Compute sparse inverse solution with mixed norm: MxNE and irMxNE

Runs an (ir)MxNE (L1/L2 [1] or L0.5/L2 [2] mixed norm) inverse solver. L0.5/L2 is done with irMxNE which allows for sparser source estimates with less amplitude bias due to the non-convexity of the L0.5/L2 mixed norm penalty.

# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
#         Daniel Strohmeier <daniel.strohmeier@tu-ilmenau.de>
#
# License: BSD-3-Clause
# Copyright the MNE-Python contributors.

import numpy as np

import mne
from mne.datasets import sample
from mne.inverse_sparse import make_stc_from_dipoles, mixed_norm
from mne.minimum_norm import apply_inverse, make_inverse_operator
from mne.viz import (
    plot_dipole_amplitudes,
    plot_dipole_locations,
    plot_sparse_source_estimates,
)

print(__doc__)

data_path = sample.data_path()
meg_path = data_path / "MEG" / "sample"
fwd_fname = meg_path / "sample_audvis-meg-eeg-oct-6-fwd.fif"
ave_fname = meg_path / "sample_audvis-ave.fif"
cov_fname = meg_path / "sample_audvis-shrunk-cov.fif"
subjects_dir = data_path / "subjects"

# Read noise covariance matrix
cov = mne.read_cov(cov_fname)
# Handling average file
condition = "Left Auditory"
evoked = mne.read_evokeds(ave_fname, condition=condition, baseline=(None, 0))
evoked.crop(tmin=0, tmax=0.3)
# Handling forward solution
forward = mne.read_forward_solution(fwd_fname)

    365 x 365 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 59) active
Reading /home/circleci/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.
Applying baseline correction (mode: mean)
Reading forward solution from /home/circleci/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 (FIFF_MNE_FORWARD_SOLUTION_GRAD)) not available
    Read MEG forward solution (7498 sources, 306 channels, free orientations)
    Desired named matrix (kind = 3523 (FIFF_MNE_FORWARD_SOLUTION_GRAD)) not available
    Read EEG forward solution (7498 sources, 60 channels, free orientations)
    Forward solutions combined: MEG, EEG
    Source spaces transformed to the forward solution coordinate frame

Run solver with SURE criterion [3]

alpha = "sure"  # regularization parameter between 0 and 100 or SURE criterion
loose, depth = 0.9, 0.9  # loose orientation & depth weighting
n_mxne_iter = 10  # if > 1 use L0.5/L2 reweighted mixed norm solver
# if n_mxne_iter > 1 dSPM weighting can be avoided.

# Compute dSPM solution to be used as weights in MxNE
inverse_operator = make_inverse_operator(
    evoked.info, forward, cov, depth=depth, fixed=True, use_cps=True
)
stc_dspm = apply_inverse(evoked, inverse_operator, lambda2=1.0 / 9.0, method="dSPM")

# Compute (ir)MxNE inverse solution with dipole output
dipoles, residual = mixed_norm(
    evoked,
    forward,
    cov,
    alpha,
    loose=loose,
    depth=depth,
    maxit=3000,
    tol=1e-4,
    active_set_size=10,
    debias=False,
    weights=stc_dspm,
    weights_min=8.0,
    n_mxne_iter=n_mxne_iter,
    return_residual=True,
    return_as_dipoles=True,
    verbose=True,
    random_state=0,
    # for this dataset we know we should use a high alpha, so avoid some
    # of the slower (lower) alpha values
    sure_alpha_grid=np.linspace(100, 40, 10),
)

t = 0.083
tidx = evoked.time_as_index(t).item()
for di, dip in enumerate(dipoles, 1):
    print(f"Dipole #{di} GOF at {1000 * t:0.1f} ms: {float(dip.gof[tidx]):0.1f}%")

