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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Overview of MEG/EEG analysis with MNE-Python
This tutorial covers the basic EEG/MEG pipeline for event-related analysis
Step1:  Loading data
^^^^^^^^^^^^
MNE-Python data structures are based around the FIF file format from
Neuromag, but there are reader functions for a wide variety of other
data formats &lt;data-formats&gt;. MNE-Python also has interfaces to a
variety of publicly available datasets &lt;datasets&gt;,
which MNE-Python can download and manage for you.
We'll start this tutorial by loading one of the example datasets (called
"sample-dataset"), which contains EEG and MEG data from one subject
performing an audiovisual experiment, along with structural MRI scans for
that subject. The 
Step2:  By default, 
Step3:  
Step4:  Preprocessing
^^^^^^^^^^^^^
MNE-Python supports a variety of preprocessing approaches and techniques
(maxwell filtering, signal-space projection, independent components analysis,
filtering, downsampling, etc); see the full list of capabilities in the
Step5:  Once we're confident about which component(s) we want to remove, we pass them
as the exclude parameter and then apply the ICA to the raw signal. The
Step6:  Detecting experimental events
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The sample dataset includes several 
Step7:  The resulting events array is an ordinary 3-column 
Step8:  Event dictionaries like this one are used when extracting epochs from
continuous data; the / character in the dictionary keys allows pooling
across conditions by requesting partial condition descriptors (i.e.,
requesting 'auditory' will select all epochs with Event IDs 1 and 2;
requesting 'left' will select all epochs with Event IDs 1 and 3). An
example of this is shown in the next section. There is also a convenient
Step9:  For paradigms that are not event-related (e.g., analysis of resting-state
data), you can extract regularly spaced (possibly overlapping) spans of data
by creating events using 
Step10:  We'll also pass the event dictionary as the event_id parameter (so we can
work with easy-to-pool event labels instead of the integer event IDs), and
specify tmin and tmax (the time relative to each event at which to
start and end each epoch). As mentioned above, by default
Step11:  Next we'll pool across left/right stimulus presentations so we can compare
auditory versus visual responses. To avoid biasing our signals to the
left or right, we'll use 
Step12:  Like 
Step13:  <div class="alert alert-info"><h4>Note</h4><p>Both 
Step14:  Estimating evoked responses
^^^^^^^^^^^^^^^^^^^^^^^^^^^
Now that we have our conditions in aud_epochs and vis_epochs, we can
get an estimate of evoked responses to auditory versus visual stimuli by
averaging together the epochs in each condition. This is as simple as calling
the 
Step15:  We can also get a more detailed view of each 
Step16:  Evoked objects can also be combined to show contrasts between conditions,
using the 
Step17:  Inverse modeling
^^^^^^^^^^^^^^^^
Finally, we can estimate the origins of the evoked activity by projecting the
sensor data into this subject's 
Step18:  Finally, in order to plot the source estimate on the subject's cortical
surface we'll also need the path to the sample subject's structural MRI files
(the subjects_dir)
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Python Code:
import os
import numpy as np
import mne
Explanation: Overview of MEG/EEG analysis with MNE-Python
This tutorial covers the basic EEG/MEG pipeline for event-related analysis:
loading data, epoching, averaging, plotting, and estimating cortical activity
from sensor data. It introduces the core MNE-Python data structures
:class:~mne.io.Raw, :class:~mne.Epochs, :class:~mne.Evoked, and
:class:~mne.SourceEstimate, and covers a lot of ground fairly quickly (at the
expense of depth). Subsequent tutorials address each of these topics in greater
detail.
   :depth: 1
We begin by importing the necessary Python modules:
End of explanation
sample_data_folder = mne.datasets.sample.data_path()
sample_data_raw_file = os.path.join(sample_data_folder, 'MEG', 'sample',
                                    'sample_audvis_filt-0-40_raw.fif')
raw = mne.io.read_raw_fif(sample_data_raw_file)
Explanation: Loading data
^^^^^^^^^^^^
MNE-Python data structures are based around the FIF file format from
Neuromag, but there are reader functions for a wide variety of other
data formats &lt;data-formats&gt;. MNE-Python also has interfaces to a
variety of publicly available datasets &lt;datasets&gt;,
which MNE-Python can download and manage for you.
