A. Vüllings, J. Hizanidis, I. Omelchenko, P. Hövel, New J. Phys. 16, 123039 (2014) CrossRefGoogle Scholar
  18. 18.
    I. Omelchenko, A. Provata, J. Hizanidis, E. Schöll, P. Hövel, Phys. Rev. E 91, 022917 (2015) ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    N.D. Tsigkri-DeSmedt, J. Hizanidis, P. Hövel, A. Provata, Eur. Phys. J. Special Topics 225, 1149 (2016) ADSCrossRefGoogle Scholar
  20. 20.
    N.D. Tsigkri-DeSmedt, J. Hizanidis, E. Schöll, P. Hövel, A. Provata, Eur. Phys. J. B 90, 139 (2017) ADSCrossRefGoogle Scholar
  21. 21.
    T. Chouzouris, I. Omelchenko, A. Zakharova, J. Hlinka, P. Jiruska, E. Schöll, Chaos 28, 045112 (2018) ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    D.P. Rosin, D. Rontani, D.J. Gauthier, Phys. Rev. E 89, 042907 (2014) ADSCrossRefGoogle Scholar
  23. 23.
    A. Buscarino, M. Frasca, L.V. Gambuzza, P. Hövel, Phys. Rev. E 91, 022817 (2015) ADSCrossRefGoogle Scholar
  24. 24.
    J. Hizanidis, E. Panagakou, I. Omelchenko, E. Schöll, P. Hövel, A. Provata, Phys. Rev. E 92, 012915 (2015) ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    T. Banerjee, P.S. Dutta, A. Zakharova, E. Schöll, Phys. Rev. E 94, 032206 (2016) ADSCrossRefGoogle Scholar
  26. 26.
    V.M. Bastidas, I. Omelchenko, A. Zakharova, E. Schöll, T. Brandes, Phys. Rev. E 92, 062924 (2015) ADSCrossRefGoogle Scholar
  27. 27.
    O.E. Omel’chenko, M. Wolfrum, S. Yanchuk, Y. Maistrenko, O. Sudakov, Phys. Rev. E 85, 036210 (2012) ADSCrossRefGoogle Scholar
  28. 28.
    S. Shima, Y. Kuramoto, Phys. Rev. E 69, 036213 (2004) ADSCrossRefGoogle Scholar
  29. 29.
    Y. Maistrenko, O. Sudakov, O. Osiv, V.L. Maistrenko, New J. Phys. 17, 073037 (2015) ADSCrossRefGoogle Scholar
  30. 30.
    J. Xie, E. Knobloch, H.C. Kao, Phys. Rev. E 92, 042921 (2015) ADSCrossRefGoogle Scholar
  31. 31.
    J.F. Totz, J. Rode, M.R. Tinsley, K. Showalter, H. Engel, Nat. Phys. 14, 282 (2017) CrossRefGoogle Scholar
  32. 32.
    D. Brunner, B. Penkovsky, R. Levchenko, E. Schöll, L. Larger, Y. Maistrenko, Chaos 28, 103106 (2018) ADSCrossRefGoogle Scholar
  33. 33.
    S.W. Haugland, L. Schmidt, K. Krischer, Sci. Rep. 5, 9883 (2015) ADSCrossRefGoogle Scholar
  34. 34.
    G.C. Sethia, A. Sen, G.L. Johnston, Phys. Rev. E 88, 042917 (2013) ADSCrossRefGoogle Scholar
  35. 35.
    G.C. Sethia, A. Sen, Phys. Rev. Lett. 112, 144101 (2014) ADSCrossRefGoogle Scholar
  36. 36.
    A. Zakharova, M. Kapeller, E. Schöll, Phys. Rev. Lett. 112, 154101 (2014) ADSCrossRefGoogle Scholar
  37. 37.
    A. Zakharova, M. Kapeller, E. Schöll, J. Phys. Conf. Ser. 727, 012018 (2016) MathSciNetCrossRefGoogle Scholar
  38. 38.
    T. Banerjee, Europhys. Lett. 110, 60003 (2015) ADSCrossRefGoogle Scholar
  39. 39.
    F.P. Kemeth, S.W. Haugland, L. Schmidt, Y.G. Kevrekidis, K. Krischer, Chaos 26, 094815 (2016) ADSCrossRefGoogle Scholar
  40. 40.
    N.C. Rattenborg, C.J. Amlaner, S.L. Lima, Neurosci. Biobehav. Rev. 24, 817 (2000) CrossRefGoogle Scholar
  41. 41.
    N.C. Rattenborg, B. Voirin, S.M. Cruz, R. Tisdale, G. Dell’Omo, H.P. Lipp, M. Wikelski, A.L. Vyssotski, Nat. Commun. 7, 12468 (2016) ADSCrossRefGoogle Scholar
  42. 42.
    C.R. Laing, C.C. Chow, Neural Comput. 13, 1473 (2001) CrossRefGoogle Scholar
  43. 43.
    H. Sakaguchi, Phys. Rev. E 73, 031907 (2006) ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    A. Rothkegel, K. Lehnertz, New J. Phys. 16, 055006 (2014) ADSCrossRefGoogle Scholar
  45. 45.
