School of Aerospace Engineering, Xi'an Jiaotong University, No. 28 West Xianning Road, Xi'an, Shaanxi 710049, People's Republic of China.
Chaos. 2022 Nov;32(11):113113. doi: 10.1063/5.0118046.
In this paper, analytical predictions of the firing cascades formed by stable and unstable firings in a Hindmarsh-Rose (HR) neuron system are completed via an implicit mapping method. The semi-analytical firing cascades present the route from periodic firings to chaos. For such cascades, the continuous firing flow of the nonlinear neuron system is discretized to form a special mapping structure for nonlinear firing activities. Stability and bifurcation analysis of the nonlinear firings are performed based on resultant eigenvalues of the global mapping structures. Stable and unstable firing solutions in the bifurcation tree exhibit clear period-doubling firing cascades toward chaos. Bifurcations are predicted accurately on the connections. Phase bifurcation trees are observed, which provide deep cognitions of neuronal firings. Harmonic dynamics of the period-doubling firing cascades are obtained and discussed for a better understanding of the contribution of the harmonics in frequency domains. The route into chaos is illustrated by the firing chains from period-1 to period-16 firings and verified by numerical solutions. The applied methods and obtained results provide new perspectives to the complex firing dynamics of the HR neuron system and present a potential strategy to regulate the firings of neurons.
本文通过隐式映射方法对 Hindmarsh-Rose（HR）神经元系统中稳定和不稳定点火形成的点火级联进行了分析预测。半解析点火级联呈现了从周期点火到混沌的路径。对于这种级联，非线性神经元系统的连续点火流被离散化，形成了用于非线性点火活动的特殊映射结构。基于全局映射结构的总特征值对非线性点火的稳定性和分岔进行了分析。在分支树中可以观察到稳定和不稳定点火解，表现出清晰的倍周期点火级联通向混沌。在连接点上可以准确地预测分岔。观察到了相位分岔树，为神经元点火提供了深刻的认识。获得了倍周期点火级联的谐波动力学，并对其进行了讨论，以更好地理解频域中谐波的贡献。通过从周期 1 到周期 16 的点火链来阐明进入混沌的路径，并通过数值解进行验证。所应用的方法和获得的结果为 HR 神经元系统的复杂点火动力学提供了新的视角，并为调节神经元的点火提供了一种潜在的策略。
