Delay-induced chimeras in neural networks with fractal topology

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

This is a preview of subscription content, log in to check access.

References

  1. 1.

    A. Pikovsky, M.G. Rosenblum, J. Kurths, Synchronization: a universal concept in nonlinear sciences (Cambridge University Press, Cambridge, 2001)

  2. 2.

    S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.U. Hwang, Phys. Rep. 424, 175 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    S. Stein, S. Luther, U. Parlitz, New J. Phys. 19, 063040 (2017)

    ADS  Article  Google Scholar 

  4. 4.

    M.J. Panaggio, D.M. Abrams, Nonlinearity 28, R67 (2015)

    ADS  Article  Google Scholar 

  5. 5.

    E. Schöll, Eur. Phys. J. Special Topics 225, 891 (2016)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  8. 8.

    I. Omelchenko, Y. Maistrenko, P. Hövel, E. Schöll, Phys. Rev. Lett. 106, 234102 (2011)

    ADS  Article  Google Scholar 

  9. 9.

    N. Semenova, A. Zakharova, E. Schöll, V.S. Anishchenko, Europhys. Lett. 112, 40002 (2015)

    ADS  Article  Google Scholar 

  10. 10.

    T.E. Vadivasova, G. Strelkova, S.A. Bogomolov, V.S. Anishchenko, Chaos 26, 093108 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    S. Bogomolov, A. Slepnev, G. Strelkova, E. Schöll, V.S. Anishchenko, Commun. Nonlinear Sci. Numer. Simul. 43, 25 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    A. Bukh, E. Rybalova, N. Semenova, G. Strelkova, V. Anishchenko, Chaos 27, 111102 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    N. Semenova, G. Strelkova, V.S. Anishchenko, A. Zakharova, Chaos 27, 061102 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    I. Omelchenko, B. Riemenschneider, P. Hövel, Y. Maistrenko, E. Schöll, Phys. Rev. E 85, 026212 (2012)

    ADS  Article  Google Scholar 

  15. 15.

    I. Omelchenko, O.E. Omel’chenko, P. Hövel, E. Schöll, Phys. Rev. Lett. 110, 224101 (2013)

    ADS  Article  Google Scholar 

  16. 16.

    J. Hizanidis, V. Kanas, A. Bezerianos, T. Bountis, Int. J. Bifurcat. Chaos 24, 1450030 (2014)

    Article  Google Scholar 

  17. 17.

    A. Vüllings, J. Hizanidis, I. Omelchenko, P. Hövel, New J. Phys. 16, 123039 (2014)

    Article  Google Scholar 

  18. 18.

    I. Omelchenko, A. Provata, J. Hizanidis, E. Schöll, P. Hövel, Phys. Rev. E 91, 022917 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    N.D. Tsigkri-DeSmedt, J. Hizanidis, P. Hövel, A. Provata, Eur. Phys. J. Special Topics 225, 1149 (2016)

    ADS  Article  Google Scholar 

  20. 20.

    N.D. Tsigkri-DeSmedt, J. Hizanidis, E. Schöll, P. Hövel, A. Provata, Eur. Phys. J. B 90, 139 (2017)

    ADS  Article  Google Scholar 

  21. 21.

    T. Chouzouris, I. Omelchenko, A. Zakharova, J. Hlinka, P. Jiruska, E. Schöll, Chaos 28, 045112 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    D.P. Rosin, D. Rontani, D.J. Gauthier, Phys. Rev. E 89, 042907 (2014)

    ADS  Article  Google Scholar 

  23. 23.

    A. Buscarino, M. Frasca, L.V. Gambuzza, P. Hövel, Phys. Rev. E 91, 022817 (2015)

    ADS  Article  Google Scholar 

  24. 24.

    J. Hizanidis, E. Panagakou, I. Omelchenko, E. Schöll, P. Hövel, A. Provata, Phys. Rev. E 92, 012915 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    T. Banerjee, P.S. Dutta, A. Zakharova, E. Schöll, Phys. Rev. E 94, 032206 (2016)

    ADS  Article  Google Scholar 

  26. 26.

