The deep brain nuclei play an important role in many brain functions and particularly motor control. Damage to these structures result in movement disorders such as in Parkinson's disease or Huntington's disease, or behavioural disorders such as Tourette syndrome. In this paper, we propose to study the connectivity profile of the deep nuclei to the motor, associative or limbic areas and we introduce a novel tool to build a probabilistic atlas of these connections to the cortex directly on the surface of the cortical mantel, as it corresponds to the space of functional interest. The tool is then applied on two populations of healthy volunteers and patients suffering from severe Huntington's disease to produce two surface atlases of the connectivity of the basal ganglia to the cortical areas. Finally, robust statistics are used to characterize the differences of that connectivity between the two populations, providing new connectivity-based biomarkers of the pathology.