DICS for power mapping

In this tutorial, we’ll simulate two signals originating from two locations on the cortex. These signals will be sinusoids, so we’ll be looking at oscillatory activity (as opposed to evoked activity).

We’ll use dynamic imaging of coherent sources (DICS) 1 to map out spectral power along the cortex. Let’s see if we can find our two simulated sources.

# Author: Marijn van Vliet <w.m.vanvliet@gmail.com>
#
# License: BSD (3-clause)

Setup

We first import the required packages to run this tutorial and define a list of filenames for various things we’ll be using.

import os.path as op
import numpy as np
from scipy.signal import welch, coherence, unit_impulse
from matplotlib import pyplot as plt

import mne
from mne.simulation import simulate_raw, add_noise
from mne.datasets import sample
from mne.minimum_norm import make_inverse_operator, apply_inverse
from mne.time_frequency import csd_morlet
from mne.beamformer import make_dics, apply_dics_csd

# We use the MEG and MRI setup from the MNE-sample dataset
data_path = sample.data_path(download=False)
subjects_dir = op.join(data_path, 'subjects')

# Filenames for various files we'll be using
meg_path = op.join(data_path, 'MEG', 'sample')
raw_fname = op.join(meg_path, 'sample_audvis_raw.fif')
fwd_fname = op.join(meg_path, 'sample_audvis-meg-eeg-oct-6-fwd.fif')
cov_fname = op.join(meg_path, 'sample_audvis-cov.fif')
fwd = mne.read_forward_solution(fwd_fname)

# Seed for the random number generator
rand = np.random.RandomState(42)

Out:

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) 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

Data simulation

The following function generates a timeseries that contains an oscillator, whose frequency fluctuates a little over time, but stays close to 10 Hz. We’ll use this function to generate our two signals.

sfreq = 50.  # Sampling frequency of the generated signal
n_samp = int(round(10. * sfreq))
times = np.arange(n_samp) / sfreq  # 10 seconds of signal
n_times = len(times)


def coh_signal_gen():
    """Generate an oscillating signal.

    Returns
    -------
    signal : ndarray
        The generated signal.
    """
    t_rand = 0.001  # Variation in the instantaneous frequency of the signal
    std = 0.1  # Std-dev of the random fluctuations added to the signal
    base_freq = 10.  # Base frequency of the oscillators in Hertz
    n_times = len(times)

    # Generate an oscillator with varying frequency and phase lag.
    signal = np.sin(2.0 * np.pi *
                    (base_freq * np.arange(n_times) / sfreq +
                     np.cumsum(t_rand * rand.randn(n_times))))

    # Add some random fluctuations to the signal.
    signal += std * rand.randn(n_times)

    # Scale the signal to be in the right order of magnitude (~100 nAm)
    # for MEG data.
    signal *= 100e-9

    return signal

Let’s simulate two timeseries and plot some basic information about them.

signal1 = coh_signal_gen()
signal2 = coh_signal_gen()

fig, axes = plt.subplots(2, 2, figsize=(8, 4))

# Plot the timeseries
ax = axes[0][0]
ax.plot(times, 1e9 * signal1, lw=0.5)
ax.set(xlabel='Time (s)', xlim=times[[0, -1]], ylabel='Amplitude (Am)',
       title='Signal 1')
ax = axes[0][1]
ax.plot(times, 1e9 * signal2, lw=0.5)
ax.set(xlabel='Time (s)', xlim=times[[0, -1]], title='Signal 2')

# Power spectrum of the first timeseries
f, p = welch(signal1, fs=sfreq, nperseg=128, nfft=256)
ax = axes[1][0]
# Only plot the first 100 frequencies
ax.plot(f[:100], 20 * np.log10(p[:100]), lw=1.)
ax.set(xlabel='Frequency (Hz)', xlim=f[[0, 99]],
       ylabel='Power (dB)', title='Power spectrum of signal 1')

# Compute the coherence between the two timeseries
f, coh = coherence(signal1, signal2, fs=sfreq, nperseg=100, noverlap=64)
ax = axes[1][1]
ax.plot(f[:50], coh[:50], lw=1.)
ax.set(xlabel='Frequency (Hz)', xlim=f[[0, 49]], ylabel='Coherence',
       title='Coherence between the timeseries')
fig.tight_layout()
../../_images/sphx_glr_plot_dics_001.png

Now we put the signals at two locations on the cortex. We construct a mne.SourceEstimate object to store them in.

