Ghostburster神经元系统的稳定性与Hopf分岔
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摘要
考虑一类弱电鱼椎体的神经元细胞Ghostburster系统模型,首先用数值计算方法给出该神经元系统的平衡点,通过分析平衡点附近Jacobi矩阵对应的特征值,分析平衡点附近的稳定性及其类型.其次,用Hopf分岔存在性理论及其分析方法给出该系统模型Hopf分岔的方向及分岔周期近似解和近似周期.结果表明,当系统参数控制在一定范围内时,系统模型产生了亚临界Hopf分岔,并出现周期逐渐增加且不稳定的周期解轨道.最后,利用MATLAB等数学软件给出理论分析对应的数值模拟结果,模拟选取树突膜钾离子电流最大电导和胞体膜注入电流的相关参数作为分岔参数,考察系统在单参变化下的动力学行为.
We considered the Ghostburster system model of a class of weak electric fish neuron cells.Firstly,the equilibrium point of the neuron system was given by numerical calculation method.By analyzing the eigenvalues of the Jacobian matrix near the equilibrium point,the stability and its type near the equilibrium point were analyzed.Secondly,using the Hopf bifurcation existence theory and its analysis method,the direction of the model Hopf bifurcation and the approximate solution and the approximate period of the bifurcation period were given.The results show that when the system parameters are controlled within a certain range,and the system model produces subcritical Hopf bifurcation,the periodic solution orbit is gradually increased and unstable.Finally,the numerical simulation results of theoretical analysis were given by using MATLAB and other mathematical software.The parameters of the maximum conductance of potassium ion current of dendritic membrane and the injection current of the cell membrane were selected as the bifurcation parameters,and the dynamic behavior of the system under single parameter changes was investigated.
We considered the Ghostburster system model of a class of weak electric fish neuron cells.Firstly,the equilibrium point of the neuron system was given by numerical calculation method.By analyzing the eigenvalues of the Jacobian matrix near the equilibrium point,the stability and its type near the equilibrium point were analyzed.Secondly,using the Hopf bifurcation existence theory and its analysis method,the direction of the model Hopf bifurcation and the approximate solution and the approximate period of the bifurcation period were given.The results show that when the system parameters are controlled within a certain range,and the system model produces subcritical Hopf bifurcation,the periodic solution orbit is gradually increased and unstable.Finally,the numerical simulation results of theoretical analysis were given by using MATLAB and other mathematical software.The parameters of the maximum conductance of potassium ion current of dendritic membrane and the injection current of the cell membrane were selected as the bifurcation parameters,and the dynamic behavior of the system under single parameter changes was investigated.
引文
[1]陈良泉.Hodgkin-Huxley模型的分岔分析与控制[D].天津:天津大学,2006.(CHEN Liangquan.Analysis and Control of the Bifurcation of the Hodgkin-Huxley Model[D].Tianjin:Tianjin University,2006.)
[2]车艳秋.Hodgkin-Huxley模型的分岔与混沌分析[D].天津:天津大学,2005.(CHE Yanqiu.Analysis of Bifurcation and Chaos in Hodgkin-Huxley Model[D].Tianjin:Tianjin University,2005.)
[3]耿建明.HHM模型的Hopf分岔分析[D].天津:天津大学,2005.(GENG Jianming.Analysis of Hopf Bifurcation in the Hodgkin-Huxley Equations for Muscles[D].Tianjin:Tianjin University,2005.)
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[7]陈颖源,王江,曾启明,等.Ghostburster模型鞍结分岔点附近放电模式的转迁控制[J].天津大学学报,2012,45(5):440-447.(CHEN Yingyuan,WANG Jiang,TSANG Kaiming,et al.Controlling the Firing Patterns around a Saddle-Node Bifurcation in the Ghostburster Model[J].Journal of Tianjin University,2012,45(5):440-447.)
[8]周思思.神经元的混沌同步分析与控制[D].天津:天津大学,2008.(ZHOU Sisi.Chaos Synchronizations Analysis and Control of Neurons[D].Tianjin:Tianjin University,2008.)
[9]陈立松.Ghostburster神经元的混沌同步与控制[D].天津:天津大学,2007.(CHEN Lisong.Chaos Synchronization and Control of Ghostburster Neuron[D].Tianjin:Tianjin University,2007.)
[10]张镇.Ghostburster模型的混沌分析与控制[D].天津:天津大学,2007.(ZHANG Zhen.Analysis and Control of the Chaos of Ghostburster Model[D].Tianjin:Tianjin University,2007.)
