When performing source localization in a distributed manner (MNE/dSPM/sLORETA), the source space is defined as a grid of dipoles that spans a large portion of the cortex. These dipoles have both a position and an orientation. In this tutorial, we will look at the various options available to restrict the orientation of the dipoles and the impact on the resulting source estimate.

Load everything we need to perform source localization on the sample dataset.

```
from mayavi import mlab
import mne
from mne.datasets import sample
from mne.minimum_norm import make_inverse_operator, apply_inverse
data_path = sample.data_path()
evokeds = mne.read_evokeds(data_path + '/MEG/sample/sample_audvis-ave.fif')
left_auditory = evokeds[0].apply_baseline()
fwd = mne.read_forward_solution(
data_path + '/MEG/sample/sample_audvis-meg-eeg-oct-6-fwd.fif',
surf_ori=True)
noise_cov = mne.read_cov(data_path + '/MEG/sample/sample_audvis-cov.fif')
subjects_dir = data_path + '/subjects'
```

Out:

```
Successfully extracted to: [u'/home/ubuntu/mne_data/MNE-sample-data']
Reading /home/ubuntu/mne_data/MNE-sample-data/MEG/sample/sample_audvis-ave.fif ...
Read a total of 4 projection items:
PCA-v1 (1 x 102) active
PCA-v2 (1 x 102) active
PCA-v3 (1 x 102) active
Average EEG reference (1 x 60) active
Found the data of interest:
t = -199.80 ... 499.49 ms (Left Auditory)
0 CTF compensation matrices available
nave = 55 - aspect type = 100
Projections have already been applied. Setting proj attribute to True.
No baseline correction applied
Read a total of 4 projection items:
PCA-v1 (1 x 102) active
PCA-v2 (1 x 102) active
PCA-v3 (1 x 102) active
Average EEG reference (1 x 60) active
Found the data of interest:
t = -199.80 ... 499.49 ms (Right Auditory)
0 CTF compensation matrices available
nave = 61 - aspect type = 100
Projections have already been applied. Setting proj attribute to True.
No baseline correction applied
Read a total of 4 projection items:
PCA-v1 (1 x 102) active
PCA-v2 (1 x 102) active
PCA-v3 (1 x 102) active
Average EEG reference (1 x 60) active
Found the data of interest:
t = -199.80 ... 499.49 ms (Left visual)
0 CTF compensation matrices available
nave = 67 - aspect type = 100
Projections have already been applied. Setting proj attribute to True.
No baseline correction applied
Read a total of 4 projection items:
PCA-v1 (1 x 102) active
PCA-v2 (1 x 102) active
PCA-v3 (1 x 102) active
Average EEG reference (1 x 60) active
Found the data of interest:
t = -199.80 ... 499.49 ms (Right visual)
0 CTF compensation matrices available
nave = 58 - aspect type = 100
Projections have already been applied. Setting proj attribute to True.
No baseline correction applied
Applying baseline correction (mode: mean)
surf_ori is deprecated and will be removed in 0.16. For handling transformations, apply convert_forward_solution after read_forward_solution instead.
Reading forward solution from /home/ubuntu/mne_data/MNE-sample-data/MEG/sample/sample_audvis-meg-eeg-oct-6-fwd.fif...
Reading a source space...
Computing patch statistics...
Patch information added...
Distance information added...
[done]
Reading a source space...
Computing patch statistics...
Patch information added...
Distance information added...
[done]
2 source spaces read
Desired named matrix (kind = 3523) not available
Read MEG forward solution (7498 sources, 306 channels, free orientations)
Desired named matrix (kind = 3523) not available
Read EEG forward solution (7498 sources, 60 channels, free orientations)
MEG and EEG forward solutions combined
Source spaces transformed to the forward solution coordinate frame
Average patch normals will be employed in the rotation to the local surface coordinates....
Converting to surface-based source orientations...
[done]
366 x 366 full covariance (kind = 1) found.
Read a total of 4 projection items:
PCA-v1 (1 x 102) active
PCA-v2 (1 x 102) active
PCA-v3 (1 x 102) active
Average EEG reference (1 x 60) active
```

