A random dipole model for spontaneous brain activity

The statistical properties of the EEG and the MEG are described mathematically as the result of randomly distributed dipoles. These dipoles represent the interactions of cortical neurons. For certain dipole distributions, the first- and second-order moments of the electric and magnetic fields are derived analytically. If the dipoles are in a spherical volume conductor and have no preference for any direction, the variance of a differentially measured EEG-signal is only a function of the electrode distance. In this paper, the theoretically derived variance function will be compared with EEG- and MEG-measurements. It is shown that a dipole with a fixed position and a randomly fluctuating amplitude is an adequate model for the alpha-rhythm. An expression for the covariance between the magnetic field and a differentially measured EEG-signal is derived. This covariance is considered as a function of the magnetometer position, and is compared with the measurements of Chapman et al. [23]. The theory can be used to obtain a (spatial) covariance matrix of the background noise, which occurs in evoked potential measurements. Such a covariance matrix can be used to obtain a maximum likelihood estimator of the dipole parameters in evoked potential studies, to evaluate the merits of the so-called "Laplacian derivation," and for the interpolation of electromagnetic data.