Theory of the EEG Inverse Problem - Quantitative EEG Analysis Methods and Clinical Applications - page 136

(panel B). Panel C shows orthogonal slices through the point of maximum current
density. Panel D shows the posterior three-dimensional cortex. Panel E shows the
average reference scalp electric potential map.

5.4.7 Other Formulations and Methods
A variety of very fruitful approaches to the inverse EEG problem exist that lie out-
side the class of discrete, three-dimensional distributed, linear imaging methods. In
what follows, some noteworthy exemplary cases are mentioned.
The beamformer methods [46, 47, 50, 51] have mostly been employed in MEG
studies, but are readily applicable to EEG measurements. Beamformers can be seen
as a spatial filtering approach to source localization. Mathematically, the
beamformer estimate of activity is based on a weighted sum of the scalp potentials.
This might appear to be a linear method, but the weights require and depend on the
time-varying EEG measurements themselves, which implies that the method is not a
linear one. The method is particularly well suited to the case in which EEG activity is
generated by a small number of dipoles whose time series have low correlation. The
method tends to fail in the case of correlated sources. It must also be stressed that
this method is an imaging technique that does not estimate the current density,
which means that there is no control over how well the image complies with the
actual EEG measurements.
The functionals in (5.24) and (5.28) have a dual interpretation. On the one
hand, they are conventional forms studied in mathematical functional analyses [25].
On the other hand, they can be derived from a Bayesian formulation of the inverse
problem [52]. Recently, the Bayesian approach has been used in setting up very com-
plicated and rich forms of the inverse problem, in which many conditions can be
imposed (in a soft or hard fashion) on the properties of the inverse solution at many
levels. An interesting example with many layers of conditions on the solution and its
properties can be studied in [53]. In general, this technique does not directly estimate
the current density, but instead gives some probability measure of the current den-
sity. In addition, these methods are nonlinear and are very computer intensive (a
problem that is less important with the development of faster CPUs).
Another noteworthy approach to the inverse problem is to consider models that
take into account the temporal properties of the current density. If the assumptions
on dynamics are correct, the model will very likely perform better than the simple
instantaneous models considered in the previous sections. One example of such an
approach is [54].

5.5 Selecting the Inverse Solution

We are in a situation in which many possible tomographies are available from which
to choose. The question of selecting the best solution is now essential. For instance:

1. Is there any way to know which method is correct?
2. If we cannot answer the first question, then at least is there any way to know
which method is best?