  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
one thousand 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. These results suggest that a sparse backbone of excitatory
interactions may play an important role in driving collective neuronal
activity.
