Nonlinear PCA: characterizing interactions between modes of brain activity.

This paper presents a nonlinear principal component analysis (PCA) that identifies underlying sources causing the expression of spatial modes or patterns of activity in neuroimaging time-series. The critical aspect of this technique is that, in relation to conventional PCA, the sources can interact to produce (second-order) spatial modes that represent the modulation of one (first-order) spatial mode by another. This nonlinear PCA uses a simple neural network architecture that embodies a specific form for the nonlinear mixing of sources that cause observed data. This form is motivated by a second-order approximation to any general nonlinear mixing and emphasizes interactions among pairs of sources. By introducing these nonlinearities principal components obtain with a unique rotation and scaling that does not depend on the biologically implausible constraints adopted by conventional PCA. The technique is illustrated by application to functional (positron emission tomography and functional magnetic resonance imaging) imaging data where the ensuing first- and second-order modes can be interpreted in terms of distributed brain systems. The interactions among sources render the expression of any one mode context-sensitive, where that context is established by the expression of other modes. The examples considered include interactions between cognitive states and time (i.e. adaptation or plasticity in PET data) and among functionally specialized brain systems (using a fMRI study of colour and motion processing).