Fig. 6

Second-order motifs dominate pairwise correlations in a wide range of networks; inhibitory common input is the dominant second-order motif. (A) Fraction of variance explained (\(R^{2}\)) from linear regressions of total correlation (\(\tilde{\mathbf {C}}_{ij}/\sqrt{\tilde {\mathbf {C}}_{ii}\tilde{\mathbf {C}}_{jj}}\)) against contributions from first order (blue), second order (red), third-order (green), and fourth-order (magenta) motifs. (B) Contributions up to fourth order (\(\tilde{\mathbf {R}}^{k}_{ij}\), for \(k=1,\ldots,4\)) vs. total correlation (\(\tilde{\mathbf {C}}_{ij}/\sqrt{\tilde {\mathbf {C}}_{ii}\tilde{\mathbf {C}}_{jj}}\)) for all E-E cell pairs in a network, for two individual networks included in panel A. (C) Fraction of variance explained (\(R^{2}\)) from linear regressions of contributions to pairwise correlations from second-order motifs (\(\tilde{R}^{2}_{ij}\)) against contributions from the distinct types of second-order motifs: inhibitory common input (magenta), excitatory common input (red), decorrelating chains (green), and correlating chains (blue). Both (A,C): Each data point represents the mean value from 15 networks with \(W_{IE}\) between 5 and 11.2; error bars show standard deviation across these values. \(W_{XY}=1\) is when the presynaptic input is exactly the average population firing rate (filtered by the synapse) in an all-to-all homogeneous network