Ghostburster神经元系统的稳定性与Hopf分岔

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.

Stability and Hopf Bifurcation of Ghostburster Neuron System

We considered the Ghostburster system model of a classof weak electric fish neuron cells. Firstly, the equilibrium point of the neuron systemwas 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 ofthe 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 byusing 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 membranewere selected as the bifurcation parameters, and the dynamic behavior of the system under single parameter changes was investigated.