The European Physical Journal Special Topics

, Volume 226, Issue 9, pp 1911–1920 | Cite as

Synchronization in network motifs of delay-coupled map-based neurons

  • J. M. Sausedo-Solorio
  • A. N. Pisarchik
Regular Article
Part of the following topical collections:
  1. Recent Advances in Nonlinear Dynamics and Complex Structures: Fundamentals and Applications


We study the influence of delayed coupling on synchronization in neural network motifs. Numerical simulations based on the Rulkov map reveal different behavior in the presence and in the absence of the delay. While without delay, synchronization improves as the coupling strength is increased, in the presence of a delay, synchronization becomes worse. We also study how a feedback loop affects synchronization. An increase in the number of neurons involved in the loop leads to desynchronization in the motifs, saturating at a certain value of the synchronization index.


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  1. 1.
    R.R. Llinás, I of the Vortex: From Neurons to Self (MIT Press, 2002)Google Scholar
  2. 2.
    G. Buzsáki (ed.), Rhythms of the Brain (Oxford University Press, 2006)Google Scholar
  3. 3.
    E.R. Kandel, J.H. Schwartz, T.M. Jessell (eds.), Principles of Neural Science (Appleton & Lange, 2000)Google Scholar
  4. 4.
    B. Katz, R. Miledi, The measurement of synaptic delay, and the time course of acetylcholine release at the neuromuscular junction, Proc. R. Soc. London B 161, 483 (1965)ADSCrossRefGoogle Scholar
  5. 5.
    D.J. Amit, The Hebbian paradigm reintegrated: Local reverberations as internal representations, Behav. Brain Sci. 18, 617 (1995)CrossRefGoogle Scholar
  6. 6.
    R.C. Elson, A.I. Selverston, R. Huerta, N.F. Rulkov, M.I. Rabinovich, H.D.I. Abarbanel, Synchronous behavior of two coupled biological neurons, Phys. Rev. Lett. 81, 5692 (1998)ADSCrossRefGoogle Scholar
  7. 7.
    R.C. Elson, A.I. Selverston, H.D.I. Abarbanel, M. Rabinovich, Inhibitory synchronization of bursting in biological neurons: Dependence on synaptic time constant, J. Neurophysiol. 88, 1166 (2002)Google Scholar
  8. 8.
    D.R. Chialvo, Generic excitable dynamics on a two-dimensional map, Chaos Solitons Fractals 5, 461 (1995)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    O. Kinouchi, M.H.R. Tragtenberg, Modeling neurons by simple maps, Int. J. Bif. Chaos 6, 2343 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S.M. Kuva, G.F. Lima, O. Kinouchi, M.H.R. Tragtenberg, A.C. Roque-da-Silva, A minimal model for excitable and bursting elements, Neurocomputing 38-40, 255 (2001)CrossRefGoogle Scholar
  11. 11.
    C.R. Laing, A. Longtin, A two-variable model of somatic-dendritic interactions in a bursting neuron, Bull. Math. Biol. 64, 829 (2002)CrossRefzbMATHGoogle Scholar
  12. 12.
    N.F. Rulkov, I. Timofeev, M. Bazhenov, Oscillations in large-scale cortical networks: map-based model, J. Comput. Neurosci. 17, 203 (2004)CrossRefGoogle Scholar
  13. 13.
    A.L. Shilnikov, N.F. Rulkov, Origin of chaos in a two-dimensional map modeling spiking-bursting neural activity, Int. J. Bif. Chaos 13, 3325 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Copelli, M.H.R. Tragtenberg, O. Kinouchi, Stability diagrams for bursting neurons modeled by three-variable maps, Physica A 342, 263 (2004)ADSCrossRefGoogle Scholar
  15. 15.
    E.M. Izhikevich, F. Hoppensteadt, Classification of bursting mappings, Int. J. Bif. Chaos 14, 3847 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    X. Sun, Q. Lu, J. Kurths, Q. Wang, Spatiotemporal coherence resonance in a map lattice, Int. J. Bif. Chaos 19, 737 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A.L. Hodgkin, A.F. Huxley, A quatitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117, 500 (1952)CrossRefGoogle Scholar
  18. 18.
    J.L. Hindmarsh, R.M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. London B 221, 87 (1984)ADSCrossRefGoogle Scholar
