A Statistical Theory of Inferring Population Geometry from Large-Scale Neural Recordings

Contemporary neuroscience has witnessed an impressive expansion in the number of neurons whose activity can be recorded simultaneously. As we add more neurons, are we recording enough trials in order to infer the population geometry of neural activity? We derive new theories, based on random matrices and free probability, to answer this question, comparing the predictions made in two distinct regimes: an extensive limit where the dimensionality of data is proportional to the number of recorded neurons and trials, and an intensive limit where the dimensionality is finite. We furthermore explore scenarios in which data exhibit high-dimensional power law spectra. We successfully test the predictions of our theory on a variety of datasets, and reveal an intriguing tradeoff between two very different resources (neurons and trials) in the accurate estimation of neural geometry: recording more neurons actually allows recording fewer trials to accurately estimate this geometry. Overall our work lays a theoretical foundation for experimental design in modern large-scale neuroscience.