Weak pairwise correlations imply strongly correlated network states in a neural population.

Biological networks have so many possible states that exhaustive sampling is impossible. Successful analysis thus depends on simplifying hypotheses, but experiments on many systems hint that complicated, higher-order interactions among large groups of elements have an important role. Here we show, in the vertebrate retina, that weak correlations between pairs of neurons coexist with strongly collective behaviour in the responses of ten or more neurons. We find that this collective behaviour is described quantitatively by models that capture the observed pairwise correlations but assume no higher-order interactions. These maximum entropy models are equivalent to Ising models, and predict that larger networks are completely dominated by correlation effects. This suggests that the neural code has associative or error-correcting properties, and we provide preliminary evidence for such behaviour. As a first test for the generality of these ideas, we show that similar results are obtained from networks of cultured cortical neurons. Much of what we know about biological networks has been learned by studying one element at a time-recording the electrical activity of single neurons, the expression levels of single genes or the concentrations of individual metabolites. On the other hand, important aspects of biological function must be shared among many elements Here we address these questions in the context of the vertebrate retina, where it is possible to make long, stable recordings from many neurons simultaneously as the system responds to complex, naturalistic inputs The scale of correlations Throughout the nervous system, individual elements communicate by generating discrete pulses termed action potentials or spikes The small values of the correlation coefficients suggest an approximation in which the cells are completely independent. For most pairs, this is true with a precision of a few per cent, but if we extrapolate this approximation to the whole population of 40 cells, it fails disastrously. I