The timeseries will have a part where the signal is active and a part where it is not. The techniques we’ll be using in this tutorial depend on being able to contrast data that contains the signal of interest versus data that does not (i.e. it contains only noise).

# The locations on the cortex where the signal will originate from. These
# locations are indicated as vertex numbers.
vertices = [[146374], [33830]]

# Construct SourceEstimates that describe the signals at the cortical level.
data = np.vstack((signal1, signal2))
stc_signal = mne.SourceEstimate(
    data, vertices, tmin=0, tstep=1. / sfreq, subject='sample')
stc_noise = stc_signal * 0.

Before we simulate the sensor-level data, let’s define a signal-to-noise ratio. You are encouraged to play with this parameter and see the effect of noise on our results.

snr = 1.  # Signal-to-noise ratio. Decrease to add more noise.

Now we run the signal through the forward model to obtain simulated sensor data. To save computation time, we’ll only simulate gradiometer data. You can try simulating other types of sensors as well.

Some noise is added based on the baseline noise covariance matrix from the sample dataset, scaled to implement the desired SNR.

# Read the info from the sample dataset. This defines the location of the
# sensors and such.
info = mne.io.read_info(raw_fname)
info.update(sfreq=sfreq, bads=[])

# Only use gradiometers
picks = mne.pick_types(info, meg='grad', stim=True, exclude=())
mne.pick_info(info, picks, copy=False)

# Define a covariance matrix for the simulated noise. In this tutorial, we use
# a simple diagonal matrix.
cov = mne.cov.make_ad_hoc_cov(info)
cov['data'] *= (20. / snr) ** 2  # Scale the noise to achieve the desired SNR

# Simulate the raw data, with a lowpass filter on the noise
stcs = [(stc_signal, unit_impulse(n_samp, dtype=int) * 1),
        (stc_noise, unit_impulse(n_samp, dtype=int) * 2)]  # stacked in time
duration = (len(stc_signal.times) * 2) / sfreq
raw = simulate_raw(info, stcs, forward=fwd)
add_noise(raw, cov, iir_filter=[4, -4, 0.8], random_state=rand)

Out:

    Read a total of 3 projection items:
        PCA-v1 (1 x 102)  idle
        PCA-v2 (1 x 102)  idle
        PCA-v3 (1 x 102)  idle
Setting up raw simulation: 1 position, "cos2" interpolation
Event information stored on channel:              STI 014
Setting up forward solutions
Computing gain matrix for transform #1/1
   Simulating data for forward operator 1/0
   Interval 0.000-10.000 sec
   Interval 10.000-20.000 sec
    2 STC iterations provided
Done
Adding noise to 204/213 channels (204 channels in cov)

We create an mne.Epochs object containing two trials: one with both noise and signal and one with just noise

events = mne.find_events(raw, initial_event=True)
tmax = (len(stc_signal.times) - 1) / sfreq
epochs = mne.Epochs(raw, events, event_id=dict(signal=1, noise=2),
                    tmin=0, tmax=tmax, baseline=None, preload=True)
assert len(epochs) == 2  # ensure that we got the two expected events

# Plot some of the channels of the simulated data that are situated above one
# of our simulated sources.
picks = mne.pick_channels(epochs.ch_names, mne.read_selection('Left-frontal'))
epochs.plot(picks=picks)
../../_images/sphx_glr_plot_dics_002.png

Out:

2 events found
Event IDs: [1 2]
2 matching events found
No baseline correction applied
Not setting metadata
3 projection items activated
Loading data for 2 events and 500 original time points ...
0 bad epochs dropped

Power mapping

With our simulated dataset ready, we can now pretend to be researchers that have just recorded this from a real subject and are going to study what parts of the brain communicate with each other.