[11]LAING C R,DOIRON B,LONGTIN A,et al.Ghostbursting:The Effects of Dendrites on Spike Patterns[J].Neurocomputing,2002,44/45/46:127-132.
[12]SHIRAHATA T.Evaluation of Kinetic Properties of Dendritic Potassium Current in Ghostbursting Model of Electrosensory Neurons[J].Appl Math,2015,6(1):128-135.
[13]张素丽.神经元Chay模型的动力学分析[D].太原:太原理工大学,2013.(ZHANG Suli.Dynamical Analysis of the Neuronal Chay Model[D].Taiyuan:Taiyuan University of Technology,2013.)
[14]HASSARD B.Bifurcation of Periodic Solutions of the Hodgkin-Huxley Model for the Squid Giant Axon[J].J Theor Biol,1978,71(3):401-420.
[15]HASSARD B,WAN Y H.Bifurcation Formulae Derived from Center Manifold Theory[J].J Math Anal Appl,1978,63(1):297-312.
[2]车艳秋.Hodgkin-Huxley模型的分岔与混沌分析[D].天津:天津大学,2005.(CHE Yanqiu.Analysis of Bifurcation and Chaos in Hodgkin-Huxley Model[D].Tianjin:Tianjin University,2005.)
[3]耿建明.HHM模型的Hopf分岔分析[D].天津:天津大学,2005.(GENG Jianming.Analysis of Hopf Bifurcation in the Hodgkin-Huxley Equations for Muscles[D].Tianjin:Tianjin University,2005.)
[4]NGOUONKADI E B M,FOTSIN H B,LOUODOP F P,et al.Bifurcations and Multistability in the Extended Hindmarsh-Rose Neuronal Oscillator[J].Chaos Solitons Fractals,2016,85(1):151-163.
[5]裴利军,王永刚,范烨.神经元Chay模型的动力学分析[J].郑州大学学报(理学版),2009,41(2):7-12.(PEI Lijun,WANG Yonggang,FAN Ye.Dynamical Analysis of the Neuronal Chay Model[J].Journal of Zhengzhou University(Natural Science Edition),2009,41(2):7-12.)
[6]DOIRON B,LAING C,LONGTIN A,et al.Ghostbursting:A Novel Neuronal Burst Mechanism[J].J Comput Neurosc,2002,12(1):5-25.
[7]陈颖源,王江,曾启明,等.Ghostburster模型鞍结分岔点附近放电模式的转迁控制[J].天津大学学报,2012,45(5):440-447.(CHEN Yingyuan,WANG Jiang,TSANG Kaiming,et al.Controlling the Firing Patterns around a Saddle-Node Bifurcation in the Ghostburster Model[J].Journal of Tianjin University,2012,45(5):440-447.)
[8]周思思.神经元的混沌同步分析与控制[D].天津:天津大学,2008.(ZHOU Sisi.Chaos Synchronizations Analysis and Control of Neurons[D].Tianjin:Tianjin University,2008.)
[9]陈立松.Ghostburster神经元的混沌同步与控制[D].天津:天津大学,2007.(CHEN Lisong.Chaos Synchronization and Control of Ghostburster Neuron[D].Tianjin:Tianjin University,2007.)
[10]张镇.Ghostburster模型的混沌分析与控制[D].天津:天津大学,2007.(ZHANG Zhen.Analysis and Control of the Chaos of Ghostburster Model[D].Tianjin:Tianjin University,2007.)
[11]LAING C R,DOIRON B,LONGTIN A,et al.Ghostbursting:The Effects of Dendrites on Spike Patterns[J].Neurocomputing,2002,44/45/46:127-132.
[12]SHIRAHATA T.Evaluation of Kinetic Properties of Dendritic Potassium Current in Ghostbursting Model of Electrosensory Neurons[J].Appl Math,2015,6(1):128-135.
[13]张素丽.神经元Chay模型的动力学分析[D].太原:太原理工大学,2013.(ZHANG Suli.Dynamical Analysis of the Neuronal Chay Model[D].Taiyuan:Taiyuan University of Technology,2013.)
[14]HASSARD B.Bifurcation of Periodic Solutions of the Hodgkin-Huxley Model for the Squid Giant Axon[J].J Theor Biol,1978,71(3):401-420.
[15]HASSARD B,WAN Y H.Bifurcation Formulae Derived from Center Manifold Theory[J].J Math Anal Appl,1978,63(1):297-312.