Skip to main content
SpringerLink
Log in

Feedback linearization control of chaos synchronization in coupled map-based neurons under external electrical stimulation

  • Regular Papers
  • Control Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

In this paper, the dynamics of single and two electrically coupled map-based neurons under external electrical stimulation is studied. In order to realize the synchronization of two chaotic spiking neurons, a controller based on the idea of feedback linearization is proposed. The simulation results demonstrate the effectiveness of this developed control method. An important feature of the feedback control is that the amplitude of control signal tends to zero as soon as the synchronization is achieved.

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. M. I. Rabinovich, P. Varona, A. I. Selverston, and H. D. I. Abarbanel, “Dynamical principles of neuroscience,” Rev. Mod. Phys., vol. 78, pp. 1213–1265, 2006.

    Article  Google Scholar 

  2. R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, “Synchronous behavior of two coupled biological neurons,” Phys. Rev. Lett., vol. 81, pp. 5692–5695, 1998.

    Article  Google Scholar 

  3. A. K. Kryukov, V. S. Petrov, L. S. Averyanova, G. V. Osipov, W. Chen, O. Drugova, and C. K. Chan, “Synchronization phenomena in mixed media of passive, excitable, and oscillatory cells,” Chaos, vol. 18, 037129, 2008.

    Article  MathSciNet  Google Scholar 

  4. S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, “The synchronization of chaotic systems,” Physics Reports, vol. 366, pp. 1–101, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. M. Gray, P. König, A. K. Engel, and W. Singer, “Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties,” Nature, vol. 338, pp. 334–337, 1989.

    Article  Google Scholar 

  6. R. Eckhorn, R. Bauer, W. Jordan, M. Brosch, W. Kruse, M. Munk, and H. J. Reitboeck, “Coherent oscillations: a mechanism of feature linking in the visual cortex? multiple electrode and correlation analyses in the cat,” Biol. Cybern., vol. 60, pp. 121–130, 1988.

    Article  Google Scholar 

  7. W. Singer, “Synchronization of cortical activity and its putative role in information processing and learning,” Annu. Rev. Physiol., vol. 55, pp. 349–374, 1993.

    Article  Google Scholar 

  8. W. A. MacKay, “Synchronized neuronal oscillations and their role in motor processes,” Trends Cogn. Sci., vol. 1, pp. 176–183, 1997.

    Article  Google Scholar 

  9. H. J. Freund, “Motor unit and muscle activity in voluntary motor control,” Physiol Rev., vol. 63, pp. 387–436, 1983.

    Google Scholar 

  10. R. Levy, W. D. Hutchison, A. M. Lozano, and J. O. Dostrovsky, “High-frequency synchronization of neuronal activity in the subthalamic nucleus of Parkinsonian patients with limb tremor,” J. Neurosci., vol. 20, pp. 7766–7775, 2000.

    Google Scholar 

  11. M. La Rosa, M. I. Rabinovicha, R. Huerta, H. D. I. Abarbanel, and L. Fortuna, “Slow regularization through chaotic oscillation transfer in an unidirectional chain of Hindmarsh-Rose models,” Phys. Lett. A, vol. 266, pp. 88–93, 2000.

    Article  Google Scholar 

  12. M. Dhamala, V. K. Jirsa, and M. Ding, “Enhancement of neural synchrony by time delay,” Phys. Rev. Lett. vol. 92, 074104, 2004.

    Article  Google Scholar 

  13. J. Wang, B. Deng, and K. M. Tsang, “Chaotic synchronization of neurons coupled with gap junction under external electrical stimulation,” Chaos, Solitons and Fractals. vol. 22, pp. 469–476, 2004.

    Article  MATH  Google Scholar 

  14. Q. Y. Wang, Q. S. Lu, G. R. Chen, and D. H. Guo, “Chaos synchronization of coupled neurons with gap junctions,” Phys. Lett. A, vol. 356, pp. 17–25, 2006.

    Article  MATH  Google Scholar 

  15. R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, “Synchronous behavior of two coupled biological neurons,” Phys. Rev. Lett., vol. 81, pp. 5692–5695, 1999.

