  An emerging paradigm proposes that neural computations can be understood at
the level of dynamical systems that govern low-dimensional trajectories of
collective neural activity. How the connectivity structure of a network
determines the emergent dynamical system however remains to be clarified. Here
we consider a novel class of models, Gaussian-mixture low-rank recurrent
networks, in which the rank of the connectivity matrix and the number of
statistically-defined populations are independent hyper-parameters. We show
that the resulting collective dynamics form a dynamical system, where the rank
sets the dimensionality and the population structure shapes the dynamics. In
particular, the collective dynamics can be described in terms of a simplified
effective circuit of interacting latent variables. While having a single,
global population strongly restricts the possible dynamics, we demonstrate that
if the number of populations is large enough, a rank-R network can approximate
any R-dimensional dynamical system.
