Unified Expression of the Quasi-Static Electromagnetic Field: Demonstration With MEG and EEG Signals

Objective: Electromagnetic recordings are useful for non-invasive measurement of human brain activity. They typically sample electric potentials on the scalp or the magnetic field outside the head using electroencephalography (EEG) or magnetoencephalography (MEG), respectively. EEG and MEG are not, however, symmetric counterparts: EEG samples a scalar field via a line integral over the electric field between two points, while MEG samples projections of a vector-valued field by small sensors. Here we present a unified mathematical formalism for electromagnetic measurements, leading to useful interpretations and signal processing methods for EEG and MEG. Methods: We represent electric and magnetic fields as solutions of Laplace's equation under the quasi-static approximation, each field representable as an expansion of the same vector spherical harmonics (VSH) but differently weighted by electro- and magnetostatic multipole moments, respectively. Results: We observe that the electric and the magnetic fields are mathematically symmetric but couple to the underlying electric source distribution in distinct ways via their corresponding multipole moments, which have concise mathematical forms. The VSH model also allows us to construct linear bases for MEG and EEG for signal processing and analysis, including interference suppression methods and system calibration. Conclusion: The VSH model is a powerful and simple approach for modeling quasi-static electromagnetic fields. Significance: Our formalism provides a unified framework for interpreting resolution questions, and paves the way for new processing and analysis methods.