  We derive an approximate expression for mutual information in a broad class
of discrete-time stationary channels with continuous input, under the
constraint of vanishing input amplitude or power. The approximation describes
the input by its covariance matrix, while the channel properties are described
by the Fisher information matrix. This separation of input and channel
properties allows us to analyze the optimality conditions in a convenient way.
We show that input correlations in memoryless channels do not affect channel
capacity since their effect decreases fast with vanishing input amplitude or
power. On the other hand, for channels with memory, properly matching the input
covariances to the dependence structure of the noise may lead to almost
noiseless information transfer, even for intermediate values of the noise
correlations. Since many model systems described in mathematical neuroscience
and biophysics operate in the high noise regime and weak-signal conditions, we
believe, that the described results are of potential interest also to
researchers in these areas.
