  In this work, we first revise some extensions of the standard Hopfield model
in the low storage limit, namely the correlated attractor case and the
multitasking case recently introduced by the authors. The former case is based
on a modification of the Hebbian prescription, which induces a coupling between
consecutive patterns and this effect is tuned by a parameter $a$. In the latter
case, dilution is introduced in pattern entries, in such a way that a fraction
$d$ of them is blank. Then, we merge these two extensions to obtain a system
able to retrieve several patterns in parallel and the quality of retrieval,
encoded by the set of Mattis magnetizations ${m^{\mu}}$, is reminiscent of the
correlation among patterns. By tuning the parameters $d$ and $a$, qualitatively
different outputs emerge, ranging from highly hierarchical, to symmetric. The
investigations are accomplished by means of both numerical simulations and
statistical mechanics analysis, properly adapting a novel technique originally
developed for spin glasses, i.e. the Hamilton-Jacobi interpolation, with
excellent agreement. Finally, we show the thermodynamical equivalence of this
associative network with a (restricted) Boltzmann machine and study its
stochastic dynamics to obtain even a dynamical picture, perfectly consistent
with the static scenario earlier discussed.
