Collective Activity in Neural Networks: the Mathematical Structure of Connection Graphs and Population Codes
Correlated, or synchronized, spiking activity among pairs of neurons is widely observed across the nervous system. How do these correlations arise from the dynamics of neural networks? The interconnectivity of neurons is one likely contributor. Moreover, recent experiments found that certain connectivity patterns, or motifs, in biological neural networks occur at markedly different frequencies than what would be expected if the neurons were randomly connected. Connecting these ideas, we find that the average correlation in a network is determined by the statistical prevalence of two families of motifs. We derive an expression for correlation that reveals the contributions of each motif in order. The prevalence of a motif is quantified by a new graphical quantity we call the motif cumulant. Importantly, in practical examples motif cumulants decay quickly with the motif size. Therefore frequencies of motifs involving only a few cells are often enough to predict the average correlation. We find that this link between local connectivity structures and global correlation is strongly affected by heterogeneity in connectivity, but can be recovered by a network-partitioning method. Next, we study the impact of correlated activity on the accuracy of information encoded by neural populations --- that is, on the accuracy of the neural code. First, we generalize an existing result to prove a rule that describes accuracy-improving correlations via their sign. For large populations, however, we find that this rule is only useful for weak correlations; many diversely structured correlations can improve coding accuracy. However, there is organization within this diversity: we prove that the optimal correlations must lie on boundaries of the allowed set of correlations. Finally, we study how motifs can change the signal-filtering property of a network. We find that the transfer function for the whole network is determined by chain motif cumulants. Moreover, the way that the effects of these motifs combine has an intriguing structure. This can be illustrated with a diagram where each cumulant is a feedback link, a process that reveals how chain motifs shape the transfer function to produce varied spectral and temporal features. We provide examples where these features are employed to sustain functions such as extending the response time constant and signal de-noising.
- Applied mathematics