Nonlinear Dynamics

, Volume 68, Issue 1–2, pp 151–159 | Cite as

Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays

Original Paper


This paper reveals the dynamical behaviors of a neural network consisting of a pair of bidirectional loops each with three identical neurons and two-way couplings between neurons of each individual loop. Time delays are introduced not only in the couplings between the loops but also in the internal connections within the individual loops. The study derives the conditions for the local stability of the network equilibrium and the existence of Hopf bifurcation. Afterwards, the study turns to showing the rich dynamical behaviors of the network through numerical analysis, such as multiple stability switches of network equilibrium, synchronous/asynchronous periodic oscillations, and the coexistence of bifurcated solutions.


Neural networks Time delay Coupled loops Stability switches Oscillations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Golubitsky, M., Stewart, I.: Nonlinear dynamics of networks: The groupoid formalism. Bull. Am. Math. Soc. 43, 305–364 (2006) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Kandel, E.R., Schwartz, J.H., Jessell, T.M.: Principles of Neural Science. McGraw-Hill, New York (2000) Google Scholar
  3. 3.
    Li, C., Xu, C., Sun, W., Xu, J., Kurths, J.: Outer synchronization of coupled discrete-time networks. Chaos 19(1), 013106 (2009) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bergman, H., Feingold, A., Nini, A., Raz, A., Slovin, H., Abeles, M., Vaadia, E.: Physiological aspects of information processing in the basal ganglia of normal and Parkinsonian primates. Trends Neurosci. 21, 32–38 (1998) CrossRefGoogle Scholar
  5. 5.
    Dias, A.P.S., Lamb, J.S.W.: Local bifurcation in symmetric coupled cell networks: Linear theory. Physica D 223(1), 93–108 (2006) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Liu, C., Weaver, D.R., Strogatz, S.H., Reppert, S.M.: Cellular construction of a circadian clock: Period determination in the suprachiasmatic nuclei. Cell 91, 855–860 (1997) CrossRefGoogle Scholar
  7. 7.
    Grosse, P., Cassidy, M.J., Brown, P.: EEG-EMG, MEGEMG and EMG-EMG frequency analysis: Physiological principles and clinical applications. Clin. Neurophysiol. 113, 1523–1531 (2002) CrossRefGoogle Scholar
  8. 8.
    Tass, P.A.: Phase Resetting in Medicine and Biology: Stochastic Modeling and Data Analysis. Springer, Berlin (1999) Google Scholar
  9. 9.
    Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. P. Natl. Acad. Sci. USA 81, 3088–3092 (1984) CrossRefGoogle Scholar
  10. 10.
    Marcus, C.M., Westervelt, R.M.: Stability of analog neural network with delay. Phys. Rev. A 39, 347–359 (1989) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Guo, S.J., Huang, L.H.: Hopf bifurcating periodic orbits in a ring of neurons with delays. Physica D 183, 19–44 (2003) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Guo, S., Chen, Y., Wu, J.: Two-parameter bifurcations in a network of two neurons with multiple delays. J. Differ. Equ. 244(2), 444–486 (2008) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Liao, X., Guo, S., Li, C.: Stability and bifurcation analysis in tri-neuron model with time delay. Nonlinear Dyn. 49(1–2), 319–345 (2007) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Duan, S., Liao, X.: An electronic implementation for Liao’s chaotic delayed neuron model with non-monotonous activation function. Phys. Lett. A 369(1–2), 37–43 (2007) CrossRefGoogle Scholar
  15. 15.
    Xu, X.: Complicated dynamics of a ring neural network with time delays. J. Phys. A 41(3), 035102-1 (2008) CrossRefGoogle Scholar
  16. 16.
    Yuan, Y.: Dynamics in a delayed-neural network. Chaos Solitons Fractals 33(2), 443–454 (2007) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, Heidelberg (2002) MATHGoogle Scholar
  18. 18.
    Shayer, L.P., Campbell, S.A.: Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays. SIAM J. Appl. Math. 61(2), 673–700 (2001) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wei, J., Zhang, C.: Bifurcation analysis of a class of neural networks with delays. Nonlinear Anal., Real World Appl. 9(5), 2234–2252 (2008) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Mao, X.C., Hu, H.Y.: Stability and Hopf bifurcation of a delayed network of four neurons with a short-cut connection. Int. J. Bifurc. Chaos 18(10), 3053–3072 (2008) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Mao, X.C., Hu, H.Y.: Dynamics of a delayed four-neuron network with a short-cut connection: Analytical, numerical and experimental studies. Int. J. Nonlinear Sci. Numer. Simul. 10(4), 523–538 (2009) CrossRefGoogle Scholar
  22. 22.
    Xu, X., Wang, Z.H.: Effects of small world connection on the dynamics of a delayed ring network. Nonlinear Dyn. 56(1–2), 127–144 (2009) MATHCrossRefGoogle Scholar
  23. 23.
    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) CrossRefGoogle Scholar
  24. 24.
    Hsu, C., Yang, T.: Periodic oscillations arising and death in delay-coupled neural loops. Int. J. Bifurc. Chaos 17(11), 4015–4032 (2007) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Cheng, C.: Induction of Hopf bifurcation and oscillation death by delays in coupled networks. Phys. Lett. A 374(2), 178–185 (2009) MathSciNetCrossRefGoogle Scholar
  26. 26.
    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) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    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) MATHCrossRefGoogle Scholar
  28. 28.
    Yuan, S., Li, P.: Stability and direction of Hopf bifurcations in a pair of identical tri-neuron network loops. Nonlinear Dyn. 61(3), 569–578 (2010) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    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) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Engineering Mechanics, College of Mechanics and MaterialsHohai UniversityNanjingChina

Personalised recommendations