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Control of coherence in excitable systems by the interplay of noise and time-delay

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Abstract.

The control of coherence and spectral properties of noise-induced oscillations by time delayed feedback is studied in a FitzHugh-Nagumo system and analyzed by reduced non-Markovian models. A two-state approach is considered as an abstract simplification of this excitable system. Rest and excited states are characterized by different waiting time distributions. This non-Markovian approach allows one to predict quantitatively the increase of coherence measured by the correlation time, and the modulation of the main frequencies of the stochastic dynamics in dependence on the delay time below Hopf bifurcations. Beyond the Hopf bifurcation, bulk oscillations of an ensemble of excitable two-state rotators emerge in the onset of coherent activation in case of delayed mean field coupling of the ensemble.

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References

  1. E. Schöll, H.G. Schuster (Eds.), Handbook of Chaos Control (Wiley-VCH, Weinheim, 2008)

  2. K. Pyragas, Phys. Lett. A 170, 421 (1992)

    Article  ADS  Google Scholar 

  3. K. Pyragas, Phil. Trans. R. Soc. A 364, 2309 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. L. Schimansky-Geier, B. Fiedler, J. Kurths, E. Schöll (Eds.), Analysis and control of complex nonlinear processes in physics, chemistry and biology (World Scientific, Singapore, 2007)

  5. E. Schöll, in Nonlinear Dynamics of Nanosystems, edited by G. Radons, B. Rumpf, H.G. Schuster (Wiley-VCH, Weinheim, 2009)

  6. E. Schöll, P. Hövel, V. Flunkert, M.A. Dahlem, in Complex Time-Delay Systems: Theory and Applications, edited by F.M. Atay (Springer, Berlin, 2010)

  7. W. Just, A. Pelster, M. Schanz, E. Schöll (eds.), Delayed Complex Systems, Theme Issue of Phil. Trans. R. Soc. A 368, 301 (2010)

  8. I.Z. Kiss, C.G. Rusin, H. Kori, J.L. Hudson, Science 316, 1886 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  9. H. Kori, C.G. Rusin, I.Z. Kiss, J.L. Hudson, CHAOS 18, 026111 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  10. D. Goldobin, M. Rosenblum, A. Pikovsky, Phys. Rev. E 67, 061119 (2003)

    Article  ADS  Google Scholar 

  11. S.A. Campbell, M. Waite, Nonlin. Anal. 47, 1093 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Hasegawa, Phys. Rev. E 70, 021912 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  13. M. Sainz-Trapága, C. Masoller, H.A. Braun, M.T. Huber, Phys. Rev. E 70, 031904 (2004)

    Article  ADS  Google Scholar 

  14. L.S. Tsimring, A. Pikovsky, Phys. Rev. Lett. 87, 250602 (2001)

    Article  ADS  Google Scholar 

  15. T. Piwonski, J. Houlihan, T. Busch, G. Huyet, Phys. Rev. Lett. 95, 040601 (2005)

    Article  ADS  Google Scholar 

  16. N.B. Janson, A.G. Balanov, E. Schöll, Phys. Rev. Lett. 93, 010601 (2004)

    Article  ADS  Google Scholar 

  17. A.G. Balanov, N.B. Janson, E. Schöll, Physica D 199, 1 (2004)

    Article  MATH  ADS  Google Scholar 

  18. E. Schöll, A.G. Balanov, N.B. Janson, A. Neiman, Stoch. Dyn. 5, 281 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Pomplun, A. Amann, E. Schöll, Europhys. Lett. 71, 366 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  20. J. Pomplun, A.G. Balanov, E. Schöll, Phys. Rev. E 75, 040101 (2007)

