Tutorial 22: Source estimation [TODO]

Authors: Francois Tadel, Elizabeth Bock, Rey R Ramirez, John C Mosher, Richard M Leahy, Sylvain Baillet

You have in your database a forward model that explains how the cortical sources determine the values on the sensors. This is useful for simulations, but what we need is to build the inverse information: how to estimate the sources when we have the recordings. This tutorials introduces the tools available in Brainstorm for solving this inverse problem.

WARNING: The new interface presented here do not include all the options yet. The mixed head models are not supported: to use them, use the old interface (menu "Compute sources" instead of "Compute sources 2016", described in the old tutorials).

Ill-posed problem

Our goal is to estimate the activity of the thousands of dipoles described by our forward model. However we only have a few hundred variables in input (the number of sensors). This inverse problem is ill-posed, meaning there is an infinite number of combinations of source activity patterns that can generate exactly the same sensor topography. Inverting the forward model directly is impossible, unless we add some strong priors to our model.

Wikipedia says: "Inverse problems are some of the most important and well-studied mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe. They have wide application in optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields."

Many solutions have been proposed in the literature, based on different assumptions on the way the brain works and depending on the amount of information we already have on the effects we are studying. Among the hundreds of methods available, in Brainstorm, we initially present three general approaches to the inverse models widely used in MEG/EEG source imaging: minimum-norm solutions, beamformers, and dipole modeling.

These approaches have the advantage of being implemented in an efficient linear form: the activity of the sources is a linear recombination of the MEG/EEG recordings, such that it is possible to solve the inverse problem as a linear kernel which is easily stored. Subsequent data manipulation and source visualization is then much simpler, as are comparisons among these techniques.

Below we first describe the minimum norm imaging approach and its options, followed by the beamformer and dipole modeling, both of which are actually quite similar and only use a subset of the options found in the minimum norm.

Source estimation options [TODO]

Before we start estimating the sources for the recordings available in our database, let's start with an overview of the options available. The screen capture below represents the options for the minimum norm estimates. The options for the other methods will be described in advanced tutorials.

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Method

Selecting one of the three Methods automatically switches the input screen automatically to reveal the options available for each. Above are the options revealed for the mininum norm imaging. First we brieflly describe the Methods options:

The next several sections discuss in detail the options as revealed by selecting the "mininum norm imaing" Method above.

Measure

As directly formed and computed, the minimum norm estimate produces a measure of the current found in each point of the source grid (either volume or surface). As discussed elsewhere in the forum (link to Mosher's discuss on units of the minimum norm), the units are strictly kept in A-m, i.e. we do not attempt to divide out the area (yielding A/m, i.e. a surface density) or volume (yielding A/m^2, i.e. a volume density) of each source point; nonetheless, it is common to refer these units as a "source density" or "current density" map when displayed directly.

More commonly, however, users often prefer to "normalize" each source point in order to compensate for the rapid falloff in signal intensity for deeper dipoles. Because of their popularity, we provide directly in the Measure box two forms of standardization, the dSPM and sLORETA. Summarizing the Measures selection:

Source model: Dipole orientations

At each point in the source grid, the current dipole may point arbitrarily in three directions. In this section of the options, we prescribe if the inverse method will constrain the orientation, leave it unconstrained, or if we impose a "soft" constraint to the orientation.

Sensors

We automatically detect and display the sensors found in your head model. In the example above, two types of magnetometers are found, gradiometers and magnetometers. You can select one or all of the sensors found in your model, such as MEG and EEG.

However, cross-modality calculations are quite dependent on the accuracy by which you have provided adequate covariance calculations. As of Fall of 2016, we have also elected to NOT account for cross-covariances between different sensor types, since regularization and stability of cross-modalities is quite involved. For multiple sensor types, the recommendation is that you try each individually and then combined, to test for discordance.

Computing sources for an average

Using the above selections, we now discuss explicit directions on how to compute and visualize.

Display: Cortex surface

Why does it look so noisy?

The source maps look very noisy and discontinuous, they show a lot of disconnected patches. This is due to the orientation constraint we imposed on the dipoles orientations. Each value on the cortex has to be interpreted as a vector, oriented perpendicular to the surface. Because of the brain circumvolutions, all the sources have different orientations, two adjacent sources have very little chance to have the same orientation in this model, therefore the minimum norm method may attribute completely different values to them. This causes all these gaps we see here.

Visually, you should not always interpret disconnected colored patches as independent sources. You cannot expect a very spatial resolution with this technique (~5-10mm). Most of the time, a cluster of disconnected source patches in the same neighborhood that show the same evolution in time can be interpreted as "there is some significant activity around here, but we don't know where exactly".

To get more continuous maps for visualization or publication purposes, you can either smooth the values explicitly on the surface (process "Sources > Spatial smoothing") or use unconstrained source models.

For data exploration, this is a good enough representation of the brain activity, mostly because it is fast and efficient. You can get a better feeling of the underlying brain activity patterns by making short interactive movies: click on the figure, then hold the left or right arrows of your keyboard.

