JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (10): 13-23.doi: 10.6040/j.issn.1671-9352.0.2022.319

Previous Articles     Next Articles

Global dynamics analysis of a class of Filippov-type HR neuron model

Wenyan LIU(),Shuai QIAO,Chenghua GAO*()   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Received:2022-05-30 Online:2023-10-20 Published:2023-10-17
  • Contact: Chenghua GAO E-mail:1299427689@qq.com;gaokuguo@163.com

RichHTML

1

PDF (PC)

138

Abstract:

Based on the HR neuron model under electromagnetic radiation, a non-smooth control strategy is proposed, in which the membrane potential is utilized as the threshold to determine the switching function corresponding to the electromagnetic induction intensity and external stimulus current. Consequently, a four-dimensional Filippov-type HR neuron model is established. Firstly, the existence, stability, and global bifurcation behaviors of equilibrium points of two subsystems are discussed by using stability theory and numerical simulation. Then, the bistable behavior and evolution modes of subsystems are investigated based on the two-parameter bifurcation analysis. Further, the existence of various equilibrium points and sliding mode dynamics of the system is analyzed by Filippov convex combination method and Utkin's equivalent control method. Finally, the sliding firing modes and bistable features under threshold control strategy are revealed by the method of fast-slow variable dissection.

Key words: Filippov-type HR neuron, threshold control strategy, sliding mode dynamics, bistability, fast-slow variable dissection

CLC Number: 

  • O193

Fig.1

Global dynamic responses of subsystem S1 on the Imax∈[0, 5.5], kmax∈[-2, 3.5] plane"

Fig.2

Global basins of attraction of subsystem S1 for kmax=-0.5, 0, 0.5 on the I×x(0) plane"

Fig.3

Sliding dynamic responses of the system (3) when Imax=1.75, gmax=0.5, Imin=0.98, gmin=-0.5"

Fig.4

Sliding dynamic responses of the system (3) when Imax=1.75, gmax=0.5, Imin=0.98, gmin=-0.5"

