I'm studying a paper on (biological) Neural Networks, and the paper studies some stability properties of an $N$-sized network, and then, as $N$ tends to infinity, it is proven that a "mean-field result holds".
Now, I've already read the Wikipedia article about "mean-field Theory" and besides the concept of "averaging" the effect of many particles into 1 averaged-effect; I don't see any other similarity: In this paper there's no Hamiltonian.
So I would like what is exactly understood by "Mean-Field Theory" in the context of Probability theory. Is there any bibliography introducing this topic?
Thank you!