Skip to main content
SpringerLink
Log in

Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

  • Research Article
  • Published:
Cognitive Neurodynamics Aims and scope Submit manuscript

Abstract

In this paper we consider the Hopf bifurcation and synchronization in the two coupled Hindmarsh–Rose excitable systems with chemical coupling and time-delay. We surveyed the conditions for Hopf bifurcations by means of dynamical bifurcation analysis and numerical simulation. The results show that the coupled excitable systems with no delay have supercritical Hopf bifurcation, while the delayed system undergoes Hopf bifurcations at critical time delays when coupling strength lies in a particular region. We also investigated the effect of the delay on the transition of bursting synchronization in the coupled system. The results are helpful for us to better understand the dynamical properties of excitable systems and the biological mechanism of information encoding and cognitive activity.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Belykh I, Lange E, Hasler M (2005) Synchronization of bursting neurons: what matters in the network topology. Phys Rev Lett 94(18):188101

    Article  PubMed  Google Scholar 

  • Burić N, Ranković D (2007) Bursting neurons with coupling delays. J Phys Lett 363(4):282–289

    Article  Google Scholar 

  • Burić N, Todorović D (2003) Dynamics of FitzHugh–Nagumo excitable systems with delayed coupling. Phys Rev E 67(6):066222

    Article  Google Scholar 

  • Burić N, Grozdanović I, Vasović N (2005) Type I vs. type II excitable systems with delayed coupling. Chaos, Solitons Fractals 23(4):1221–1233

    Google Scholar 

  • Burić N, Todorović K, Vasović N (2008) Synchronization of bursting neurons with delayed chemical synapses. Phys Rev E 78(3):036211

    Article  Google Scholar 

  • Dhamala M, Jirsa VK, Ding MZ (2004) Enhancement of neural synchrony by time delay. Phys Rev Lett 92(7):74104

    Article  Google Scholar 

  • Engelborghs K, Luzyanina T, Samaey G (2007) DDE-BIFTOOL: a Matlab package for bifurcation analysis of delay differential equations. http://twr.cs.kuleuven.be/research/software/delay/ddebiftool.shtml

  • Faria T (2000) On a planer system modelling a neuron network with memory. J Differ Equ 168(1):129–149

    Article  Google Scholar 

  • Lu QS (2010) Ordinary differential equation and dynamical system. Beijing Aerospace University Press, Beijing

    Google Scholar 

  • Neefs PJ, Steur E, Nijmeijer H (2010) Network complexity and synchronous behavior—an experimental approach. Int J Neural Syst 20(3):233–247

    Article  PubMed  CAS  Google Scholar 

  • Ranković D (2011) Bifurcations of Fitzhugh–Nagumo excitable systems with chemical delayed coupling. Matematiqki Vesnik 63:103–114

    Google Scholar 

  • Shi X, Lu QS (2004) Complete synchronization of coupled Hindmarsh–Rose neurons with ring structure. Chin Phys Lett 21(9):1695–1698

    Article  Google Scholar 

  • Somers D, Kopell N (1993) Rapid synchronization through fast threshold modulation. J Biol Cybern 68(5):393–407

    Article  CAS  Google Scholar 

  • Vasović N, Burić N, Todorović K, Grozdanović I (2012) Synchronization of the minimal models of bursting neurons coupled by delayed chemical or electrical synapses. J Chin Phys 21(1):10203

    Article  Google Scholar 

  • Wang HX, Lu QS, Wang QY (2007) Bursting and synchronization transition in the coupled modified ML neurons. J Commun Nonlinear Sci Numer Simul 13(8):1668–1675

    Article  Google Scholar 

  • Wang QY, Lu QS, Chen GR, Duan LX (2009) Bifurcation and synchronization of synaptically coupled FHN models with time delay. Chaos, Solitons Fractals 39(2):918–925

    Article  Google Scholar 

  • Yuan GY, Zhang GC, Wang GR, Chen SG (2005) Synchronization and asynchronization in two coupled excitable systems. Commun Theor Phys 43(3):459–465

    Article  Google Scholar 

  • Zhen B, Xu J (2010) Simple zero singularity analysis in a coupled FitzHugh–Nagumo neural system with delay. Neurocomputing 73(4–6):874–882

    Article  Google Scholar 

Download references

Acknowledgments

The Project was supported by the NSFC (Grant No.10902001 and 11072013).

Author information

Authors and Affiliations

  1. School of Science, North China University of Technology, Beijing, 100144, China

    Lixia Duan & Denggui Fan

  2. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019, USA

    Lixia Duan

  3. Department of Dynamics and Control, Beihang University, Beijing, 100191, China

    Qishao Lu

Authors
  1. Lixia Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Denggui Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qishao Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lixia Duan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Duan, L., Fan, D. & Lu, Q. Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling. Cogn Neurodyn 7, 341–349 (2013). https://doi.org/10.1007/s11571-012-9237-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11571-012-9237-6

Keywords

Search

Navigation