Journal of Computational Neuroscience

, Volume 36, Issue 1, pp 55–66 | Cite as

Synchronization of delayed coupled neurons in presence of inhomogeneity



In principle, two directly coupled limit cycle oscillators can overcome mismatch in intrinsic rates and match their frequencies, but zero phase lag synchronization is just achievable in the limit of zero mismatch, i.e., with identical oscillators. Delay in communication, on the other hand, can exert phase shift in the activity of the coupled oscillators. In this study, we address the question of how phase locked, and in particular zero phase lag synchronization, can be achieved for a heterogeneous system of two delayed coupled neurons. We have analytically studied the possibility of inphase synchronization and near inphase synchronization when the neurons are not identical or the connections are not exactly symmetric. We have shown that while any single source of inhomogeneity can violate isochronous synchrony, multiple sources of inhomogeneity can compensate for each other and maintain synchrony. Numeric studies on biologically plausible models also support the analytic results.


Synchronization Delay Inhomogeneity Neuronal excitability Phase resetting curves 


  1. Bayati, M., & Valizadeh, A. (2012). Effect of synaptic plasticity on the structure and dynamics of disordered networks of coupled neurons. Physical Review E, 86, 011925+.CrossRefGoogle Scholar
  2. Brown, E., Moehlis, J., Holmes, P. (2004). On the phase reduction and response dynamics of neural oscillator populations. Neural Computation, 16(4), 673–715.PubMedCrossRefGoogle Scholar
  3. Canavier, C.C. (2006). Phase response curve. Scholarpedia, 1(12), 1332.CrossRefGoogle Scholar
  4. Contreras, D., Destexhe, A., Sejnowski, T.J., Steriade, M. (1996). Control of spatiotemporal coherence of a thalamic oscillation by corticothalamic feedback. Science, 274(5288), 771–774.PubMedCrossRefGoogle Scholar
  5. D’Huys, O., Vicente, R., Erneux, T., Danckaert, J., Fischer, I. (2008). Synchronization properties of network motifs: influence of coupling delay and symmetry. Chaos (Woodbury N.Y.), 18(3), 037116–037116.CrossRefGoogle Scholar
  6. Engel, A.K., Konig, P., Kreiter, A.K., Singer, W. (1991). Interhemispheric synchronization of oscillatory neuronal responses in cat visual cortex. Science, 252(5009), 1177–1179.PubMedCrossRefGoogle Scholar
  7. Ermentrout, B. (1996). Type I membranes, phase resetting curves, and synchrony. Neural Computation, 8(5), 979–1001.PubMedCrossRefGoogle Scholar
  8. Ermentrout, B., Pascal, M., Gutkin, B. (2001). The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators. Neural Computation, 13(6), 1285–1310.PubMedCrossRefGoogle Scholar
  9. Ermentrout, G.B., Glass, L., Oldeman, B.E. (2012). The shape of phase-resetting curves in oscillators with a saddle node on an invariant circle bifurcation. Neural computation, 24(12), 3111–3125.PubMedCrossRefGoogle Scholar
  10. Ernst, U., Pawelzik, K., Geisel, T. (1995). Synchronization induced by temporal delays in pulse-coupled oscillators. Physical Review Letters, 74(9), 1570–1573.PubMedCrossRefGoogle Scholar
  11. Ernst, U., Pawelzik, K., Geisel, T. (1998). Delay-induced multistable synchronization of biological oscillators. Physical Review E, 57, 2150–2162.CrossRefGoogle Scholar
  12. Fischer, I., Vicente, R., Buldú, J.M., Peil, M., Mirasso, C.R., Torrent, M.C., Ojalvo, J.G. (2006). Zero-lag long-range synchronization via dynamical relaying. Physical Review Letters, 97(12), 123902+.PubMedCrossRefGoogle Scholar
