Exact minimax entropy models of large-scale neuronal activity.

In the brain, fine-scale correlations combine to produce macroscopic patterns of activity. However, as experiments record from larger and larger populations, we approach a fundamental bottleneck: the number of correlations one would like to include in a model grows larger than the available data. In this undersampled regime, one must focus on a sparse subset of correlations; the optimal choice contains the maximum information about patterns of activity or, equivalently, minimizes the entropy of the inferred maximum entropy model. Applying this "minimax entropy" principle is generally intractable, but here we present an exact and scalable solution for pairwise correlations that combine to form a tree (a network without loops). Applying our method to over 1000 neurons in the mouse hippocampus, we find that the optimal tree of correlations reduces our uncertainty about the population activity by 14% (over 50 times more than a random tree). Despite containing only 0.1% of all pairwise correlations, this minimax entropy model accurately predicts the observed large-scale synchrony in neural activity and becomes even more accurate as the population grows. The inferred Ising model is almost entirely ferromagnetic (with positive interactions) and exhibits signatures of thermodynamic criticality. Together, these results suggest that a large amount of information may be compressed into a small number of correlations between neurons, and provide the tools for identifying the most important correlations in other complex living systems.