  Recently a daily routine for associative neural networks has been proposed:
the network Hebbian-learns during the awake state (thus behaving as a standard
Hopfield model), then, during its sleep state, optimizing information storage,
it consolidates pure patterns and removes spurious ones: this forces the
synaptic matrix to collapse to the projector one (ultimately approaching the
Kanter-Sompolinksy model). This procedure keeps the learning Hebbian-based (a
biological must) but, by taking advantage of a (properly stylized) sleep phase,
still reaches the maximal critical capacity (for symmetric interactions). So
far this emerging picture (as well as the bulk of papers on unlearning
techniques) was supported solely by mathematically-challenging routes, e.g.
mainly replica-trick analysis and numerical simulations: here we rely
extensively on Guerra's interpolation techniques developed for neural networks
and, in particular, we extend the generalized stochastic stability approach to
the case. Confining our description within the replica symmetric approximation
(where the previous ones lie), the picture painted regarding this
generalization (and the previously existing variations on theme) is here
entirely confirmed. Further, still relying on Guerra's schemes, we develop a
systematic fluctuation analysis to check where ergodicity is broken (an
analysis entirely absent in previous investigations). We find that, as long as
the network is awake, ergodicity is bounded by the Amit-Gutfreund-Sompolinsky
critical line (as it should), but, as the network sleeps, sleeping destroys
spin glass states by extending both the retrieval as well as the ergodic
region: after an entire sleeping session the solely surviving regions are
retrieval and ergodic ones and this allows the network to achieve the perfect
retrieval regime (the number of storable patterns equals the number of neurons
in the network).
