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，其中神经元可以递归连接，或通过多层堆叠。然而，初学者不易掌握该工具的本质及背后的物理机制。本文以通用随机神经网络为具体实例，对该方法进行了教学性介绍。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。我们还回顾了近期及历史上应用该工具的重要研究。此外，介绍了一种物理图像清晰的替代方法——动力学空穴方法，可推导出完全相同的结果。文中详细阐述了求解积分-微分平均场方程的数值实现方法，并举例说明了涨落耗散定理的探究过程。