Nonlinear Dynamics

, Volume 84, Issue 2, pp 1063–1078 | Cite as

Stability, bifurcation, and synchronization of delay-coupled ring neural networks

  • Xiaochen Mao
  • Zaihua Wang
Original Paper


This paper studies the nonlinear dynamics of coupled ring networks each with an arbitrary number of neurons. Different types of time delays are introduced into the internal connections and couplings. Local and global asymptotical stability of the coupled system is discussed, and sufficient conditions for the existence of different bifurcated oscillations are given. Numerical simulations are performed to validate the theoretical results, and interesting neuronal activities are observed, such as rest state, synchronous oscillations, asynchronous oscillations, and multiple switches of the rest states and different oscillations. It is shown that the number of neurons in the sub-networks plays an important role in the network characteristics.


Time delay Coupled systems Neural networks Synchronization Nonlinear dynamics 



The authors thank the financial support from the National Natural Science Foundation of China under Grant Nos. 11472097 and 11002047, the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) under Grant No. MCMS-0113G01, and the Fundamental Research Funds for the Central Universities under Grant No. 2015B18214. They thank the anonymous reviewers for their helpful comments and suggestions that have helped to improve the presentation of this paper.


  1. 1.
    Yamaguchi, S., Isejima, H., Matsuo, T., Okura, R., Yagita, K., Kobayashi, M., Okamura, H.: Synchronization of cellular clocks in the suprachiasmatic nucleus. Science 302(5649), 1408–1412 (2003)CrossRefGoogle Scholar
  2. 2.
    Nijmeijer, H., Rodriguez-Angeles, A.: Synchronization of mechanical systems. World Scientific Publishing, Singapore (2003)MATHGoogle Scholar
  3. 3.
    Mao, X.C.: Stability switches, bifurcation, and multi-stability of coupled networks with time delays. Appl. Math. Comput. 218(11), 6263–6274 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Murguia, C., Fey, R.H.B., Nijmeijer, H.: Network synchronization of time-delayed coupled nonlinear systems using predictor-based diffusive dynamic couplings. Chaos 25(2), 023108 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Emelianova, Y.P., Emelyanov, V.V., Ryskin, N.M.: Synchronization of two coupled multimode oscillators with time-delayed feedback. Commun. Nonlinear Sci. Numer. Simulat 19(10), 3778–3791 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Louzada, V.H.P., Araujo, N.A.M., Andrade, J.S., Herrmann, H.J.: Breathing synchronization in interconnected networks. Sci. Rep. 3, 3289 (2013)CrossRefGoogle Scholar
  7. 7.
    Weicker, L., Erneux, T., Keuninckx, L., Danckaert, J.: Analytical and experimental study of two delay-coupled excitable units. Phys. Rev. E 89(1), 012908 (2014)CrossRefGoogle Scholar
  8. 8.
    Flunkert, V., Fischer, I., Fischer, I.: Dynamics, control and information in delay-coupled systems: an overview. Philos. Trans. R. Soc. A 371(1999), 20120465 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hu, H.Y., Wang, Z.H.: Dynamics of controlled mechanical systems with delayed feedback. Springer, Heidelberg (2002)CrossRefMATHGoogle Scholar
  10. 10.
    Sipahi, R., Niculescu, S.-I., Abdallah, C.T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay: limitations and opportunities. IEEE Control Syst. Mag. 31(1), 38–65 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Orosz, G., Wilson, R.E., Stepan, G.: Traffic jams: dynamics and control. Philos. Trans. R. Soc. A 368(1928), 4455–4479 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Shepherd, G.M.: Neurobiology. Oxford University Press, New York (1983)Google Scholar
  13. 13.
    Murray, J.D.: Mathematical Biology. Springer, New York (1990)MATHGoogle Scholar
  14. 14.
    Marcus, C.M., Westervelt, R.M.: Stability of analog neural network with delay. Phys. Rev. A 39, 347–359 (1989)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)CrossRefGoogle Scholar
  16. 16.
    Timme, M., Wolf, F., Geisel, T.: Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators. Phys. Rev. Lett. 89, 258701 (2002)CrossRefGoogle Scholar
  17. 17.
    Punetha, N., Prasad, A., Ramaswamy, R.: Phase-locked regimes in delay-coupled oscillator networks. Chaos 24(4), 043111 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Popovych, O.V., Yanchuk, S., Tass, P.A.: Delay- and coupling-induced firing patterns in oscillatory neural loops. Phys. Rev. Lett. 107(22), 228102 (2011)CrossRefGoogle Scholar
  19. 19.
    Soriano, M.C., Flunkert, V., Fischer, I.: Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers. Chaos 23(4), 043133 (2013)CrossRefMATHGoogle Scholar
  20. 20.
