Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron models typically involve observable differences in time scales. The Hodgkin–Huxley model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the model.

中文翻译：

---

[公式：见文本]中的鞍慢流形和鸭式轨道以及在完整霍奇金-赫胥黎模型中的应用。


许多生理现象都具有某些变量比其他变量演化得快得多的特性。例如，神经元模型通常涉及可观察到的时间尺度差异。霍奇金-赫胥黎模型因解释鱿鱼巨轴突中产生动作电位的离子机制而闻名。 Rubin 和 Wechselberger (Biol. Cyber​​n. 97:5–32, 2007) 将该模型无量纲化，并获得了一个具有两个快变量、两个慢变量和显式时间尺度比 ε 的奇异扰动系统。该系统的动力学很复杂，具有周期性轨道，具有一系列由小幅度振荡（SAO）分隔的动作电位；也称为混合模式振荡 (MMO)。该系统的慢动力学由称为慢流形的二维局部不变流形组织，该流形可以是吸引流形，也可以是鞍形流形。在本文中，我们介绍了计算二维鞍形慢流形及其稳定和不稳定快速流形的通用方法。我们还开发了一种用于检测和继续相关鸭式轨道的技术，该技术是由吸引慢流形和鞍慢流形之间的相互作用产生的，并提供了一种在 $\mathbb{R}^{4}$ 中组织 SAO 的机制。我们首先使用折叠节点的扩展四维范式来测试我们的方法。我们的结果表明，我们的计算给出了该模型的慢流形和鸭式轨道的可靠近似。然后，我们的计算方法用于研究完整 Hodgkin-Huxley 模型的鞍慢流形和相关鸭翼轨道在组织 MMO 和确定动作电位的放电率中的作用。 对于足够大的 ε ，鸭式轨道排列成具有相同数量 SAO 的双鸭式轨道对。我们说明了双鸭式轨道如何将吸引慢流形分成许多条带，这些条带起到旋转扇区的作用。结果是我们能够解开慢流形的几何结构和相关的鸭式轨道，而无需缩小模型。