  Neuroimaging is the growing area of neuroscience devoted to produce data with
the goal of capturing processes and dynamics of the human brain. We consider
the problem of inferring the brain connectivity network from time dependent
functional magnetic resonance imaging (fMRI) scans. To this aim we propose the
symmetric graphical lasso, a penalized likelihood method with a fused type
penalty function that takes into explicit account the natural symmetrical
structure of the brain. Symmetric graphical lasso allows one to learn
simultaneously both the network structure and a set of symmetries across the
two hemispheres. We implement an alternating directions method of multipliers
algorithm to solve the corresponding convex optimization problem. Furthermore,
we apply our methods to estimate the brain networks of two subjects, one
healthy and the other affected by a mental disorder, and to compare them with
respect to their symmetric structure. The method applies once the temporal
dependence characterising fMRI data has been accounted for and we compare the
impact on the analysis of different detrending techniques on the estimated
brain networks.
