Explicitly Nonlinear Connectivity-Matrix Independent Component Analysis in Resting fMRI Data

ABSTRACT Independent component analysis (ICA) is a widely used data-driven technique for investigating brain structure and function to extract intrinsic networks. However, the ability of ICA, a linear mixing model, to capture nonlinear relationships is inherently limited. While nonlinear ICA can be used to estimate nonlinear+ linear mixtures, it can be useful to study the degree to which there is nonlinearity above and beyond the widely studied linear resting networks. Here, we propose a way to divide the data into sources exhibiting linear-only or explicitly nonlinear dependencies in resting functional magnetic resonance imaging (fMRI) data. Such an approach can be very informative as it allows us to evaluate the degree to which a given network might be linear, nonlinear, or both linear and nonlinear. Here, we present an enhanced connectivity-domain ICA approach, connectivity-matrix ICA, incorporating normalized mutual information (NMI) after canceling the linear effects to measure explicitly nonlinear (EN) relationships within voxel connectivity. This integration enables the identification of brain spatial maps that exhibit pronounced explicitly nonlinear dependencies while excluding linear relationships. By eliminating linear dependencies and utilizing NMI, we discover highly structured resting networks that conventional functional connectivity methods would typically overlook. The results indicate that several maps show only linear or EN relationships, and the rest of the components display both linear and nonlinear patterns. We categorized these maps as linear-only, EN-only, and linear-EN maps. We also evaluate differences in the identified networks in a schizophrenia dataset. A significant global difference has been discovered between schizophrenia and controls in some linear-EN maps, such as the frontal lobe. Moreover, the temporal lobe and thalamus display linear group differences, while the visual and motor cortex display global differences in nonlinear relationships as their primary driver of these disparities. In sum, our findings emphasize the significance of accounting for explicitly nonlinear dependencies in functional connectivity analysis and demonstrate the effectiveness of the extended cmICA approach in revealing previously unrecognized brain dynamics.