  13. Gilson, M., Burkitt, A., Grayden, D., Thomas, D., van Hemmen, J.L. (2009). Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. I. Input selectivity strengthening correlated input pathways. Biological Cybernetics, 101(2), 81–102.PubMedCrossRefGoogle Scholar
  14. Gray, C.M., Engel, A.K., König, P., Singer, W. (1990). Stimulus-dependent neuronal oscillations in cat visual cortex: receptive field properties and feature dependence. European Journal of Neuroscience, 2(7), 607–619.PubMedCrossRefGoogle Scholar
  15. Gütig, R., Aharonov, R., Rotter, S., Sompolinsky, H. (2003). Learning input correlations through nonlinear temporally asymmetric hebbian plasticity. The Journal of Neuroscience, 23(9), 3697–3714.PubMedGoogle Scholar
  16. Hansel, D., Mato, G., Meunier, C. (1995). Synchrony in excitatory neural networks. Neural Computation, 7(2), 307–337.PubMedCrossRefGoogle Scholar
  17. Hashemi, M., Valizadeh, A., Azizi, Y. (2012). Effect of duration of synaptic activity on spike rate of a hodgkin-huxley neuron with delayed feedback. Physical Review E, 85(2), 021917.CrossRefGoogle Scholar
  18. Hodgkin, A.L., & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500–544.PubMedGoogle Scholar
  19. Izhikevich, E.M. (2006). Dynamical systems in neuroscience: the geometry of excitability and bursting (computational neuroscience) (1st Ed.). Cambridge: MIT Press.Google Scholar
  20. Kopell, N. (1988). Toward a theory of modeling central pattern genertors. In A.H. Cohen, S. Rossignol, S. Grillner, N. Kopell (Eds.), Neural control of rhythmic movements in vertebrates (pp. 369-413). New York: Wiley.Google Scholar
  21. Krogh-Madsen, T., Butera, R., Ermentrout, G.B., Glass, L. (2012). Phase resetting neural oscillators: topological theory versus the realworld In N.W. Schultheiss, A.A. Prinz, R.J. Butera (Eds.), Phase response curves in neuroscience, Vol. 6 of springer series in computational neuroscience (pp. 33-51). New York: Springer.CrossRefGoogle Scholar
  22. Kuramoto, Y. (2003). Chemical oscillations, waves, and turbulence (dover books on chemistry). Dover Publications, dover ed ed.Google Scholar
  23. Masuda, N., & Kori, H. (2007). Formation of feedforward networks and frequency synchrony by spike-timing-dependent plasticity. Journal of Computational Neuroscience, 22(3), 327–345.PubMedCrossRefGoogle Scholar
  24. Melloni, L., Molina, C., Pena, M., Torres, D., Singer, W., Rodriguez, E. (2007). Synchronization of neural activity across cortical areas correlates with conscious perception. The Journal of Neuroscience Science, 27(11), 2858–2865.CrossRefGoogle Scholar
  25. Mirollo, R.E., & Strogatz, S.H. (1990). Synchronization of pulse-coupled biological oscillators. SIAM Journal on Applied Mathematics, 50(6), 1645–1662.CrossRefGoogle Scholar
  26. Pikovsky, A., Rosenblum, M., Kurths, J. (2003). Synchronization: A universal concept in nonlinear sciences (cambridge nonlinear science series) (1st Ed.). Cambridge: Cambridge University Press.Google Scholar
  27. Ringo, J.L., Doty, R.W., Demeter, S., Simard, P.Y. (1994). Time is of the essence: a conjecture that hemispheric specialization arises from interhemispheric conduction delay. Cerebral Cortex, 4(4), 331–343.PubMedCrossRefGoogle Scholar
  28. Roelfsema, P.R., Engel, A.K., König, P., Singer, W. (1997). Visuomotor integration is associated with zero time-lag synchronization among cortical areas. Nature, 385(6612), 157–161.PubMedCrossRefGoogle Scholar
  29. Salami, M., Itami, C., Tsumoto, T., Kimura, F. (2003). Change of conduction velocity by regional myelination yields constant latency irrespective of distance between thalamus and cortex. Proceedings of the National Academy of Sciences, 100(10), 6174–6179.CrossRefGoogle Scholar