We'll start this tutorial by loading one of the example datasets (called
"sample-dataset"), which contains EEG and MEG data from one subject
performing an audiovisual experiment, along with structural MRI scans for
that subject. The :func:mne.datasets.sample.data_path function will
automatically download the dataset if it isn't found in one of the expected
locations, then return the directory path to the dataset (see the
documentation of :func:~mne.datasets.sample.data_path for a list of places
it checks before downloading). Note also that for this tutorial to run
smoothly on our servers, we're using a filtered and downsampled version of
the data (:file:sample_audvis_filt-0-40_raw.fif), but an unfiltered version
(:file:sample_audvis_raw.fif) is also included in the sample dataset and
could be substituted here when running the tutorial locally.
End of explanation
print(raw)
print(raw.info)
Explanation: By default, :func:~mne.io.read_raw_fif displays some information about the
file it's loading; for example, here it tells us that there are four
"projection items" in the file along with the recorded data; those are
:term:SSP projectors &lt;projector&gt; calculated to remove environmental noise
from the MEG signals, plus a projector to mean-reference the EEG channels;
these are discussed in the tutorial tut-projectors-background.
In addition to the information displayed during loading,
you can get a glimpse of the basic details of a :class:~mne.io.Raw object
by printing it; even more is available by printing its info attribute
(a :class:dictionary-like object &lt;mne.Info&gt; that is preserved across
:class:~mne.io.Raw, :class:~mne.Epochs, and :class:~mne.Evoked
objects). The info data structure keeps track of channel locations,
applied filters, projectors, etc. Notice especially the chs entry,
showing that MNE-Python detects different sensor types and handles each
appropriately. See tut-info-class for more on the :class:~mne.Info
class.
End of explanation
raw.plot_psd(fmax=50)
raw.plot(duration=5, n_channels=30)
Explanation: :class:~mne.io.Raw objects also have several built-in plotting methods;
here we show the power spectral density (PSD) for each sensor type with
:meth:~mne.io.Raw.plot_psd, as well as a plot of the raw sensor traces with
:meth:~mne.io.Raw.plot. In the PSD plot, we'll only plot frequencies below
50 Hz (since our data are low-pass filtered at 40 Hz). In interactive Python
sessions, :meth:~mne.io.Raw.plot is interactive and allows scrolling,
scaling, bad channel marking, annotation, projector toggling, etc.
End of explanation
# set up and fit the ICA
ica = mne.preprocessing.ICA(n_components=20, random_state=97, max_iter=800)
ica.fit(raw)
ica.exclude = [1, 2]  # details on how we picked these are omitted here
ica.plot_properties(raw, picks=ica.exclude)
Explanation: Preprocessing
^^^^^^^^^^^^^
MNE-Python supports a variety of preprocessing approaches and techniques
(maxwell filtering, signal-space projection, independent components analysis,
filtering, downsampling, etc); see the full list of capabilities in the
:mod:mne.preprocessing and :mod:mne.filter submodules. Here we'll clean
up our data by performing independent components analysis
(:class:~mne.preprocessing.ICA); for brevity we'll skip the steps that
helped us determined which components best capture the artifacts (see
tut-artifact-ica for a detailed walk-through of that process).
