一类Filippov型HR神经元模型的全局动力学分析

doi:10.6040/j.issn.1671-9352.0.2022.319

摘要/Abstract

摘要：

基于电磁辐射下的HR神经元模型，引入一种以膜电位为阈值的非光滑控制策略，即确定与电磁感应强度和外部刺激电流相对应的切换函数，从而建立一类四维Filippov型HR神经元模型。利用稳定性理论和数值模拟方法讨论2个子系统平衡点的存在性、稳定性和全局分岔行为。另外，基于双参数分岔分析，研究子系统的双稳态行为和演化模式。进一步，利用Filippov凸组合方法和Utkin等度控制方法分析系统的各类平衡点和滑膜动力学的存在性。最后，采用快-慢动力学的方法揭示阈值控制策略下的滑动放电模式和多稳态特征。

关键词: Filippov型HR神经元, 阈值控制策略, 滑膜动力学, 多稳态, 快-慢动力学

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

中图分类号:

O193

刘文岩,乔帅,高承华. 一类Filippov型HR神经元模型的全局动力学分析[J]. 《山东大学学报(理学版)》, 2023, 58(10): 13-23.

Wenyan LIU,Shuai QIAO,Chenghua GAO. Global dynamics analysis of a class of Filippov-type HR neuron model[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(10): 13-23.

图/表 4

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参考文献 29

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.

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