Motif statistics and spike correlations in neuronal networks

For this model, there is an explicit expression for pairwise correlation in terms of the connectivity matrix. By expanding this expression in a series, one can relate each term to a different motif (with a different number of connections). Through a resumming technique, we show that the average correlation across the network can be closely approximated using the frequencies of only first and second order motifs. These are the diverging motif—two cells both receiving projections from another cell, its counterpart the converging motif—two cells projecting to a common cell, and the chain motif—three cells linked by two consecutive projections. Specifically, we show that the prevalence of diverging and chain motifs tends to increase correlation, while the converging motif makes no contribution to the average correlation. Moreover, we numerically show that variance of correlations across the network is largely determined by the frequency of the chain motif (see Figure 1) alone. Finally, we demonstrate potential effect of motif statistics on neural coding by showing how motif frequencies impact linear Fisher information. In particular, we find that the linear Fisher information is only affected by converging motif frequency (more information with given more converging motifs).