In this primer, we give a review of the inverse problem for EEG source localization. This is intended for the researchers new in the field to get insight in the state-of-the-art techniques used to find approximate solutions of the brain sources giving rise to a scalp potential recording. Furthermore, a review of the performance results of the different techniques is provided to compare these different inverse solutions. The authors also include the results of a Monte-Carlo analysis which they performed to compare four non parametric algorithms and hence contribute to what is presently recorded in the literature. An extensive list of references to the work of other researchers is also provided. This paper starts off with a mathematical description of the inverse problem and proceeds to discuss the two main categories of methods which were developed to solve the EEG inverse problem, mainly the non parametric and parametric methods. The main difference between the two is to whether a fixed number of dipoles is assumed a priori or not. Various techniques falling within these categories are described including minimum norm estimates and their generalizations, LORETA, sLORETA, VARETA, S-MAP, ST-MAP, Backus-Gilbert, LAURA, Shrinking LORETA FOCUSS (SLF), SSLOFO and ALF for non parametric methods and beamforming techniques, BESA, subspace techniques such as MUSIC and methods derived from it, FINES, simulated annealing and computational intelligence algorithms for parametric methods. From a review of the performance of these techniques as documented in the literature, one could conclude that in most cases the LORETA solution gives satisfactory results. In situations involving clusters of dipoles, higher resolution algorithms such as MUSIC or FINES are however preferred. Imposing reliable biophysical and psychological constraints, as done by LAURA has given superior results. The Monte-Carlo analysis performed, comparing WMN, LORETA, sLORETA and SLF, for different noise levels and different simulated source depths has shown that for single source localization, regularized sLORETA gives the best solution in terms of both localization error and ghost sources. Furthermore the computationally intensive solution given by SLF was not found to give any additional benefits under such simulated conditions.

Background: The aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.

The back-projected independent components (BICs) of single-trial, auditory P300 and contingent negative variation (CNV) evoked potentials (EPs) were derived using independent component analysis (ICA) and cluster analysis. The method was tested in simulation including a study of the electric dipole equivalents of the signal sources. P300 data were obtained from healthy and Alzheimer's disease (AD) subjects. The BICs were of approximately 100 ms duration and approximated positive- and negative-going half-sinusoids. Some positively and negatively peaking BICs constituting the P300 coincided with known peaks in the averaged P300. However, there were trial-to-trial differences in their occurrences, particularly where a positive or a negative BIC could occur with the same latency in different trials, a fact which would be obscured by averaging them. These variations resulted in marked differences in the shapes of the reconstructed, artefact-free, single-trial P300s. The latencies of the BIC associated with the P3b peak differed between healthy and AD subjects (p < 0.01). More reliable evidence than that obtainable from single-trial or averaged P300s is likely to be found by studying the properties of the BICs over a number of trials. For the CNV, BICs corresponding to both the orienting and the expectancy components were found.

Brain-computer interface (BCI) systems based on steady-state visual evoked potentials (SSVEPs) have gained considerable popularity because of the robustness and high information transfer rate these can provide. Typical SSVEP setups make use of visual targets flashing at different frequencies, where a user's choice is determined from the SSVEPs elicited by the user gazing at a specific target. The range of stimulus frequencies available for such setups is limited by a variety of factors, including the strength of the evoked potentials as well as user comfort and safety with light stimuli flashing at those frequencies. One way to tackle this limitation is by introducing targets flickering at the same frequency but with different phases. In this paper, we propose the use of the analytic common spatial patterns (ACSPs) method to discriminate between phase coded SSVEP targets, and we demonstrate that the complex-valued spatial filters used for discrimination can exceed the performance of existing techniques. Furthermore, the ACSP method also yields a set of spatial patterns, separable into amplitude and phase components, that provide insight into the underlying brain activity.

One of the most important stages in a brain–computer interface (BCI) system is that of extracting features that can reliably discriminate data recorded during different user states. A popular technique used for feature extraction in BCIs is the common spatial patterns (CSP) method, which provides a set of spatial filters that optimally discriminate between two classes of data in the least-squares sense. The method also yields a set of spatial patterns that are associated with the most relevant activity for distinguishing between the two classes. The high recognition rates that have been achieved with the method have led to its widespread adoption in the field. Here, a variant of the CSP method that considers EEG data in its complex form is described. By explicitly considering the amplitude and phase information in the data, the analytic CSP (ACSP) technique can provide a more comprehensive picture of the underlying activity, resulting in improved classification accuracies and more informative spatial patterns than the conventional CSP method. In this paper, we elaborate on the theoretical aspects of the ACSP algorithm and demonstrate the advantages of the method through a number of simulations and through tests on EEG data.