AI 中文总结
本研究采用三时间尺度生物物理神经元振荡器模型，结合几何奇异摄动理论与全系统分岔分析，揭示混合模式爆发振荡的组织机制及多动力学转变规律，凸显三时间尺度视角的重要性。
AI 中文摘要
混合模式振荡（MMOs）以小振幅振荡（SAOs）和大振幅振荡（LAOs）交替出现为特征，爆发振荡是多时间尺度系统中常见的复杂振荡动力学形式，已在多个科学领域被广泛研究。混合模式爆发振荡（MMBOs）结合了MMOs和爆发的特征，其LAOs被组织成爆发事件。现有多数研究将MMBOs视为两时间尺度现象，根据时间尺度的分组方式识别不同的几何机制。本研究采用三时间尺度生物物理皮层神经元振荡器模型，证明几何奇异摄动理论（GSPT）的三时间尺度实现对MMBO动力学具有更强的预测洞察力。该视角统一了先前从快慢分析中识别的机制，同时揭示了可延迟Hopf（CDH）奇异性作为奇异极限附近MMBOs的组织中心。我们对完整的八维模型进行详细分岔分析，以确定MMBOs如何沿同宿轨道族组织。随后，我们将GSPT与全系统分岔分析结合，证明调整相对时间尺度如何诱导MMOs、MMBOs和爆发动力学之间的转变。研究结果强调了从三时间尺度视角研究MMBOs的重要性，并证明将GSPT与全系统分岔分析结合可揭示单独使用任一方法都不明显的组织结构和机制。
英文摘要
Mixed-mode oscillations (MMOs), characterized by the alternation of small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs), and bursting oscillations are common forms of complex oscillatory dynamics observed in systems with multiple timescales and have been widely studied across scientific disciplines. Mixed-mode bursting oscillations (MMBOs) combine features of MMOs and bursting, with LAOs organized into burst events. Most existing studies treat MMBOs as two-timescale phenomena, identifying distinct geometric mechanisms depending on how the timescales are grouped. In this work, we use a three-timescale biophysical cortical neuronal oscillator model to demonstrate that a three-timescale implementation of geometric singular perturbation theory (GSPT) provides stronger predictive insight into MMBO dynamics. This perspective unifies mechanisms previously identified from fast-slow analysis, while revealing the canard-delayed-Hopf (CDH) singularity as an organizing center for MMBOs near the singular limit. We perform a detailed bifurcation analysis of the full eight-dimensional model to determine how MMBOs are organized along families of isolas. We then combine GSPT with full-system bifurcation analysis to show how tuning the relative timescales induces transitions among MMOs, MMBOs, and bursting dynamics. Our results highlight the importance of studying MMBOs from a three-timescale perspective and demonstrate that combining GSPT with full-system bifurcation analysis can reveal organizing structures and mechanisms that are not apparent from either approach alone.
