  Neural networks encode information through their collective spiking activity
in response to external stimuli. This population response is noisy and strongly
correlated, with complex interplay between correlations induced by the
stimulus, and correlations caused by shared noise. Understanding how these
correlations affect information transmission has so far been limited to pairs
or small groups of neurons, because the curse of dimensionality impedes the
evaluation of mutual information in larger populations. Here we develop a
small-correlation expansion to compute the stimulus information carried by a
large population of neurons, yielding interpretable analytical expressions in
terms of the neurons' firing rates and pairwise correlations. We validate the
approximation on synthetic data and demonstrate its applicability to
electrophysiological recordings in the vertebrate retina, allowing us to
quantify the effects of noise correlations between neurons and of memory in
single neurons.