info["bads"] and noise_cov["bads"] do not match, excluding bad channels from both
Computing inverse operator with 364 channels.
    364 out of 366 channels remain after picking
Selected 364 channels
Creating the depth weighting matrix...
    203 planar channels
    limit = 7262/7498 = 10.020866
    scale = 2.58122e-08 exp = 0.9
    Picked elements from a free-orientation depth-weighting prior into the fixed-orientation one
    Average patch normals will be employed in the rotation to the local surface coordinates....
    Converting to surface-based source orientations...
    [done]
Whitening the forward solution.
    Created an SSP operator (subspace dimension = 4)
Computing rank from covariance with rank=None
    Using tolerance 3.5e-13 (2.2e-16 eps * 305 dim * 5.2  max singular value)
    Estimated rank (mag + grad): 302
    MEG: rank 302 computed from 305 data channels with 3 projectors
    Using tolerance 1.1e-13 (2.2e-16 eps * 59 dim * 8.7  max singular value)
    Estimated rank (eeg): 58
    EEG: rank 58 computed from 59 data channels with 1 projector
    Setting small MEG eigenvalues to zero (without PCA)
    Setting small EEG eigenvalues to zero (without PCA)
Creating the source covariance matrix
Adjusting source covariance matrix.
Computing SVD of whitened and weighted lead field matrix.
    largest singular value = 6.21995
    scaling factor to adjust the trace = 6.82623e+18 (nchan = 364 nzero = 4)
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 noise covariance matrix with rank 360 (4 small eigenvalues omitted)
    Computing noise-normalization factors (dSPM)...
[done]
Applying inverse operator to "Left Auditory"...
    Picked 364 channels from the data
    Computing inverse...
    Eigenleads need to be weighted ...
    Computing residual...
    Explained  69.4% variance
    dSPM...
[done]
Converting forward solution to surface orientation
    Average patch normals will be employed in the rotation to the local surface coordinates....
    Converting to surface-based source orientations...
    [done]
info["bads"] and noise_cov["bads"] do not match, excluding bad channels from both
Computing inverse operator with 364 channels.
    364 out of 366 channels remain after picking
Selected 364 channels
Creating the depth weighting matrix...
Applying loose dipole orientations to surface source spaces: 0.9
Whitening the forward solution.
    Created an SSP operator (subspace dimension = 4)
Computing rank from covariance with rank=None
    Using tolerance 3.5e-13 (2.2e-16 eps * 305 dim * 5.2  max singular value)
    Estimated rank (mag + grad): 302
    MEG: rank 302 computed from 305 data channels with 3 projectors
    Using tolerance 1.1e-13 (2.2e-16 eps * 59 dim * 8.7  max singular value)
    Estimated rank (eeg): 58
    EEG: rank 58 computed from 59 data channels with 1 projector
    Setting small MEG eigenvalues to zero (without PCA)
    Setting small EEG eigenvalues to zero (without PCA)
Creating the source covariance matrix
Adjusting source covariance matrix.
Reducing source space to 543 sources
Whitening data matrix.
Warm starting...
alpha: 100.0
alpha: 93.33333333333333
alpha: 86.66666666666667
alpha: 80.0
alpha: 73.33333333333333
alpha: 66.66666666666666
alpha: 60.0
alpha: 53.33333333333333
alpha: 46.666666666666664
alpha: 40.0
Fitting SURE on grid.
alpha: 100.0
alpha: 93.33333333333333
alpha: 86.66666666666667
alpha: 80.0
alpha: 73.33333333333333
alpha: 66.66666666666666
alpha: 60.0
alpha: 53.33333333333333
alpha: 46.666666666666664
alpha: 40.0
Computing SURE values on grid.
alpha 100.0 :: sure -60279.60369182901
alpha 93.33333333333333 :: sure -59855.33381573209
alpha 86.66666666666667 :: sure -60434.87086631408
alpha 80.0 :: sure -60386.80654520041
alpha 73.33333333333333 :: sure -60336.58240546244
alpha 66.66666666666666 :: sure -60303.8315552053
alpha 60.0 :: sure -60157.013442516116
alpha 53.33333333333333 :: sure -59974.67525579427
alpha 46.666666666666664 :: sure -59597.8691366681
alpha 40.0 :: sure -59084.52688080946
Selected alpha: 86.66666666666667
Explained  22.4% variance
[done]
Dipole #1 GOF at 83.0 ms: 8.1%
Dipole #2 GOF at 83.0 ms: 33.3%

