Defining "success" for pairwise maximum entropy models

Statistical physics approaches to understanding complex systems have been remarkably successful. Numerous scientific fields are undergoing a data revolution, and high accuracy simple models for complex phenomena are in high demand. In particular, the maximum entropy principle as formulated by Jaynes provides a powerful framework to capture the collective nature of an ensemble while keeping all but a few measurement constraints maximally random. For binary variables, these models are equivalent to Ising models with competing interactions. Despite their ubiquity, deciding whether these models "work" remains a challenge. Here, we study the activity of 1000+ simultaneously active cells in the brain of mice as they navigate a virtual environment. Leveraging the massive scale of these data, we compared 900 different models for different subgroups of 100 neurons each. All neurons are selected out of the same hippocampal population but randomly from gradually increasing spatial field. We find that the more spatially contiguous the subgroup is, the better the pairwise model captures its collective behavior. Systematic comparison of many different predictions across these many examples allows us to draw the boundaries for success of these models.