In this work we demonstrate how Finsler geometry---and specifically the related geodesic tracto-graphy---can be levied to analyze structural connections between different brain regions. We present new theoretical developments which support the definition of a novel Finsler metric and associated connectivity measures, based on closely related works on the Riemannian framework for diffusion MRI. Using data from the Human Connectome Project, as well as population data from an autism spectrum disorder study, we demonstrate that this new Finsler metric, together with the new connectivity measures, results in connectivity maps that are much closer to known tract anatomy compared to previous geodesic connectivity methods. Our implementation can be used to compute geodesic distance and connectivity maps for segmented areas and is publicly available.