Activity patterns will also look sharper when we compute normalized measures (later in this tutorial). In most of the screen captures in this following sections, the contrast of the figures has been enhanced for illustration purposes. Don't worry if it looks a lot less colorful on your screen.

Display: MRI Viewer

Display: MRI 3D

Sign of constrained maps

You should pay attention to the sign of the current amplitudes that are given by the minimum norm method: they can be positive or negative and they oscillate around zero. Display the sources on the surface, set the amplitude threshold to 0%, then configure the colormap to show relative values (uncheck the "Absolute values" option), you would see those typical stripes of positive and negative values around the sulci. Double-click on the colorbar after testing this to reset the colormap.

This pattern is due to the orientation constraint imposed on the dipoles. On both sides of a sulcus, we have defined dipoles that are very close to each other, but with opposite orientations. If we have a pattern of activity on one side of a suclus that can be assimilated to an electric dipole (green arrow), the minimum norm model will try to explain it with the dipoles that are available in the head model (red and blue arrows). Because of the dipoles orientations, it translates into positive values (red arrows) on one side of the sulcus and negative on the other side (blue arrows).

When displaying the cortical maps at one time point, we are usually not interested by the sign of the minimum norm values but rather by their amplitude. This is why we always display them by default with the colormap option "absolute values" selected.

However, we cannot simply discard the sign of these values because we need them for other types of analysis, typically time-frequency decompositions and connectivity analysis. For estimating frequency measures on the source maps, we need to keep the oscillations around zero.

Unconstrained orientations

In the cases where the orientation constraint imposed on the dipoles orientations looks too strong, it is possible to relax it partially (option "loose constraints") or completely (option "unconstrained"). This is typically something to consider when using a MNI template instead of the subject's anatomy, or when studying deeper or non-cortical brain regions for which the normal to the FreeSurfer cortex surface is unlikely to match any physiological reality.

In terms of data representation, the option "unconstrained" and "loose constraints" are very similar. Instead of using one dipole at each cortical location, a base of three orthogonal dipoles is used.
Here we will only illustrate the fully unconstrained case.

Source map normalization

The current density values returned by the minimum norm method have a few problems:

Normalizing the current density maps with respect to a reference level (estimated from noise recordings, pre-stimulus baseline or resting state recordings) can help with all these issues at the same time. Some normalizations can be computed independently from the recordings, and added to the linear inverse operator (dSPM or sLORETA). Another way of proceeding is to divide the current density maps by the standard deviation estimated over a baseline (Z-score).

The normalization options do not change the temporal dynamics of your results, they are just different ways for looking at the same minimum norm maps. If you look at the time series associated with one given source, it would be exactly the same for all the normalizations, except for a scaling factor. Only the relative weights change between the sources, and these weights do not change over time.

dSPM, sLORETA

Z-score

Typical recommendations

Delete your experiments

Computing sources for single trials

Because the minimum norm model is linear, we can compute an inverse model independently from the recordings and apply it on the recordings when needed. We will now illustrate how to compute a shared inverse model for all the imported epochs.

Averaging in source space

Computing the average

Visualization filters

Low-pass filter

Z-score normalization

Advanced

Averaging normalized values

Averaging normalized source maps within a single subject requires more attention than averaging current density maps. The amplitude of the normalized measures increase with the SNR of the signal, the higher the SNR the higher the normalized score. For instance, the average of the dSPM for the single trials is lower than the dSPM of the averaged trials.

dSPM

Z-score

sLORETA

Advanced

Display: Contact sheets and movies

A good way to represent what is happening in time is to generate contact sheets or videos. Right-click on any figure and go to the menu Snapshot to check out all the possible options. For a nicer result, take some time to adjust the size of the figure, the amplitude threshold and the colormap options (hiding the colorbar can be a good option for contact sheets).

A time stamp is added to the captured figure. The size of the text font is fixed, so if you want it to be readable in the contact sheet, you should make you figure very small before starting the capture. The screen captures below where produced with the colormap "hot".

Advanced

Model evaluation

One way to evaluate the accuracy of the source reconstruction if to simulate recordings using the estimated source maps. This is done simply by multiplying the source time series with the forward model:
MEG_simulated [Nmeg x Ntime] = Forward_model [Nmeg x Nsources] * MN_sources [Nsources x Ntime]
Then you can compare visually the original MEG recordings with the simulated ones. More formally, you can compute an error measure from the residuals (recordings - simulated).

To simulate MEG recordings from a minimum norm source model, right-click on the source file, then select the menu "Model evaluation > Simulate recordings".

Open side-by-side the original and simulated MEG recordings for the same condition:

Advanced

Advanced minimum norm options

Right-click on the deviant average for Run#01 > Compute sources [2016].
Click on the button [Show details] to bring up all the advanced minimum norm options.