End of explanation
orig_raw = raw.copy()
raw.load_data()
ica.apply(raw)
# show some frontal channels to clearly illustrate the artifact removal
chs = ['MEG 0111', 'MEG 0121', 'MEG 0131', 'MEG 0211', 'MEG 0221', 'MEG 0231',
       'MEG 0311', 'MEG 0321', 'MEG 0331', 'MEG 1511', 'MEG 1521', 'MEG 1531',
       'EEG 001', 'EEG 002', 'EEG 003', 'EEG 004', 'EEG 005', 'EEG 006',
       'EEG 007', 'EEG 008']
chan_idxs = [raw.ch_names.index(ch) for ch in chs]
orig_raw.plot(order=chan_idxs, start=12, duration=4)
raw.plot(order=chan_idxs, start=12, duration=4)
Explanation: Once we're confident about which component(s) we want to remove, we pass them
as the exclude parameter and then apply the ICA to the raw signal. The
:meth:~mne.preprocessing.ICA.apply method requires the raw data to be
loaded into memory (by default it's only read from disk as-needed), so we'll
use :meth:~mne.io.Raw.load_data first. We'll also make a copy of the
:class:~mne.io.Raw object so we can compare the signal before and after
artifact removal side-by-side:
End of explanation
events = mne.find_events(raw, stim_channel='STI 014')
print(events[:5])  # show the first 5
Explanation: Detecting experimental events
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The sample dataset includes several :term:"STIM" channels &lt;stim channel&gt;
that recorded electrical
signals sent from the stimulus delivery computer (as brief DC shifts /
squarewave pulses). These pulses (often called "triggers") are used in this
dataset to mark experimental events: stimulus onset, stimulus type, and
participant response (button press). The individual STIM channels are
combined onto a single channel, in such a way that voltage
levels on that channel can be unambiguously decoded as a particular event
type. On older Neuromag systems (such as that used to record the sample data)
this summation channel was called STI 014, so we can pass that channel
name to the :func:mne.find_events function to recover the timing and
identity of the stimulus events.
End of explanation
event_dict = {'auditory/left': 1, 'auditory/right': 2, 'visual/left': 3,
              'visual/right': 4, 'smiley': 5, 'buttonpress': 32}
Explanation: The resulting events array is an ordinary 3-column :class:NumPy array
&lt;numpy.ndarray&gt;, with sample number in the first column and integer event ID
in the last column; the middle column is usually ignored. Rather than keeping
track of integer event IDs, we can provide an event dictionary that maps
the integer IDs to experimental conditions or events. In this dataset, the
mapping looks like this:
+----------+----------------------------------------------------------+
| Event ID | Condition                                                |
+==========+==========================================================+
| 1        | auditory stimulus (tone) to the left ear                 |
+----------+----------------------------------------------------------+
| 2        | auditory stimulus (tone) to the right ear                |
+----------+----------------------------------------------------------+
| 3        | visual stimulus (checkerboard) to the left visual field  |
+----------+----------------------------------------------------------+
| 4        | visual stimulus (checkerboard) to the right visual field |
+----------+----------------------------------------------------------+
| 5        | smiley face (catch trial)                                |
+----------+----------------------------------------------------------+
| 32       | subject button press                                     |
+----------+----------------------------------------------------------+
End of explanation
fig = mne.viz.plot_events(events, event_id=event_dict, sfreq=raw.info['sfreq'])
fig.subplots_adjust(right=0.7)  # make room for the legend
Explanation: Event dictionaries like this one are used when extracting epochs from
continuous data; the / character in the dictionary keys allows pooling
across conditions by requesting partial condition descriptors (i.e.,
requesting 'auditory' will select all epochs with Event IDs 1 and 2;
requesting 'left' will select all epochs with Event IDs 1 and 3). An
example of this is shown in the next section. There is also a convenient
:func:~mne.viz.plot_events function for visualizing the distribution of
events across the duration of the recording (to make sure event detection
worked as expected). Here we'll also make use of the :class:~mne.Info
attribute to get the sampling frequency of the recording (so our x-axis will
be in seconds instead of in samples).
End of explanation
reject_criteria = dict(mag=4000e-15,     # 4000 fT
                       grad=4000e-13,    # 4000 fT/cm
                       eeg=150e-6,       # 150 μV
                       eog=250e-6)       # 250 μV
Explanation: For paradigms that are not event-related (e.g., analysis of resting-state
data), you can extract regularly spaced (possibly overlapping) spans of data
by creating events using :func:mne.make_fixed_length_events and then
proceeding with epoching as described in the next section.