    R.G. Andrzejak, C. Rummel, F. Mormann, K. Schindler, Sci. Rep. 6, 23000 (2016) ADSCrossRefGoogle Scholar
  46. 46.
    A.E. Motter, S.A. Myers, M. Anghel, T. Nishikawa, Nat. Phys. 9, 191 (2013) CrossRefGoogle Scholar
  47. 47.
    J.C. Gonzalez-Avella, M.G. Cosenza, M.S. Miguel, Physica A 399, 24 (2014) ADSCrossRefGoogle Scholar
  48. 48.
    A.M. Hagerstrom, T.E. Murphy, R. Roy, P. Hövel, I. Omelchenko, E. Schöll, Nat. Phys. 8, 658 (2012) CrossRefGoogle Scholar
  49. 49.
    M.R. Tinsley, S. Nkomo, K. Showalter, Nat. Phys. 8, 662 (2012) CrossRefGoogle Scholar
  50. 50.
    S. Nkomo, M.R. Tinsley, K. Showalter, Phys. Rev. Lett. 110, 244102 (2013) ADSCrossRefGoogle Scholar
  51. 51.
    E.A. Martens, S. Thutupalli, A. Fourriere, O. Hallatschek, Proc. Natl. Acad. Sci. U.S.A. 110, 10563 (2013) ADSCrossRefGoogle Scholar
  52. 52.
    T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski, Y. Maistrenko, Sci. Rep. 4, 6379 (2014) ADSCrossRefGoogle Scholar
  53. 53.
    L. Larger, B. Penkovsky, Y. Maistrenko, Phys. Rev. Lett. 111, 054103 (2013) ADSCrossRefGoogle Scholar
  54. 54.
    L.V. Gambuzza, A. Buscarino, S. Chessari, L. Fortuna, R. Meucci, M. Frasca, Phys. Rev. E 90, 032905 (2014) ADSCrossRefGoogle Scholar
  55. 55.
    L. Larger, B. Penkovsky, Y. Maistrenko, Nat. Commun. 6, 7752 (2015) ADSCrossRefGoogle Scholar
  56. 56.
    M. Wickramasinghe, I.Z. Kiss, PLoS ONE 8, e80586 (2013) ADSCrossRefGoogle Scholar
  57. 57.
    L. Schmidt, K. Schönleber, K. Krischer, V. García-Morales, Chaos 24, 013102 (2014) ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    E.A. Viktorov, T. Habruseva, S.P. Hegarty, G. Huyet, B. Kelleher, Phys. Rev. Lett. 112, 224101 (2014) ADSCrossRefGoogle Scholar
  59. 59.
    S. Loos, J.C. Claussen, E. Schöll, A. Zakharova, Phys. Rev. E 93, 012209 (2016) ADSCrossRefGoogle Scholar
  60. 60.
    V. Semenov, A. Zakharova, Y. Maistrenko, E. Schöll, Europhys. Lett. 115, 10005 (2016) ADSCrossRefGoogle Scholar
  61. 61.
    N. Semenova, A. Zakharova, V.S. Anishchenko, E. Schöll, Phys. Rev. Lett. 117, 014102 (2016) ADSCrossRefGoogle Scholar
  62. 62.
    A. Zakharova, N. Semenova, V.S. Anishchenko, E. Schöll, Patterns of Dynamics, in Springer Proceedings in Mathematics & Statistics, edited by P. Gurevich, J. Hell, B. Sandstede (Springer, 2018), Vol. 205, p. 44 Google Scholar
  63. 63.
    A. Yeldesbay, A. Pikovsky, M. Rosenblum, Phys. Rev. Lett. 112, 144103 (2014) ADSCrossRefGoogle Scholar
  64. 64.
    F. Böhm, A. Zakharova, E. Schöll, K. Lüdge, Phys. Rev. E 91, 040901 (R) (2015) CrossRefGoogle Scholar
  65. 65.
    L. Schmidt, K. Krischer, Phys. Rev. Lett. 114, 034101 (2015) ADSCrossRefGoogle Scholar
  66. 66.
    L. Schmidt, K. Krischer, Chaos 25, 064401 (2015) ADSCrossRefGoogle Scholar
  67. 67.
    P. Katsaloulis, D.A. Verganelakis, A. Provata, Fractals 17, 181 (2009) CrossRefGoogle Scholar
  68. 68.
    P. Expert, T.S. Evans, V.D. Blondel, R. Lambiotte, Proc. Natl. Acad. Sci. U.S.A. 108, 7663 (2011) ADSCrossRefGoogle Scholar
  69. 69.
    P. Katsaloulis, A. Ghosh, A.C. Philippe, A. Provata, R. Deriche, Eur. Phys. J. B 85, 1 (2012) CrossRefGoogle Scholar
  70. 70.
    P. Katsaloulis, J. Hizanidis, D.A. Verganelakis, A. Provata, Fluct. Noise Lett. 11, 1250032 (2012) CrossRefGoogle Scholar
  71. 71.
