Learning Lead Field Corrections for EEG–fMRI Functional Connectivity Alignment

Joint analysis of EEG and BOLD-fMRI signals is a route to studying how brain structure relates to brain function. This paper presents a model that learns a low-rank correction to the T1-derived lead field per subject. The correction is fitted by matching the EEG-side parcel covariance and the fMRI-derived parcel covariance, each estimated within-subject from the corresponding modality's own resting data. We call this construction Lead-Field Low-Rank Correction (LF-LoRA). The covariance matching has the form ${\Sigma }_z\leftrightarrow {\hat{\Sigma }}_z=B{\Sigma }_y{B}^{\top }$, where ${\Sigma }_{\{y,z\}}$ represents the EEG/BOLD-parcel signal covariance and $B$ is a linear EEG-to-parcel operator built from the corrected lead field $A=A_0+U{V}^{\top }$. The simple model structure is consistent with the linear forward-modeling assumption from which the lead field is defined, and its solution may help explain subject-specific structure in the EEG forward model (Hämäläinen et al., 1993). The linear model admits a decoupled temporal operator which aligns ${\Sigma }_y$ and ${\Sigma }_z$ at lag $\tau >0$. The temporal operator provides a means to investigate the dynamics of neural activity from EEG. We discuss the solution space of LF-LoRA. We evaluate the construction on the LEMON cohort (Babayan et al., 2019). Experiments validate the subject specificity of the learned model and the utility of reading $U{V}^{\top }$ on cortex.