Multi-dimensional connectivity: a conceptual and mathematical review

The estimation of functional connectivity between regions of the brain, for example based on statistical dependencies between the time series of activity in each region, has become increasingly important in neuroimaging. Typically, multiple time series (e.g. from each voxel in fMRI data) are first reduced to a single time series that summarises the activity in a region of interest, e.g. by averaging across voxels or by taking the first principal component; an approach we call one-dimensional connectivity. However, this summary approach ignores potential multi-dimensional connectivity between two regions, and a number of recent methods have been proposed to capture such complex dependencies. Here we review the most common multi-dimensional connectivity methods, from an intuitive perspective, from a formal (mathematical) point of view, and through a number of simulated and real (fMRI and MEG) data examples that illustrate the strengths and weaknesses of each method. The paper is accompanied with both functions and scripts, which implement each method and reproduce all the examples.