1 QIAO Shuai , GAO Chenghua , AN Xinlei , et al. Electrical activities, excitability and multistability transitions of the hybrid neuronal model induced by electromagnetic induction and autapse[J]. Modern Physics Letters B, 2022, 36 (12): 2250006.
doi: 10.1142/S0217984922500063
2 MA Jun , TANG Jun . A review for dynamics in neuron and neuronal network[J]. Nonlinear Dynamics, 2017, 89 (3): 1569- 1578.
doi: 10.1007/s11071-017-3565-3
3 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.
doi: 10.1113/jphysiol.1952.sp004764
4 MORRIS C , LECAR H . Voltage oscillations in the barnacle giant muscle fiber[J]. Biophysical Journal, 1981, 35 (1): 193- 213.
doi: 10.1016/S0006-3495(81)84782-0
5 FITZHUGH R . Impulses and physiological states in theoretical models of nerve membrane[J]. Biophysical Journal, 1961, 1 (6): 445- 466.
doi: 10.1016/S0006-3495(61)86902-6
6 CHAYT R . Chaos in a three-variable model of an excitable cell[J]. Physica D: Nonlinear Phenomena, 1985, 16 (2): 233- 242.
doi: 10.1016/0167-2789(85)90060-0
7 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.
doi: 10.1038/296162a0
8 乔帅, 张莉, 安新磊, 等. 电磁辐射下HR神经元模型的分岔分[J]. 河北师范大学学报(自然科学版), 2019, 43 (4): 122- 128.
QIAO Shuai , ZHANG Li , AN Xinlei , et al. Bifurcation analysis on HR neuron models under electromagnetic radiation[J]. Journal of Hebei Normal University(Natural Science Edition), 2019, 43 (4): 122- 128.
9 MA Jun , TANG Jun . A review for dynamics of collective behaviors of network of neurons[J]. Science China Technological Sciences, 2015, 58 (12): 2038- 2045.
doi: 10.1007/s11431-015-5961-6
10 祁慧敏, 张莉, 安新磊, 等. 电磁感应下一类新皮层神经元的放电及同步[J]. 山东大学学报(理学版), 2021, 56 (5): 1- 11.
QI Huimin , ZHANG Li , AN Xinlei , et al. Firing and synchronization of a neocortical neuron under electromagnetic induction[J]. Journal of Shandong University(Natural Science), 2021, 56 (5): 1- 11.
11 高月月, 李新颖, 李宁. 电磁感应下Chay神经元的放电分岔特性与同步[J]. 山东大学学报(理学版), 2021, 56 (1): 43- 51.
GAO Yueyue , LI Xinying , LI Ning . Firing bifurcation characteristics and synchronization of Chay neuron under electromagnetic induction[J]. Journal of Shandong University(Natural Science), 2021, 56 (1): 43- 51.
12 LYU Mi , WANG Chunni , REN Guodong , et al. Model of electrical activity in a neuron under magnetic flow effect[J]. Nonlinear Dynamics, 2016, 85 (3): 1479- 1490.
doi: 10.1007/s11071-016-2773-6
13 WU Fuqiang , GU Huaguang . Bifurcations of negative responses to positive feedback current mediated by memristor in a neuron model with bursting patterns[J]. International Journal of Bifurcation and Chaos, 2020, 30 (4): 2030009.
doi: 10.1142/S0218127420300098
14 安新磊, 乔帅, 张莉. 基于麦克斯韦电磁场理论的神经元动力学响应与隐藏放电控制[J]. 物理学报, 2021, 70 (5): 40- 59.
AN Xinlei , QIAO Shuai , ZHANG Li . Dynamic response and control of neuros based on electromagnetic field theory[J]. Acta Physica Sinica, 2021, 70 (5): 40- 59.
15 张秀芳, 马军, 徐莹, 等. 光电管耦合FitzHugh-Nagumo神经元的同步[J]. 物理学报, 2021, 70 (9): 090502.
ZHANG Xiufang , MA Jun , XU Ying , et al. Synchronization between FitzHugh-Nagumo neurons coupled with phototube[J]. Acta Physica Sinica, 2021, 70 (9): 090502.
16 WANG Guowei , YU Dong , DING Qianming , et al. Effects of electric field on multiple vibrational resonances in Hindmarsh-Rose neuronal systems[J]. Chaos, Solitons & Fractals, 2021, 150, 111210.
17 GE Mengyan , LU Lulu , XU Ying , et al. Vibrational mono-/bi-resonance and wave propagation in FitzHugh-Nagumo neural systems under electromagnetic induction[J]. Chaos Solitons & Fractals, 2020, 133, 109645.
18 LI Tianyu , WANG Guowei , YU Dong , et al. Synchronization mode transitions induced by chaos in modified Morris-Lecar neural systems with weak coupling[J]. Nonlinear Dynamics, 2022, 108 (3): 2611- 2625.
doi: 10.1007/s11071-022-07318-5
19 乔帅, 安新磊, 王红梅, 等. 电磁感应下HR神经元模型的分岔分析与控制[J]. 山东大学学报(理学版), 2020, 55 (9): 1- 9.
QIAO Shuai , AN Xinlei , WANG Hongmei , et al. Bifurcation analysis and control of HR neuron model under electromagnetic induction[J]. Journal of Shandong University (Natural Science), 2020, 55 (9): 1- 9.
20 ZHANG Ge , GUO Daqing , WU Fuqiang , et al. Memristive autapse involving magnetic coupling and excitatory autapse enhance firing[J]. Neurocomputing, 2020, 379, 296- 304.
doi: 10.1016/j.neucom.2019.10.093
21 BAO Han , HU Aihuang , LIU Wenbo , et al. Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction[J]. IEEE Transactions on Neural Networks and Learning Systems, 2020, 31 (2): 502- 511.
doi: 10.1109/TNNLS.2019.2905137
22 LIN Hairong , WANG Chunhua , SUN Yichuang , et al. Firing multistability in a locally active memristive neuron model[J]. Nonlinear Dynamics, 2020, 100 (4): 3667- 3683.
doi: 10.1007/s11071-020-05687-3
23 张蒙, 成宇, 张艺, 等. 忆阻器的温度效应改进模型及其仿生神经突触传递[J]. 动力学与控制学报, 2021, 19 (6): 67- 75.
ZHANG Meng , CHENG Yu , ZHANG Yi , et al. An improved temperature-based memristor model and its bionic synaptic transmission[J]. Journal of Dynamics and Control, 2021, 19 (6): 67- 75.
24 王义波, 闵富红, 张雯, 等. 忆阻FitzHugh-Nagumo神经元电路有限时间同步[J]. 南京师范大学学报(工程技术版), 2020, 20 (2): 7- 14.
WANG Yibo , MIN Fuhong , ZHANG Wen , et al. Finite-time synchronization of memristor-based FitzHugh-Nagumo circuit[J]. Journal of Nanjing Normal University (Engineering and Technology Edition), 2020, 20 (2): 7- 14.