Plot dipole activations

plot_dipole_amplitudes(dipoles)

# Plot dipole location of the strongest dipole with MRI slices
idx = np.argmax([np.max(np.abs(dip.amplitude)) for dip in dipoles])
plot_dipole_locations(
    dipoles[idx],
    forward["mri_head_t"],
    "sample",
    subjects_dir=subjects_dir,
    mode="orthoview",
    idx="amplitude",
)

# Plot dipole locations of all dipoles with MRI slices
for dip in dipoles:
    plot_dipole_locations(
        dip,
        forward["mri_head_t"],
        "sample",
        subjects_dir=subjects_dir,
        mode="orthoview",
        idx="amplitude",
    )

• 
• 
• 
• 

Plot residual

ylim = dict(eeg=[-10, 10], grad=[-400, 400], mag=[-600, 600])
evoked.pick(picks=["meg", "eeg"], exclude="bads")
evoked.plot(ylim=ylim, proj=True, time_unit="s")
residual.pick(picks=["meg", "eeg"], exclude="bads")
residual.plot(ylim=ylim, proj=True, time_unit="s")

• 
• 

Generate stc from dipoles

stc = make_stc_from_dipoles(dipoles, forward["src"])

Converting dipoles into a SourceEstimate.
[done]

View in 2D and 3D (“glass” brain like 3D plot)

solver = "MxNE" if n_mxne_iter == 1 else "irMxNE"
plot_sparse_source_estimates(
    forward["src"],
    stc,
    bgcolor=(1, 1, 1),
    fig_name=f"{solver} (cond {condition})",
    opacity=0.1,
)

True
Total number of active sources: [ 70819 235846]

Morph onto fsaverage brain and view

morph = mne.compute_source_morph(
    stc,
    subject_from="sample",
    subject_to="fsaverage",
    spacing=None,
    sparse=True,
    subjects_dir=subjects_dir,
)
stc_fsaverage = morph.apply(stc)
src_fsaverage_fname = subjects_dir / "fsaverage" / "bem" / "fsaverage-ico-5-src.fif"
src_fsaverage = mne.read_source_spaces(src_fsaverage_fname)

plot_sparse_source_estimates(
    src_fsaverage,
    stc_fsaverage,
    bgcolor=(1, 1, 1),
    fig_name=f"Morphed {solver} (cond {condition})",
    opacity=0.1,
)

surface source space present ...
    Left-hemisphere map read.
    Right-hemisphere map read.
[done]
    Reading a source space...
    [done]
    Reading a source space...
    [done]
    2 source spaces read
True
Total number of active sources: [ 47678 300147]

References

[1]

Alexandre Gramfort, Matthieu Kowalski, and Matti S. Hämäläinen. Mixed-norm estimates for the M/EEG inverse problem using accelerated gradient methods. Physics in Medicine and Biology, 57(7):1937–1961, 2012. doi:10.1088/0031-9155/57/7/1937.

[2]

Daniel Strohmeier, Jens Haueisen, and Alexandre Gramfort. Improved MEG/EEG source localization with reweighted mixed-norms. In Proceedings of PRNI-2014, 1–4. Tübingen, 2014. IEEE. doi:10.1109/PRNI.2014.6858545.

[3]

Charles-Alban Deledalle, Samuel Vaiter, Jalal Fadili, and Gabriel Peyré. Stein unbiased gradient estimator of the risk (sugar) for multiple parameter selection. SIAM Journal on Imaging Sciences, 7(4):2448–2487, 2014. doi:10.1137/140968045.