Chapter 1 - Introduction
3 EEG and Electrical source imaging (ESI)
3.2 Electrical Source Imaging (ESI)
3.2.2 The Inverse Problem
The inverse problem estimates the electrical source distribution given the lead field matrix and the measured scalp potentials. The resolution of the inverse problem requires the inversion of the lead field matrix in order to get the source activity from the scalp potentials. However, one cannot simply invert this matrix in order to estimate the sources, because a given scalp electric potential map can be explained by the activity of an infinite number of source configurations - the inverse problem is ill-posed. Indeed, the maximal activity in an electrode at a given moment does not necessarily mean that the sources of the underlying scalp map are immediately underneath this electrode. However, it is noteworthy that different scalp potential fields (maps) must have been generated by different source configurations [11].
Therefore, in order to solve the inverse problem a priori assumptions have to be made. There are several inverse solution algorithms to solve this problem that incorporate different assumptions [11]. Thus, in order to obtain the current source densities:
c K
T
J
( ).
(5)where T
(K )
is the inverse algorithm used and is expressed as a function of the lead field matrix, K.These inverse solution methods are either based on a dipole source model, where the number of unknowns is much smaller than the measured data, i.e. number of scalp electrodes (overdetermined models), or on distributed source models, where the number of unknowns is much larger than the number of measured data (underdetermined models). They have been previously compared using simulated EEG interictal spike
distributed inverse solutions are more suitable for estimating extended sources [65]. Therefore, one should always carefully consider the goal of the ESI localisation when choosing the inverse solution algorithm. For a review on the comparison between several inverse solution methods, we refer the reader to [11]. In the next section, we will provide a brief overview of several inverse solution algorithms currently available with special focus on distributed linear inverse solutions.
3.2.2.1 Dipole Models
Equivalent Current Dipole (ECD) models aim to explain the scalp electric field with one or a few ECD (single or multiple dipole model, respectively), whose location and orientation are unknown. This turns that, in equation (5), K and J are expressed as a function of the position and orientation of the dipole. In this approach, a search is made for the best dipole position(s) and orientation(s).
In the single dipole model, it is assumed that at a given moment in time, the observed scalp map is due to the activation of a single, small cortical area. This model is quite simplistic because many brain areas can be active a given moment. For the multiple dipole model, two or more dipoles are allowed to vary their position, orientation and intensity at the same time.
ECD approaches are based on iterative algorithms that estimate the source parameters that minimize the residual error, i.e., the best location and dipole moment (six parameters for each dipole) of the sources are found by minimizing the difference between the electrical potential map generated by these dipoles using a certain forward model, and the actual measured scalp map. Non-linear optimization methods based on directed search algorithms are normally used. A drawback of these methods is that they can get trapped in undesirable local minima. Each iterative step requires several forward solution calculations using test dipole parameters to compare the fit produced by the test dipole with that of the previous step. To account for this problem, a fix number of solution points can be considered a priori.
The conditions under which a dipole model makes sense are those where electric neuronal activity is dominated by a small brain area, such as the case of focal epileptic spike. However, they are not the most appropriate to describe activation of widespread neuronal networks, such as the propagation of IEDs. The advantage of current dipole source models is that there are less unknowns than measured data. However, the solution is very sensitive to the a priori defined number of sources and initial parameters (dipole locations and orientations), which can bias or corrupt the results.
3.2.2.2 Distributed Source Models: Minimum norm estimates, LORETA and LAURA
In the distributed source models, the current source density is estimated for a 3D grid of solution points with fixed positions. Thus, each solution point can be seen as an equivalent dipole with fix position and variable strength and orientation. Each solution point in this grid is considered as a possible location of the current source, and thus, there is no a priori assumption of the number of the active sources (dipoles) in the brain, in contrast to dipolar models. Moreover, since the current source density is estimated for each solution point in this grid, these models allow the comparison of the results of ESI with local field potentials recorded with intracranial EEG. They are also better suited than ECD to describe the involvement of several brain regions at a given moment. However, the number of solution points (unknowns) is much larger than the
number of scalp electrodes (measured data), leading to a highly underdetermined problem: an infinite number of sources configurations can lead to the same scalp potential. Thus, some assumptions have to be made in order to identify the best possible solution. Depending on the algorithm used, the assumptions made can be purely mathematical, based on physiological known phenomena or on knowledge derived from other structural or functional imaging modalities. Distributed inverse solutions are linear since they linearly relate the measured data to the estimated solution.
