Group analysis of self-organizing maps based on functional MRI using restricted Frechet means

Studies of functional MRI data are increasingly concerned with the estimation of differences in spatio-temporal networks across groups of subjects or experimental conditions. Unsupervised clustering and independent component analysis (ICA) have been used to identify such spatio-temporal networks. While these approaches have been useful for estimating these networks at the subject-level, comparisons over groups or experimental conditions require further methodological development. In this paper, we tackle this problem by showing how self-organizing maps (SOMs) can be compared within a Frechean inferential framework. Here, we summarize the mean SOM in each group as a Frechet mean with respect to a metric on the space of SOMs. The advantage of this approach is twofold. Firstly, it allows the visualization of the mean SOM in each experimental condition. Secondly, this Frechean approach permits one to draw inference on group differences, using permutation of the group labels. We consider the use of different distance functions, and introduce one extension of the classical sum of minimum distance (SMD) between two SOMs, which take into account the spatial pattern of the fMRI data. The validity of these methods is illustrated on synthetic data. Through these simulations, we show that the two distance functions of interest behave as expected, in the sense that the ones capturing temporal and spatial aspects of the SOMs are more likely to reach significance under simulated scenarios characterized by temporal, spatial [and spatio-temporal] differences, respectively. In addition, a re-analysis of a classical experiment on visually-triggered emotions demonstrates the usefulness of this methodology. In this study, the multivariate functional patterns typical of the subjects exposed to pleasant and unpleasant stimuli are found to be more similar than the ones of the subjects exposed to emotionally neutral stimuli. In this re-analysis, the group-level SOM output units with the smallest sample Jaccard indices were compared with standard GLM group-specific z-score maps, and provided considerable levels of agreement. Taken together, these results indicate that our proposed methods can cast new light on existing data by adopting a global analytical perspective on functional MRI paradigms.