  The standard Hopfield model for associative neural networks accounts for
biological Hebbian learning and acts as the harmonic oscillator for pattern
recognition, however its maximal storage capacity is $\alpha \sim 0.14$, far
from the theoretical bound for symmetric networks, i.e. $\alpha =1$. Inspired
by sleeping and dreaming mechanisms in mammal brains, we propose an extension
of this model displaying the standard on-line (awake) learning mechanism (that
allows the storage of external information in terms of patterns) and an
off-line (sleep) unlearning$\&$consolidating mechanism (that allows
spurious-pattern removal and pure-pattern reinforcement): this obtained daily
prescription is able to saturate the theoretical bound $\alpha=1$, remaining
also extremely robust against thermal noise. Both neural and synaptic features
are analyzed both analytically and numerically. In particular, beyond obtaining
a phase diagram for neural dynamics, we focus on synaptic plasticity and we
give explicit prescriptions on the temporal evolution of the synaptic matrix.
We analytically prove that our algorithm makes the Hebbian kernel converge with
high probability to the projection matrix built over the pure stored patterns.
Furthermore, we obtain a sharp and explicit estimate for the "sleep rate" in
order to ensure such a convergence. Finally, we run extensive numerical
simulations (mainly Monte Carlo sampling) to check the approximations
underlying the analytical investigations (e.g., we developed the whole theory
at the so called replica-symmetric level, as standard in the
Amit-Gutfreund-Sompolinsky reference framework) and possible finite-size
effects, finding overall full agreement with the theory.
