Skip to main content
SpringerLink
Log in

Stabilizing control of Hopf bifurcation in the Hodgkin–Huxley model via washout filter with linear control term

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

It is a significant issue to control bifurcation because many neuronal diseases have close relevance to bifurcation of neuron system. Some studies have been done on bifurcation control in the Hodgkin–Huxley (HH) model, but there is no clear mathematical criterion for bifurcation stabilization. In this paper, according to Routh–Hurwitz stability criterion, we employ linear control term of washout filter-aided dynamic feedback controller to stabilize bifurcation of the HH model. As a result, we can deduce linear control gain based on the criterion, and simulation shows the method is effective for making the HH model stable. The controller designs described here are achieved by electrical stimulus, so it may have potential applications in the diagnosis and therapy of dynamical diseases.

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. Durand, D.M., Bikson, M.: Suppression and control of epileptiform activity by electrical stimulation: A review. Proc. IEEE 89, 1065–1082 (2001)

    Article  Google Scholar 

  2. Fröhlich, F., Jezernik, S.: Feedback control of Hodgkin–Huxley nerve cell dynamics. Control Eng. Pract. 10, 008–019 (2004)

    Google Scholar 

  3. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)

    Google Scholar 

  4. Hodgkin, A.L., Huxley, A.F.: Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J. Physiol. 116, 449–472 (1952)

    Google Scholar 

  5. Hodgkin, A.L., Huxley, A.F.: The components of membrane conductance in the giant axon of Loligo. J. Physiol. 116, 473–496 (1952)

    Google Scholar 

  6. Brandt, M.E., Chen, G.: Bifurcation control of two nonlinear models of cardiac activity. IEEE Trans. Circ. Syst.—I 44, 1031–1034 (1997)

    Article  Google Scholar 

  7. Ito, K.: Stabilization of the dynamic elastica only by damping torque. Nonlinear Anal.: Real World Appl. 10, 3122–3131 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Messaoudi, S.A., Mustafa, M.I.: On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal.: Real World Appl. 10, 3132–3140 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Wang, H.O., Abed, E.H.: Bifurcation control of a chaotic system. Automatica 31, 1213–1226 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, Z., Yu, P.: Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions. Chaos Solitons Fractals 26, 1231–1248 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Berns, D.W., Moiola, J.L., Chen, G.: Feedback control of limit cycle amplitudes from a frequency domain approach. Automatica 34, 1567–1573 (1998)

    Article  MATH  Google Scholar 

  12. Luo, G.W., Lv, X.H.: Controlling bifurcation and chaos of a plastic impact oscillator. Nonlinear Anal.: Real World Appl. 10, 2048–2061 (2009)

    Google Scholar 

  13. Xie, Y., Chen, L.N., Kang, Y.M., Aihara, K.: Controlling the onset of Hopf bifurcation in the Hodgkin–Huxley model. Phys. Rev. E 77, 061921 (2008)

    Article  MathSciNet  Google Scholar 

  14. Wang, J., Chen, L.Q., Fei, X.Y.: Bifurcation control of the Hodgkin–Huxley equations. Chaos Solitons Fractals 33, 217–224 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wang, J., Tsang, K.M., Zhang, H.: Hopf bifurcation in the Hodgkin–Huxley model exposed to ELF electrical field. Chaos Solitons Fractals 20, 759–764 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Wang, J., Che, Y.Q., Fei, X.Y., Li, L.: Multi-parameter Hopf-bifurcation in Hodgkin–Huxley model exposed to ELF external electric field. Chaos Solitons Fractals 26, 1221–1229 (2005)

    Article  MATH  Google Scholar 

  17. Hidekazui, F., Doi, S., Nomura, T., Sato, S.: Hopf bifurcations in multiple-parameter space of the Hodgkin–Huxley equations I. Global organization of bistable periodic solutions. Biol. Cybern. 82, 215–222 (2000)

    Article  Google Scholar 

  18. Stuchly, M.A., Dawson, T.W.: Interaction of low-frequency electric and magnetic fields with the human body. Proc. IEEE 47, 1974–1981 (2000)

    Google Scholar 

  19. Wang, J., Zhang, T., Che, Y.Q.: Chaos control and synchronization of two neurons exposed to ELF external electric field. Chaos Solitons Fractals 34, 839–850 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lee, H.C., Abed, E.H.: Washout filters in the bifurcation control of high alpha flight dynamics. In: Proceedings of the American Control Conference, Boston, pp. 206–211 (1991)

  21. Liu, W.M.: Criterion of Hopf bifurcation without using eigenvalues. J. Math. Anal. Appl. 182, 250–255 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hurwitz, A.: On the conditions under which an equation has only roots with negative real parts. Math. Ann. 46, 273–284 (1895)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electronic and Information Eng., Tianjin University, Tianjin, 300072, China

    L. Ding & C. Hou

Authors
  1. L. Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Ding.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ding, L., Hou, C. Stabilizing control of Hopf bifurcation in the Hodgkin–Huxley model via washout filter with linear control term. Nonlinear Dyn 60, 131–139 (2010). https://doi.org/10.1007/s11071-009-9585-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-009-9585-x

Keywords

Search

Navigation