Nonlinear Dynamics

, Volume 92, Issue 4, pp 1695–1706 | Cite as

AC-induced coexisting asymmetric bursters in the improved Hindmarsh–Rose model

  • Bocheng BaoEmail author
  • Aihuang Hu
  • Quan Xu
  • Han Bao
  • Huagan Wu
  • Mo Chen
Original Paper


In this paper, an external alternating current (AC) is injected into the Hindmarsh–Rose (HR) neuron model to imitate the periodic stimulus effect on the membrane potential in the axon of a neuron and then an improved HR model is proposed. The AC equilibrium point and its stability in the proposed model are investigated theoretically, and the AC-induced coexisting behaviors of asymmetric bursters are revealed by MATLAB numerical simulations. Due to the injection of the AC item, the stability distribution of the unique AC equilibrium point in the improved HR model varies between unstable and stable intervals with the periodic evolution of the time, which leads to the emergence of various types of coexisting asymmetric bursters under different initial conditions of the bursting variable, such as hyperchaotic and periodic bursters, chaotic and periodic bursters, quasiperiodic and periodic bursters, two periodic bursters with different periodicities, and so on. Additionally, a simulated circuit model is designed and PSIM circuit simulations are performed to exhibit coexisting behaviors of asymmetric bursters, which effectively confirm the numerically simulated results.


Improved Hindmarsh–Rose (HR) model Alternating current (AC) Coexisting asymmetric bursters Stability distribution 



This work was supported by the grants from the National Natural Science Foundations of China under Grant Nos. 51777016, 51607013, 61601062, and 11602035, and the Natural Science Foundations of Jiangsu Province, China, under Grant No. BK20160282.