First, we’ll create a source estimate of the MEG data. We’ll use both a straightforward MNE-dSPM inverse solution for this, and the DICS beamformer which is specifically designed to work with oscillatory data.

Computing the inverse using MNE-dSPM:

# Compute the inverse operator
fwd = mne.read_forward_solution(fwd_fname)
inv = make_inverse_operator(epochs.info, fwd, cov)

# Apply the inverse model to the trial that also contains the signal.
s = apply_inverse(epochs['signal'].average(), inv)

# Take the root-mean square along the time dimension and plot the result.
s_rms = np.sqrt((s ** 2).mean())
title = 'MNE-dSPM inverse (RMS)'
brain = s_rms.plot('sample', subjects_dir=subjects_dir, hemi='both', figure=1,
                   size=600, time_label=title, title=title)

# Indicate the true locations of the source activity on the plot.
brain.add_foci(vertices[0][0], coords_as_verts=True, hemi='lh')
brain.add_foci(vertices[1][0], coords_as_verts=True, hemi='rh')

# Rotate the view and add a title.
brain.show_view(view={'azimuth': 0, 'elevation': 0, 'distance': 550,
                'focalpoint': [0, 0, 0]})
../../_images/sphx_glr_plot_dics_003.png

Out:

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) 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
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]
Computing inverse operator with 204 channels.
    204 out of 366 channels remain after picking
Selected 204 channels
Creating the depth weighting matrix...
    204 planar channels
    limit = 7261/7498 = 10.004929
    scale = 2.59947e-08 exp = 0.8
Applying loose dipole orientations. Loose value of 0.2.
Whitening the forward solution.
Computing data rank from covariance with rank=None
    Using tolerance 4.5e-14 (2.2e-16 eps * 204 dim * 1  max singular value)
    Estimated rank (grad): 204
    GRAD: rank 204 computed from 204 data channels with 0 projectors
    Setting small GRAD 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 = 5.60587
    scaling factor to adjust the trace = 2.91651e+18
Preparing the inverse operator for use...
    Scaled noise and source covariance from nave = 1 to nave = 1
    Created the regularized inverter
    The projection vectors do not apply to these channels.
    Created the whitener using a noise covariance matrix with rank 204 (0 small eigenvalues omitted)
    Computing noise-normalization factors (dSPM)...
[done]
Applying inverse operator to "signal"...
    Picked 204 channels from the data
    Computing inverse...
    Eigenleads need to be weighted ...
    Computing residual...
    Explained  75.0% variance
    Combining the current components...
    dSPM...
[done]
Using control points [1.41101364 1.65792783 3.61406449]

We will now compute the cortical power map at 10 Hz. using a DICS beamformer. A beamformer will construct for each vertex a spatial filter that aims to pass activity originating from the vertex, while dampening activity from other sources as much as possible.

The mne.beamformer.make_dics() function has many switches that offer precise control over the way the filter weights are computed. Currently, there is no clear consensus regarding the best approach. This is why we will demonstrate two approaches here:

  1. The approach as described in 2, which first normalizes the forward solution and computes a vector beamformer.

  2. The scalar beamforming approach based on 3, which uses weight normalization instead of normalizing the forward solution.

# Estimate the cross-spectral density (CSD) matrix on the trial containing the
# signal.
csd_signal = csd_morlet(epochs['signal'], frequencies=[10])

# Compute the spatial filters for each vertex, using two approaches.
filters_approach1 = make_dics(
    info, fwd, csd_signal, reg=0.05, pick_ori='max-power', normalize_fwd=True,
    inversion='single', weight_norm=None)
print(filters_approach1)

filters_approach2 = make_dics(
    info, fwd, csd_signal, reg=0.1, pick_ori='max-power', normalize_fwd=False,
    inversion='matrix', weight_norm='unit-noise-gain')
print(filters_approach2)

