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Beamforming methods

Authors: Hui-Ling Chan, Francois Tadel, Sylvain Baillet

The estimation of source distribtion is an important step to understand the brain activity from EEG and MEG data. Dipole fitting, minimum norm estimation, and beamformer are three commonly used methods. It has been proved that beamforming methods provide good spatial resolution. This tutorial will show how to apply beamforming methods to MEG data and obtain the statistic map of source activation.

We are going to use the protocol TutorialRaw created in the previous tutorial ?Epoching and Averaging. If you have not followed this tutorial yet, please do it now.

Contents

Introduction
Linearly-constrained minimum variance (LCMV) beamformer
Maximum constrast beamformer (MCB)
Beamformer-based correlation/coherence imaging

Introduction

Beamfoming methods scan each targeted voxel/vertex position $\bf r$ and estimate the spatial filter ${\bf w}_{\bf r,q}$ . By multiplying with the MEG recordings ${\bf m}(t)$ , the spatial filter ${\bf w}_{\bf r,q}$ outputs the temporal waveform $y_{\bf r,q}(t)$ of the dipole source at that position with the dipole orientation $\bf q$ as below:

$y_{\bf r,\bf q}(t) = {{\bf w}_{\bf r,\bf q}}^{\rm T}{\bf m}(t)\enspace,$

where 'T' indicates the transpose of a matrix or vector. The beamforming spatial filter can be vector type or scalar type.

Vector-type beamformer

For each position $\mathbf{r}$ , three orthogonal spatial filters $\mathbf{W}_{\mathbf{r}}=\left[\mathbf{w}_{\mathbf{r},\mathbf{q}_x},\mathbf{w}_{\mathbf{r},\mathbf{q}_y},\mathbf{w}_{\mathbf{r},\mathbf{q}_z}\right]$ are computed by applying the unit-gain constraint as well as the minimum norm and minimum variance criteria as below:

$\hat{\mathbf{W}}_{\mathbf r}=(\mathbf{C}_{\rm s}+\alpha\mathbf{I})^{-1}\mathbf{L}_\mathbf{r}\left(\mathbf{L}_\mathbf{r}^{\rm T}(\mathbf{C}_{\rm s}+\alpha\mathbf{I})^{-1}\mathbf{L}_\mathbf{r}\right)^{-1} \enspace,$

where ${\bf C}_{\rm s}$ is the covariance matrix of MEG recordings during window $T_{\rm_s}$ , $\mathbf{I}$ is the identity matrix, $\mathbf{L}_{\mathbf{r}}$ is the gain matrix for the dipole located at position $\bf{r}$ , and $\alpha$ is the regularization parameter which compromises the minimum norm and minimum variance criteria.

Scalar-type beamformer

For each position $\mathbf{r}$ , the source orientation $\mathbf{q}$ is first estimated to enable the spatial filter to output the source activity with maxmum power or fitting some other criteria.The estiamted dipole orientation $\hat{\mathbf{q}}$ is then applied to calculate the spatial filter as follows:

$\hat{\mathbf{w}}_{\mathbf{r,q}}=(\mathbf{C}_{\rm s}+\alpha\mathbf{I})^{-1}\mathbf{L}_\mathbf{r}\hat{\bf{q}}\left(\mathbf{L}_\mathbf{r}^{\rm T}\hat{\bf q}(\mathbf{C}_{\rm s}+\alpha\mathbf{I})^{-1}\mathbf{L}_\mathbf{r}\hat{\bf q}\right)^{-1} \enspace.$

Linearly-constrained minimum variance (LCMV) beamformer

LCMV beamformer is vector-type beamformer. For each position $\mathbf{r}$ , this method calculates the neural activity index, which is interpreted as the estimate of source to noise variance.

Neural activity index

For each position $\mathbf{r}$ , the variance of source activity during active state $T_{\rm a}$ is calculated as follows:

$\rm{Var}_{\rm a}(\bf r) =trace \left \{\left[{\bf L}_{\bf r}^{\rm T}{\bf C}_{\rm a}{\bf L}_{\bf r}\right]^{-1}\right \}\enspace,$

$\rm{Var}_{\rm a}(\bf r) =\lambda_1 \left \{\left[{\bf L}_{\bf r}^{\rm T}{\bf C}_{\rm a}{\bf L}_{\bf r}\right]^{-1}\right \}\enspace,$

where ${\bf C}_{\rm a}$ is the covariance matrix computed from MEG recordings during active state $T_{\rm a}$ and $\lambda_1\{\}$ indicates the maximum eigenvalue of the expression in braces.

