Journal of Statistical Physics

, Volume 145, Issue 3, pp 713–733 | Cite as

Nonlocal Mechanism for Synchronization of Time Delay Networks

  • Ido Kanter
  • Evi Kopelowitz
  • Roni Vardi
  • Meital Zigzag
  • Dana Cohen
  • Wolfgang Kinzel
Article

Abstract

We present the interplay between synchronization of networks with heterogeneous delays and the greatest common divisor (GCD) of loops composing the network. We distinguish between two types of networks; (I) chaotic networks and (II) population dynamic networks with periodic activity driven by external stimuli. For type (I), in the weak chaos region, the units of a chaotic network characterized by GCD=1 are in a chaotic zero-lag synchronization, whereas for GCD>1, the network splits into GCD-clusters in which clustered units are in zero-lag synchronization. These results are supported by simulations of chaotic systems, self-consistent and mixing arguments, as well as analytical solutions of Bernoulli maps. Type (II) is exemplified by simulations of Hodgkin Huxley population dynamic networks with unidirectional connectivity, synaptic noise and distribution of delays within neurons belonging to a node and between connecting nodes. For a stimulus to one node, the network splits into GCD-clusters in which cluster neurons are in zero-lag synchronization. For complex external stimuli, the network splits into clusters equal to the greatest common divisor of loops composing the network (spatial) and the periodicity of the external stimuli (temporal). The results suggest that neural information processing may take place in the transient to synchronization and imply a much shorter time scale for the inference of a perceptual entity.