# You can save these to disk with:
# filters_approach1.save('filters_1-dics.h5')

# Compute the DICS power map by applying the spatial filters to the CSD matrix.
power_approach1, f = apply_dics_csd(csd_signal, filters_approach1)
power_approach2, f = apply_dics_csd(csd_signal, filters_approach2)

# Plot the DICS power maps for both approaches.
for approach, power in enumerate([power_approach1, power_approach2], 1):
    title = 'DICS power map, approach %d' % approach
    brain = power.plot('sample', subjects_dir=subjects_dir, hemi='both',
                       figure=approach + 1, size=600, time_label=title,
                       title=title)

    # Indicate the true locations of the source activity on the plot.
    brain.add_foci(vertices[0][0], coords_as_verts=True, hemi='lh')
    brain.add_foci(vertices[1][0], coords_as_verts=True, hemi='rh')

    # Rotate the view and add a title.
    brain.show_view(view={'azimuth': 0, 'elevation': 0, 'distance': 550,
                    'focalpoint': [0, 0, 0]})
  • ../../_images/sphx_glr_plot_dics_004.png
  • ../../_images/sphx_glr_plot_dics_005.png

Out:

Computing cross-spectral density from epochs...
    Computing CSD matrix for epoch 1
[done]
Computing inverse operator with 204 channels.
    204 out of 366 channels remain after picking
Selected 204 channels
Creating the depth weighting matrix...
Whitening the forward solution.
Computing data rank from covariance with rank=None
    Using tolerance 4.5e+08 (2.2e-16 eps * 204 dim * 1e+22  max singular value)
    Estimated rank (grad): 204
    GRAD: rank 204 computed from 204 data channels with 0 projectors
    Setting small GRAD eigenvalues to zero (without PCA)
Creating the source covariance matrix
Adjusting source covariance matrix.
Computing DICS spatial filters...
Computing beamformer filters for 7498 sources
Filter computation complete
<Beamformer  |  DICS, subject "sample", 7498 vert, 204 ch, max-power ori, single inversion>
Computing inverse operator with 204 channels.
    204 out of 366 channels remain after picking
Selected 204 channels
Whitening the forward solution.
Computing data rank from covariance with rank=None
    Using tolerance 4.5e+08 (2.2e-16 eps * 204 dim * 1e+22  max singular value)
    Estimated rank (grad): 204
    GRAD: rank 204 computed from 204 data channels with 0 projectors
    Setting small GRAD eigenvalues to zero (without PCA)
Creating the source covariance matrix
Adjusting source covariance matrix.
Computing DICS spatial filters...
Computing beamformer filters for 7498 sources
Filter computation complete
<Beamformer  |  DICS, subject "sample", 7498 vert, 204 ch, max-power ori, unit-noise-gain norm, matrix inversion>
Computing DICS source power...
[done]
Computing DICS source power...
[done]
Using control points [6.08968634e-25 7.00369866e-25 1.52085242e-24]
Using control points [6.79228505e-25 8.35522906e-25 1.87088038e-24]

Excellent! All methods found our two simulated sources. Of course, with a signal-to-noise ratio (SNR) of 1, is isn’t very hard to find them. You can try playing with the SNR and see how the MNE-dSPM and DICS approaches hold up in the presence of increasing noise. In the presence of more noise, you may need to increase the regularization parameter of the DICS beamformer.

References

1

Gross et al. (2001). Dynamic imaging of coherent sources: Studying neural interactions in the human brain. Proceedings of the National Academy of Sciences, 98(2), 694-699. https://doi.org/10.1073/pnas.98.2.694

2

van Vliet, et al. (2018) Analysis of functional connectivity and oscillatory power using DICS: from raw MEG data to group-level statistics in Python. bioRxiv, 245530. https://doi.org/10.1101/245530

3

Sekihara & Nagarajan. Adaptive spatial filters for electromagnetic brain imaging (2008) Springer Science & Business Media

Total running time of the script: ( 0 minutes 38.556 seconds)

Estimated memory usage: 184 MB

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