Felix Darvas and Richard M Leahy
Signal & Image Processing Institute, University of Southern California, Los Angeles CA90089
We present a survey of imaging and signal processing methods that use data from magnetoencephalographic (MEG) or electroencephalographic (EEG) measurements to produce spatiotemporal maps of neuronal activity as well as measures of functional connectivity between active brain regions. During the course of the chapter, we give a short introduction to the basic bioelectro-magnetic inverse problem and present a number of methods that have been developed to solve this problem. We discuss methods to address the statistical relevance of inverse solutions, which is especially important if imaging methods are used to compute the inverse. For such solutions, permutation methods can be used to identify regions of interest, which can subsequently be used for the analysis of functional connectivity. The third section of the chapter reviews a collection of methods commonly used in EEG and MEG connectivity analysis, emphasizing their restrictions and advantages and their applicability to time series extracted from inverse solutions.
1 The Inverse Problem in MEG/EEG
Magnetoencephalography (MEG) measures non-invasively the magnetic fields produced by electrical activity in the human brain at a millisecond temporal resolution. The generators of these magnetic fields are dendritic currents in the pyramidal cells of the cerebral cortex. Since the currents produced by individual neurons are exceedingly weak, thousands of neurons have to be coherently active to produce a field that can be measured by MEG. The macroscopic fields generated by such ensembles of coherent neurons have strengths on the order of a few picotesla and are still one billion times smaller than the magnetic field of the earth.
A common electrical model for an ensemble of coherent neurons is the equivalent current dipole (ECD), which idealizes the ensemble as a single point source of electrical current. Due to the columnar organization of the cortex the ECD can be assumed to be oriented normally to the cortical surface
(Dale, and Serano, 1993, Okada, et al., 1997) and its location to be constrained to cortex. However, the model of an ECD with free orientation, which can be located anywhere within in the brain volume, remains common (Fuchs, et al., 1999).
The inverse problem is to find the neuronal activity, i.e. the location and strength of the associated ECDs, on the cerebral cortex or throughout the brain volume from noninvasive measurements of the magnetic fields produced outside the head. Likewise, if electroencephalographic (EEG) measurements are recorded, the change in scalp potentials due to the ECD inside the head volume is used to determine its strength and location. The solution of the inverse problem first requires the solution of a forward problem, which involves computation of the magnetic fields or electric potential changes outside the head due to an ECD in the brain volume. The basic physical laws from which the forward model for either MEG or EEG can be computed are Maxwell's equations under the assumption of stationarity (Hamalainen, et al., 1993). This assumption is valid for the typical frequencies produced by the human brain, i.e. for frequencies on the order of 100 Hz, where the respective electromagnetic wavelengths 300 m) far exceed the size of the head and thus changes in the fields produced by the neural currents inside the head can be considered instantaneous. Analytical solutions in geometries with spherical symmetry for the MEG/EEG forward problem have been discussed extensively by (Mosher, et al., 1999a, Zhang, 1995, Berg and Scherg, 1994, Sarvas, 1987). While for MEG the fields are only minimally distorted by biological tissue, anisotropies and inhomogeneities of the volume conductor have a strong impact on the scalp surface potentials. A simple spherical homogenous volume conductor model can be used for MEG with little impact on localization accuracy in the inverse solution (Leahy, et al., 1998), whereas for EEG it has been shown (Darvas, et al., 2006, Fuchs, et al., 2002, Baillet, et al., 2001), that numerical solutions of the forward model using a realistic head geometry can significantly improve the inverse solution over spherical models. Numerical methods such as the boundary element method (BEM), finite element method (FEM) and finite difference method (FDM) have been described in detail elsewhere (e.g. Fuchs, et al., 2001, Johnson, 1997). The forward model is solely dependent on the electromagnetic properties of the human head and the sensor geometry and is therefore independent of individual data recordings and need only be computed once per subject. Another important property of the forward model is that it is linear in terms of the strength of the neuronal currents. Consequently the summation of two source configurations produces the sum of the fields of the individual sources. For an individual ECD, the forward problem can be cast as a simple matrix-vector product as follows:
d (ti)= g (rk) • q (r_k,U) ,d G Rn,g G Rnx3,q,r G R3, (1)
Where d is the vector of measurements collected for each of the n detectors, g is the forward field for each detector for a source at location _k, and q is an ECD at location _k. For multiple sources eq. (1) can be expanded to
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