Canard Theory and Excitability

Martin Wechselberger, John Mitry, and John Rinzel

Abstract An important feature of many physiological systems is that they evolve on multiple scales. From a mathematical point of view, these systems are modeled as singular perturbation problems. It is the interplay of the dynamics on different temporal and spatial scales that creates complicated patterns and rhythms. Many important physiological functions are linked to time-dependent changes in the forcing which leads to nonautonomous behaviour of the cells under consideration. Transient dynamics observed in models of excitability are a prime example. Recent developments in canard theory have provided a new direction for understanding these transient dynamics. The key observation is that canards are still well defined in nonautonomous multiple scales dynamical systems, while equilibria of an autonomous system do, in general, not persist in the corresponding driven, nonautonomous system. Thus canards have the potential to significantly shape the nature of solutions in nonautonomous multiple scales systems. In the context of neuronal excitability, we identify canards of folded saddle type as firing threshold manifolds. It is remarkable that dynamic information such as the temporal evolution of an external drive is encoded in the location of an invariant manifold—the canard.

翻译原文

Martin Wechselberger 和 John Mitry

摘要：许多生理系统的一个重要特征是它们在多个尺度上进化。从数学的角度看，这些系统被建模为奇异摄动问题。正是不同时空尺度上动力学的相互作用，创造了复杂的模式和节律。许多重要的生理功能与强迫的时间依赖性变化有关，这种变化导致所考虑的细胞的非自主性行为。在兴奋性模型中观察到的瞬态动力学就是一个很好的例子。鸭解理论的最新发展为理解这些瞬态动力学提供了一个新的方向。关键的观察结果是，鸭解在非自治多尺度动力系统中仍然是良定义的，而自治系统的平衡点一般情况下并不存在于相应的驱动的非自治系统中。因此，在非自治多尺度系统中，鸭解有可能显著地影响解的性质。在神经兴奋性的背景下，我们将折鞍型鸭解识别为发放阈值流形。值得注意的是，诸如外部驱动的时间演化等动态信息被编码在不变流形的位置—鸭解。
关键词：鸭解；几何奇异摄动理论；兴奋性；神经动力学；发放阈值流行；分界线；瞬态吸引子。

翻译稿件