Let’s start by examining the source space as constructed by the
`mne.setup_source_space()`

function. Dipoles are placed along fixed
intervals on the cortex, determined by the `spacing`

parameter. The source
space does not define the orientation for these dipoles.

```
lh = fwd['src'][0] # Visualize the left hemisphere
verts = lh['rr'] # The vertices of the source space
tris = lh['tris'] # Groups of three vertices that form triangles
dip_pos = lh['rr'][lh['vertno']] # The position of the dipoles
white = (1.0, 1.0, 1.0) # RGB values for a white color
gray = (0.5, 0.5, 0.5) # RGB values for a gray color
red = (1.0, 0.0, 0.0) # RGB valued for a red color
mlab.figure(size=(600, 400), bgcolor=white)
# Plot the cortex
mlab.triangular_mesh(verts[:, 0], verts[:, 1], verts[:, 2], tris, color=gray)
# Mark the position of the dipoles with small red dots
mlab.points3d(dip_pos[:, 0], dip_pos[:, 1], dip_pos[:, 2], color=red,
scale_factor=1E-3)
mlab.view(azimuth=180, distance=0.25)
```

While the source space defines the position of the dipoles, the inverse operator defines the possible orientations of them. One of the options is to assign a fixed orientation. Since the neural currents from which MEG and EEG signals originate flows mostly perpendicular to the cortex [1], restricting the orientation of the dipoles accordingly places a useful restriction on the source estimate.

By specifying `fixed=True`

when calling
`mne.minimum_norm.make_inverse_operator()`

, the dipole orientations are
fixed to be orthogonal to the surface of the cortex, pointing outwards. Let’s
visualize this:

```
mlab.figure(size=(600, 400), bgcolor=white)
# Plot the cortex
mlab.triangular_mesh(verts[:, 0], verts[:, 1], verts[:, 2], tris, color=gray)
# Show the dipoles as arrows pointing along the surface normal
normals = lh['nn'][lh['vertno']]
mlab.quiver3d(dip_pos[:, 0], dip_pos[:, 1], dip_pos[:, 2],
normals[:, 0], normals[:, 1], normals[:, 2],
color=red, scale_factor=1E-3)
mlab.view(azimuth=180, distance=0.1)
```

Restricting the dipole orientations in this manner leads to the following source estimate for the sample data:

```
# Compute the source estimate for the 'left - auditory' condition in the sample
# dataset.
inv = make_inverse_operator(left_auditory.info, fwd, noise_cov, fixed=True)
stc = apply_inverse(left_auditory, inv, pick_ori=None)
# Visualize it at the moment of peak activity.
_, time_max = stc.get_peak(hemi='lh')
brain = stc.plot(surface='white', subjects_dir=subjects_dir,
initial_time=time_max, time_unit='s', size=(600, 400))
```

Out:

```
Computing inverse operator with 364 channels.
Created an SSP operator (subspace dimension = 4)
estimated rank (mag + grad): 302
Setting small MEG eigenvalues to zero.
Not doing PCA for MEG.
estimated rank (eeg): 58
Setting small EEG eigenvalues to zero.
Not doing PCA for EEG.
Total rank is 360
Creating the depth weighting matrix...
203 planar channels
limit = 7262/7498 = 10.020865
scale = 2.58122e-08 exp = 0.8
Picked elements from a free-orientation depth-weighting prior into the fixed-orientation one
The default settings controlling the application of cortical patch statistics (cps) in the creation of forward operators with fixed orientation will be modified in 0.16. The cps (if available) will then be applied by default. To avoid this warning, set use_cps explicitly to False (the current default) or True (the new default).
Changing to fixed-orientation forward solution with surface-based source orientations...
[done]
Computing inverse operator with 364 channels.
Creating the source covariance matrix
Whitening the forward solution.
Adjusting source covariance matrix.
Computing SVD of whitened and weighted lead field matrix.
largest singular value = 5.49738
scaling factor to adjust the trace = 1.2178e+19
Preparing the inverse operator for use...
Scaled noise and source covariance from nave = 1 to nave = 55
Created the regularized inverter
Created an SSP operator (subspace dimension = 4)
Created the whitener using a full noise covariance matrix (4 small eigenvalues omitted)
Computing noise-normalization factors (dSPM)...
[done]
Picked 364 channels from the data
Computing inverse...
(eigenleads need to be weighted)...
(dSPM)...
[done]
```

