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：

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[公式：见正文]中的鞍形慢流形和Canard轨道及其在Full Hodgkin-Huxley模型中的应用。

许多生理现象具有某些变量比其他变量进化得快得多的特性。例如，神经元模型通常涉及可观察到的时标差异。霍奇金-赫克斯利（Hodgkin-Huxley）模型因解释在鱿鱼巨型轴突中产生动作电位的离子机理而闻名。Rubin和Wechselberger（Biol。Cyber​​n。97：5–32，2007）对该模型进行了无量纲化处理，并获得了具有两个快速，两个慢变量和明确的时标比ε的奇异摄动系统。该系统的动力学是复杂的，具有周期性的轨道，具有一系列由小幅度振荡（SAO）隔开的动作电位。也称为混合模式振荡（MMO）。该系统的慢动力学由二维局部不变的流形（称为慢流形）组织，该流形可以是吸引型也可以是鞍型。在本文中，我们介绍了一种用于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。我们还开发了一种技术，该技术用于检测和继续由吸引流和鞍形慢流形之间的相互作用引起的相关卡纳德轨道，并提供了一种在$ \ mathbb {R} ^ {4} $中组织SAO的机制。我们首先用扩展的折叠节点的四维法线形式测试我们的方法。我们的结果表明，我们的计算给出了该模型的慢流形和卡纳德轨道的可靠近似值。然后，我们的计算方法被用于研究整个Hodgkin-Huxley模型的鞍形慢流形和相关的卡纳德轨道在组织MMO和确定动作电位发射率方面的作用。对于ε足够大的情况，卡纳德轨道以成对的双卡纳德轨道成对排列，且具有相同的SAO。我们说明双胞胎卡纳德轨道如何将吸引缓慢的流形划分为许多带，这些带起到旋转扇区的作用。结果是我们可以解开慢流形和相关的卡纳德轨道的几何形状，而无需简化模型。卡纳德轨道以成对的双卡纳德轨道排列，并具有相同的SAO。我们说明双胞胎卡纳德轨道如何将吸引缓慢的流形划分为许多带，这些带起到旋转扇区的作用。结果是我们可以解开慢流形和相关的卡纳德轨道的几何形状，而无需简化模型。卡纳德轨道以成对的双卡纳德轨道排列，并具有相同的SAO。我们说明双胞胎卡纳德轨道如何将吸引缓慢的流形划分为许多带，这些带起到旋转扇区的作用。结果是我们可以解开慢流形和相关的卡纳德轨道的几何形状，而无需简化模型。