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

Abstract

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.

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Figure 1: Weak pairwise cross-correlations and the failure of the independent approximation.
Figure 2: A maximum entropy model including all pairwise interactions gives an excellent approximation of the full network correlation structure.
Figure 3: Pairwise interactions and individual cell biases, as in equation (1).
Figure 4: Interactions and local fields in networks of different size.
Figure 5: Extrapolation to larger networks.

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Acknowledgements

We thank G. Stephens and G. Tkačik for discussions, N. Tkachuk for help with the experiments, S. Marom for sharing his lab's cultured cortical networks data with us, and E. J. Chichilnisky for sharing his lab's primate retina results with us. This work was supported in part by the NIH and by the E. Matilda Zeigler Foundation.

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Correspondence to Elad Schneidman.

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Supplementary information

Supplementary Figure 1

Effect of bin size discretization on the second order maximum entropy model. The success of the second order maximum entropy model is not sensitive to the temporal discretization bin size. (PDF 77 kb)

Supplementary Figure 2

The success of the second order maximum entropy model is not sensitive to the size of the network. The success of the second order maximum entropy model is not sensitive to the size of the sub-network modeled. (PDF 72 kb)

Supplementary Figure 3

Average interaction field vs. average local field in maximum entropy models of different size sub-networks. The average interaction field versus the average local field, in the second order maximum entropy models. For larger sub-networks the interaction field dominates the local field. (DOC 40 kb)

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Schneidman, E., Berry, M., Segev, R. et al. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature 440, 1007–1012 (2006). https://doi.org/10.1038/nature04701

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