    V.M. Bastidas, I. Omelchenko, A. Zakharova, E. Schöll, T. Brandes, Phys. Rev. E 92, 062924 (2015)

    ADS  Article  Google Scholar 

  27. 27.

    O.E. Omel’chenko, M. Wolfrum, S. Yanchuk, Y. Maistrenko, O. Sudakov, Phys. Rev. E 85, 036210 (2012)

    ADS  Article  Google Scholar 

  28. 28.

    S. Shima, Y. Kuramoto, Phys. Rev. E 69, 036213 (2004)

    ADS  Article  Google Scholar 

  29. 29.

    Y. Maistrenko, O. Sudakov, O. Osiv, V.L. Maistrenko, New J. Phys. 17, 073037 (2015)

    ADS  Article  Google Scholar 

  30. 30.

    J. Xie, E. Knobloch, H.C. Kao, Phys. Rev. E 92, 042921 (2015)

    ADS  Article  Google Scholar 

  31. 31.

    J.F. Totz, J. Rode, M.R. Tinsley, K. Showalter, H. Engel, Nat. Phys. 14, 282 (2017)

    Article  Google Scholar 

  32. 32.

    D. Brunner, B. Penkovsky, R. Levchenko, E. Schöll, L. Larger, Y. Maistrenko, Chaos 28, 103106 (2018)

    ADS  Article  Google Scholar 

  33. 33.

    S.W. Haugland, L. Schmidt, K. Krischer, Sci. Rep. 5, 9883 (2015)

    ADS  Article  Google Scholar 

  34. 34.

    G.C. Sethia, A. Sen, G.L. Johnston, Phys. Rev. E 88, 042917 (2013)

    ADS  Article  Google Scholar 

  35. 35.

    G.C. Sethia, A. Sen, Phys. Rev. Lett. 112, 144101 (2014)

    ADS  Article  Google Scholar 

  36. 36.

    A. Zakharova, M. Kapeller, E. Schöll, Phys. Rev. Lett. 112, 154101 (2014)

    ADS  Article  Google Scholar 

  37. 37.

    A. Zakharova, M. Kapeller, E. Schöll, J. Phys. Conf. Ser. 727, 012018 (2016)

    MathSciNet  Article  Google Scholar 

  38. 38.

    T. Banerjee, Europhys. Lett. 110, 60003 (2015)

    ADS  Article  Google Scholar 

  39. 39.

    F.P. Kemeth, S.W. Haugland, L. Schmidt, Y.G. Kevrekidis, K. Krischer, Chaos 26, 094815 (2016)

    ADS  Article  Google Scholar 

  40. 40.

    N.C. Rattenborg, C.J. Amlaner, S.L. Lima, Neurosci. Biobehav. Rev. 24, 817 (2000)

    Article  Google 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)

    ADS  Article  Google Scholar 

  42. 42.

    C.R. Laing, C.C. Chow, Neural Comput. 13, 1473 (2001)

    Article  Google Scholar 

  43. 43.

    H. Sakaguchi, Phys. Rev. E 73, 031907 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  44. 44.

    A. Rothkegel, K. Lehnertz, New J. Phys. 16, 055006 (2014)

    ADS  Article  Google Scholar 

  45. 45.

    R.G. Andrzejak, C. Rummel, F. Mormann, K. Schindler, Sci. Rep. 6, 23000 (2016)

    ADS  Article  Google Scholar 

  46. 46.

    A.E. Motter, S.A. Myers, M. Anghel, T. Nishikawa, Nat. Phys. 9, 191 (2013)

    Article  Google Scholar 

  47. 47.

    J.C. Gonzalez-Avella, M.G. Cosenza, M.S. Miguel, Physica A 399, 24 (2014)

    ADS  Article  Google Scholar 

  48. 48.

    A.M. Hagerstrom, T.E. Murphy, R. Roy, P. Hövel, I. Omelchenko, E. Schöll, Nat. Phys. 8, 658 (2012)

    Article  Google Scholar 

  49. 49.

    M.R. Tinsley, S. Nkomo, K. Showalter, Nat. Phys. 8, 662 (2012)

    Article  Google Scholar 

  50. 50.

    S. Nkomo, M.R. Tinsley, K. Showalter, Phys. Rev. Lett. 110, 244102 (2013)

    ADS  Article  Google Scholar 

  51. 51.

