一类分段线性自突触神经元模型的分岔分析

Bifurcation Analysis of A Piecewise Linear Self-synaptic Neuron Model

研究了带有突触电导和门控阈值的McKean模型,给出系统平衡点存在与稳定的参数条件,理论分析系统的Persistence边界平衡点分岔和non-smooth fold边界平衡点分岔,并通过在切换面附近引入广义Jacobi矩阵,理论推导系统发生不连续Hopf分岔的参数条件,进而讨论系统跨边界极限环的存在性,数值仿真验证理论推导的可靠性.基于跨边界极限环半径随参数变化而变化,为了解变化过程中极限环与边界的位置关系,该文通过数值分析得到极限环与边界发生擦边分岔的参数阈值.

The McKean model with synaptic conductance and gated thresholds is studied.The existence and stability of the equilibrium point of the system are given.The persistence and non-smooth fold bifurcation of the boundary equilibrium point of the system are theoretically analyzed.And by introducing the generalized Jacobi matrix near the switching manifold,the parameter conditions for the occurrence of discontinuous Hopf bifurcation in the system are theoretically deduced,and then the existence of the cross-boundary limit cycles for the system is discussed.The numerical simulation verifies the reliability of the theoretical derivation.The radius of the cross-boundary limit cycle varies with the changes of parameters.In order to understand the positional relationship between the limit cycle and the boundary during the process of change,the parameter threshold of grazing bifurcation of the limit cycle is obtained by numerical analysis.