When the location $\mathbf{r}$ is far from sensors, the elements of lead field matrix $\mathbf{L}_{\bf r}$ are small. So the elements of $\left[{\bf L}_{\bf r}^{\rm T}{\bf C}_{\rm a}{\bf L}_{\bf r}\right]^{-1}$ are generally large and the estimated variance for the deep source becomes large. When the location $\mathbf{r}$ is close to sensors, the elements of lead field matrix $\mathbf{L}_{\bf r}$ are large. It results in small values of the elements of $\left[{\bf L}_{\bf r}^{\rm T}{\bf C}_{\rm a}{\bf L}_{\bf r}\right]^{-1}$ and small estimated variance for the superficial source. To reduce the effect caused by the depth of source location, the estimated source variance $\rm{Var}_{\rm a}(\bf r)$ is normalized by the noise variance $\rm{Var}_{\rm c}(\bf r)$ as follows:

$\rm{Var}_{\rm N}(\bf r) =\frac{\rm{trace} \left \{\left[{\bf L}_{\bf r}^{\rm T}{\bf C}_{\rm a}{\bf L}_{\bf r}\right]^{-1}\right \}}{\rm{trace} \left \{\left[{\bf L}_{\bf r}^{\rm T}{\bf C}_{\rm c}{\bf L}_{\bf r}\right]^{-1}\right \}}\enspace,$

$\rm{Var}_{\rm N}(\bf r) =\frac{\lambda_1 \left \{\left[{\bf L}_{\bf r}^{\rm T}{\bf C}_{\rm a}{\bf L}_{\bf r}\right]^{-1}\right \}}{\lambda_1 \left \{\left[{\bf L}_{\bf r}^{\rm T}{\bf C}_{\rm c}{\bf L}_{\bf r}\right]^{-1}\right \}}\enspace,$

where ${\bf C}_{\rm c}$ is the covariance matrix computed from MEG recordings during control state $T_{\rm c}$ . The normalized variance $\rm{Var}_{\rm N}(\bf r)$ is called neural activity index (NAI).

Compute Neural Activty Index for a sets of epoches

Drag and drop all of the epoches in Subject01 / right / right | timeoffset (98 files) into Process1.
Left click on RUN and choose Sources > LCMV Beamformer' in Pipeline editor.
With this window you can select the method you want to use to estimate the source variance and noise variance, the time window of active state, and the sensors you are going to use for this estimation. You can edit the following options:
- Comment: This field contains what is going to be displayed in the database explorer.
- Method: Please select Unconstrained (max eigenvalue). This method calculates the variances of source activity and noise by using the maximum eigenvalue. The method Unconstrained (trace) calculates the variances of source activity and noise by using trace. The method Cortial Constrained use the orientation of cortical surface as dipole orientation. The unconstrained methods may give more smooth results than the cortical constrained method.
- Time window of active state:
- Regularization parameter: This value will be multiplied with the maximum eigenvalue of the covariance matrix computed from the recordings during Time range of interest. The larger value of regularization parameter may give the more smooth results.
- Sensors type: Modalities that are used for the reconstruction. Here we only have one type of MEG sensors, so nothing to change.
- Click on Run.
Two new files are available in the database explorer.
- right: LCMV: spatial filter(Unconstr)|0.0_295.8ms: This field
- right: LCMV: neural activity index(Unconstr)|0.0_295.8ms(ws:296ms): Please

Double click on this file. Right click on anywhere of the pop-up window. Select Colormap:Sources > Maximum: Custom.... Set the Minimum to be 1.2 and the Maximum to be 2.0. Click Ok.
<<attachment:Colormaplimit.png>>

Set the time to be anytime between 0 and 295.8 ms. Then click Surface on the right panel of database explorer. Set the Amplitude to be 60%. The distribution of neural activity index displays strong activation in the left sensory area.

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-  * right: LCMV: neural activity index(Unconstr)|0.0_295.8ms(ws:296ms):  Please <<BR>><<BR>>
  {{attachment:NAIright2.png}}
<<BR>> <<BR>>Double click on this file. Right click on anywhere of the pop-up window. Select Colormap:Sources > Maximum: Custom.... Set the Minimum to be 1.2 and the Maximum to be 2.0. Click Ok. <<BR>> [[attachment:Colormaplimit|]]<<attachment:Colormaplimit.png>>
<<BR>> <<BR>>Set the time to be anytime between 0 and 295.8 ms. Then click Surface on the right panel of database explorer. Set the Amplitude to be 60%. The distribution of neural activity index displays strong activation in the left sensory area. <<BR>>
 {{attachment:NAIrightmap4.png}} <<BR>><<BR>>
+  * right: LCMV: neural activity index(Unconstr)|0.0_295.8ms(ws:296ms):  Please <<BR>><<BR>> {{attachment:NAIright2.png}}

  <<BR>> <<BR>>
Double click on this file. Right click on anywhere of the pop-up window. Select Colormap:Sources > Maximum: Custom.... Set the Minimum to be 1.2 and the Maximum to be 2.0. Click Ok. <<BR>>
<<attachment:Colormaplimit.png>>
<<BR>> <<BR>>
Set the time to be anytime between 0 and 295.8 ms. Then click Surface on the right panel of database explorer. Set the Amplitude to be 60%. The distribution of neural activity index displays strong activation in the left sensory area. <<BR>>
{{attachment:NAIrightmap4.png}} <<BR>><<BR>>

Software

Users

Development

Beamforming methods

Introduction

Vector-type beamformer

Scalar-type beamformer

Linearly-constrained minimum variance (LCMV) beamformer

Neural activity index

Compute Neural Activty Index for a sets of epoches

Maximum constrast beamformer (MCB)

Section 1

Section 2

Beamformer-based correlation/coherence imaging

Dynamic imaging of coherent sources (DICS)

Spatiotemporal imaging of linearly-related source components (SILSC)