The direction of the estimated current is now restricted to two directions: inward and outward. In the plot, blue areas indicate current flowing inwards and red areas indicate current flowing outwards. Given the curvature of the cortex, groups of dipoles tend to point in the same direction: the direction of the electromagnetic field picked up by the sensors.

Forcing the source dipoles to be strictly orthogonal to the cortex makes the source estimate sensitive to the spacing of the dipoles along the cortex, since the curvature of the cortex changes within each ~10 square mm patch. Furthermore, misalignment of the MEG/EEG and MRI coordinate frames is more critical when the source dipole orientations are strictly constrained [2]. To lift the restriction on the orientation of the dipoles, the inverse operator has the ability to place not one, but three dipoles at each location defined by the source space. These three dipoles are placed orthogonally to form a Cartesian coordinate system. Let’s visualize this:

```
mlab.figure(size=(600, 400), bgcolor=white)
# Define some more colors
green = (0.0, 1.0, 0.0)
blue = (0.0, 0.0, 1.0)
# Plot the cortex
mlab.triangular_mesh(verts[:, 0], verts[:, 1], verts[:, 2], tris, color=gray)
# Make an inverse operator with loose dipole orientations
inv = make_inverse_operator(left_auditory.info, fwd, noise_cov, fixed=False,
loose=1.0)
# Show the three dipoles defined at each location in the source space
dip_dir = inv['source_nn'].reshape(-1, 3, 3)
dip_dir = dip_dir[:len(dip_pos)] # Only select left hemisphere
for ori, color in zip((0, 1, 2), (red, green, blue)):
mlab.quiver3d(dip_pos[:, 0], dip_pos[:, 1], dip_pos[:, 2],
dip_dir[:, ori, 0], dip_dir[:, ori, 1], dip_dir[:, ori, 2],
color=color, scale_factor=1E-3)
mlab.view(azimuth=180, distance=0.1)
```

Out:

```
Computing inverse operator with 364 channels.
Created an SSP operator (subspace dimension = 4)
estimated rank (mag + grad): 302
Setting small MEG eigenvalues to zero.
Not doing PCA for MEG.
estimated rank (eeg): 58
Setting small EEG eigenvalues to zero.
Not doing PCA for EEG.
Total rank is 360
Creating the depth weighting matrix...
203 planar channels
limit = 7262/7498 = 10.020865
scale = 2.58122e-08 exp = 0.8
Computing inverse operator with 364 channels.
Creating the source covariance matrix
Whitening the forward solution.
Adjusting source covariance matrix.
Computing SVD of whitened and weighted lead field matrix.
largest singular value = 5.2188
scaling factor to adjust the trace = 3.44205e+19
```

When computing the source estimate, the activity at each of the three dipoles is collapsed into the XYZ components of a single vector, which leads to the following source estimate for the sample data:

```
# Compute the source estimate, indicate that we want a vector solution
stc = apply_inverse(left_auditory, inv, pick_ori='vector')
# Visualize it at the moment of peak activity.
_, time_max = stc.magnitude().get_peak(hemi='lh')
brain = stc.plot(subjects_dir=subjects_dir, initial_time=time_max,
time_unit='s', size=(600, 400), overlay_alpha=0)
```

Out:

```
Preparing the inverse operator for use...
Scaled noise and source covariance from nave = 1 to nave = 55
Created the regularized inverter
Created an SSP operator (subspace dimension = 4)
Created the whitener using a full noise covariance matrix (4 small eigenvalues omitted)
Computing noise-normalization factors (dSPM)...
[done]
Picked 364 channels from the data
Computing inverse...
(eigenleads need to be weighted)...
(dSPM)...
[done]
```

