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.

This review highlights the strengths and weaknesses of the three categories of SSVEP training methods. Training-free systems are more practical but their performance is limited due to inter-subject variability resulting from the complex EEG activity. Feature extraction methods that incorporate some training data address this issue and in fact have outperformed training-free methods: subject-specific BCIs are tuned to the individual yielding the best performance at the cost of long, tiring training sessions making these methods unsuitable for everyday use; subject-independent BCIs that make use of training data from various subjects offer a good trade-off between training effort and performance, making these BCIs better suited for practical use.

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.

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There is an important evidence of differences in the EEG frequency spectrum of control subjects as compared to epileptic subjects. In particular, the study of children presents difficulties due to the early stages of brain development and the various forms of epilepsy indications. In this study, we consider children that developed epileptic crises in the past but without any other clinical, psychological, or visible neurophysiological findings. The aim of the paper is to develop reliable techniques for testing if such controlled epilepsy induces related spectral differences in the EEG. Spectral features extracted by using nonparametric, signal representation techniques (Fourier and wavelet transform) and a parametric, signal modeling technique (ARMA) are compared and their effect on the classification of the two groups is analyzed. The subjects performed two different tasks: a control (rest) task and a relatively difficult math task. The results show that spectral features extracted by modeling the EEG signals recorded from individual channels by an ARMA model give a higher discrimination between the two subject groups for the control task, where classification scores of up to 100% were obtained with a linear discriminant classifier.