Dimension of activity in random neural networks

Neural networks are high-dimensional nonlinear dynamical systems that process information through the coordinated activity of many connected units. Understanding how biological and machine-learning networks function and learn requires knowledge of the structure of this coordinated activity, information contained, for example, in cross covariances between units. Self-consistent dynamical mean field theory (DMFT) has elucidated several features of random neural networks -- in particular, that they can generate chaotic activity -- however, a calculation of cross covariances using this approach has not been provided. Here, we calculate cross covariances self-consistently via a two-site cavity DMFT. We use this theory to probe spatiotemporal features of activity coordination in a classic random-network model with independent and identically distributed (i.i.d.) couplings, showing an extensive but fractionally low effective dimension of activity and a long population-level timescale. Our formulae apply to a wide range of single-unit dynamics and generalize to non-i.i.d. couplings. As an example of the latter, we analyze the case of partially symmetric couplings.