  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 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.
