Data may now be recorded concurrently from EEG and functional MRI, using the Simultaneous Imaging for Tomographic Electrophysiology (SITE) method. As yet, there is no established means to integrate the analysis of the combined data set. Recognizing that the hemodynamically convolved time-varying EEG spectrum, S, is intrinsically multidimensional in space, frequency, and time motivated us to use multiway Partial Least-Squares (N-PLS) analysis to decompose EEG (independent variable) and fMRI (dependent variable) data uniquely as a sum of "atoms". Each EEG atom is the outer product of spatial, spectral, and temporal signatures and each fMRI atom the product of spatial and temporal signatures. The decomposition was constrained to maximize the covariance between corresponding temporal signatures of the EEG and fMRI. On all data sets, three components whose spectral peaks were in the theta, alpha, and gamma bands appeared; only the alpha atom had a significant temporal correlation with the fMRI signal. The spatial distribution of the alpha-band atom included parieto-occipital cortex, thalamus, and insula, and corresponded closely to that reported by Goldman et al. [NeuroReport 13(18) (2002) 2487] using a more conventional analysis. The source reconstruction from EEG spatial signature showed only the parieto-occipital sources. We interpret these results to indicate that some electrical sources may be intrinsically invisible to scalp EEG, yet may be revealed through conjoint analysis of EEG and fMRI data. These results may also expose brain regions that participate in the control of brain rhythms but may not themselves be generators. As of yet, no single neuroimaging method offers the optimal combination of spatial and temporal resolution; fusing fMRI and EEG meaningfully extends the spatio-temporal resolution and sensitivity of each method.