Inverse Problem The general EEG inverse problem can be stated as follows: given a set of electric potentials from discrete sites on the surface of the head and the associated positions of those measurements, as well as the geometry and conductivity of the different regions within the head, calculate the locations and magnitudes of the electric current sources within the brain. Mathematically, it is an inverse source problem in terms of the primary electric current sources within the brain and can be described by the same Poisson's equation as the forward problem, Eq. (1), but with a different set of boundary conditions on the scalp:    where  is the electrostatic potential on the surface of the head known at discrete points (electrode locations) and  in Eq. (1) are now unknown current sources. The solution to this inverse problem can be formulated as the non-linear optimization problem of finding a least squares fit of a set of current dipoles to the observed data over the entire time series, or minimization with respect to the model parameters of the following cost function:    where  is the value of the measured electric potential on the  electrode at time instant  and  is the result of the forward model computation for a particular source configuration; the sum extends over all channels and time frames. A brute-force implementation of the above method would require solving the forward problem for every possible configuration of dipoles in order to find the configuration that minimizes Eq. (6). Each dipole in the model has six parameters: location coordinates (x, y, z), orientation (  ,  ) and time-dependent dipole strength P(t). The number of dipoles is usually determined by iteratively adding one dipole at a time until a ``reasonable'' fit to the data has been found. Even when restricting the location of the dipole to a lattice of sites, the configuration space is factorially large. This is a bottleneck of many localization procedures [12, 29]. Assume now that we could decompose the signals on the electrodes, such that we know electrode potentials due to each dipole separately. Then for every set of electrode potentials we would need to search for only one dipole, thus dramatically reducing our search space. We will discuss this useful filtering technique in the next section.          Fri Oct 8 13:55:47 MDT 1999

 
Revised: March , 2005