Dynamic source localization and functional connectivity estimation with multiple penalized state-space models

State-space models are extensively utilized across a variety of research fields to explore hidden dynamics. While traditional estimation methods are strongly theoretical supported, such as Kalman filtering and expectation maximization, they often involve significant computational demands when applied to models with a large number of parameters. Although sparse inverse covariance estimators can reduce these computational burdens, they introduce a nuanced trade-off: increasing estimation sparsity may inadvertently heighten estimation bias, particularly in scenarios with low signal-to-noise ratio (SNR). As alternative, to tackle these complexities, we propose a comprehensive three-forked approach: 1) We introduce multiple penalized state-space (MPSS) models that incorporate data-driven regularization to adaptively manage the trade-off between sparsity and bias; 2) We develop innovative algorithms inspired by backpropagation, such as state-space gradient descent and alternating least squares, specifically tailored for efficient computation within MPSS frameworks; 3) We enhance model validation through an extension of the K-fold cross-validation technique for state-space models, designed to optimize regularization parameters effectively. We demonstrate the robustness of our MPSS regularization framework through both simpler and more intricate simulations under varied SNR conditions, including analyses of large-scale synthetic magneto/electro-encephalography (MEG/EEG) data. Additionally, we apply our MPSS models to tackle both brain source localization and functional connectivity issues simultaneously in real event-related MEG/EEG data, covering thousands of cortical sources. Our methodology addresses and surpasses the limitations of existing approaches, which are often restricted to smaller scales and specific regions of interest, thereby potentially facilitating a more precise and comprehensive understanding of cognitive brain functions.