  19. 19.
    M.V. Ivanchenko, T. Nowotny, A.I. Selverston, M.I. Rabinovich, Pacemaker and network mechanisms of rhythm generation: Cooperation and competition, Theor. Biol. 253, 452 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M.I. Rabinovich, H.D.I. Abarbanel, The role of chaos in neural systems, Neuroscience 87, 5 (1999)CrossRefGoogle Scholar
  21. 21.
    X. Lang, Q. Lu, J. Kurths, Phase synchronization in noise-driven bursting neurons, Phys. Rev. E 82, 021909 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    F.S. Matias, P.V. Carelli, C. Mirasso, R.M. Copelli, Anticipated synchronization in a biologically plausible model of neuronal motifs, Phys. Rev. E 84, 021922 (2011)ADSCrossRefGoogle Scholar
  23. 23.
    N.F. Rulkov, Modeling of spiking-bursting neural behavior using two-dimensional map, Phys. Rev. E 65, 041922 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    M.V. Ivanchenko, G.V. Osipov, V.D. Shalfeev, J. Kurths, Phase synchronization of chaotic intermittent oscillations, Phys. Rev. Lett. 92, 134101 (2004)ADSCrossRefGoogle Scholar
  25. 25.
    I. Franoviĉ, V. Miljkoviĉ, Power law behavior related to mutual synchronization of chemically coupled map neurons, Eur. Phys. J. B 76, 613 (2010)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    C. Mayol, C.R. Mirasso, R. Toral, Anticipated synchronization and the predict-prevent control method in the FitzHugh-Nagumo model system, Phys. Rev. E 85, 056216 (2012)ADSCrossRefGoogle Scholar
  27. 27.
    S.S. Shen-Orr, R. Milo, S. Mangan, U. Alon, Network motifs in the transcriptional regulation network of Escherichia coli, Nat. Genet. 31, 64 (2002)CrossRefGoogle Scholar
  28. 28.
    S. Valverde, R.V. Solé, Network motifs in computational graphs: A case study in software architecture, Phys. Rev. E 72, 026107 (2005)ADSCrossRefGoogle Scholar
  29. 29.
    I. Lodato, S. Boccaletti, V. Latora, Synchronization properties of network motifs, Europhys. Lett. 78, 28001 (2007)ADSCrossRefGoogle Scholar
  30. 30.
    G. de Vries, Bursting as an emergent phenomenon in coupled chaotic maps, Phys. Rev. E 64, 051914 (2001)ADSCrossRefGoogle Scholar
  31. 31.
    N.F. Rulkov, Regularization of synchronized chaotic bursts, Phys. Rev. Lett. 86, 183 (2001)ADSCrossRefGoogle Scholar
  32. 32.
    A. Shilnikov, N.F. Rulkov, Subthreshold oscillations in a map-based neuron model, Phys. Lett. A 328, 177 (2004)ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    A. Shilnikov, R. Gordon, I. Belykh, Polyrhythmic synchronization in bursting networking motifs, Chaos 18, 037120 (2008)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    I. Belykh, E. de Lange, M. Hasler, Synchronization of bursting neurons: What matters in the network topology, Phys. Rev. Lett. 94, 188101 (2005)ADSCrossRefGoogle Scholar
  35. 35.
    H. Wang, Q. Lu, Q. Wang, Bursting and synchronization transition in the coupled modified ML neurons, Commun. Nonlinear Sci. Numer. Simulat. 13, 1668 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    J.M. Sausedo-Solorio, A.N. Pisarchik, Synchronization of map-based neurons with memory and synaptic delay, Phys. Lett. A 378, 2108 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    Q. Wang, M. Perc, Z. Duan, G. Chen, Synchronization transitions on scale-free neuronal networks due to finite information transmission delays, Phys. Rev. E 80, 026206 (2009)ADSCrossRefGoogle Scholar
  38. 38.
    Q. Wang, G. Chen, M. Perc, Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling, Plos One 6, e15851 (2011)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Universidad Autónoma del Estado de HidalgoPachuca, HidalgoMexico
  2. 2.Center for Biomedical Technology, Technical University of MadridMadridSpain
  3. 3.Centro de Investigaciones en ÓpticaLeón, GuanajuatoMexico
  4. 4.Yuri Gagarin Saratov State Technical UniversitySaratovRussia

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