Exactly solvable statistical physics models for large neuronal populations

Maximum-entropy methods provide a principled path connecting measurements of neural activity directly to statistical physics models, and this approach has been successful for populations of $N~100$ neurons. As $N$ increases in new experiments, we enter an undersampled regime where we have to choose which observables should be constrained in the maximum-entropy construction. The best choice is the one that provides the greatest reduction in entropy, defining a "minimax entropy" principle. This principle becomes tractable if we restrict attention to correlations among pairs of neurons that link together into a tree; we can find the best tree efficiently, and the underlying statistical physics models are exactly solved. We use this approach to analyze experiments on $N~1500$ neurons in the mouse hippocampus, and we find that the resulting model captures key features of collective activity in the network.