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Synchronization of a set of coupled chaotic FitzHugh–Nagumo and Hindmarsh–Rose neurons with external electrical stimulation

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Abstract

This paper addresses problem of synchronization of n-neurons which are electrically coupled to each other. Gap junction coupled chaotic FitzHugh–Nagumo (FHN) neurons and linearly coupled Hindmarsh–Rose (H–R) neurons are considered with each neuron subjected to external electrical stimulation. The neurons are assumed to be linked in a chain-like structure. It is shown that the suitable choice of coupling strength of gap junction is sufficient to meet out the synchronization condition. Simple stability analysis based on partial contraction approach is used to establish the exponential convergence of states of different neurons to each other. The conditions for complete synchronization are derived analytically for a general system having n-electrically coupled neurons. The advantage of the proposed approach lies in its simplicity and provides an alternative method of achieving synchronization. Further, in comparison with Lyapunov-based analysis, the formulation of error dynamics is avoided while establishing synchronizing conditions. Numerical simulations are shown for both FHN and Hindmarsh–Rose neurons separately to justify the effectiveness of the proposed approach to meet out synchronization requirements.

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Notes

  1. The system given in (12) represents auxiliary system for the actual system in (11) because replacement of \(\left( \xi _{i}, \eta _{i} \right) \) pair of states by \(\left( x_{i},y_{i} \right) \) pair results in actual system. In other words, actual system (11) represents one particular solution of auxiliary system in (12).

References

  1. Ott, E., Grebogi, C., York, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Jackson, E.A., Hubler, A.: Periodic entrainment of chaotic logistic map dynamics. Phys. D 44, 404–409 (1990)

    MathSciNet  MATH  Google Scholar 

  3. Ogorzalek, M.J.: Taming chaos: Part II- control scheme via robust asymptotic feedback. IEEE Trans. Circ. Syst. I 40(10), 700–706 (1993)

    Article  MATH  Google Scholar 

  4. Femat, R., Alvarez, J., Castillo-Toledo, B., Gonzalez, J.: On robust chaos suppression in a class of no driven oscillators: application to the Chua’s circuit. IEEE Trans. Circ. Syst. 46, 1150–52 (1999)

    Article  MATH  Google Scholar 

  5. Femat, R., Jauregui-Ortiz, R., Solys-Perales, G.: A chaos based communication scheme via robust asymptotic feedback. IEEE Trans. Circ. Syst. 48(10), 1161–1169 (2001)

    Article  Google Scholar 

  6. Isidori, A.: Non-linear Control Systems. Springer, Berlin (1995)

    MATH  Google Scholar 

  7. Jayaram, A., Tadi, M.: Synchronization of chaotic systems based on SDRE method. Chaos Solitons Fractals 28(3), 707–715 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sharma, B.B., Kar, I.N.: Parametric convergence and control of chaotic system using adaptive feedback linearization. Chaos Solitons Fractals 40, 1475–1483 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yassen, M.T.: Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos Solitons Fractals 26, 913–920 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu, J., Chen, S., Chen, L.: Adaptive control for anti-synchronization of Chua’s chaotic system. Phys. Lett. A 339, 455–460 (2005)

    Article  MATH  Google Scholar 

  12. Li, D., Lu, J.A., Wu, X.: Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos Solitons Fractals 23, 79–85 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grassi, G., Mascolo, S.: Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal. IEEE Trans. Cric. Syst. I 44(10), 1011–1014 (1997)

    Google Scholar 

  14. Liao, T.L., Huang, N.S.: An observer based approach for chaotic synchronization and secure communication. IEEE Trans. Circ. Syst. I 46(9), 1144–1149 (1999)

    Article  MATH  Google Scholar 

  15. Boutayeb, M., Darouach, M., Rafaralahy, H.: Generalized state-space observers for chaotic synchronization with applications to secure communication. IEEE Trans. Circ. Syst. I 49(3), 345–349 (2002)

    Article  MathSciNet  Google Scholar 

  16. Park, J.H.: Stability criterion for synchronization of linearly coupled unified chaotic systems. Chaos Solitons Fractals 23, 1319–1325 (2005)

    Article  MATH  Google Scholar 

  17. Wang, Y., Guan, Z.H., Wang, H.O.: Feedback and adaptive control for the synchronization of Chen system via a single variable. Phys. Lett. A 312, 34–40 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, M., Han, Z.: Controlling and synchronizing chaotic Genesio system via nonlinear feedback control. Chaos Solitons Fractals 17, 709–716 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lu, J., Zhou, T., Zhang, S.: Chaos synchronization between linearly coupled chaotic systems. Chaos Solitons Fractals 14, 529–541 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Huang, X., Cao, J., Li, Y.: Takagi-Sugeno fuzzy-model-based control of hyperchaotic Chen system with norm-bounded uncertainties. J. Syst. Control Eng. 224(03), 223–234 (2010)

    Google Scholar 

  21. Pai, M.C.: Robust synchronization of chaotic systems using adaptive sliding mode output feedback control. J. Syst. Control Eng. 226(05), 598–605 (2012)

    Google Scholar 

  22. Mirollo, R.E., Strogatz, S.H.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50, 1645–1662 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nischwitz, A., Glunder, H.: Local lateral inhibition-a key to spike synchronization. Biol. Cyber-net 73, 389–400 (1995)

    Article  MATH  Google Scholar 

  24. Bennett, M.V.L., Verselis, V.K.: Biophysics of gap junction. Seminars Cell Biol. 3, 29–47 (1992)

    Article  Google Scholar 

  25. Baigent, S., Stark, J., Warner, A.: Modeling the effect of gap junction nonlinearities in systems of coupled cells. J. Theor. Biol. 186, 223–239 (1997)

