A scale-dependent measure of system dimensionality
A fundamental problem in science is uncovering the effective number of dynamical degrees of freedom in a complex system, a quantity that depends on the spatio-temporal scale at which the system is observed. Here, we propose a scale-dependent generalization of a classic enumeration of latent variables, the Participation Ratio. We show how this measure relates to conventional quantities such as the Correlation dimension and Principal Component Analysis, and demonstrate its properties in dynamical systems such as the Lorentz attractor. We apply the method to neural population recordings in multiple brain areas and brain states, and demonstrate fundamental differences in the effective dimensionality of neural activity in behaviorally engaged states versus spontaneous activity. Our method applies broadly to multi-variate data across fields of science.