  1. 1.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)CrossRefGoogle Scholar
  2. 2.
    Ozer, M., Uzuntarla, M., Perc, M., Graham, L.J.: Spike latency and jitter of neuronal membrane patches with stochastic Hodgkin–Huxley channels. J. Theor. Biol. 261(1), 83–92 (2009)CrossRefGoogle Scholar
  3. 3.
    Ozer, M., Perc, M., Uzuntarla, M.: Stochastic resonance on Newman–Watts networks of Hodgkin–Huxley neurons with local periodic driving. Phys. Lett. A 373(10), 964–968 (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Xu, Y., Jia, Y., Ge, M.Y., Lu, L.L., Yang, L.J., Zhan, X.: Effects of ion channel blocks on electrical activity of stochastic Hodgkin–Huxley neural network under electromagnetic induction. Neurocomputing (2017).
  5. 5.
    Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B Biol. Sci. 221(1222), 87–102 (1984)CrossRefGoogle Scholar
  6. 6.
    Gu, H.G., Pan, B.B.: A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model. Nonlinear Dyn. 81(4), 2107–2126 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Wu, X.Y., Ma, J., Yuan, L.H., Liu, Y.: Simulating electric activities of neurons by using PSPICE. Nonlinear Dyn. 75(1–2), 113–126 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Upadhyay, R.K., Mondal, A., Teka, W.W.: Mixed mode oscillations and synchronous activity in noise induced modified Morris–Lecar neural system. Int. J. Bifurc. Chaos 27(5), 1730019 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ren, G.D., Zhou, P., Ma, J., Cai, N., Alsaedi, A., Ahmad, B.: Dynamical response of electrical activities in digital neuron circuit driven by autapse. Int. J. Bifurc. Chaos 27(12), 1750187 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ren, G.D., Xu, Y., Wang, C.N.: Synchronization behavior of coupled neuron circuits composed of memristors. Nonlinear Dyn. 88(2), 893–901 (2017)CrossRefGoogle Scholar
  11. 11.
    Ma, J., Xu, Y., Wang, C.N., Jin, W.Y.: Pattern selection and self-organization induced by random boundary initial values in a neuronal network. Phys. A 461, 586–594 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    González-Miranda, J.M.: Complex bifurcation structures in the Hindmarsh–Rose neuron model. Int. J. Bifurc. Chaos 17(9), 3071–3083 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Innocenti, G., Morelli, A., Genesio, R., Torcini, A.: Dynamical phases of the Hindmarsh–Rose neuronal model: studies of the transition from bursting to spiking chaos. Chaos 17(4), 043128 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gu, H.G., Pan, B.B., Chen, G.R., Duan, L.X.: Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn. 78(1), 391–407 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    González-Miranda, J.M.: Observation of a continuous interior crisis in the Hindmarsh–Rose neuron model. Chaos 13(3), 845–852 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Thottil, S.K., Ignatius, R.P.: Nonlinear feedback coupling in Hindmarsh–Rose neurons. Nonlinear Dyn. 87, 1879–1899 (2017)CrossRefGoogle Scholar
  17. 17.
    Innocenti, G., Genesio, R.: On the dynamics of chaotic spiking–bursting transition in the Hindmarsh–Rose neuron. Chaos 19(2), 023124 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ngouonkadi, E.B.M., Fotsin, H.B., Fotso, P.L., Tamba, V.K., Cerdeira, H.A.: Bifurcations and multistability in the extended Hindmarsh–Rose neuronal oscillator. Chaos Solitons Fractals 85, 151–163 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wu, K.J., Luo, T.Q., Lu, H.W., Wang, Y.: Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model. Neural Comput. Appl. 27(3), 739–747 (2016)CrossRefGoogle Scholar
  20. 20.
    Gu, H.: Biological experimental observations of an unnoticed chaos as simulated by the Hindmarsh–Rose model. PLoS ONE 8(12), e81759 (2013)CrossRefGoogle Scholar
  21. 21.
    Djeundam, S.R.D., Yamapi, R., Kofane, T.C., Azizalaoui, M.A.: Deterministic and stochastic bifurcations in the Hindmarsh–Rose neuronal model. Chaos 23(3), 033125 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kaslik, E.: Analysis of two- and three-dimensional fractional-order Hindmarsh–Rose type neuronal models. Frac. Calc. Appl. Anal. 20(3), 623–645 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Dong, J., Zhang, G.J., Xie, Y., Yao, H., Wang, J.: Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model. Cogn. Neurodyn. 8(2), 167–175 (2014)CrossRefGoogle Scholar
  24. 24.
    Lakshmanan, S., Lim, C.P., Nahavandi, S., Prakash, M., Balasubramaniam, P.: Dynamical analysis of the Hindmarsh–Rose neuron with time delays. IEEE Trans. Neural Netw. Learn. 28(8), 1953–1958 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lv, M., Wang, C., Ren, G., Ma, J., Song, X.: Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn. 85(3), 1479–1490 (2016)CrossRefGoogle Scholar
  26. 26.
    Lv, M., Ma, J.: Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205, 375–381 (2016)CrossRefGoogle Scholar
  27. 27.
    Lu, L.L., Jia, Y., Liu, W.H., Yang, L.J.: Mixed stimulus-induced mode selection in neural activity driven by high and low frequency current under electromagnetic radiation. Complexity 2017, 7628537 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ge, M.Y., Jia, Y., Xu, Y., Yang, L.J.: Mode transition in electrical activities of neuron driven by high and low frequency stimulus in the presence of electromagnetic induction and radiation. Nonlinear Dyn. 91(1), 515–523 (2018)CrossRefGoogle Scholar
  29. 29.
    Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dyn. 89(3), 1569–1578 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Spitzer, N.C.: Electrical activity in early neuronal development. Nature 444, 707–712 (2006)CrossRefGoogle Scholar
  31. 31.
    Wang, Y., Ma, J., Xu, Y., Wu, F., Zhou, P.: The electrical activity of neurons subject to electromagnetic induction and Gaussian white noise. Int. J. Bifurc. Chaos 27(2), 1750030 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Faisal, A.A., Selen, L.P.J., Wolpert, D.M.: Noise in the nervous system. Nat. Rev. Neurosci. 9(4), 292–303 (2008)CrossRefGoogle Scholar
  33. 33.
    Willott, J.F., Lu, S.M.: Noise-induced hearing loss can alter neural coding and increase excitability in the central nervous system. Science 216(4552), 1331–1334 (1982)CrossRefGoogle Scholar
  34. 34.
    Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10(6), 1171–1266 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Bao, B.C., Jiang, P., Wu, H.G., Hu, F.W.: Complex transient dynamics in periodically forced memristive chua’s circuit. Nonlinear Dyn. 79(4), 2333–2343 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Xu, Q., Zhang, Q.L., Bao, B.C., Hu, Y.H.: Non-autonomous second-order memristive chaotic circuit. IEEE Access 5, 21039–21045 (2017)CrossRefGoogle Scholar
  37. 37.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16(3), 285–317 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Bao, B.C., Qian, H., Xu, Q., Chen, M., Wang, J., Yu, Y.J.: Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network. Front. Comput. Neurosci. 11, 81 (2017)CrossRefGoogle Scholar
  39. 39.
    Bao, H., Wang, N., Bao, B.C., Chen, M., Jin, P.P., Wang, G.Y.: Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria. Commun. Nonlinear Sci. Numer. Simul. 57, 264–275 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Kuznetsov, N.V., Leonov, G.A., Yuldashev, M.V., Yuldashev, R.V.: Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE. Commun. Nonlinear Sci. Numer. Simul. 51, 39–49 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Information Science and EngineeringChangzhou UniversityChangzhouChina

Personalised recommendations