    Article  Google Scholar 

  26. Baigent, S., Stark, J., Warner, A.: Convergent dynamics of two cells coupled by a nonlinear gap junction. Nonlinear Anal. 47, 257–268 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Baigent, S.: Cells coupled by voltage-dependent gap junctions: the asymptotic dynamical limit. Biosystems 68, 213–222 (2003)

    Article  Google Scholar 

  28. Thompson, C.J., Bardos, D.C., Yang, Y.S., Joyner, K.H.: Nonlinear cable model for cells exposed to electric fields I. Chaos Solitons Fractals 10(11), 1825–1842 (1999)

    Article  MATH  Google Scholar 

  29. Shuai, J.W., Durand, D.M.: Phase synchronization in two coupled chaotic neurons. Phys. Lett. A 264, 289–297 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, J., Bin, D., Tsang, K.M.: Chaotic synchronization of neuron coupled with gap junction under external electrical stimulation. Chaos Solitons Fractals 22, 469–476 (2004)

    Article  MATH  Google Scholar 

  31. Bin, D., Jiang, W., Xiangyang, F.: Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control. Chaos Solitons Fractals 29, 182–189 (2006)

    Article  Google Scholar 

  32. Che, Y.Q., Wang, J., Zhou, S.S., Bin, D.: Robust synchronization control of coupled chaotic neurons under external electrical stimulation. Chaos Solitons Fractals 40, 1333–1342 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yu, H., Wang, J., Liu, C., Deng, B., Wei, X.: Vibrational resonance in excitable neuronal systems. Chaos (2011). doi:10.1063/1.3644390

  34. Yu, H., Wang, J., Deng, B., Wei, X., Wong, Y.K., Chan, W.L., Tsang, K.M.: Chaotic phase synchronization in small-world networks of bursting neurons. Chaos 21, 013127 (2011). doi:10.1063/1.3565027

    Article  MathSciNet  Google Scholar 

  35. Yang, D.: Self-synchronization of coupled chaotic FitzHugh–Nagumo systems with unreliable communication links. Commun. Nonlinear Sci. Numer. Simul. 18(10), 2783–2789 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ciszak, M., Euzzor, S., Gelrude, A., Tito, F., Meucci, R.: Noise and coupling induced synchronization in a network of chaotic neurons. Commun. Nonlinear Sci. Numer. Simul. 18(4), 938–945 (2013)

    Article  MathSciNet  Google Scholar 

  37. Iqbal, M., Rehan, M., Khaliq, A., Rehman, S.U., Hong, K.S.: Synchronization of coupled different chaotic FitzHugh–Nagumo neurons with unknown parameters under communication-direction-dependent coupling. Commun. Nonlinear Sci. Numer. Simul. 2014, 1–12 (2014)

    MATH  Google Scholar 

  38. Sharma, B.B., Kar, I.N.: Observer-based synchronization scheme for a class of chaotic systems using contraction theory. Nonlinear Dyn. 63(3), 429–445 (2011)

    Article  MathSciNet  Google Scholar 

  39. Aminzarey, Z., Sontag, E.D.: Contraction methods for nonlinear systems: A brief introduction and some open problems. IEEE Conference on Decision and Control, pp. 3835–3847 (2014)

  40. Aylward, E.M., Parrilo, P.A., Slotine, J.J.E.: Stability and robustness analysis of nonlinear systems via contraction metrics and sos programming. Automatica 44(8), 2163–2170 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Angeli, D.: A Lyapunov approach to incremental stability properties. IEEE Trans. Aut. Control 47(3), 410–421 (2002)

    Article  MathSciNet  Google Scholar 

  42. Fromian, V., Scorletti, G., Ferreres, G.: Nonlinear performance of a PI controlled missile: an explanation. Int. J. Robust Nonlinear Control 9(8), 485–518 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109–2112 (1998)

    Article  Google Scholar 

  44. Heagy, J.F., Carroll, T.L., Pecora, L.M.: Synchronous chaos in coupled oscillator systems. Phys. Rev. E 50(3), 1874–1885 (1994)

    Article  Google Scholar 

  45. Wang, W., Slotine, J.J.E.: On partial contraction analysis for coupled nonlinear oscillators. Biol. Cybern. 92, 1 (2004)

    MathSciNet  Google Scholar 

  46. Sanjaya, M., Mamat, M., Salleh, Z., Mohd, I.: Bidirectional chaotic synchronization of Hindmarsh–Rose neuron model. Appl. Math. Sci. 5(54), 2685–2695 (2011)

    MathSciNet  MATH  Google Scholar 

  47. Linaro, D., Righero, M., Biey, M., Storace, M.: Synchronization properties in networks of Hindmarsh–Rose neurons and their PWL approximations with linear symmetric coupling, IEEE international symposium on circuits and systems, pp. 1685–1688 (2009)

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  1. Electrical Engineering Department, National Institute of Technology, Hamirpur, 177005, Himachal Pradesh, India

    Himesh Handa & B. B. Sharma

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  1. Himesh Handa
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Correspondence to Himesh Handa.

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Handa, H., Sharma, B.B. Synchronization of a set of coupled chaotic FitzHugh–Nagumo and Hindmarsh–Rose neurons with external electrical stimulation. Nonlinear Dyn 85, 1517–1532 (2016). https://doi.org/10.1007/s11071-016-2776-3

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