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Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model

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Abstract

In this paper, the effects of different parameters on the dynamic behavior of the nonlinear dynamical system are investigated based on modified Hindmarsh–Rose neural nonlinear dynamical system model. We have calculated and analyzed dynamic characteristics of the model under different parameters by using single parameter bifurcation diagram, time response diagram and two parameter bifurcation diagram. The results show that the period-adding bifurcation (with or without chaos), period-doubling bifurcation and intermittent chaos phenomenon (periodic and intermittent chaotic) can be observed more clearly and directly from the two parameter bifurcation diagram, and the optimal parameters matching interval can also be found easily.

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References

  1. Shou TD (2006) Neurobiology. Higher Education Press, Beijing

    Google Scholar 

  2. Hodgkin AL, Huxley AF (1952) The components of membrane conductance in the giant axon of Loligo. J Physiol 116:473–496

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT Press, Cambridge

    Google Scholar 

  6. Morris C, Lecar H (1981) Voltage oscillations in the blamable giant muscle fiber. Biophys J 35:193–213

    Article  Google Scholar 

  7. Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic beta-cell. Biophys J 42:181–190

    Article  Google Scholar 

  8. Hindmarsh JL, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Proc R Soc Lond Ser B 221:87–102

    Article  Google Scholar 

  9. Rinzel J, Lee YS (1987) Dissection of a model for neuronal parabolic bursting. J Math Biol 25:653–675

    Article  MathSciNet  MATH  Google Scholar 

  10. Grubelnk V, Larsen AZ, Kummer U et al (2001) Mitochondria regulates the amplitude of simple and complex calcium oscillations. Biophys Chem 94:59–74

    Article  Google Scholar 

  11. Ge ML, Guo HY, Wang GJ, Yan WL (2003) Research for synchronous oscillation on electrically coupled Chay neurons. Acta Biophys Sin 19(2):135–140

    Google Scholar 

  12. Perc M, Marhl M (2004) Synchronization of regular and chaotic oscillations: the role of local divergence and the slow passage effect—a case study on calcium oscillations. Int J Bifurc Chaos Appl Sci Eng 14(8):2735–2751

    Article  MathSciNet  MATH  Google Scholar 

  13. Izhikevich EM, Desai NS, Walcott EC et al (2003) Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci 26(3):161–167

    Article  Google Scholar 

  14. Ricardo AL, Rafael MG, Héctor P et al (2010) High order sliding-mode dynamic control for chaotic intracellular calcium oscillations. Nonlinear Anal Real World Appl 11:217–231

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu XL, Liu SQ (2012) Codimension-two bifurcations analysis in two-dimensional Hindmarsh–Rose model. Nonlinear Dyn 67:847–857

    Article  MATH  Google Scholar 

  16. Zhou J, Wu QJ, Xiang L (2012) Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization. Nonlinear Dyn 69:1393–1403

    Article  MathSciNet  MATH  Google Scholar 

  17. Ji QB, Lu QS, Yang ZQ et al (2008) Bursting Ca2+ oscillations and synchronization in coupled cells. Chin Phys Lett 25:3879–3883

    Article  Google Scholar 

  18. Nchange AK, Kepseu WD, Woafo P (2008) Noise induced intercellular propagation of calcium waves. Phys A 387:2519–2525

    Article  Google Scholar 

  19. Zhang F, Lu QS, Su JZ (2009) Transition in complex calcium bursting induced by IP3 degradation. Chaos Solitons Fractals 5:2285–2290

    Article  Google Scholar 

  20. Zheng YH, Lu QS (2008) Synchronization in ring coupled chaotic neurons with time delay. J Dyn Control 6(3):208–212

    MathSciNet  Google Scholar 

  21. Wang HX, Wang QY, Lu QS (2011) Bursting oscillations, bifurcation and synchronization in neuronal systems. Chaos Solitons Fractals 44(8):667–675

    Article  MATH  Google Scholar 

  22. Nimet D, Imail Ö, Recai K (2012) Experimental realizations of the HR neuron model with programmable hardware and synchronization applications. Nonlinear Dyn 70:2343–2358

    Article  MathSciNet  Google Scholar 

  23. Ma SQ, Lu QS, Feng ZS (2010) Synchrony and lag synchrony on a neuron model coupling with time delay. Int J Nonlinear Mech 45(6):659–665

    Article  Google Scholar 

  24. Li B, He ZM (2014) Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model. Nonlinear Dyn 76:697–715

    Article  MathSciNet  MATH  Google Scholar 

  25. Shilnikov A, Marina K (2008) Methods of the qualitative theory for the Hindmarsh–Rose model: a case study—a tutorial. Int J Bifurc Chaos 18(8):2141–2168

    Article  MathSciNet  MATH  Google Scholar 

  26. Djeundam Dtchetgnia SR, Yamapi R, Filatrella G, Kofane TC (2015) Stability of the synchronized network of Hindmarsh–Rose neuronal models with nearest and global couplings. Commun Nonlinear Sci Numer Simul 22(1):545–563

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang DG, Zhang Q, Zhu XY (2015) Exploring a type of central pattern generator based on Hindmarsh–Rose model: from theory to application. Int J Neural Syst 25(1):1450028 (15 pages)

    Article  Google Scholar 

  28. Tsumoto K, Kitajima H, Yoshinaga T (2006) Bifurcations in Morris–Lecar neuron model. Neurocomputing 69:293–316

    Article  Google Scholar 

  29. Keplinger K, Wackerbauer R (2014) Transient spatiotemporal chaos in the Morris–Lecar neuronal ring network. Chaos 24(1):385–397

    Article  MathSciNet  Google Scholar 

  30. Song SL (2010) The transition rules of injured nerve spontaneous discharge rhythm under the dual parameters. Shaanxi Normal University, Xi’an

    Google Scholar 

  31. Krasimira TA, Hinke MO, Thorsten R et al (2010) Full system bifurcation analysis of endocrine bursting models. J Theor Biol 264(4):1133–1146

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This work was supported by the by the National Social Science foundation of China (No. 12CGL004), the Science and Technology Support Project of Gansu Province (No. 1304FKCA097) and the Basic Scientific Research Expenses of Finance Department of Gansu Province (No. 214150).

Conflict of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. School of Electronic and Information Engineering, LanZhou JiaoTong University, Lanzhou, 730070, China

    Kaijun Wu, Tianqi Luo & Yang Wang

  2. School of Mathematics and Physics, LanZhou JiaoTong University, Lanzhou, 730070, China

    Huaiwei Lu

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  1. Kaijun Wu
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  2. Tianqi Luo
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  3. Huaiwei Lu
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  4. Yang Wang
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Correspondence to Kaijun Wu.

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Wu, K., Luo, T., Lu, H. et al. Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model. Neural Comput & Applic 27, 739–747 (2016). https://doi.org/10.1007/s00521-015-1892-1

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  • DOI: https://doi.org/10.1007/s00521-015-1892-1

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