    Sadeghi, S., Valizadeh, A.: Synchronization of delayed coupled neurons in presence of inhomogeneity. J. Comput. Neurosci. 36(1), 55–66 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ge, J.H., Xu, J.: Computation of synchronized periodic solution in a BAM network with two delays. IEEE Trans. Neural Netw. 21(3), 439–450 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ying, J., Guo, S., He, Y.: Multiple periodic solutions in a delay-coupled system of neural oscillators. Nonlinear Anal. Real World Appl. 12(5), 2767–2783 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Song, Y.L.: Hopf bifurcation and spatio-temporal patterns in delay-coupled van der Pol oscillators. Nonlinear Dyn. 63(1–2), 223–237 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wirkus, S., Rand, R.: The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dyn. 30(3), 205–221 (2002)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Correa, D.P.F., Wulff, C., Piqueira, J.R.C.: Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 793–820 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Shen, Z., Zhang, C.: Double Hopf bifurcation of coupled dissipative Stuart–Landau oscillators with delay. Appl. Math. Comput. 227, 553–566 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Usacheva, S.A., Ryskin, N.M.: Phase locking of two limit cycle oscillators with delay coupling. Chaos 24(2), 023123 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Caceres, M.O.: Time-delayed coupled logistic capacity model in population dynamics. Phys. Rev. E 90(2), 022137 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Campbell, S.A., Edwards, R., Van Den Driessche, P.: Delayed coupling between two neural network loops. SIAM. J. Appl. Math. 65(1), 316–335 (2005)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Cheng, C.: Induction of Hopf bifurcation and oscillation death by delays in coupled networks. Phys. Lett. A 374(2), 178–185 (2009)Google Scholar
  31. 31.
    Song, Z.G., Xu, J.: Stability switches and multistability coexistence in a delay-coupled neural oscillators system. J. Theor. Biol. 313, 98–114 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Jiang, Y., Guo, S.: Linear stability and Hopf bifurcation in a delayed two-coupled oscillator with excitatory-to-inhibitory connection. Nonlinear Anal. Real World Appl. 11(3), 2001–2015 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Peng, Y., Song, Y.: Stability switches and Hopf bifurcations in a pair of identical tri-neuron network loops. Phys. Lett. A 373(20), 1744–1749 (2009)CrossRefMATHGoogle Scholar
  34. 34.
    Mao, X.C.: Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays. Nonlinear Dynam. 68(1), 151–159 (2012)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Song, Y., Tade, M.O., Zhang, T.: Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling. Nonlinearity 22(5), 975–1001 (2009)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Kandel, E.R., Schwartz, J.H., Jessell, T.M.: Principles of neural science. McGraw-Hill, New York (2000)Google Scholar
  37. 37.
    Graybiel, A.M.: Basal ganglia-input, neural activity, and relation to the cortex. Curr. Opin. Neurobiol. 1(4), 644–651 (1991)CrossRefGoogle Scholar
  38. 38.
    Nana, B., Woafo, P.: Synchronization in a ring of four mutually coupled van der Pol oscillators: theory and experiment. Phys. Rev. E 74(4), 046213 (2006)CrossRefGoogle Scholar
  39. 39.
    Hoppensteadt, F.C., Izhikevich, E.M.: Weakly connected neural networks. Springer, New York (1997)CrossRefMATHGoogle Scholar
  40. 40.
    Hisi, A.N.S., Guimaraes, P.R., de Aguiar, M.A.M.: The role of predator overlap in the robustness and extinction of a four species predator–prey network. Phys. A 389(21), 4725–4733 (2010)CrossRefGoogle Scholar
  41. 41.
    Xu, X.: Complicated dynamics of a ring neural network with time delays. J. Phys. A 41(3), 035102 (2008)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Guo, S.J., Huang, L.H.: Hopf bifurcating periodic orbits in a ring of neurons with delays. Phys. D 183, 19–44 (2003)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Campbell, S.A., Yuan, Y., Bungay, S.D.: Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18(6), 2827–2846 (2005)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Burić, N., Grozdanović, I., Vasović, N.: Excitable systems with internal and coupling delays. Chaos Soliton Fractals 36(4), 853–861 (2008)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Song, Y., Makarov, V.A., Velarde, M.G.: Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks. Biol. Cybern. 101(2), 147–167 (2009)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Tass, P.A., Hauptmann, C.: Therapeutic modulation of synaptic connectivity with desynchronizing brain stimulation. Int. J. Psychophysiol. 64(1), 53–61 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Engineering Mechanics, College of Mechanics and MaterialsHohai UniversityNanjingChina
  2. 2.State Key Laboratory of Mechanics and Control of Mechanical StructuresNanjing University of Aeronautics and AstronauticsNanjingChina
  3. 3.Institute of SciencePLA University of Science and TechnologyNanjingChina

Personalised recommendations