We derive the mean-field equations of completely connected networks of excitatory/inhibitory Hodgkin-Huxley and Fitzhugh-Nagumo neurons and prove that there is propagation to chaos, i.e. that in the limit the neurons become a) independent (this is the propagation to chaos) and b) a copy (with the same law) of a new individual, the mean field limit. This is related to some recently published experimental work by Eker et al., Science 2010. We show the results of numerical experiments that confirm the propagation to chaos and indicate, through the notion of Fischer information, that this is optimal in terms of information processing. We also consider finite size effects, i.e. the difference between the mean field situation when neuronal populations are of infinite size and the real situation, when the size is finite and show that the mean field approximation is very good for populations of reasonable size.