We explored the mathematical foundations of Recurrent Neural Networks ($\mathtt{RNN}$s) and three fundamental procedures: temporal rescaling, discretisation and linearisation. These techniques provide essential tools for characterizing $\mathtt{RNN}$s behaviour, enabling insights into temporal dynamics, practical computational implementation, and linear approximations for analysis. We discuss the flexible order of application of these procedures, emphasizing their significance in modelling and analyzing $\mathtt{RNN}$s for neuroscience and machine learning applications. We explicitly describe here under what conditions these procedures can be interchangeable.

翻译：我们探究了循环神经网络（$\mathtt{RNN}$s）的数学基础及其三种基本过程：时间重新标度、离散化和线性化。这些技术为刻画$\mathtt{RNN}$的行为提供了重要工具，能够洞悉时间动态特性、实现实际计算以及提供用于分析的线性近似。我们讨论了这些过程应用的灵活顺序，强调了它们在神经科学与机器学习应用中建模与分析$\mathtt{RNN}$s的重要性，并在此明确描述了这些过程可互换的条件。