    A. Provata, P. Katsaloulis, D.A. Verganelakis, Chaos S 45, 174 (2012) CrossRefGoogle Scholar
  72. 72.
    S. Ulonska, I. Omelchenko, A. Zakharova, E. Schöll, Chaos 26, 094825 (2016) ADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    A. zur Bonsen, I. Omelchenko, A. Zakharova, E. Schöll, Eur. Phys. J. B 91, 65 (2018) ADSCrossRefGoogle Scholar
  74. 74.
    J. Sieber, O.E. Omel’chenko, M. Wolfrum, Phys. Rev. Lett. 112, 054102 (2014) ADSCrossRefGoogle Scholar
  75. 75.
    C. Bick, E.A. Martens, New J. Phys. 17, 033030 (2015) ADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    I. Omelchenko, O.E. Omel’chenko, A. Zakharova, M. Wolfrum, E. Schöll, Phys. Rev. Lett. 116, 114101 (2016) ADSCrossRefGoogle Scholar
  77. 77.
    I. Omelchenko, O.E. Omel’chenko, A. Zakharova, E. Schöll, Phys. Rev. E 97, 012216 (2018) ADSCrossRefGoogle Scholar
  78. 78.
    A. Ahlborn, U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004) ADSCrossRefGoogle Scholar
  79. 79.
    V. Flunkert, I. Fischer, E. Schöll, Philo. Trans. R. Soc. A 371, 20120465 (2013) ADSCrossRefGoogle Scholar
  80. 80.
    A. Saha, U. Feudel, Phys. Rev. E 95, 062219 (2017) ADSMathSciNetCrossRefGoogle Scholar
  81. 81.
    A. Gjurchinovski, E. Schöll, A. Zakharova, Phys. Rev. E 95, 042218 (2017) ADSMathSciNetCrossRefGoogle Scholar
  82. 82.
    A. Zakharova, N. Semenova, V.S. Anishchenko, E. Schöll, Chaos 27, 114320 (2017) ADSMathSciNetCrossRefGoogle Scholar
  83. 83.
    J. Sawicki, I. Omelchenko, A. Zakharova, E. Schöll, Eur. Phys. J. Special Topics 226, 1883 (2017) ADSCrossRefGoogle Scholar
  84. 84.
    R. FitzHugh, Biophys. J. 1, 445 (1961) ADSCrossRefGoogle Scholar
  85. 85.
    J. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IRE 50, 2061 (1962) CrossRefGoogle Scholar
  86. 86.
    D.P. Rosin, K.E. Callan, D.J. Gauthier, E. Schöll, Europhys. Lett. 96, 34001 (2011) ADSCrossRefGoogle Scholar
  87. 87.
    M. Heinrich, T. Dahms, V. Flunkert, S.W. Teitsworth, E. Schöll, New J. Phys. 12, 113030 (2010) ADSCrossRefGoogle Scholar
  88. 88.
    B.B. Mandelbrot, The fractal geometry of nature, 3rd ed. (W.H. Freeman and Comp., New York, 1983) Google Scholar
  89. 89.
    J. Feder, Fractals (Plenum Press, New York, 1988) Google Scholar
  90. 90.
    I. Omelchenko, A. Zakharova, P. Hövel, J. Siebert, E. Schöll, Chaos 25, 083104 (2015) ADSMathSciNetCrossRefGoogle Scholar
  91. 91.
    D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998) ADSCrossRefGoogle Scholar
  92. 92.
    C.A. Bachmair, E. Schöll, Eur. Phys. J. B 87, 276 (2014) ADSCrossRefGoogle Scholar
  93. 93.
    P. Hövel, E. Schöll, Phys. Rev. E 72, 046203 (2005) CrossRefGoogle Scholar
  94. 94.
    S. Yanchuk, M. Wolfrum, P. Hövel, E. Schöll, Phys. Rev. E 74, 026201 (2006) ADSMathSciNetCrossRefGoogle Scholar
  95. 95.
    S.A. Brandstetter, M.A. Dahlem, E. Schöll, Philos. Trans. Roy. Soc. A 368, 391 (2010) ADSCrossRefGoogle Scholar
  96. 96.
    S. Yanchuk, P. Perlikowski, Phys. Rev. E 79, 046221 (2009) ADSMathSciNetCrossRefGoogle Scholar
  97. 97.
    W. Just, T. Bernard, M. Ostheimer, E. Reibold, H. Benner, Phys. Rev. Lett. 78, 203 (1997) ADSCrossRefGoogle Scholar
  98. 98.
    W. Just, E. Reibold, K. Kacperski, P. Fronczak, J.A. Holyst, H. Benner, Phys. Rev. E 61, 5045 (2000) ADSCrossRefGoogle Scholar
  99. 99.
    W. Just, H. Benner, C.V. Loewenich, Physica D 199, 33 (2004) ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Jakub Sawicki
    • 1
    Email author
  • Iryna Omelchenko
    • 1
  • Anna Zakharova
    • 1
  • Eckehard Schöll
    • 1
  1. 1.Institut für Theoretische Physik, Technische Universität BerlinBerlinGermany

Personalised recommendations