  In this paper we study, via equilibrium statistical mechanics, the properties
of the internal energy of an Hopfield neural network whose patterns are stored
continuously (Gaussian distributed). The model is shown to be equivalent to a
bipartite spin glass in which one party is given by dichotomic neurons and the
other party by Gaussian spin variables. Dealing with replicated systems, beyond
the Mattis magnetization, we introduce two overlaps, one for each party, as
order parameters of the theory: The first is a standard overlap among neural
configurations on different replicas, the second is an overlap among the
Gaussian spins of different replicas. The aim of this work is to show the
existence of constraints for these order parameters close to ones found in many
other complex systems as spin glasses and diluted networks: we find a class of
Ghirlanda-Guerra-like identities for both the overlaps, generalizing the well
known constraints to the neural networks, as well as new identities where noise
is involved explicitly.
