  We present a systematic study to test a recently introduced phenomenological
renormalization group, proposed to coarse-grain data of neural activity from
their correlation matrix. The approach allows, at least in principle, to
establish whether the collective behavior of the network of spiking neurons is
described by a non-Gaussian critical fixed point. We test this renormalization
procedure in a variety of models focusing in particular on the contact process,
which displays an absorbing phase transition at $\lambda = \lambda_c$ between a
silent and an active state. We find that the results of the coarse-graining do
not depend on the presence of long-range interactions, but some scaling
features persist in the super-critical system up to a distance of $10\%$ from
$\lambda_c$. Our results provide insights on the possible subtleties that one
needs to consider when applying such phenomenological approaches directly to
data to infer signatures of criticality.