Epoching continuous data
^^^^^^^^^^^^^^^^^^^^^^^^
The :class:~mne.io.Raw object and the events array are the bare minimum
needed to create an :class:~mne.Epochs object, which we create with the
:class:mne.Epochs class constructor. Here we'll also specify some data
quality constraints: we'll reject any epoch where peak-to-peak signal
amplitude is beyond reasonable limits for that channel type. This is done
with a rejection dictionary; you may include or omit thresholds for any of
the channel types present in your data. The values given here are reasonable
for this particular dataset, but may need to be adapted for different
hardware or recording conditions. For a more automated approach, consider
using the autoreject package_.
End of explanation
epochs = mne.Epochs(raw, events, event_id=event_dict, tmin=-0.2, tmax=0.5,
                    reject=reject_criteria, preload=True)
Explanation: We'll also pass the event dictionary as the event_id parameter (so we can
work with easy-to-pool event labels instead of the integer event IDs), and
specify tmin and tmax (the time relative to each event at which to
start and end each epoch). As mentioned above, by default
:class:~mne.io.Raw and :class:~mne.Epochs data aren't loaded into memory
(they're accessed from disk only when needed), but here we'll force loading
into memory using the preload=True parameter so that we can see the
results of the rejection criteria being applied:
End of explanation
conds_we_care_about = ['auditory/left', 'auditory/right',
                       'visual/left', 'visual/right']
epochs.equalize_event_counts(conds_we_care_about)  # this operates in-place
aud_epochs = epochs['auditory']
vis_epochs = epochs['visual']
del raw, epochs  # free up memory
Explanation: Next we'll pool across left/right stimulus presentations so we can compare
auditory versus visual responses. To avoid biasing our signals to the
left or right, we'll use :meth:~mne.Epochs.equalize_event_counts first to
randomly sample epochs from each condition to match the number of epochs
present in the condition with the fewest good epochs.
End of explanation
aud_epochs.plot_image(picks=['MEG 1332', 'EEG 021'])
Explanation: Like :class:~mne.io.Raw objects, :class:~mne.Epochs objects also have a
number of built-in plotting methods. One is :meth:~mne.Epochs.plot_image,
which shows each epoch as one row of an image map, with color representing
signal magnitude; the average evoked response and the sensor location are
shown below the image:
End of explanation
frequencies = np.arange(7, 30, 3)
power = mne.time_frequency.tfr_morlet(aud_epochs, n_cycles=2, return_itc=False,
                                      freqs=frequencies, decim=3)
power.plot(['MEG 1332'])
Explanation: <div class="alert alert-info"><h4>Note</h4><p>Both :class:`~mne.io.Raw` and :class:`~mne.Epochs` objects have
    :meth:`~mne.Epochs.get_data` methods that return the underlying data
    as a :class:`NumPy array <numpy.ndarray>`. Both methods have a ``picks``
    parameter for subselecting which channel(s) to return; ``raw.get_data()``
    has additional parameters for restricting the time domain. The resulting
    matrices have dimension ``(n_channels, n_times)`` for
    :class:`~mne.io.Raw` and ``(n_epochs, n_channels, n_times)`` for
    :class:`~mne.Epochs`.</p></div>
Time-frequency analysis
^^^^^^^^^^^^^^^^^^^^^^^
The :mod:mne.time_frequency submodule provides implementations of several
algorithms to compute time-frequency representations, power spectral density,
and cross-spectral density. Here, for example, we'll compute for the auditory
epochs the induced power at different frequencies and times, using Morlet
wavelets. On this dataset the result is not especially informative (it just
shows the evoked "auditory N100" response); see here
&lt;inter-trial-coherence&gt; for a more extended example on a dataset with richer
frequency content.