    E.A. Martens, S. Thutupalli, A. Fourriere, O. Hallatschek, Proc. Natl. Acad. Sci. U.S.A. 110, 10563 (2013)

    ADS  Article  Google Scholar 

  52. 52.

    T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski, Y. Maistrenko, Sci. Rep. 4, 6379 (2014)

    ADS  Article  Google Scholar 

  53. 53.

    L. Larger, B. Penkovsky, Y. Maistrenko, Phys. Rev. Lett. 111, 054103 (2013)

    ADS  Article  Google Scholar 

  54. 54.

    L.V. Gambuzza, A. Buscarino, S. Chessari, L. Fortuna, R. Meucci, M. Frasca, Phys. Rev. E 90, 032905 (2014)

    ADS  Article  Google Scholar 

  55. 55.

    L. Larger, B. Penkovsky, Y. Maistrenko, Nat. Commun. 6, 7752 (2015)

    ADS  Article  Google Scholar 

  56. 56.

    M. Wickramasinghe, I.Z. Kiss, PLoS ONE 8, e80586 (2013)

    ADS  Article  Google Scholar 

  57. 57.

    L. Schmidt, K. Schönleber, K. Krischer, V. García-Morales, Chaos 24, 013102 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  58. 58.

    E.A. Viktorov, T. Habruseva, S.P. Hegarty, G. Huyet, B. Kelleher, Phys. Rev. Lett. 112, 224101 (2014)

    ADS  Article  Google Scholar 

  59. 59.

    S. Loos, J.C. Claussen, E. Schöll, A. Zakharova, Phys. Rev. E 93, 012209 (2016)

    ADS  Article  Google Scholar 

  60. 60.

    V. Semenov, A. Zakharova, Y. Maistrenko, E. Schöll, Europhys. Lett. 115, 10005 (2016)

    ADS  Article  Google Scholar 

  61. 61.

    N. Semenova, A. Zakharova, V.S. Anishchenko, E. Schöll, Phys. Rev. Lett. 117, 014102 (2016)

    ADS  Article  Google 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

  63. 63.

    A. Yeldesbay, A. Pikovsky, M. Rosenblum, Phys. Rev. Lett. 112, 144103 (2014)

    ADS  Article  Google Scholar 

  64. 64.

    F. Böhm, A. Zakharova, E. Schöll, K. Lüdge, Phys. Rev. E 91, 040901 (R) (2015)

    Article  Google Scholar 

  65. 65.

    L. Schmidt, K. Krischer, Phys. Rev. Lett. 114, 034101 (2015)

    ADS  Article  Google Scholar 

  66. 66.

    L. Schmidt, K. Krischer, Chaos 25, 064401 (2015)

    ADS  Article  Google Scholar 

  67. 67.

    P. Katsaloulis, D.A. Verganelakis, A. Provata, Fractals 17, 181 (2009)

    Article  Google Scholar 

  68. 68.

    P. Expert, T.S. Evans, V.D. Blondel, R. Lambiotte, Proc. Natl. Acad. Sci. U.S.A. 108, 7663 (2011)

    ADS  Article  Google Scholar 

  69. 69.

    P. Katsaloulis, A. Ghosh, A.C. Philippe, A. Provata, R. Deriche, Eur. Phys. J. B 85, 1 (2012)

    Article  Google Scholar 

  70. 70.

    P. Katsaloulis, J. Hizanidis, D.A. Verganelakis, A. Provata, Fluct. Noise Lett. 11, 1250032 (2012)

    Article  Google Scholar 

  71. 71.

    A. Provata, P. Katsaloulis, D.A. Verganelakis, Chaos S 45, 174 (2012)

    Article  Google Scholar 

  72. 72.

    S. Ulonska, I. Omelchenko, A. Zakharova, E. Schöll, Chaos 26, 094825 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  73. 73.

    A. zur Bonsen, I. Omelchenko, A. Zakharova, E. Schöll, Eur. Phys. J. B 91, 65 (2018)

    ADS  Article  Google Scholar 

  74. 74.

