具有重置的非线性神经元模型中的野振荡：（II）混合模式振荡
Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations
摘要：
本文继续研究一类具有重置的二维非线性混合动力系统的复杂动力学问题，该系统模拟了具有自适应和尖峰发射的神经元电压动力学。我们表明，这些模型可以一般地显示一种形式的混合模式振荡（MMO），这是具有交替的小振荡与尖峰或突发（多个连续尖峰）的轨迹。这些产生的机制从根本上依赖于混合结构的流动：不变流形的连续动态管理小振荡，而离散复位管理发射的尖峰或突发，与经典MMO机制的常微分方程涉及超过三个维度，一般依赖于一个时标分离。机制的分解揭示了MMO的几何起源，允许对与特定数量的小振荡相关的重置流形上的点进行相对简单的分类。我们表明，MMO模式可以通过研究轨道的离散适应映射，这是奇异的，因为它具有离散的不连续性与无界的左，右导数。我们研究轨道的地图通过旋转理论的不连续的圆地图和详细阐明复杂的行为所产生的情况下，MMO显示在每个连续的尖峰之间的最多一个小的振荡。
英文摘要：
This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left- and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes.
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