25 QIAO Shuai , AN Xinlei . Dynamic expression of a HR neuron model under an electric field[J]. International Journal of Modern Physics B, 2021, 35 (2): 2150024.
26 YANG Yi , LIAO Xiaofeng . Filippov Hindmarsh-Rose neuronal model with threshold policy control[J]. IEEE Transactions on Neural Networks & Learning Systems, 2018, 30 (1): 306- 311.
27 FILIPPOV A F . Differential equations with discontinuous righthand sides[M]. Dordrecht: Kluwer Academic Publishers, 1988.
28 UTKIN V I , CHANG H C . Sliding mode control on electro-mechanical systems[J]. Mathematical Problems in Engineering, 2002, 8 (4/5): 451- 473.
29 WANG Zhixiang , ZHANG Chun , ZHANG Zhengdi , et al. Bursting oscillations with boundary homoclinic bifurcations in a Filippov-type Chua's circuit[J]. Pramana, 2020, 94 (1): 95.
[1] Shengqiang LIU,Ningjuan MA. Peak value and peak time of nonlinear heterogeneous epidemic model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(10): 1-12.
[2] ZHANG Mei-jiao, ZHANG Jian-gang, NAN Meng-ran, WEI Li-xiang. Bifurcation of hydraulic turbine governing system under color noise excitation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(12): 94-99.
[3] . [J]. J4, 2009, 44(3): 81-83 .
[4] LIU Ling-xia . Existence of an analytic solution for a first order iterative functional differential equation [J]. J4, 2007, 42(12): 82-86 .
[5] QIAO Shuai, AN Xin-lei, WANG Hong-mei, ZHANG Wei. Bifurcation analysis and control of HR neuron model under electromagnetic induction [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(9): 1-9.
[6] ZHANG Jinke, ZHANG Jiangang. Signal detection and fault diagnosis based on improved particle swarm optimization algorithm [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(5): 63-75.
[7] WANG Hong-mei, AN Xin-lei, QIAO Shuai, ZHANG Wei. Bifurcation analysis and synchronous control of e-HR neuron model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(9): 10-18.
[8] GAO Yue-yue, LI Xin-ying, LI Ning. Firing bifurcation characteristics and synchronization of Chay neuron under electromagnetic induction [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(1): 43-51.
[9] WEI Li-xiang, ZHANG Jian-gang, NAN Meng-ran, ZHANG Mei-jiao. Stability and Hopf bifurcation of a flux neuron model with time delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(5): 12-22.
[10] LU Zheng-yu, ZHOU Wei, YU Huan-huan, ZHAO Na. Dynamic analysis of game model considering advertising spillover effect [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(11): 63-70.
[11] QI Hui-min, ZHANG Li, AN Xin-lei, QIAO Shuai, XIAO Ran. Firing and synchronization of a neocortical neuron under electromagnetic induction [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(5): 1-11.
[12] DAI Li-hua, HUI Yuan-xian. Almost automorphic solutions for shunting inhibitory cellular neural networks with leakage delays on time scales [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(10): 97-108.
[13] ZHU Yan-lan, ZHOU Wei, CHU Tong, LI Wen-na. Complex dynamic analysis of the duopoly game under management delegation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(7): 32-45.
[14] LYU Ning. Dynamics of the stage-structured population system with two kinds of pulses [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(12): 100-110.
[15] CAO Hui-rong , ZHOU Wei, CHU Tong, ZHOU Jie. Dynamic analysis of Bertrand game model about taxation of government and subsidy [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(11): 52-62.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] TANG Feng-qin1, BAI Jian-ming2. The precise large deviations for a risk model with extended negatively upper orthant dependent claim  sizes[J]. J4, 2013, 48(1): 100 -106 .
[2] CHENG Zhi1,2, SUN Cui-fang2, WANG Ning1, DU Xian-neng1. On the fibre product of Zn and its property[J]. J4, 2013, 48(2): 15 -19 .
[3] TANG Xiao-hong1, HU Wen-xiao2*, WEI Yan-feng2, JIANG Xi-long2, ZHANG Jing-ying2, SHAO Xue-dong3. Screening and biological characteristics studies of wide wine-making yeasts[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 12 -17 .
[4] Ming-Chit Liu. THE TWO GOLDBACH CONJECTURES[J]. J4, 2013, 48(2): 1 -14 .
[5] ZHAO Tong-xin1, LIU Lin-de1*, ZHANG Li1, PAN Cheng-chen2, JIA Xing-jun1. Pollinators and pollen polymorphism of  Wisteria sinensis (Sims) Sweet[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 1 -5 .
[6] WANG Kai-rong, GAO Pei-ting. Two mixed conjugate gradient methods based on DY[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 16 -23 .
[7] YANG Jun. Characterization and structural control of metalbased nanomaterials[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 1 -22 .
[8] DONG Wei-wei. A new method of DEA efficiency ranking for decision making units with independent subsystems[J]. J4, 2013, 48(1): 89 -92 .
[9] ZHANG Jing-you, ZHANG Pei-ai, ZHONG Hai-ping. The application of evolutionary graph theory in the design of knowledge-based enterprises’ organization strucure[J]. J4, 2013, 48(1): 107 -110 .
[10] MAO Ai-qin1,2, YANG Ming-jun2, 3, YU Hai-yun2, ZHANG Pin1, PAN Ren-ming1*. Study on thermal decomposition mechanism of  pentafluoroethane fire extinguishing agent[J]. J4, 2013, 48(1): 51 -55 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd