Neural subspaces, minimax entropy, and mean-field theory for networks of neurons

Recent advances in experimental techniques enable the simultaneous recording of activity from thousands of neurons in the brain, presenting both an opportunity and a challenge: to build meaningful, scalable models of large neural populations. Correlations in the brain are typically weak but widespread, suggesting that a mean-field approach might be effective in describing real neural populations, and we explore a hierarchy of maximum entropy models guided by this idea. We begin with models that match only the mean and variance of the total population activity, and extend to models that match the experimentally observed mean and variance of activity along multiple projections of the neural state. Confronted by data from several different brain regions, these models are driven toward a first-order phase transition, characterized by the presence of two nearly degenerate minima in the energy landscape, and this leads to predictions in qualitative disagreement with other features of the data. To resolve this problem we introduce a novel class of models that constrain the full probability distribution of activity along selected projections. We develop the mean-field theory for this class of models and apply it to recordings from 1000+ neurons in the mouse hippocampus. This 'distributional mean--field' model provides an accurate and consistent description of the data, offering a scalable and principled approach to modeling complex neural population dynamics.