Skip to main content
SpringerLink
Log in

Stability switches and Bogdanov-Takens bifurcation in an inertial two-neuron coupling system with multiple delays

  • Article
  • Special Topic: Neurodynamics
  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

In this paper, we investigate an inertial two-neural coupling system with multiple delays. We analyze the number of equilibrium points and demonstrate the corresponding pitchfork bifurcation. Results show that the system has a unique equilibrium as well as three equilibria for different values of coupling weights. The local asymptotic stability of the equilibrium point is studied using the corresponding characteristic equation. We find that multiple delays can induce the system to exhibit stable switching between the resting state and periodic motion. Stability regions with delay-dependence are exhibited in the parameter plane of the time delays employing the Hopf bifurcation curves. To obtain the global perspective of the system dynamics, stability and periodic activity involving multiple equilibria are investigated by analyzing the intersection points of the pitchfork and Hopf bifurcation curves, called the Bogdanov-Takens (BT) bifurcation. The homoclinic bifurcation and the fold bifurcation of limit cycle are obtained using the BT theoretical results of the third-order normal form. Finally, numerical simulations are provided to support the theoretical analyses.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hopfield J J. Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci (USA), 1984, 81: 3088–3092

    Article  Google Scholar 

  2. Schieve W C, Bulsara A R, Davis G M. Single effective neuron. Phys Rev A, 1991, 43: 2613–2623

    Article  Google Scholar 

  3. Song Z G, Xu J. Bursting near Bautin bifurcation in a neural network with delay coupling. Int J Neural Syst, 2009, 19: 359–373

    Article  MathSciNet  Google Scholar 

  4. Song Z G, Xu J. Codimension-two bursting analysis in the delayed neural system with external stimulations. Nonlinear Dyn, 2012, 67: 309–328

    Article  MATH  Google Scholar 

  5. Ashmore J F, Attwell D. Models for electrical tuning in hair cells. Proc Royal Soc London B, 1985, 226: 325–334

    Article  Google Scholar 

  6. Angelaki D E, Correia M J. Models of membrane resonance in pigeon semicircular canal type II hair cells. Biol Cybernet, 1991, 65: 1–10

    Article  Google Scholar 

  7. Mauro A, Conti F, Dodge F, et al. Subthreshold behavior and phenomenological impedance of the squid giant axon. J General Physiol, 1970, 55: 497–523

    Article  Google Scholar 

  8. Badcock K L, Westervelt R M. Dynamics of simple electronic neural networks. Physical D, 1987, 28: 305–316

    Article  Google Scholar 

  9. Wheeler D W, Schieve W C. Stability and chaos in an inertial two-neuron system. Physical D, 1997, 105: 267–284

    Article  MATH  Google Scholar 

  10. Tani J, Fujita M. Coupling of memory search and mental rotation by a nonequilibrium dynamics neural network. IEICF Trans Fund Electron Commun Comput Sci E, 1992, 75-A(5): 578–585

    Google Scholar 

  11. Tani J. Proposal of chaotic steepest descent method for neural networks and analysis of their dynamics. Electron Commun Jpn, 1992, 75(4): 62–70

    Article  MathSciNet  Google Scholar 

  12. Campbell S A. Time delays in neural systems. In: McIntosh A R, Jirsa V K, eds. Handbook of Brain Connectivity. Springer, 2007, 65–90

    Chapter  Google Scholar 

  13. Song Z G, Xu J. Bifurcation and chaos analysis for a delayed two-neural network with a variation slope ratio in the activation function. Int J Bifurcat Chaos, 2012, 22: 1250105

    Article  Google Scholar 

  14. Liu Q, Liao X F, Yang D G, et al. The research for Hopf bifurcation in a single inertial neuron model with external forcing. In: Lin T Y, Hu X, Han J, et al, eds. IEEE Proceeding of International Conference on Granular Computing, San Jose, California, 2007. 528–533

    Google Scholar 

  15. Li C G, Chen G R, Liao X F, et al. Hopf bifurcation and chaos in a single inertial neuron model with time delays. Eur Phys J B, 2004, 41: 337–343

