  In this paper we introduce and investigate the statistical mechanics of
hierarchical neural networks: First, we approach these systems \`a la Mattis,
by thinking at the Dyson model as a single-pattern hierarchical neural network
and we discuss the stability of different retrievable states as predicted by
the related self-consistencies obtained from a mean-field bound and from a
bound that bypasses the mean-field limitation. The latter is worked out by
properly reabsorbing fluctuations of the magnetization related to higher levels
of the hierarchy into effective fields for the lower levels. Remarkably, mixing
Amit's ansatz technique (to select candidate retrievable states) with the
interpolation procedure (to solve for the free energy of these states) we prove
that (due to gauge symmetry) the Dyson model accomplishes both serial and
parallel processing. One step forward, we extend this scenario toward multiple
stored patterns by implementing the Hebb prescription for learning within the
couplings. This results in an Hopfield-like networks constrained on a
hierarchical topology, for which, restricting to the low storage regime (where
the number of patterns grows at most logarithmical with the amount of neurons),
we prove the existence of the thermodynamic limit for the free energy and we
give an explicit expression of its mean field bound and of the related improved
bound
