Interpreting high-dimensional datasets requires new computational and analytical methods. We developed such methods to extract and analyze neural activity from 20,000 neurons recorded simultaneously in awake, behaving mice. The neural activity was not low-dimensional as commonly thought, but instead was high-dimensional and obeyed a power-law scaling across its eigenvalues. We developed a theory that proposes that neural responses to external stimuli maximize information capacity while maintaining a smooth neural code. We then observed power-law eigenvalue scaling in many real-world datasets, and therefore developed a nonlinear manifold embedding algorithm called Rastermap that can capture such high-dimensional structure.