  We propose a novel model for a glass-forming liquid which allows to switch in
a continuous manner from a standard three-dimensional liquid to a fully
connected mean-field model. This is achieved by introducing k additional
particle-particle interactions which thus augments the effective number of
neighbors of each particle. Our computer simulations of this system show that
the structure of the liquid does not change with the introduction of these
pseudo neighbours and by means of analytical calculations, we determine the
structural properties related to these additional neighbors. We show that the
relaxation dynamics of the system slows down very quickly with increasing k and
that the onset and the mode-coupling temperatures increase. The systems with
high values of k follow the MCT power law behaviour for a larger temperature
range compared to the ones with lower values of k. The dynamic susceptibility
indicates that the dynamic heterogeneity decreases with increasing k whereas
the non-Gaussian parameter is independent of it. Thus we conclude that with the
increase in the number of pseudo neighbours the system becomes more mean-field
like. By comparing our results with previous studies on mean-field like system
we come to the conclusion that the details of how the mean-field limit is
approached are important since they can lead to different dynamical behavior in
this limit.