    J. Sieber, O.E. Omel’chenko, M. Wolfrum, Phys. Rev. Lett. 112, 054102 (2014)

    ADS  Article  Google Scholar 

  75. 75.

    C. Bick, E.A. Martens, New J. Phys. 17, 033030 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  76. 76.

    I. Omelchenko, O.E. Omel’chenko, A. Zakharova, M. Wolfrum, E. Schöll, Phys. Rev. Lett. 116, 114101 (2016)

    ADS  Article  Google Scholar 

  77. 77.

    I. Omelchenko, O.E. Omel’chenko, A. Zakharova, E. Schöll, Phys. Rev. E 97, 012216 (2018)

    ADS  Article  Google Scholar 

  78. 78.

    A. Ahlborn, U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004)

    ADS  Article  Google Scholar 

  79. 79.

    V. Flunkert, I. Fischer, E. Schöll, Philo. Trans. R. Soc. A 371, 20120465 (2013)

    ADS  Article  Google Scholar 

  80. 80.

    A. Saha, U. Feudel, Phys. Rev. E 95, 062219 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  81. 81.

    A. Gjurchinovski, E. Schöll, A. Zakharova, Phys. Rev. E 95, 042218 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  82. 82.

    A. Zakharova, N. Semenova, V.S. Anishchenko, E. Schöll, Chaos 27, 114320 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  83. 83.

    J. Sawicki, I. Omelchenko, A. Zakharova, E. Schöll, Eur. Phys. J. Special Topics 226, 1883 (2017)

    ADS  Article  Google Scholar 

  84. 84.

    R. FitzHugh, Biophys. J. 1, 445 (1961)

    ADS  Article  Google Scholar 

  85. 85.

    J. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IRE 50, 2061 (1962)

    Article  Google Scholar 

  86. 86.

    D.P. Rosin, K.E. Callan, D.J. Gauthier, E. Schöll, Europhys. Lett. 96, 34001 (2011)

    ADS  Article  Google Scholar 

  87. 87.

    M. Heinrich, T. Dahms, V. Flunkert, S.W. Teitsworth, E. Schöll, New J. Phys. 12, 113030 (2010)

    ADS  Article  Google Scholar 

  88. 88.

    B.B. Mandelbrot, The fractal geometry of nature, 3rd ed. (W.H. Freeman and Comp., New York, 1983)

  89. 89.

    J. Feder, Fractals (Plenum Press, New York, 1988)

  90. 90.

    I. Omelchenko, A. Zakharova, P. Hövel, J. Siebert, E. Schöll, Chaos 25, 083104 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  91. 91.

    D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)

    ADS  Article  Google Scholar 

  92. 92.

    C.A. Bachmair, E. Schöll, Eur. Phys. J. B 87, 276 (2014)

    ADS  Article  Google Scholar 

  93. 93.

    P. Hövel, E. Schöll, Phys. Rev. E 72, 046203 (2005)

    Article  Google Scholar 

  94. 94.

    S. Yanchuk, M. Wolfrum, P. Hövel, E. Schöll, Phys. Rev. E 74, 026201 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  95. 95.

    S.A. Brandstetter, M.A. Dahlem, E. Schöll, Philos. Trans. Roy. Soc. A 368, 391 (2010)

    ADS  Article  Google Scholar 

  96. 96.

    S. Yanchuk, P. Perlikowski, Phys. Rev. E 79, 046221 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  97. 97.

    W. Just, T. Bernard, M. Ostheimer, E. Reibold, H. Benner, Phys. Rev. Lett. 78, 203 (1997)

    ADS  Article  Google Scholar 

  98. 98.

    W. Just, E. Reibold, K. Kacperski, P. Fronczak, J.A. Holyst, H. Benner, Phys. Rev. E 61, 5045 (2000)

    ADS  Article  Google Scholar 

  99. 99.

    W. Just, H. Benner, C.V. Loewenich, Physica D 199, 33 (2004)

    ADS  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jakub Sawicki.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sawicki, J., Omelchenko, I., Zakharova, A. et al. Delay-induced chimeras in neural networks with fractal topology. Eur. Phys. J. B 92, 54 (2019). https://doi.org/10.1140/epjb/e2019-90309-6

Download citation

Keywords

  • Statistical and Nonlinear Physics