  In a recent work we have introduced a novel approach to study the effect of
weak non-linearities in the transfer function on the information transmitted by
an analogue channel, by means of a perturbative diagrammatic expansion. We
extend here the analysis to all orders in perturbation theory, which allows us
to release any constraint concerning the magnitude of the expansion parameter
and to establish the rules to calculate easily the contribution at any order.
As an example we explicitly compute the information up to the second order in
the non-linearity, in presence of random gaussian connectivities and in the
limit when the output noise is not small. We analyze the first and second order
contributions to the mutual information as a function of the non-linearity and
of the number of output units. We believe that an extensive application of our
method via the analysis of the different contributions at distinct orders might
be able to fill a gap between well known analytical results obtained for linear
channels and the non trivial treatments which are required to study highly
non-linear channels.
