Skip to main content
SpringerLink
Log in

Different types of bursting in Chay neuronal model

  • Published:
Science in China Series G: Physics, Mechanics and Astronomy Aims and scope Submit manuscript

Abstract

Based on actual neuronal firing activities, bursting in the Chay neuronal model is considered, in which V K, reversal potentials for K+, V C, reversal potentials for Ca2+, time kinetic constant λ n and an additional depolarized current I are considered as dynamical parameters. According to the number of the Hopf bifurcation points on the upper branch of the bifurcation curve of fast subsystem, which is associated with the stable limit cycle corresponding to spiking states, different types of bursting and their respective dynamical behavior are surveyed by means of fast-slow dynamical bifurcation analysis.

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. Zhou J, Meng R, Sui X H. Various tolerances to arsenic trioxide between human cortical neurons and leukemic cells. Sci China Ser C-Life Sci, 2006, 49(6): 567–572

    Article  Google Scholar 

  2. Liu X M, Chen S, Zhang Y X. Modulation of dragon’s blood on tetrodotoxin-resistant sodium currents in dorsal root ganglion neurons and identification of its material basis for efficacy. Sci China Ser C-Life Sci, 2006, 49(3): 274–285

    Article  Google Scholar 

  3. Deschenes M, Roy J P, Steriade M. Thalamic bursting mechanism: An invariant slow current revealed by membrane hyperpolarization. Brain Res, 1982, 239: 289–293

    Article  Google Scholar 

  4. Harris-Warrick R M, Flamm R E. Multiple mechanisms of bursting in a conditional bursting neuron. J Neurosci, 1987, 7: 2113–2128

    Google Scholar 

  5. Ashcroft F, Rorsman P. Electrophysiology of the pancreatic β-cell. Prog Biophys Molec Biol, 1989, 54: 87–143

    Article  Google Scholar 

  6. Johnson S W, Seutin V, North R A. Burst firing in dopamine neurons induced by N-Methyl-D-Aspartate: Role of electrogenic sodium pump. Science, 1992, 258: 665–667

    Article  ADS  Google Scholar 

  7. Rinzel J. Bursting oscillation in an excitable membrane model. In: Sleeman B D, Jarvis R J, eds. Ordinary and Partial Differential Equations. Berlin: Springer-Verlag, 1985. 304–316

    Chapter  Google Scholar 

  8. Rinzel J. A formal classification of bursting mechanisms in excitable systems. In: Teramoto E, Yamaguti M, eds. Mathematical Topics in Population Biology, Morphogenesis and Neurosciences. Berlin: Springer-Verlag, 1987. 267–281

    Google Scholar 

  9. Sherman A, Rinzel J. Rhythmogenic effects of weak electrotonic coupling in neuronal model. Proc Natl Acad Sci USA, 1992, 89: 2471–2474

    Article  ADS  Google Scholar 

  10. Rinzel J, Lee Y S. Dissection of a model for neuronal parabolic bursting. J Math Biol, 1987, 25: 653–675

    Article  MATH  MathSciNet  Google Scholar 

  11. Av-Ron E, Parnas H, Segel L. A basic biophysical model for bursting neurons. Biol Cybern, 1993, 69: 87–95

    Article  Google Scholar 

  12. Bertram R, Butte M J, Kiemel T, et al. Topological and phenomenological classification of bursting oscillations. Bull Math Biol, 1995, 57: 413–439

    MATH  Google Scholar 

  13. Holden L, Erneux T. Slow passage through a Hopf bifurcation: Form oscillatory to steady state solutions. SIAM J Appl Math, 1993, 53: 1045–1058

    Article  MATH  MathSciNet  Google Scholar 

  14. Holden L, Erneux T. Understanding bursting oscillations as periodic slow passages through bifurcation and limit points. J Math Biol, 1993, 31: 351–365

    Article  MATH  MathSciNet  Google Scholar 

  15. Smolen P, Terman D, Rinzel J. Properties of a bursting model with two slow inhibitory variables. SIAM J Appl Math, 1993, 53: 861–892

    Article  MATH  MathSciNet  Google Scholar 

  16. Pernarowski M. Fast subsystem bifurcations in a slowly varied Lienard system exhibiting bursting. SIAM J Appl Math, 1994, 54: 814–832

    Article  MATH  MathSciNet  Google Scholar 

  17. De Vries G. Multiple bifurcations in a polynomial model of bursting oscillations. J Nonlin Sci, 1998, 8: 281–316

    Article  MATH  Google Scholar 

  18. Rush M E, Rinzel J. Analysis of bursting in thalamic neuron model. Biol Cybern, 1994, 71: 281–291

    Article  MATH  Google Scholar 

  19. Kepecs A, Wang X J. Analysis of complex bursting in cortical pyramidal neuron models. Neurocomputing, 2000, 32–33: 81–187

    Google Scholar 

  20. Soto-Trevino C, Kopell N, Watson D. Parabolic bursting revisited. J Mat Biol, 1996, 35: 114–128

    Article  MATH  MathSciNet  Google Scholar 

  21. Chay T R, Fan Y S, Lee Y S. Bursting, spiking, chaos, fractals, and university in biological rhythms, Int J Bif Chaos, 1995, 5: 595–635

    Article  MATH  Google Scholar 

  22. Booth V, Carr T W, Erneux T. Near-threshold bursting is delayed by a slow passage near a limit point. SIAM J Appl Math, 1997, 57: 1406–1420

    Article  MATH  MathSciNet  Google Scholar 

  23. Izhikevich E M. Neural excitability, spiking and bursting. Int J Bif Chaos, 2000, 10: 1171–1266

    Article  MATH  MathSciNet  Google Scholar 

  24. Yang Z Q, Lu Q S. Gu H G, et al. Gwn-induced bursting, spiking, and random subthreshold impulsing oscillation before Hopf bifurcations in the Chay model. Int J Bif Chaos, 2004, 14: 4143–4159

    Article  MATH  MathSciNet  Google Scholar 

  25. Yang Z Q, Lu Q S. The integer multiple “fold/homoclinic” bursting induced by noise in the Chay neuronal model. Int J Nonlinear Sci Numer Simul, 2005, 6: 1–6

    MathSciNet  Google Scholar 

  26. Yang Z Q, Lu Q S. Transitions from bursting to spiking due to depolarizing current in Chay neuronal model. Commun Nonlinear Sci Numer Simul, 2007, 12: 357–365

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Chay T R. Chaos in a three-variable model of an excitable cell. Physica D, 1985, 16: 233–242

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science & LMIB of Ministry of Education of China, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China

    ZhuoQin Yang & QiShao Lu

Authors
  1. ZhuoQin Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. QiShao Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to ZhuoQin Yang or QiShao Lu.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 10432010, 10526002 and 10702002)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, Z., Lu, Q. Different types of bursting in Chay neuronal model. Sci. China Ser. G-Phys. Mech. Astron. 51, 687–698 (2008). https://doi.org/10.1007/s11433-008-0069-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11433-008-0069-7

Keywords

Search

Navigation