    Article  ADS  Google Scholar 

  21. V. Flunkert, E. Schöll, Phys. Rev. E 76, 066202 (2007)

    Article  ADS  Google Scholar 

  22. R. Aust, P. Hövel, J. Hizanidis, E. Schöll, Eur. Phys. J. Special Topics 187, 77 (2010)

    Article  ADS  Google Scholar 

  23. G. Stegemann, A.G. Balanov, E. Schöll, Phys. Rev. E 73, 016203 (2006)

    Article  ADS  Google Scholar 

  24. J. Hizanidis, E. Schöll, Phys. Rev. E 78, 066205 (2008)

    Article  ADS  Google Scholar 

  25. N. Majer, E. Schöll, Phys. Rev. E 79, 011109 (2009)

    Article  ADS  Google Scholar 

  26. B. Lindner, J. García-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)

    Article  ADS  Google Scholar 

  27. T. Prager, H.P. Lerch, L. Schimansky-Geier, E. Schöll, J. Phys. A 40, 11045 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. N. Kouvaris, F. Müller, L. Schimansky-Geier, Phys. Rev. E 82, 061124 (2010)

    Article  ADS  Google Scholar 

  29. A. Pototsky, N.B. Janson, Phys. Rev. E 77, 031113 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  30. S.A. Brandstetter, M.A. Dahlem, E. Schöll, Phil. Trans. R. Soc. A 368, 391 (2010)

    Article  MATH  ADS  Google Scholar 

  31. B. Hauschildt, N.B. Janson, A.G. Balanov, E. Schöll, Phys. Rev. E 74, 051906 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  32. P. Hövel, M.A. Dahlem, E. Schöll, Int. J. Bifur. Chaos 20, 813 (2010)

    Article  MATH  Google Scholar 

  33. P. Hövel, S.A. Shah, M.A. Dahlem, E. Schöll, Feedback-dependent control of stochastic synchronization in coupled neural systems, in From Physics to Control through an Emergent View, edited by L. Fortuna, A.L. Fradkov, M. Frasca (World Scientific, Singapore, 2010), Vol. 15 of Nonlinear Science Seris B, p. 35

  34. E.M. Izhikevich, Int. J. Bifur. Chaos 10, 1171 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. E. Schöll, G. Hiller, P. Hövel, M.A. Dahlem, Phil. Trans. R. Soc. A 367, 1079 (2009)

    Article  MATH  ADS  Google Scholar 

  36. F.M. Schneider, E. Schöll, M.A. Dahlem, Chaos 19, 015110 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  37. 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)

    Article  MATH  ADS  Google Scholar 

  38. S.J. Schiff, K. Jerger, D.H. Duong, T. Chang, M.L. Spano, W.L. Ditto, Nature (London) 370, 615 (1994)

    Article  ADS  Google Scholar 

  39. K.A. Richardson, S.J. Schiff, B.J. Gluckman, Phys. Rev. Lett. 94, 028103 (2005)

    Article  ADS  Google Scholar 

  40. M.G. Rosenblum, A.S. Pikovsky, Phys. Rev. Lett. 92, 114102 (2004)

    Article  ADS  Google Scholar 

  41. O.V. Popovych, C. Hauptmann, P.A. Tass, Phys. Rev. Lett. 94, 164102 (2005)

    Article  ADS  Google Scholar 

  42. M. Gassel, E. Glatt, F. Kaiser, Fluct. Noise Lett. 7, L225 (2007)

    Article  Google Scholar 

  43. V.S. Anishchenko, V. Astakhov, A. Neiman, T. Vadivasova, L. Schimansky–Geier, in Nonlinear Dynamics of Chaotic and Stochastic Systems (Springer, Berlin, 2007)

  44. G. Hu, T. Ditzinger, C.Z. Ning, H. Haken, Phys. Rev. Lett. 71, 807 (1993)

    Article  ADS  Google Scholar 

  45. A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 775 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. A. Longtin, Phys. Rev. E 55, 868 (1997)

