Deep Linear Modeling of Hierarchical Functional Connectivity in the Human Brain

The human brain exhibits hierarchical modular organization, which is not depicted by conventional fMRI functional connectivity reconstruction methods such as independent component analysis (ICA). To map hierarchical brain connectivity networks (BCNs), we propose a novel class of deep (multilayer) linear models that are constructed such that each successive layer decomposes the features of the preceding layer. Three of these are multilayer variants of Sparse Dictionary Learning (SDL), Non-Negative Matrix Factorization (NMF) and Fast ICA (FICA). We present a fourth deep linear model, Deep Matrix Fitting (MF), which incorporates both rank reduction for data-driven hyperparameter determination as well as a distributed optimization function. We also introduce a novel framework for theoretical comparison of these deep linear models based on their combination of mathematical operators, the predictions of which are tested using simulated resting state fMRI data with known ground truth BCNs. Consistent with the theoretical predictions, Deep MF and Deep SDL performed best for connectivity estimation of 1st layer networks, whereas Deep FICA and Deep NMF were modestly better for spatial mapping. Deep MF provided the best overall performance, including computational speed. These deep linear models can efficiently map hierarchical BCNs without requiring the manual hyperparameter tuning, extensive fMRI training data or high-performance computing infrastructure needed by deep nonlinear models, such as convolutional neural networks (CNNs) or deep belief networks (DBNs), and their results are also more explainable from their mathematical structure. These benefits gain in importance as continual improvements in the spatial and temporal resolution of fMRI reveal more of the hierarchy of spatiotemporal brain architecture. These new models of hierarchical BCNs may also advance the development of fMRI diagnostic and prognostic biomarkers, given the recent recognition of disparities between low-level vs high-level network connectivity across a wide range of neurological and psychiatric disorders.