A class of families of marginal probabilities on sets of discrete random variables is studied and a necessary and sufficient condition for the consistency of the given marginals is provided. This result allows one to verify the consistency of the marginals through a Boltzmann statistical analysis. The procedure is then applied in order to verify the hypotheses assumed in a recent model of neocortical associative areas, according to which connected modules of neurons are simultaneously active with probability higher than chance, and inter-modular connections are very diluted. The verification becomes a typical problem of extremely diluted spin systems in Boltzmann-Gibbs ensemble. The results presented here justify the assumptions made in the neuroscientific theory, and an upper bound to the inter-modular activity correlation is found.