    Article  ADS  Google Scholar 

  47. B. Lindner, L. Schimansky-Geier, Phys. Rev. E 61, 6103 (2000)

    Article  ADS  Google Scholar 

  48. J. García-Ojalvo, J.M. Sancho, Noise in Spatially Extended Systems (Springer, New York, 1999)

  49. C. Masoller, Phys. Rev. Lett. 88, 034102 (2002)

    Article  ADS  Google Scholar 

  50. J. Hizanidis, A.G. Balanov, A. Amann, E. Schöll, Phys. Rev. Lett. 96, 244104 (2006)

    Article  ADS  Google Scholar 

  51. F. Sagués, J.M. Sancho, J. García-Ojalvo, Rev. Mod. Phys. 79, 829 (2007)

    Article  ADS  Google Scholar 

  52. B. Lindner, L. Schimansky-Geier, Phys. Rev. E 60, 7270 (1999)

    Article  ADS  Google Scholar 

  53. V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, E. Schöll, Phys. Rev. E 79, 065201 (2009)

    Article  ADS  Google Scholar 

  54. C.-U. Choe, T. Dahms, P. Hvel, E. Schöll, Phys. Rev. E 81, 025205 (2010)

    Article  ADS  Google Scholar 

  55. M. Falcke, Adv. Phys. 53, 255 (2004)

    Article  ADS  Google Scholar 

  56. L. Meinhold, L. Schimansky-Geier, Phys. Rev. E 66, 050901 (2002)

    Article  ADS  Google Scholar 

  57. S. Rüdiger, L. Schimansky-Geier, J. Theor. Biol. 259, 96 (2009)

    Article  Google Scholar 

  58. T. Prager, B. Naundorf, L. Schimansky-Geier, Physica A 325, 176 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  59. T. Prager, L. Schimansky-Geier, Phys. Rev. Lett. 91, 230601 (2003)

    Article  ADS  Google Scholar 

  60. T Prager, L. Schimansky-Geier, Phys. Rev. E 71, 031112 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  61. T. Prager, M. Falcke, L. Schimansky-Geier, M.A. Zaks, Phys. Rev. E 76, 011118 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  62. K. Wood, C. Van De Broeck, R. Kawai, K. Lindenberg, Phys. Rev. E 74, 031113 (2006)

    Article  ADS  Google Scholar 

  63. T. Schwalger, B. Lindner, Phys. Rev. E 78, 210603 (2008)

    Article  Google Scholar 

  64. T. Ohira, T. Yamane, Phys. Rev. E 61, 1247 (2000)

    Article  ADS  Google Scholar 

  65. D. Huber, L.S. Tsimring, Phys. Rev. Lett. 91, 260601 (2003)

    Article  ADS  Google Scholar 

  66. D. Huber, L.S. Tsimring, Phys. Rev. E 71, 036150 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  67. M. Kimizuka, T. Munakata, Phys. Rev. E 80, 021139 (2009)

    Article  ADS  Google Scholar 

  68. M.K.S. Yeung, S.H. Strogatz, Phys. Rev. Lett. 82, 648 (1999)

    Article  ADS  Google Scholar 

  69. S. Trimper, K. Zabrocki, Phys. Lett. A 321, 205 (2004)

    Article  MATH  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  72. R.L. Stratonovich, Topics in the Theory of Random Noise I (Gordon and Breach, 1962)

  73. V.M. Kenkre, E.W. Montroll, M.F. Schlesinger, J. Stat. Phys. 9, 45 (1973)

    Article  ADS  Google Scholar 

  74. D.T. Gillespie, Phys. Lett. 64A, 22 (1977)

    MathSciNet  ADS  Google Scholar 

  75. P. Jung, U. Behn, E. Pantazelou, F. Moss, Phys. Rev. A 46, R1709 (1992)

    Article  ADS  Google Scholar 

  76. M. Borkovec, P. Hänggi, P. Talkner, Rev. Mod. Phys. 62, 251 (1990)

    Article  ADS  Google Scholar 

  77. G.T. Gurija, M.A. Livshits, Z. Phys. B - Condensed Matter 47, 71 (1982)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  79. M. Wolfrum, S. Yanchuk, P. Hövel, E. Schöll, Eur. Phys. J. Special Topics 191, 91 (2010)

    Google Scholar 

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Authors and Affiliations

  1. Humboldt-Universität zu Berlin, Institut für Physik, Newtonstr. 15, 12489, Berlin, Germany

    N. Kouvaris & L. Schimansky-Geier

  2. Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, 15310, Athens, Greece

    N. Kouvaris

  3. Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623, Berlin, Germany

    E. Schöll

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  1. N. Kouvaris
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Kouvaris, N., Schimansky-Geier, L. & Schöll, E. Control of coherence in excitable systems by the interplay of noise and time-delay. Eur. Phys. J. Spec. Top. 191, 29–51 (2010). https://doi.org/10.1140/epjst/e2010-01340-x

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