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Patterns of activity in networks of neurons are a prototypical complex system. Here we analyze data on the retina to show that information shared between a single neuron and the rest of the network is compressible, through a combination of the information bottleneck and an iteration scheme inspired by the renormalization group. The result is that the number of parameters needed to describe the distribution of joint activity scales with the square of the number of neurons, even though the interactions are not well approximated as pairwise. Our results also show that the shared information is essentially equal to the information that individual neurons carry about natural visual inputs, which has implications for the structure of the neural code.