It has become evident that connectivity is a major factor influencing the brain's computation power and stability and it is not adequate to investigate functional specialisation without considering how different brain areas interact. In fact, disturbances of brain connectivity have been implicated in a number of diseases including schizophrenia ADHD, autism AD, stroke and brain trauma. This has resulted in recent interest in network organisation and dynamics. Magnetic Resonance Imaging (MRI) can be used to derive structural and functional brain networks from diffusion weighted MRI (DWI) and resting-state functional MRI (rs-fMRI). Several tractography techniques have been developed that exploit voxel-based directional information to extract structural networks from DWI. On the other hand, functional networks are defined based on the temporal correlations between spatially remote neurophysiological events. The goal of combining these approaches is to provide a whole-brain connectivity description that reflects structure and function. Integrating measures of structural and functional brain connectivity holds the promise of dramatically improving our understanding of brain function and malfunction and could lead to the development of clinically useful biomarkers.

I have developed a systematic framework to learn across several subjects a mapping from brain anatomical connectivity to functional connectivity based on inference. I demonstrated the knowledge we gained by a number of different approaches and how this led us to a structured-output learning task in order to account for the strongly correlated parameters. The key advantage of our latest approach is that it accounts for indirect connectivity and it utilises a generative model based on graphical models of autoregressive Gaussian processes. A graphical model of the fMRI time series is an undirected graph with nodes equal to the number of ROIs. Each pair of nodes is connected with an edge if the underline time series are conditionally dependent, given the other time series. The problem to solve is known as covariance selection problem, which is the problem of computing the maximum likelihood estimate of the inverse covariance matrix of a multivariate Gaussian variable, subject to conditional independence constrains. We used the common structure based on the structural connectivity to impose conditional independence and thus to enhance the robustness of the estimation of the covariance matrix. This natural parameterization of functional connectivity also enforces the positive-definiteness of the predicted covariance and thus matches the structure of the output space. Our results show that functional connectivity can be explained by anatomical connectivity on a rigorous statistical basis.