JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (5): 12-22.doi: 10.6040/j.issn.1671-9352.0.2020.586

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Stability and Hopf bifurcation of a flux neuron model with time delay

WEI Li-xiang, ZHANG Jian-gang*, NAN Meng-ran, ZHANG Mei-jiao   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Published:2021-05-13

Abstract: A time-delayed flux neuron model with magnetically controlled memristor to study the effects of time delay and external stimulation current on the dynamic behavior of the model is put forward. The stability of the model at the equilibrium point is discussed by using the Routh-Hurwitz criterion and the stability of the Hopf bifurcation at the critical point is further studied by using the central manifold theorem. By numerical stimulation, the time series and single-double bifurcation diagrams of the model with different time delays. When the time lag and external forcing current are changed, it is found that there are many discharge patterns in the model. By selecting appropriate time lag or external forcing current, the resting state, spiking state and periodic bursting state are obtained, which is conducive to explain the abnormal discharge behavior of brain or nerve center caused by electromagnetic radiation.

Key words: magnetically controlled memristors, time delay, external stimulation current, Hopf bifurcation, electromagnetic radiation, two parameter bifurcation

CLC Number: 

  • O441.4
[1] HODGKIN A L, HUXLEY A F. A quantitative description of membrane current and its application to conduction and excitation in nerve[J]. The Journal of Physiology, 1952, 117(4):500-544.
[2] HINDMARSH J L, ROSE R M. A model of the nerve impulse using two first-order differential equations[J]. Nature, 1982, 296(5853):162-164.
[3] MOUJAHID A, DANJOU A, TORREALDEA F J, et al. Efficient synchronization of structurally adaptive coupled Hindmarsh-Rose neurons[J]. Chaos, Solitons and Fractals, 2011, 44(11):929-933.
[4] RECH C P. Dynamics in theparameter space of a neuron model[J]. Chinese Physics Letters, 2012, 29(6):60506-60509.
[5] GU Huaguang, PAN Baobao. A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model[J]. Nonlinear Dynamics, 2015, 81(4):2107-2126.
[6] LV Mi, WANG Chunni, MA Jun, et al. Model of electrical activity in a neuron under magnetic flow effect[J]. Nonlinear Dynamics, 2016, 85(3):1479-1490.
[7] WU Fuqiang, WANG Chunni, MA Jun, et al. Model of electrical activity in cardiac tissue under electromagnetic induction[J]. Scientific Reports, 2016, 6(1):8-19.
[8] WANG Ya, MA Jun, XU Ying, et al. The electrical activity of neurons subject to electromagnetic induction and Gaussian white noise[J]. International Journal of Bifurcation and Chaos, 2017, 27(2):1750030-1750041.
[9] 乔帅,安新磊,王红梅,等. 磁通e-HR神经元隐藏放电与分岔行为的研究[J]. 云南大学学报(自然科学版), 2020, 42(4):685-694. QIAO Shuai, AN Xinlei, WANG Hongmei, et al. Hidden discharge and bifurcation behavior of magnetic flux e-HR neurons[J]. Journal of Yunnan University(Natural Sciences Edition), 2020, 42(4):685-694.
[10] TANG Keming, WANG Zuolei, SHI Xuerong. Electrical activity in a time-delay four-variable neuron model under electromagnetic induction[J]. Frontiers in Computational Neuroscience, 2017, 11:105-112.
[11] 张艳娇,李美生,陆启韶. ML神经元的放电模式及时滞对神经元同步的影响[J]. 动力学与控制学报, 2009, 7(1):19-23. ZHANG Yanjiao, LI Meisheng, LU Qishao. Firing patterns and the effect of time-delay coupling on synchronization of two coupled chaotic ML neurons[J]. Journal of Dynamics Control, 2009, 7(1):19-23.
[12] AL-HUSSEIN A B A, RAHMA F, JAFARI S. Hopf bifurcation and chaos in time-delay model of glucose-insulin regulatory system[J]. Chaos, Solitons and Fractals, 2020, 137:109845-109852.
[13] 于欢欢,安新磊,路正玉,等. 具有时滞磁通神经元模型的Hopf分岔分析[J].吉林大学学报(理学版), 2019, 57(5):1111-1121. YU Huanhuan, AN Xinlei, LU Zhengyu, et al. Hopf bifurcation analysis of flux neuron model with time delays[J]. Journal of Jilin University(Science Edition), 2019, 57(5):1111-1121.
[14] ZHANG Z Z, KUNDU S, TRIPATHI J P, et al. Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays[J]. Chaos, Solitons and Fractals, 2020, 131:109483-109499.
[15] JUNGES L, GALLAS A C J. Stability diagrams for continuous wide-range control of two mutually delay-coupled semiconductor lasers[J]. New Journal of Physics, 2015, 17(5):53038-53049.
[16] JUNGES L, PÖSCHEL T, GALLAS A C J. Characterization of the stability of semiconductor lasers with delayed feedback according to the Lang-Kobayashi model[J]. The European Physical Journal D, 2013, 67(7):1-9.
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