  30. Sieling, F.H., Canavier, C.C., Prinz, A.A. (2009). Predictions of phase-locking in excitatory hybrid networks: excitation does not promote phase-locking in pattern-generating networks as reliably as inhibition. Journal of Neurophysiology, 102(1), 69–84.PubMedCrossRefGoogle Scholar
  31. Singer, W. (1993). Synchronization of cortical activity and its putative role in information processing and learning. Annual Review of Physiology, 55(1), 349–374.PubMedCrossRefGoogle Scholar
  32. Smeal, R.M., Ermentrout, G.B., White, J.A. (2010). Phase-response curves and synchronized neural networks. Philosophical Transactions of the Royal Society B: Biological Sciences, 365(1551), 2407–2422.CrossRefGoogle Scholar
  33. Song, S., Miller, K.D., Abbott, L.F. (2000). Competitive Hebbian learning through spike-timing-dependent synaptic plasticity. Nature Neuroscience, 3(9), 919–926.PubMedCrossRefGoogle Scholar
  34. Traub, R.D., Whittington, M.A., Jefferys, J.G.R. (1999). Fast oscillations in cortical circuits (computational neuroscience). A Bradford Book.Google Scholar
  35. Traub, R.D., Whittington, M.A., Stanford, I.M., Jefferys, J.G.R. (1996). A mechanism for generation of long-range synchronous fast oscillations in the cortex. Nature, 383(6601), 621–624.PubMedCrossRefGoogle Scholar
  36. Tsodyks, M., Mitkov, I., Sompolinsky, H. (1993). Pattern of synchrony in inhomogeneous networks of oscillators with pulse interactions. Physical Review Letters, 71, 1280–1283.PubMedCrossRefGoogle Scholar
  37. Vicente, R., Gollo, L.L., Mirasso, C.R., Fischer, I., Pipa, G. (2008). Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays. Proceedings of the National Academy of Sciences, 105(44), 17157–17162.CrossRefGoogle Scholar
  38. Viriyopase, A., Bojak, I., Zeitler, M., Gielen, S. (2012). When Long-Range Zero-Lag Synchronization is Feasible in Cortical Networks. Frontiers in Computational Neuroscience, 6, (49).Google Scholar
  39. Vreeswijk, C., Abbott, L.F., Bard, EG. (1994). When inhibition not excitation synchronizes neural firing. Journal of Computational Neuroscience, 1(4), 313–321.PubMedCrossRefGoogle Scholar
  40. Wang, S., Chandrasekaran, L., Fernandez, F.R. White, J.A., Canavier, C.C. (2012). Short conduction delays cause inhibition rather than excitation to favor synchrony in hybrid neuronal networks of the entorhinal cortex. PLoS Computational Biology, 8(1), e1002306+.PubMedCentralPubMedCrossRefGoogle Scholar
  41. Wang, X.-J., & Buzsáki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. The Journal of Neuroscience, 16(20), 6402–6413.PubMedGoogle Scholar
  42. White, J., Chow, C., Rit, J., Soto-Treviño, C., Kopell, N. (1998). Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. Journal of Computational Neuroscience, 5(1), 5–16.PubMedCrossRefGoogle Scholar
  43. Winfree, A.T. (2010). The geometry of biological time (interdisciplinary applied mathematics). Springer, softcover reprint of hardcover 2nd ed. 2001 ed.Google Scholar
  44. Womelsdorf, T., Schoffelen, J.-M., Oostenveld, R., Singer, W., Desimone, R., Engel, A.K., Fries, P. (2007). Modulation of neuronal interactions through neuronal synchronization. Science, 316(5831), 1609–1612.PubMedCrossRefGoogle Scholar
  45. Woodman, M., & Canavier, C. (2011). Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators. Journal of computational neuroscience, 31(2), 401–418.PubMedCentralPubMedCrossRefGoogle Scholar
  46. Zeitler, M., Daffertshofer, A., Gielen, C. (2009). Asymmetry in pulse-coupled oscillators with delay. Physical Review E, 79(6), 065203.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Advanced Studies in Basic Sciences (IASBS)ZanjanIran

Personalised recommendations