    Article  Google Scholar 

  16. Liu Q, Liao X F, Guo S T, et al. Stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation. Nonlinear Anal-Real World Appl, 2009, 10: 2384–2395

    Article  MATH  MathSciNet  Google Scholar 

  17. Liu Q, Liao X F, Wang G Y, et al. Research for Hopf bifurcation of an inertial two-neuron system with time delay. In: Zhang Y Q, Lin T Y, eds. IEEE Proceeding of International Conference on Granular Computing, Atlanta, USA, 2006. 420–423

    Google Scholar 

  18. Liu Q, Liao X F, Liu Y, et al. Dynamics of an inertial two-neuron system with time delay. Nonlinear Dyn, 2009, 58: 573–609

    Article  MATH  MathSciNet  Google Scholar 

  19. Song Z G, Xu J. Stability switches and multistability coexistence in a delay-coupled neural oscillators system. J Theor Biol, 2012, 313(21): 98–114

    Article  MathSciNet  Google Scholar 

  20. Zhen B, Xu J. Fold-Hopf bifurcation analysis for a coupled Fitz-Hugh-Nagumo neural system with time delay. Int J Bifurcation Chaos, 2010, 20: 3919–3934

    Article  MATH  MathSciNet  Google Scholar 

  21. Song Z G, Xu J. Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays. Cogn Neurodyn, 2013, 7: 505–521

    Article  MathSciNet  Google Scholar 

  22. Jiang W, Yuan Y. Bogdanov-Takens singularity in Van der Pol’s oscillator with delayed feedback. Physica D, 2007, 227: 149–161

    Article  MATH  MathSciNet  Google Scholar 

  23. Jiang J, Song Y. Bogdanov-Takens bifurcation in an oscillator with negative damping and delayed position feedback. Appl Math Modell, 2013, 37: 8091–8105

    Article  MathSciNet  Google Scholar 

  24. He X, Li C, Shu Y. Bogdanov-Takens bifurcation in a single inertial neuron model with delay. Neurocomputing, 2012, 89: 193–201

    Article  Google Scholar 

  25. Dong T, Liao X F, Huang T W, et al. Hopf-pitchfork bifurcation in an inertial two-neuron system with time delay. Neurocomputing, 2012, 97: 223–232

    Article  Google Scholar 

  26. Ge J H, Xu J. Weak resonant double Hopf bifurcations in an inertial four-neuron model with time delay. Int J Neural Syst, 2012, 22(1): 63–75

    Article  Google Scholar 

  27. Xu X. Local and global Hopf bifurcation in a two-neuron network with multiple delays. Int J Bifurcation Chaos, 2008, 18(4): 1015–1028

    Article  MATH  Google Scholar 

  28. Campbell S A, Ncube I, Wu J. Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system. Physica D, 2006, 214(2):101–119

    Article  MATH  MathSciNet  Google Scholar 

  29. Liao X F, Wong K-W, Wu Z F. Asymptotic stability criteria for a two-neuron network with different time delays. IEEE Trans. Neural Networks, 2003, 14(1): 222–227

    Article  Google Scholar 

  30. Gopalsamy K, Leung Issic K C. Convergence under dynamical thresholds with delays. IEEE Trans. Neural Networks, 1997, 8(2): 341–348

    Article  Google Scholar 

  31. Kuznetsov Y A. Elements of Applied Bifurcation Theory. Springer, New York, 1995

    Book  MATH  Google Scholar 

  32. Dong T, Liao X. Bogdanov-Takens bifurcation in a tri-neuron BAM neural network model with multiple delays. Nonlinear Dyn, 2013, 71: 583–595

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Information Technology, Shanghai Ocean University, Shanghai, 201306, China

    ZiGen Song

  2. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    Jian Xu

Authors
  1. ZiGen Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Song, Z., Xu, J. Stability switches and Bogdanov-Takens bifurcation in an inertial two-neuron coupling system with multiple delays. Sci. China Technol. Sci. 57, 893–904 (2014). https://doi.org/10.1007/s11431-014-5536-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-014-5536-y

Keywords

Search

Navigation