In the context of computational anatomy, one aims at understanding and modelling the anatomy of the brain and its variations across a population. This geometrical variability is often measured from precisely defined anatomical landmarks such as sulcal lines or meshes of brain structures. This requires (1) to compare geometrical objects without introducing too many non realistic priors and (2) to retrieve the variability of the whole brain from the variability of the landmarks. We propose, in this paper, to infer a statistical brain model from the consistent integration of variability of sulcal lines. The similarity between two sets of lines is measured by a distance on currents that does not assume any type of point correspondences and it is not sensitive to the sampling of lines. This shape similarity measure is used in a diffeomorphic registrations which retrieves a single deformation of the whole 3D space. This diffeomorphism integrates the variability of all lines in a as spatially consistent manner as possible. Based on repeated pairwise registrations on a large database, we learn how the mean anatomy varies in a population by computing statistics on diffeomorphisms. Whereas usual methods lead to descriptive measures of variability, such as variability maps or statistical tests, our model is generative: we can simulate new observations according to the learned probability law on deformations. In practice, this variability captured by the model is synthesized in the principal modes of deformations. As a deformation is dense, we can also apply it to other anatomical structures defined in the template space. This is illustrated the action of the principal modes of deformations to a mean cortical surface. Eventually, our current-based diffeomorphic registration (CDR) approach is carefully compared to a pointwise line correspondences (PLC) method. Variability measures are computed with both methods on the same dataset of sulcal lines. The results suggest that we retrieve more variability with CDR than with PLC, especially in the direction of the lines. Other differences also appear which highlight the different methodological assumptions each method is based on.