Review on solving the inverse problem in EEG source analysis.
- Authors
- Grech, Roberta; Cassar, Tracey; Muscat, Joseph; Camilleri, Kenneth P; Fabri, Simon G; Zervakis, Michalis; Xanthopoulos, Petros; Sakkalis, Vangelis; Vanrumste, Bart
- Year
- 2008
- Journal
- Journal of neuroengineering and rehabilitation
- PMID
- 18990257
- DOI
- 10.1186/1743-0003-5-25
- PMCID
- PMC2605581
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.
General block diagram for an artificial neural network system used for inverse source localization.
Individual Layers in which the simulated dipoles lie. Red crosses represent sources lying close to the surface (57 in total), black crosses represent sources lying in the middle of the spherical cortex model (37 in total) and blue crosses represent sources lying deep within the cortex (14 in total).
Box-whisker diagrams. These show the median (horizontal line within each box), the interquartile range (between the bottom and top of each box) and the range of scores (shown by the whiskers). Circles represent outliers. Plots (a) and (b) show the results for each of the four inverse solutions (horizontal axis) for error measure ED2 with a SNR of 5 dB. (a) shows the results without regularization and (b) shows the results with regularization.
| # | Section | Preview |
|---|---|---|
| 160 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.1 The Bayesian framework β 3.1.2 The Backus-Gilbert method | TuΞ³=EuΞ³β LuLuTEuΞ³β Lu |
| 161 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.1 The Bayesian framework β 3.1.2 The Backus-Gilbert method | where: |
| 162 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.1 The Bayesian framework β 3.1.2 The Backus-Gilbert method | Lu=Gu1p,EuΞ³=CuΞ³+βv=13(1βΞ΄uv)Fv,CuΞ³=GuWΞ³BGGuT,Fv=GvGvT. |
| 163 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.1 The Bayesian framework β 3.1.2 The Backus-Gilbert method | 'β ' denotes the Moore-Penrose pseudoinverse. |
| 164 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | An extension of the Backus-Gilbert method is called the Weighted Resolution Optimization (WROP)β¦ |
| 165 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | [W1Ξ³GdeP]ll=||vlβvΞ³||2+Ξ²GdeP. |
| 166 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | The second part of the functional to be minimzed is replaced by |
| 167 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | βv=13(1βΞ΄uv)TuΞ³TGvW2Ξ³GdePGvTTuΞ³ |
| 168 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | where |
| 169 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | [W2Ξ³GdeP]ll=||vlβvΞ³||2+Ξ²GdeP+Ξ±GdeP, |
| 170 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | Ξ± GdeP and Ξ² GdeP are scalars greater than zero. In practice this means that there is more tradeβ¦ |
| 171 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | In this case the inverse operator is: |
| 172 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | TuΞ³=Ξ²GdeP{GuW1Ξ³GdePGuT+βv=13(1βΞ΄uv)GvW2Ξ³GdePGvT}β GuIΞ³. |
| 173 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | In [5] five different inverse methods (the class of instantaneous, 3D, discrete linear solutions forβ¦ |
| 174 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization | The WROP method is a family of linear distributed solutions including all weighted minimum normβ¦ |
| 175 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization β LAURA | As stated in [39] in a norm minimization approach we make several assumptions in order to choose theβ¦ |
| 176 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization β LAURA | According to Maxwell's laws of electromagnetic field, the strength of each source falls off with theβ¦ |
| 177 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization β LAURA | In LAURA the current estimate is given by the following equation: |
| 178 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization β LAURA | D^LAURA=WjGT(GWjβ1GT+Ξ±IN)β1M |
| 179 | 3 Inverse solutions β 3.1 Non parametric optimization methods β 3.1.3 The weighted resolution optimization β LAURA | The W j matrix is constructed as follows: |
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| Title | Year | PMID |
|---|---|---|
| Advances in Electrophysiological Research. | 2015 | 26259089 |
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