The general framework to estimate the unknown dipole intensities JNSx1with NS >>NE for distributed linear inverse solution methods is defined by:
)
The solution of equation (6) is given by:
equation (6): the first term represents the difference between the measured and the estimated data (i.e., the data fit) and the second term represents the importance given to a priori defined constraints (necessary to have a unique solution) by the regularisation parameter. This regularisation parameter also accounts for the noise in the data and provides stability to the solution such that small variations in the data do not lead to large variations in the source configuration. However, a too high value of α can smooth too much the solution. Thus, an appropriate value should be found since this parameter can strongly influence the inverse solution results [48]. A common way to find the optimal regularisation value is to use the L-curve method [68], but other methods exist [57]. only one combination of sources can have both the lowest intensity and at the same time exactly fit the data.However, minimum norm does not localize deep sources because it punishes solutions where there is a strong activation of a large number of solution points, favouring thus weak and superficial sources since less activity is required in a superficial source to give a certain scalp voltage map [11, 70].
The weighted minimum norm algorithm compensates for this tendency of the minimum norm solution to favour superficial sources. It includes a weight matrix, W=D-1, where D is a diagonal matrix containing the weights, which should be larger for deep sources than for more superficial ones. There are other weighting functions that have been proposed and we refer the reader to [48] for further information.
In the Low Resolution Electromagnetic Tomography (LORETA) [71], it is assumed that the current
neighbours. The method minimizes the Laplacian of the weighted sources to produce the smoothest possible inverse solution. This smoothness property of LORETA is based on the physiological knowledge that the activity a large population of pyramidal cells must be active simultaneously, synchronously and with the same type of postsynaptic potentials, as referred in section 2.1.1. However, the limited spatial resolution of the EEG and the distance between solution points leads to a much larger spatial scale where these constraints might no longer hold. In LORETA, the weight matrix W is not restricted to be diagonal and the weights are constructed such that:
s
s s
TWJ j AveNeihgb j
J ( )2 (9)
where
AveNeihgb ( j
s)
is an informal notation to indicate the average of current densities in the immediate neighbourhood of the source s.The Local Auto-Regressive Averages (LAURA) incorporates biophysical laws as constraints: it assumes that the source activity will decrease with the inverse of the cubic distance for vector fields, and with the inverse of the squared distance for potential fields [72]. LAURA uses this law in terms of a local autoregressive average in which the coefficients depend on the distances between each solution point and the others in their immediate neighbourhood. In LAURA, the following construction of the second term on the right-hand side of equation (5) holds:
s
s s
TWJ j WeightedAveNeihgb j
J ( )2 (10)
where
WeightedAv eNeihgb ( j
s)
is an informal notation for the distance-weighted average of current densities in the immediate neighbourhood of source s.LAURA was the inverse solution method used in our work.
3.2.2.3 Other approaches: Beamformers and Bayesian approaches
In beamforming, the activity at one cortical area is estimated by minimizing the interference of other possible simultaneously active sources. Spatial filters are designed for each dipole source such that only the signal coming from the location of the source dipole of interest is kept. The basic assumption in beamforming is that the sources are uncorrelated, and therefore the activity of each source in the distributed source model is estimated independent of the others. Beamforming algorithms optimize a goal function that represents the ratio between activity and noise at source, for instance, described by the covariance matrices.
Beamformers have the advantage that the number of dipoles must not be assumed a priori. We refer the reader to [57, 73, 74] for more details on beamformers.
In Bayesian approaches, an estimator of J, Jˆ, that maximizes the posterior source activity distribution of J given the measured data
, is found:)) ( ( ˆ max p J
J x (11)
where, according to Bayes' law:
) (
) ( ) ) (
(
p J p J J p
p (12)
) (
p is the probability of the given EEG data and thus it is constant, p
(
J)
is the probability of fitting the EEG data given the source distribution, p(
J)
is the prior probability of the source activity (all the prior knowledge is incorporated here). These two are directly related to the forward model and the magnitudes of the sources can be recovered by applying the expectation operator: J E
p(
J ))
.There are many Bayesian approaches that can include linear and non-linear estimators. Bayesian methods can also be used to estimate a probability distribution of solutions rather than a single best solution. They can incorporate several a priori information for the estimation of the sources, which allows a more complete description of the anatomical and/or functional information [75, 76]. These prior information can be about the neural current, combined spatial and temporal constraints, and others [75, 76]. For a more detailed review on Bayesian approaches and other inverse solution methods, please see [57].