一类离散Hindmarsh-Rose模型的分支延拓

In this program we would carry out the bifurcation continuation for one-parameter bifurcation points of discrete Hindmarsh-Rose model and get one-parameter bifurcation curves and two-parameter bifurcation points. In a neighborhood of the two-parameter bifurcation point, we would give the distribution of bifurcations in two-parameter plane, especially illustrating the periodicity of dynamical behaviors and global orbits induced by two-parameter bifurcation point. Moreover, we would apply the obtained results from discrete Hindmarsh-Rose model to more discrete neural models. The research would reveal the periodicity and the transitions between different dynamical behaviors in two-parameter plane. The research would enhance the insights into the periodicity and the transitions between different dynamical behaviors of the model, such as the possibilities and ways of the transitions between different dynamical behaviors.

本项目拟对具体的离散Hindmarsh-Rose模型单参数分支点进行分支延拓，从而得到单参数分支曲线和双参数分支点，并将在双参数分支点的邻域内作出双参数平面上分支的局部分布图，尤其是对双参数分支诱导出的全局轨道进行数值模拟。并且将作出模型在双参数平面上的周期分布图。另外，本项目拟把对具体离散Hindmarsh-Rose模型的研究结果应用到更多离散神经学模型中，研究常见神经系统在双参数平面上的周期分布问题以及如何对双参数分支诱导的全局轨道进行数值模拟。对离散Hindmarsh-Rose模型的研究将会深刻揭示这一模型不同神经运动状态的周期性以及它们之间的转化问题，比如不同的神经运动状态是否会发生转化以及如何转化。