These comments below have been written by me, Rey Ramirez, in response to the reviewers' comments and suggestions. I highly appreciate all suggestions, comments, and constructive criticism. I didn't know where else to respond. If this is the wrong place to add responses, please let me know.
1. I agree, the only difference between iEEG and LFP is in the type and size (that is spatial scale) of electrode used and where they are placed (e.g., laminar electrodes are often used to get the LFPs through the layers of the cortex), but that might be a tangential detail that goes beyond the scope of a scholarpedia article, so I will eliminate the mentioning of LFPs. Also, usually people don't solve inverse problem for LFPs, so I agree, it's not so relevant.
2. The use of virtual 3rd order gradiometers is an important method to remove noise used often by researchers that have the CTF system. It is just mentioned for completeness. People that want to learn more can read the paper by Vrba. But I will follow advice and crop the sentence.
3. Just to elaborate on the issue of non-uniqueness. Non-uniqueness is complicated problem and full articles have been written about it. Most people distinguish physical from mathematical nonuniqueness, physical being caused by the intrinsic EM physics of silent sources, and the mathematical due to the undetermined nature of the equations (assuming that you frame it as an underdetermined problem). In this case, all the vectors residing in the null space are silent too, since L*j_null=0. Even if everything is framed as an overdetermined system (e.g., dipole fitting), any of these vectors of the null space can always be added.
4. An explanation of why and how the sensors are coregistered with the MRI is now included. Note that I just mention that the coregistration involves a translation and rotation. No details are given since this transformation is done in different ways by different people. Some use a matrix transformation using the lpa, rpa, and nas fiducials, but others use minimization algorithms that make a tighter transformation using more fiducials, or the headshape file and the segmented skin surface from the MRI. Going into all of this would go beyond the scope of this article.
5. Equation 3 has been numbered and all subsequent equations have been updated.
6. The resolution matrix has now been defined.
7. The two p's in equation 13 are actually the same p. The sign(p) is used to allow for negative p as used in the original FOCUSS algorithm and the MFT algorithm.
8. The sections on MAP and SBL/ARD (source-space based methods) have been shortened. However, I have left enough mathematical detail so that people can actually implement these algorithms with their own code. Since these algorithms are extremely powerful, I think shortening these sections further would be unwise. Although this part is dense and complicated, the benefits of this algorithms are worth it, and people can always learn further details by reading the cited articles. For now I will leave it at this level of detail since I still have to wait for the comments of the second reviewer. However, I hope the article will change in the future and eventually hit the perfect balance of detail and clarity.
Comments by Reviewer 2
Dear Rey Ramirez, thanks for your effort in writing this article. It will be a very useful resource for people to learn about solutions to the inverse problem. You have done a good job in summarizing the different source localization techniques (certainly a challenging task). I have some comments that could potentially improve the article further.
- I suggest to state the general problem and introduce the general terms and mathematical variables in a separate section before the "Localization inaccuracy" section. In this section I would cover the following:
- introduce the forward problem
- introduce the inverse problem (with B=LJ+N and explanation of leadfields)
- describe that solving the inverse problem consists of the two steps of specifying the volume conductor model (sphere, BEM, FEM) and the source model (dipole, scanning, distributed source model).
- In the section "Data pre-processing" I suggest to specify that only linear transformations of the data are allowed to preserve the structure required for source localization. (You can not use standard source localization algorithms for amplitude or phase data).
- Just before the section "Parametric dipole modeling" it would be nice to have a short introduction/overview to the 3 main sections
- A very important part of the manuscript is (of course) the description of source localization algorithms. I think this part could be improved in 3 ways:
- In the beamforming section I suggest to have 2 subsections (Non-adaptive beamformers, Adaptive beamformers). In the adaptive beamformer section I would describe the different algorithms without further subsubsections to highlight the similarities (e.g. between LCMV and SAM).
- The source space based methods are presented in a common mathematical framework. After introducing the framework it would be very helpful to inform the reader about the decisions one has to make (e.g. leadfield normalization, type of smoothness constraint, use of source covariance etc).(Ideally, even the beamforming algorithms could be incorporated in the same framework).
- In my opinion there is too much detail in the MAP, ARD sections. It makes it very difficult for the naive reader to follow the article. I suggest to remove most of the mathematical, technical details. These details could be added as appendix.
- I suggest to include the (rather obvious) statement in the first paragraph that sources refer to activated neuronal populations (to explain the term source localization to naive readers)
- In the MEG subsection (section: Measurement modalities) I suggest to remove the part "...of magnetometers and/or gradiometers (planar or radial)." It is too detailed and the differences between magnetometers and gradiometers is not explained and not relevant in this context.
<review> This is Rey Ramirez letting Reviewer 2 that most if not all of his/her suggestions have been incorporated into the new text. This involved re-organizing some sections completely, and adding some new text. After the intro, the neuronal origin of EM signals starts, followed by the nature of the EM signals, and followed by a general overview section of several aspects of source localization. I did not change too much on the presentation of beamforming vs source space based methods, but clarified why it is that these two approaches are very different and should be kept separate. In a grand unification paper, I would put them in the same context, but this is not a grand unification paper. Such unification could confuse non-experts. The adaptive vs non-adaptive dichotomy was not change too much because all that 'adaptive' means is that the noise covariance is modeled from the data. This is not only true for LCMV and SAM but also for dSPM and sLORETA when they use a non-diagional noise covariance matrix obtained from the data. Thus, both source-space based methods or spatial filters can be made adaptive or non-adaptive. It's just a matter of whether the user wants to model correlated noise or not. What really distinguishes these two approaches is whether the "localization vector or spatial filter" is built from only local gain vectors, or whether the "localizing vector" is computed with a gram matrix built from gain vectors associated from the whole source space (source-space based methods). I clarify this throughout the paper. Thanks for all the suggestions to make it all more clear. Keep in touch. <review></review>