Studying brain connectivity is important in neuroscience. In this talk, we investigate the connectivity patterns between two brain structures - the neocortex and the cerebellum - by analyzing functional magnetic resonance imaging (fMRI) data across a wide range of mental tasks. We build models to predict the cerebellar activity patterns from the cortical activity patterns, and evaluate their prediction performance. To handle the challenge of high dimensionality, we constrain the problem using knowledge of the underlying biological connectivity, such as spatial smoothness, low-dimensional structure, and non-negativity of connectivity weights, and implement smoothness-penalized, low-rank, matrix regression procedures. Empirical studies demonstrate the promising performance of the analysis procedures.