Keywords

Synchronization Networks Chaos Neural networks 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization. Cambridge University Press, Cambridge (2001) MATHCrossRefGoogle Scholar
  2. 2.
    Schuster, H.G., Just, W.: Deterministic Chaos. Wiley-VCH Verlag GmbH & Co. KGaA, Germany (2005) MATHCrossRefGoogle Scholar
  3. 3.
    Albert, R., Barabási, A.-L.: Rev. Mod. Phys. 74, 47 (2002) ADSMATHCrossRefGoogle Scholar
  4. 4.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks, Oxford University Press, London (2003) MATHCrossRefGoogle Scholar
  5. 5.
    Strogatz, S.H.: Nature 410, 268 (2001) ADSCrossRefGoogle Scholar
  6. 6.
    Barabási, A.-L., Albert, R.A.: Science 286, 509–512 (1999) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Phys. Rep. 469, 93 (2008) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Pecora, L.M., Carroll, T.L.: Phys. Rev. Lett. 80, 2109 (1998) ADSCrossRefGoogle Scholar
  9. 9.
    Jost, J., Joy, M.P.: Phys. Rev. E 65, 016201 (2002) MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Atay, F.M., Biyikoglu, T.: Phys. Rev. E 72, 016217 (2005) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Nishikawa, T., Motter, A.E., Lai, Y.-C., Hoppensteadt, F.C.: Phys. Rev. Lett. 91, 014101 (2003) ADSCrossRefGoogle Scholar
  12. 12.
    Masoller, C., Marti, A.C.: Phys. Rev. Lett. 94, 134102 (2005) ADSCrossRefGoogle Scholar
  13. 13.
    Wu, X., Wang, B., Zhou, T., Wang, W., Zhao, M., Yang, H.: Chin. Phys. Lett. 23, 1046 (2006) ADSCrossRefGoogle Scholar
  14. 14.
    Sorrentino, F.: Chaos 17, 033101 (2007) MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Hong, H., Kim, B.J., Choi, M.Y., Park, H.: Phys. Rev. E 65, 067105 (2002) Google Scholar
  16. 16.
    Gawne, T.J., Richmond, B.J.: J. Neurosci. 13, 2758 (1993) Google Scholar
  17. 17.
    Zohary, E., Shadlen, M.N., Newsome, W.T.: Nature 370, 140 (1994) ADSCrossRefGoogle Scholar
  18. 18.
    Vaadia, E., et al.: Nature 373, 515 (1995) ADSCrossRefGoogle Scholar
  19. 19.
    Tchumatchenko, T., Malyshev, A., Geisel, T., Volgushev, M., Wolf, F.: Phys. Rev. Lett. 104, 058102 (2010) ADSCrossRefGoogle Scholar
  20. 20.
    Shadlen, M.N., Newsome, W.T.: J. Neurosci. 18, 3870 (1998) Google Scholar
  21. 21.
    Kriener, B., Tetzlaff, T., Aertsen, A., Diesmann, M.: Neural Comput. 20, 2185 (2008) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Steriade, M., McCormick, D.A., Sejnowski, T.J.: Science 262, 679 (1993) ADSCrossRefGoogle Scholar
  23. 23.
    Gray, C.M., Konig, P., Engel, A.K., Singer, W.: Nature 338, 334 (1989) ADSCrossRefGoogle Scholar
  24. 24.
    Morison, R.S., Dempsey, E.W.: Am. J. Physiol. 135, 281 (1942) Google Scholar
  25. 25.
    Eckhorn, R., et al.: Biol. Cybern. 60, 121 (1988) CrossRefGoogle Scholar
  26. 26.
    Singer, W.C.M.: Annu. Rev. Neurosci. 18, 555 (1995) CrossRefGoogle Scholar
  27. 27.
    Nunez, P.L., Srinavasan, R.: Electric Fields of the Brain: The Neurophysics of EEG, 2nd edn. Oxford University Press, London (2006) CrossRefGoogle Scholar
  28. 28.
    Kanter, I., Zigzag, M., Englert, A., Geissler, F., Kinzel, W.: Europhys. Lett. 93, 60003 (2011) ADSCrossRefGoogle Scholar
  29. 29.
    Englert, A., Heiligenthal, S., Kinzel, W., Kanter, I.: Phys. Rev. E 83, 046222 (2011) ADSCrossRefGoogle Scholar
  30. 30.
    Kanter, I., Kopelowitz, E., Vardi, R., Zigzag, M., Kinzel, W., Abeles, M., Cohen, D.: Europhys. Lett. 93, 66001 (2011) ADSCrossRefGoogle Scholar
  31. 31.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Science. Academic Press, New York (1979) Google Scholar
  32. 32.
    Abrams, D.M., Mirollo, R., Strogatz, S.H., Wiley, D.A.: Phys. Rev. Lett. 101, 084103 (2008) ADSCrossRefGoogle Scholar
  33. 33.
    Ma, R., Wang, J., Liu, Z.: Europhys. Lett. 91, 40006 (2010) ADSCrossRefGoogle Scholar
  34. 34.
    Kestler, J., Kinzel, W., Kanter, I.: Phys. Rev. E 76, 035202 (2007) ADSCrossRefGoogle Scholar
  35. 35.
    Lang, R., Kobayashi, K.: IEEE J. Quantum Electron. 16, 347 (1980) ADSCrossRefGoogle Scholar
  36. 36.
    Ahlers, V., Parlitz, U., Lauterborn, W.: Phys. Rev. E 58, 7208 (1998) ADSCrossRefGoogle Scholar
  37. 37.
    Klein, E., et al.: Phys. Rev. E 73, 066214 (2006) ADSCrossRefGoogle Scholar
  38. 38.
    Murphy, T.E., Roy, R.: Nat. Photonics 2(12), 714–715 (2008) ADSCrossRefGoogle Scholar
  39. 39.
    Uchida, A., et al.: Nat. Photonics 2(12), 728–732 (2008) ADSCrossRefGoogle Scholar
  40. 40.
    Reidler, I., Aviad, Y., Rosenbluh, M., Kanter, I.: Phys. Rev. Lett. 103(2), 024102 (2009) ADSCrossRefGoogle Scholar
  41. 41.
    Kanter, I., Aviad, Y., Reidler, I., Cohen, E., Rosenbluh, M.: Nat. Photonics 4(1), 58–61 (2010) ADSCrossRefGoogle Scholar
  42. 42.
    Kestler, J., Kopelowitz, E., Kanter, I., Kinzel, W.: Phys. Rev. E 77, 046209 (2008) MathSciNetADSCrossRefGoogle Scholar
  43. 43.
    Zigzag, M., Butkovski, M., Englert, A., Kinzel, W., Kanter, I.: Europhys. Lett. 85, 60005 (2009) ADSCrossRefGoogle Scholar
  44. 44.
    Englert, A., et al.: Phys. Rev. Lett. 104, 114102 (2010) ADSCrossRefGoogle Scholar
  45. 45.
    Zigzag, M., Butkovski, M., Englert, A., Kinzel, W., Kanter, I.: Phys. Rev. E 81, 036215 (2010) ADSCrossRefGoogle Scholar
  46. 46.
    Flunkert, V., Yanchuk, S., Dahms, T., Schöll, E.: Phys. Rev. Lett. 105, 254101 (2010) ADSCrossRefGoogle Scholar
  47. 47.
    Hodgkin, A.L., Huxley, A.F.: J. Physiol. 117, 500 (1952) Google Scholar
  48. 48.
    Abeles, M.: Corticonics. Cambridge University Press, Cambridge (1991) CrossRefGoogle Scholar
  49. 49.
    Braitenberg, V., Shuz, A.: Cortex: Statistics and Geometry of Neuronal Connectivity, 2nd edn. Springer, Berlin (1998) Google Scholar
  50. 50.
    Scannell, J.W., Blakemore, C., Young, M.P.: J. Neurosci. 15, 1463 (1995) Google Scholar
  51. 51.
    Braitenberg, V.: J. Comput. Neurosci. 10, 71 (2001) CrossRefGoogle Scholar
  52. 52.
    Evarts, E.V.: J. Neurophysiol. 27, 152 (1964) Google Scholar
  53. 53.
    Frank, H.: Graph Theory. Addison-Wesley, Reading (1995) Google Scholar
  54. 54.
    Shinomoto, S., et al.: PLoS Comput. Biol. 5, e1000433 (2009) MathSciNetCrossRefGoogle Scholar
  55. 55.
    Shimokawa, T., Shinomoto, S.: Neural Comput. 21, 1931 (2009) MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Thomson, A.M., Deuchars, J.: Trends Neurosci. 17, 119 (1994) CrossRefGoogle Scholar
  57. 57.
    Tsodyks, M.V., Markram, H.: Proc. Natl. Acad. Sci. USA 94, 719 (1997) ADSCrossRefGoogle Scholar
  58. 58.
    Milo, R., et al.: Science 303, 1538 (2004) ADSCrossRefGoogle Scholar
  59. 59.
    Barabasi, A.-L., Albert, R.A.: Science 286, 509 (1999) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Ido Kanter
    • 1
  • Evi Kopelowitz
    • 1
  • Roni Vardi
    • 2
  • Meital Zigzag
    • 1
  • Dana Cohen
    • 2
  • Wolfgang Kinzel
    • 3
  1. 1.Department of PhysicsBar-Ilan UniversityRamat-GanIsrael
  2. 2.Gonda Interdisciplinary Brain Research Center, and the Goodman Faculty of Life SciencesBar-Ilan UniversityRamat-GanIsrael
  3. 3.Institute for Theoretical PhysicsUniversity of Wuerzburg, Am HublandWuerzburgGermany

Personalised recommendations