We review some widely studied models and firing dynamics for neuronal systems, both at the single cell and network level, and dynamical systems techniques to study them. In particular, we focus on two topics in mathematical neuroscience that have attracted the attention of mathematicians for decades: single-cell excitability and bursting. We review the mathematical framework for three types of excitability and onset of repetitive firing behavior in single-neuron models and their relation with Hodgkin's classification in 1948 of repetitive firing properties. We discuss the mathematical dissection of bursting oscillations using fast/slow analysis and demonstrate the approach using single-cell and mean-field network models. Finally, we illustrate the properties of Type III excitability in which case repetitive firing for constant or slow inputs is absent. Rather, firing is in response only to rapid enough changes in the stimulus. Our case study involves neuronal computations for sound localization for which neurons in the auditory brain stem perform extraordinarily precise coincidence detection with submillisecond temporal resolution.

中文翻译：

---

神经元兴奋性、振荡和符合检测的非线性动力学

我们回顾了一些广泛研究的神经元系统模型和放电动力学，包括单细胞和网络级别，以及研究它们的动力系统技术。我们特别关注数学神经科学中几十年来引起数学家关注的两个主题：单细胞兴奋性和爆发性。我们回顾了单神经元模型中三种类型的兴奋性和重复放电行为的数学框架，以及它们与 1948 年霍奇金重复放电特性分类的关系。我们使用快速/慢速分析讨论突发振荡的数学剖析，并使用单细胞和平均场网络模型演示该方法。最后，我们说明了 III 型兴奋性的特性，在这种情况下，不存在对恒定或缓慢输入的重复激发。相反，发射只是对刺激足够快速的变化做出反应。我们的案例研究涉及声音定位的神经元计算，听觉脑干中的神经元以亚毫秒时间分辨率执行非常精确的符合检测。