    Article  Google Scholar 

  16. Y. Q. Che, J. Wang, S. S. Zhou, and B. Deng, “Robust synchronization control of coupled chaotic neurons under external electrical stimulation,” Chaos, Solitons and Fractals, vol. 40, pp. 1333–1342, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  17. O. Cornejo-Pérezand and R. Femat, “Unidirectional synchronization of Hodgkin-Huxley neurons,” Chaos, Solitons and Fractals, vol. 25, pp. 43–53, 2005.

    Article  MathSciNet  Google Scholar 

  18. J. Wang, B. Deng, and X. Y. Fei, “Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control,” Chaos, Solitons and Fractals, vol. 29, pp. 182–189, 2006.

    Article  Google Scholar 

  19. J. Wang, B. Deng, and X. Y. Fei, “Chaotic synchronization of two coupled neurons via nonlinear control in external electrical stimulation,” Chaos, Solitons and Fractals, vol. 27, pp. 1272–1278, 2006.

    Article  MATH  Google Scholar 

  20. C. A. S. Batista, S. R. Lopes, R. L. Viana, and A. M. Batista, “Delayed feedback control of bursting synchronization in a scale-free neuronal network,” Neural Networks, vol. 23, pp. 114–124, 2010.

    Article  Google Scholar 

  21. O. V. Popovych, C. Hauptmann, and P. A. Tass, “Control of neuronal synchrony by nonlinear delayed feedback,” Biol. Cybern., vol. 95, pp. 69–85, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” J. Physiol., vol. 117, pp. 500–544, 1952.

    Google Scholar 

  23. R. FitzHugh, “Mathematical models of threshold phenomena in the nerve membrane,” Bull. Math. Biophysics, vol. 17, pp. 257–278, 1955.

    Article  Google Scholar 

  24. R. M. Rose and J. L. Hindmarsh, “The assembly of ionic currents in a thalamic neuron I. the threedimensional model,” Proc. R. Soc. Lond. B, vol. 237, pp. 267–288, 1989.

    Article  Google Scholar 

  25. N. F. Rulkov, “Regularization of synchronized chaotic bursts,” Phys. Rev. Lett., vol. 86, pp. 183–186 2001

    Article  Google Scholar 

  26. N. F. Rulkov, “Modeling of spiking-bursting neural behavior using two-dimensional map,” Phys. Rev. E, vol. 65, 041922, 2002.

    Article  MathSciNet  Google Scholar 

  27. B. Ibarz, J. M. Casado, and M. A. F. Sanjuan, “Map-based models in neuronal dynamics,” Physics Reports, vol. 501, pp. 1–74, 2011.

    Article  Google Scholar 

  28. N. F. Rulkov, I Timofeev, and M. Bazhenov, “Oscillations in large-scale cortical networks: mapbased model,” J. Comput. Neurosci., vol. 17, pp. 203–223, 2004.

    Article  Google Scholar 

  29. M. Bazhenov, N. F. Rulkov, and I. Timofeev, “Effect of synaptic connectivity on long-range synchronization of fast cortical oscillations,” J. Neurophysiolgy, vol. 100, pp. 1562–1575, 2008.

    Article  Google Scholar 

  30. X. Shi and Q. Lu, “Burst synchronization of electrically and chemically coupled map-based neurons,” Physica A, vol. 388, pp. 2410–2419, 2009.

    Article  Google Scholar 

  31. H. Cao and M. A. F. Sanjuán, “A mechanism for elliptic-like bursting and synchronization of bursts in a map-based neuron network,” Cogn. Process., vol. 10, pp. 23–31, 2009.

    Article  Google Scholar 

  32. B. Ibarz, H. Cao, and M. A. F. Sanjuán, “Bursting regimes in map-based neuron models coupled through fast threshold modulation,” Phys. Rev. E, vol. 77, 051918, 2008.

    Article  MathSciNet  Google Scholar 

  33. G. Tanaka, B. Ibarz, M. A. F. Sanjuán, and K. Aihara, “Synchronization and propagation of bursts in network of map neurons,” Chaos, vol. 16, 013113, 2006.

    Article  MathSciNet  Google Scholar 

  34. B. Ibarz, G. Tanaka, M. A. F. Sanjuán, and K. Aihara, “Sensitivity versus resonance in twodimensional spiking-bursting neuron models,” Phys. Rev. E, vol. 75, 041902, 2007.

    Article  MathSciNet  Google Scholar 

  35. M. A. Stuchly and T. W. Dawson, “Interaction of low-frequency electric and magnetic fields with the human body,” Proc. IEEE, vol. 88, pp. 643–664, 2000.