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Depth weighting

Briefly, the use of various depth weightings was far more debated in the 1990s, before the introduction of MNE normalization via dSPM, sLORETA, and other "z-scoring" methods, which mostly negate the effects of depth weighting. At each point in the source grid, the deeper points are "boosted" to increase their signal strength relative to the shallower dipoles; otherwise, the resulting MNE current density maps are too dominated by the shallower sources. If using dSPM or sLORETA, little difference in using depth weighting should be noted. To understand how to set these parameters, please refer to the MNE manual. (options --depth, --weightexp and --weightlimit).

Noise covariance regularization

MNE and dipole modeling are best done with an accurate model of the noise covariance, which is generally computed from experimental data. As such, these estimates are themselves prone to errors that arise from relatively too few data points, weak sensors, and strange data dependencies that cause the eigenspectrum of the covariance matrix to be somewhat deficient. In Brainstorm, we provide several means to "stabilize" or "regularize" the noise covariance matrix, so that source estimation calculations are more robust to small errors.

Regularization parameter

In MNE, as mentioned above in the comparisons among Methods, the data covariance matrix is essentially synthesized by adding the noise covariance matrix to a modeled signal covariance matrix. The signal covariance matrix is generated by passing the source prior through the forward (head) model. The source prior is in turn prescribed by the source model orientation and the depth weighting. A final regularization parameter, however, is how much weight the signal model should be given relative to the noise model, i.e. the "signal to noise ratio" (SNR). In Brainstorm, as of Fall 2016, we follow the definition of SNR as first defined in the original MNE software of Hamalainen. The signal covariance matrix is "whitened" by the noise covariance matrix, such that the whitened eigenspectrum has elements in terms of SNR (power). We find the mean of this spectrum, then take the square root to yield the average SNR (amplitude). The default in MNE and in Brainstorm is "3", i.e. the average SNR (power) is 9.

Output mode

As mentioned above, these methods create a convenient linear imaging kernel that is "tall" in the number of elemental dipoles (one or three per grid point) and "wide" only in the number of sensors. At subsequent visualzation time, we efficiently multiply the kernel with the data matrix.

For some custom purposes, however, a user may find it convenient to pre-multiply the data matrix and generate the full source estimation matrix. This would only be recommended in small data sets, since the full results can become quite large.

LCMV beamformer

As mentioned in the introduction above, two other Methods can be selected for source estimation, a beamformer and dipole modeling. In this section, we review the options for the beamformer. By selecting "LCMV beamformer, the options automatically switch to the below figure (shown with details).

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Measure

The Measure has switched to a single option, the "Pseudo Neural Activity Index," (PNAI), after the Van Veen definition of the Neural Activity Index (NAI). We have altered the definition to rely strictly on the data covariance, without need for a noise covariance matrix, but the basic premise is the same as in dSPM, sLORETA, and other normalizations. Viewing the resulting "map," identically as is done with MNE, dSPM, and sLORETA above, reveals possibly multiple sources as peaks in the map, scored analogous to z-scoring. The PNAI can then be dropped into the Process Box and the optimal dipole and orientations extracted for every time instance.

Source Model: Dipole orientations

The definitions here are identical to the MNE above; however, we do not recommend you select "constrained" (although technically allowed). Choose only "unconstrained" and let the "dipole scanning" process optimize the orientation with respect to the data.

Sensors

Same considerations as in MNE.

Data covariance regularization

Same definitions as in MNE, only applied to the data covariance matrix, rather than the noise covariance marix. Recommendation is to use median eigenvalue.

Output mode

Recommend to use kernel only.

Dipole modeling

Although not widely recognized, dipole modeling and beamforming are more alike than they are different. Below is the options screen when dipole modeling is selected.

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You will notice that "Measure" is now missing, but the resulting imaging kernel file is directly analogous to the PNAI result of LCMV beamforming. The user may identically display this scanning measure as well, where again the units are a form of z-scoring.

Source model, Sensors, Noise covariance regularization, and output mode all follow from the explanations given above for the LCMV and for the MNE.

Again, the recommendation is to ALWAYS use "unconstrained" source modeling and let the Process "dipole scanning" optimize the orientation of the dipole for every time instance. Similarly, use "median eigenvalue" for the optimization.

The tutorial "MEG current phantom (Elekta)" demonstrates dipole modeling of 32 individual dipoles under realistic experimental noise conditions.

Equations [TODO]

...

Advanced

On the hard drive

Constrained shared kernel

Right-click on a shared inverse file in the database explorer > File > View file contents.

Structure of the source files: results_*.mat

Mandatory fields:

Optional fields:

Full source maps

In Intra-subject, right-click on one of the normalized averages > File > View file contents.

This file has the same structure as a shared inverse kernel, with the following differences:

Filename tags

Useful functions

Additional documentation [TODO]

Articles [TODO]

Tutorials

Forum discussions








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Tutorials/SourceEstimation (last edited 2016-09-09 00:58:19 by JohnMosher)