Estructura de PECAM-

The neuromagnetic inverse problem states [2, 5]: Given a known distribution of magnetic field outside the head, can we calculate the current density distribution inside the head? This is an ill-defined problem in physics because due to field cancellation effects, any single measured magnetic field distribution could be caused by an infinite number of current distributions in the brain. Despite the difficulties, several methods have been proposed to solve the neuromagnetic inverse problem, all of which make assumptions on the type of current distribution observed. The most common assumption is that an area of active cortex can be modelled by an equivalent current dipole (as described above). Generally this assumption is reasonable given that the active area is small when compared to the distance from which it is observed. Beyond this, each of the inverse problem solutions has its own unique assumption set, which must be understood if that solution is to be applied successfully. It is beyond the scope of this chapter to mention explicitly all of the inverse problem solutions, and for an excellent review, the reader is referred to Baillet et al., (2001) [10]. Here, three of the most successful inverse solutions are briefly discussed.

The oldest, and simplest inverse problem solution in MEG is known as dipole fitting [11]. Here, a dipole is placed in the source space and an iterative algorithm allows that dipole to shift in position, orientation and magnitude. On each iteration, the resulting magnetic field is calculated using the forward solution and compared to the actual field measured. Optimal solutions are derived based on a minimisation of the difference between the measured and modelled fields. The major assumption is that the number of active neuronal sources is small, and known a-priori, meaning that this model becomes more realistic when dipole fitting is applied over small time windows. (Typically one attempts

to explain activity in small —10 ms blocks during which only one or two dipoles are

assumed to be active.)

In minimum norm estimation [2, 12], the assumption is that the source configuration with the least energy that also minimises the differences between the measured and estimated fields accounts for the recorded data. Such algorithms suffer because these source configurations must always exist in the most superficial layer of cortex, meaning that an arbitrary depth weighting must be factored in (weighted minimum norm estimation). A number of techniques by which to do this have been proposed, [2] including one in which a depth bias is introduced based on active locations derived from fMRI data. The problem with this approach is that it necessarily assumes a linear relationship between the effects measured in MEG and those measured in fMRI, which may not be the case in general.

A third inverse problem solution is known as the MEG beamformer [13]. The beamformer effectively represents a special case of the weighted minimum norm approach, where the weighting is applied based on the assumption that timecourses of neuronal sources remain uncorrelated for the duration over which the fit occurs. Unlike dipole fitting and minimum norm, which have been used mainly to explain field patterns measured during a short time window in averaged responses, beamformers have been most successful in identifying induced changes in cortical oscillatory power. Since these induced changes occur on a longer time scale, this means that beamformers perform better in blocked experimental designs, making them ideal for use in stimulus paradigms similar to those used in fMRI. In the next section, the mathematical principles behind the beamformer approach are put forward and its solution is derived.

5.2.1 Lead Fields

The lead field is derived from the solution to the forward problem (see Equation 5.26), and describes the sensitivity of the ith MEG sensor to a current distribution at position r' in the brain. If b; is the output of the i

th

MEG sensor then the lead field /i(

r' ) must satisfy the equation:

b

= 11

(

For a simple dipolar current distribution in a spherical head model this simplifies to give:

Q,r0 )=1;(ro ).Q [5.28]

Thus under these assumptions, the lead field is readily obtained from Equation 5.26 and a knowledge of the geometry of he MEG system.

5.2.2 The Linearly Constrained Minimum Variance Beamformer Problem

The dipolar source strength (or dipole moment) at any position in the brain can be written as a three-component vector Q(rt2 ) In the MEG beamformer approach [13], the idea is

to approximate that dipolar source strength to 6(r(2) using a weighted sum of the

magnetic field measurements at each of the N MEG sensors i.e.

Q(r' )=WT (r' )x(t) [5.29]

Where W is the N x 3 weights matrix and x(t) is the N — dimensional vector of field

measurements at each of the MEG sensors taken at time t.

Ideally, it would be possible to define the matrix W such that:

I i f ' = o

WT ( ro )L( r' )= r [5.30]

0 if r' r ro

Where the matrix L( r' ) is the (N x 3) matrix of lead fields formulated using Equations 5.26 and 5.28. The first column of LW ) represents the detectable magnetic field strength bi at each of the MEG sensors due to a current dipole at position (r') having

unit moment in the z direction, and zero moment in the 9 and i directions. Similarly, the second and third columns represent the fields created by unit dipoles in the 9, and directions respectively. For a situation in which the columns of W(r' ), and W(ro) are

linearly independent, it is possible to solve Equation 5.30 mathematically. However for an experimental situation, this will not be the case, and one must derive a solution for 1V( r' ) that is in some way optimal. Using the Linearly Constrained Minimum Variance (LCMV) beamformer [13], this optimisation is achieved by the minimisation of the variance of 6( r' ) subject to the linear constraint that:

W (ro )L( ro )= I [5.31]

This linear constraint ensures that the energy originating from the source position remains in the estimate whilst the minimisation suppresses energy from all other locations in the

brain. Furthermore, the LCMV technique forces the suppression of energy from position

r'

only if there is significant energy originating there. The unity passband ensures the estimate

afr

)

Mathematically, the LCMV beamformer problem can be posed as: