  We focus on a linear chain of $N$ first-neighbor-coupled logistic maps at
their edge of chaos in the presence of a common noise. This model,
characterised by the coupling strength $\epsilon$ and the noise width
$\sigma_{max}$, was recently introduced by Pluchino et al [Phys. Rev. E {\bf
87}, 022910 (2013)]. They detected, for the time averaged returns with
characteristic return time $\tau$, possible connections with $q$-Gaussians, the
distributions which optimise, under appropriate constraints, the nonadditive
entropy $S_q$, basis of nonextensive statistics mechanics. We have here a
closer look on this model, and numerically obtain probability distributions
which exhibit a slight asymmetry for some parameter values, in variance with
simple $q$-Gaussians. Nevertheless, along many decades, the fitting with
$q$-Gaussians turns out to be numerically very satisfactory for wide regions of
the parameter values, and we illustrate how the index $q$ evolves with $(N,
\tau, \epsilon, \sigma_{max})$. It is nevertheless instructive on how careful
one must be in such numerical analysis. The overall work shows that physical
and/or biological systems that are correctly mimicked by the Pluchino et al
model are thermostatistically related to nonextensive statistical mechanics
when time-averaged relevant quantities are studied.
