Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of generic random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.

翻译：动力学平均场理论是一种强大的物理学工具，用于分析神经网络的典型行为，其中神经元可以循环连接，或者多层神经元可以堆叠。然而，初学者不易掌握该工具的本质及其背后的物理学原理。本文通过一个特定实例——通用随机神经网络，对该方法进行了教学式介绍。在该网络中，神经元通过相关突触随机且全连接，因此网络展现出丰富的涌现集体动力学行为。同时，我们还回顾了运用该工具的相关过去及近期重要研究。此外，另一种物理意义明确且可替代的方法，即动力学空穴法，也被引入以推导出完全相同的结果。文中还详细介绍了求解积分微分平均场方程的数值实现方法，并辅以探索涨落耗散定理的示例说明。