    Article  Google Scholar 

  36. C. F. Blackman, S. G. Benane, D. J. Elliott, D. E. House, and M. M. Pollock, “Influence of electromagnetic fields on the efflux of calcium ions from brain tissue in vitro: a three-model analysis consistent with the frequency response up to 510 Hz,” Bioelectromagnetics, vol. 9, pp. 215–327, 1988.

    Article  Google Scholar 

  37. T. Kotnik, D. Miklavčič, and T. Slivnik, “Time course of transmembrane voltage induced by timevarying electric fields-a method for theoretical analysis and its application,” Bioelectrochem. Bioenerg., vol. 45, pp. 3–16, 1998.

    Article  Google Scholar 

  38. T. Kotnik and D. Miklavcic, “Theoretical evaluation of the distributed power dissipation in biological cells exposed to electric fields,” Bioelectromagnetics, 2000, vol. 21, pp. 385–394, 2000.

    Article  Google Scholar 

  39. T. Zhang, J. Wang, X. Y. Fei, and B. Deng, “Synchronization of coupled FitzHugh-Nagumo systems via MIMO feedback linearization control,” Chaos, Solitons and Fractals, vol. 33, pp. 194–202, 2007.

    Article  Google Scholar 

  40. C. C. Fuh and H. H. Tsai, “Control of discrete-time chaotic systems via feedback linearization,” Chaos, Solitons and Fractals, vol. 13, pp. 285–294, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  41. K. Nam, “Linearization of discrete-time nonlinear systems and a canonical structure,” IEEE Trans Automatic Control, vol. 34, pp. 119–122, 1989.

    Article  MathSciNet  Google Scholar 

  42. B. Jakubczyk, “Feedback linearization of discretetime systems,” Systems and Control Letters, vol. 9, pp. 411–416, 1987.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electrical Engineering and Automation, Tianjin University, Tianjin, China

    Hai-Tao Yu & Jiang Wang

  2. Department of Electrical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

    Yiu Kwong Wong, Wai Lok Chan & Kai Ming Tsang

Authors
  1. Hai-Tao Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yiu Kwong Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wai Lok Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kai Ming Tsang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yiu Kwong Wong.

Additional information

Recommended by Editorial Board member Yoshito Ohta under the direction of Editor Jae Weon Choi. This work was supported by the Key National Natural Science Foundation of China Grant 50537030, the National Natural Science Foundation of China Grant 50707020, the Postdoctoral Science Foundation of China Grant 20070410756 and the Hong Kong Polytechnic University Grant G-U488.

Hai-Tao Yu received his B.S. degree from the Hebei University of Technology, Tianjin, China, and his M.S. degree from the Tianjin University, Tianjin, China. He is currently a Ph.D. candidate at the School of Electrical Engineering and Automation, Tianjin University, Tianjin, China. His current research interests include nonlinear control and neural control engineering.

Yiu Kwong Wong received his BSc and MSc degrees from the University of London, and his Ph.D. degree from the Heriot-Watt University, UK. His current research interests include modeling, simulation, nonlinear control and intelligent control.

Wai Lok Chan received his BSc(Eng) and MPhil degrees from University of Hong Kong, and his Ph.D. degree from City University London. He is now an Associate Professor in the Department of Electrical Engineering, The Hong Kong Polytechnic University. His major research interests are in microprocessor applications and applications of artificial intelligence.

Kai Ming Tsang received his B.Eng. and Ph.D. degrees in Control Engineering from the University of Sheffield, U.K. At present, he is an Associate Professor in the Department of Electrical Engineering of the Hong Kong Polytechnic University. His research interests include system identification, fuzzy logic, adaptive control and pattern recognition.

Jiang Wang received his B.S., M.S. and Ph.D. degrees from the Tianjin University, Tianjin, China. He is currently a Professor at the School of Electrical Engineering and Automation, Tianjin University, Tianjin, China. His current research interests include process control, nonlinear control and neural control engineering.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yu, HT., Wong, Y.K., Chan, W.L. et al. Feedback linearization control of chaos synchronization in coupled map-based neurons under external electrical stimulation. Int. J. Control Autom. Syst. 9, 867–874 (2011). https://doi.org/10.1007/s12555-011-0507-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-011-0507-6

Keywords

Search

Navigation