  We perform an {\it ab initio} modeling of amorphous copper-doped alumina
(a-Al$_2$O$_3$:Cu), a prospective memory material based on resistance
switching, and study the structural origin of electronic conduction in this
material. We generate molecular dynamics based models of a-Al$_2$O$_3$:Cu at
various Cu-concentrations and study the structural, electronic and vibrational
properties as a function of Cu-concentration. Cu atoms show a strong tendency
to cluster in the alumina host, and metallize the system by filling the band
gap uniformly for higher Cu-concentrations. We also study thermal fluctuations
of the HOMO-LUMO energy splitting and observe the time evolution of the size of
the band gap, which can be expected to have an important impact on the
conductivity. We perform a numerical computation of conduction pathways, and
show its explicit dependence on Cu connectivity in the host. We present an
analysis of ion dynamics and structural aspects of localization of classical
normal modes in our models.