End of explanation
aud_evoked = aud_epochs.average()
vis_evoked = vis_epochs.average()
mne.viz.plot_compare_evokeds(dict(auditory=aud_evoked, visual=vis_evoked),
                             legend='upper left', show_sensors='upper right')
Explanation: Estimating evoked responses
^^^^^^^^^^^^^^^^^^^^^^^^^^^
Now that we have our conditions in aud_epochs and vis_epochs, we can
get an estimate of evoked responses to auditory versus visual stimuli by
averaging together the epochs in each condition. This is as simple as calling
the :meth:~mne.Epochs.average method on the :class:~mne.Epochs object,
and then using a function from the :mod:mne.viz module to compare the
global field power for each sensor type of the two :class:~mne.Evoked
objects:
End of explanation
aud_evoked.plot_joint(picks='eeg')
aud_evoked.plot_topomap(times=[0., 0.08, 0.1, 0.12, 0.2], ch_type='eeg')
Explanation: We can also get a more detailed view of each :class:~mne.Evoked object
using other plotting methods such as :meth:~mne.Evoked.plot_joint or
:meth:~mne.Evoked.plot_topomap. Here we'll examine just the EEG channels,
and see the classic auditory evoked N100-P200 pattern over dorso-frontal
electrodes, then plot scalp topographies at some additional arbitrary times:
End of explanation
evoked_diff = mne.combine_evoked([aud_evoked, -vis_evoked], weights='equal')
evoked_diff.pick_types('mag').plot_topo(color='r', legend=False)
Explanation: Evoked objects can also be combined to show contrasts between conditions,
using the :func:mne.combine_evoked function. A simple difference can be
generated by negating one of the :class:~mne.Evoked objects passed into the
function. We'll then plot the difference wave at each sensor using
:meth:~mne.Evoked.plot_topo:
End of explanation
# load inverse operator
inverse_operator_file = os.path.join(sample_data_folder, 'MEG', 'sample',
                                     'sample_audvis-meg-oct-6-meg-inv.fif')
inv_operator = mne.minimum_norm.read_inverse_operator(inverse_operator_file)
# set signal-to-noise ratio (SNR) to compute regularization parameter (λ²)
snr = 3.
lambda2 = 1. / snr ** 2
# generate the source time course (STC)
stc = mne.minimum_norm.apply_inverse(vis_evoked, inv_operator,
                                     lambda2=lambda2,
                                     method='MNE')  # or dSPM, sLORETA, eLORETA
Explanation: Inverse modeling
^^^^^^^^^^^^^^^^
Finally, we can estimate the origins of the evoked activity by projecting the
sensor data into this subject's :term:source space (a set of points either
on the cortical surface or within the cortical volume of that subject, as
estimated by structural MRI scans). MNE-Python supports lots of ways of doing
this (dynamic statistical parametric mapping, dipole fitting, beamformers,
etc.); here we'll use minimum-norm estimation (MNE) to generate a continuous
map of activation constrained to the cortical surface. MNE uses a linear
:term:inverse operator to project EEG+MEG sensor measurements into the
source space. The inverse operator is computed from the
:term:forward solution for this subject and an estimate of the
covariance of sensor measurements &lt;tut_compute_covariance&gt;. For this
tutorial we'll skip those computational steps and load a pre-computed inverse
operator from disk (it's included with the sample data
&lt;sample-dataset&gt;). Because this "inverse problem" is underdetermined (there
is no unique solution), here we further constrain the solution by providing a
regularization parameter specifying the relative smoothness of the current
estimates in terms of a signal-to-noise ratio (where "noise" here is akin to
baseline activity level across all of cortex).
End of explanation
# path to subjects' MRI files
subjects_dir = os.path.join(sample_data_folder, 'subjects')
# plot
stc.plot(initial_time=0.1, hemi='split', views=['lat', 'med'],
         subjects_dir=subjects_dir)
Explanation: Finally, in order to plot the source estimate on the subject's cortical
surface we'll also need the path to the sample subject's structural MRI files
(the subjects_dir):
End of explanation
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