Phan Ngoc Anh, Wang Yangyang
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, USA.
Department of Mathematics, Brandeis University, Waltham, Massachusetts 02453, USA.
Chaos. 2024 May 1;34(5). doi: 10.1063/5.0181308.
Mixed-mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov-Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris-Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.
混合模式振荡（MMOs）是多时间尺度动力系统的复杂振荡行为，其中存在大幅度振荡和小幅度振荡的交替。众所周知，双时间尺度系统中的MMOs可以源于与折叠节点奇点相关的鸭机制或快速子系统的延迟安德罗诺夫-霍普夫分岔（DHB）。虽然双时间尺度系统中的MMOs已得到广泛研究，但对于三时间尺度系统中出现的MMOs了解较少。在这项工作中，我们研究了具有三个不同时间尺度的耦合莫里斯-莱卡神经元中MMOs的机制。我们研究了在存在一种称为鸭延迟霍普夫（CDH）奇点的情况下以及不存在CDH的情况下出现的两种MMOs。在这两种情况下，我们研究了MMOs的特征和机制如何随时间尺度的变化而变化。我们的分析表明，由CDH支持的MMOs表现出比不存在CDH时更强的鲁棒性。此外，我们表明仅仅存在CDH并不能保证MMOs的发生。这项工作为双时间尺度背景下的两种独立机制，即鸭机制和DHB，在三时间尺度环境中相互作用并产生更强鲁棒性的MMOs，特别是抵抗时间尺度变化的条件提供了重要见解。
