Exactly-solvable statistical physics model of large neuronal populations

In networks of neurons, fine-scale interactions build upon one another to produce large-scale patterns of activity. But inferring these interactions from state-of-the-art experiments poses a fundamental challenge: As the number of cells increases, the number of possible interactions quickly outpaces the number of available statistics. Thus, rather than inferring all interactions, one must focus on a sparse network. Here, we seek to identify the most informative network of effective interactions in large neuronal populations. We show that the optimal network -- in a precise information-theoretic sense -- is the one that provides the most information about the population while remaining maximally ignorant about all other interactions. For general networks, solving this "minimax entropy" problem is computationally prohibitive. However, for networks with tree topologies, we present an exact solution; that is, we propose a method for efficiently inferring the maximally informative tree of interactions in large populations. We apply our method to over one thousand neurons in the mouse hippocampus. Despite containing only 0.1% of all possible interactions, the optimal tree reduces the entropy of independent neurons by 14% (over 50 times more than a random tree). In fact, the optimal tree accurately predicts the distribution of population-wide activity and even increases in accuracy as the population grows. Moreover, while random interactions are nearly evenly split between positive and negative, the optimal tree is almost entirely positive, with activity in one neuron inducing activity in the others. Together, these results suggest that a sparse network of strong excitatory interactions may guide the collective activity. Broadly, our framework allows principled models from statistical physics to keep pace with the explosion of large-scale neural recordings.