Often, the best results will be obtained by allowing the dipoles to have
somewhat free orientation, but not stray too far from a orientation that is
perpendicular to the cortex. The `loose`

parameter of the
`mne.minimum_norm.make_inverse_operator()`

allows you to specify a value
between 0 (fixed) and 1 (unrestricted or “free”) to indicate the amount the
orientation is allowed to deviate from the surface normal.

```
# Set loose to 0.2, the default value
inv = make_inverse_operator(left_auditory.info, fwd, noise_cov, fixed=False,
loose=0.2)
stc = apply_inverse(left_auditory, inv, pick_ori='vector')
# Visualize it at the moment of peak activity.
_, time_max = stc.magnitude().get_peak(hemi='lh')
brain = stc.plot(subjects_dir=subjects_dir, initial_time=time_max,
time_unit='s', size=(600, 400), overlay_alpha=0)
```

Out:

```
Computing inverse operator with 364 channels.
Created an SSP operator (subspace dimension = 4)
estimated rank (mag + grad): 302
Setting small MEG eigenvalues to zero.
Not doing PCA for MEG.
estimated rank (eeg): 58
Setting small EEG eigenvalues to zero.
Not doing PCA for EEG.
Total rank is 360
Creating the depth weighting matrix...
203 planar channels
limit = 7262/7498 = 10.020865
scale = 2.58122e-08 exp = 0.8
Computing inverse operator with 364 channels.
Creating the source covariance matrix
Applying loose dipole orientations. Loose value of 0.2.
Whitening the forward solution.
Adjusting source covariance matrix.
Computing SVD of whitened and weighted lead field matrix.
largest singular value = 5.49264
scaling factor to adjust the trace = 1.64e+19
Preparing the inverse operator for use...
Scaled noise and source covariance from nave = 1 to nave = 55
Created the regularized inverter
Created an SSP operator (subspace dimension = 4)
Created the whitener using a full noise covariance matrix (4 small eigenvalues omitted)
Computing noise-normalization factors (dSPM)...
[done]
Picked 364 channels from the data
Computing inverse...
(eigenleads need to be weighted)...
(dSPM)...
[done]
```

Often, further analysis of the data does not need information about the
orientation of the dipoles, but rather their magnitudes. The `pick_ori`

parameter of the `mne.minimum_norm.apply_inverse()`

function allows you
to specify whether to return the full vector solution (`'vector'`

) or
rather the magnitude of the vectors (`None`

, the default) or only the
activity in the direction perpendicular to the cortex (`'normal'`

).

```
# Only retain vector magnitudes
stc = apply_inverse(left_auditory, inv, pick_ori=None)
# Visualize it at the moment of peak activity.
_, time_max = stc.get_peak(hemi='lh')
brain = stc.plot(surface='white', subjects_dir=subjects_dir,
initial_time=time_max, time_unit='s', size=(600, 400))
```

Out:

```
Preparing the inverse operator for use...
Scaled noise and source covariance from nave = 1 to nave = 55
Created the regularized inverter
Created an SSP operator (subspace dimension = 4)
Created the whitener using a full noise covariance matrix (4 small eigenvalues omitted)
Computing noise-normalization factors (dSPM)...
[done]
Picked 364 channels from the data
Computing inverse...
(eigenleads need to be weighted)...
combining the current components...
(dSPM)...
[done]
```

[1] | Hämäläinen, M. S., Hari, R., Ilmoniemi, R. J., Knuutila, J., & Lounasmaa, O. V. “Magnetoencephalography - theory, instrumentation, and applications to noninvasive studies of the working human brain”, Reviews of Modern Physics, 1993. http://dx.doi.org/10.1103/RevModPhys.65.413 |

[2] | Lin, F. H., Belliveau, J. W., Dale, A. M., & Hämäläinen, M. S. (2006). Distributed current estimates using cortical orientation constraints. Human Brain Mapping, 27(1), 1–13. http://doi.org/10.1002/hbm.20155 |

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