.backtick {font-size: 16px;})>><abbr {font-weight: bold;})>> <em strong {font-weight: normal; font-style: normal; padding: 2px; border-radius: 5px; background-color: #EEE; color: #111;})>>
= Connectivity =
''Authors: Hossein Shahabi, Raymundo Cassani, Takfarinas Medani, François Tadel, [[https://www.neurospeed-bailletlab.org/sylvain-baillet|Sylvain Baillet]]''
Brain functions (e.g., in cognition, behavior and perception) stem from the coordinated activity of multiple regions. [[http://www.scholarpedia.org/article/Brain_connectivity|Brain connectivity]] investigates how these different regions (or nodes) interact as a network. Depending on which connectivity characteristic is studied, a distinction is made between '''structural''' (fiber pathways), '''functional''' (non-directed statistical associations) and '''effective''' (causal interactions) connectivity between regions. Effective connectivity is often referred as directed functional connectivity. In this tutorial we will see how to compute different connectivity metrics for non-directed and directed functional analyses using Brainstorm, first with simulated data and later with real data.
We encourage the interested reader to [[https://pubmed.ncbi.nlm.nih.gov/34906715/|learn more]] about the specific aspects of electrophysiology for studying human ''connectomics''.
<>
== Introduction ==
=== Definitions ===
Connectivity analyses are commonly performed by computing a bivariate measure between pairs of regional time series of interest. The outcome (a.k.a ''connectome'') can be presented as a '''connectivity graph''' (left image), where each region is represented as a node (x, y, z,...), and the values of the connectivity metric shown next to the edge linking between two nodes. The ''connectome ''can also be represented by a '''connectivity matrix''', a.k.a. adjacency matrix (right image).
{{attachment:cnx_graph_matrix.png}}
'''Sensors or sources:''' The signals used for connectivity analyses can be from '''sensor data''' (EEG/MEG signals) or from '''source time seriess''' (voxels or scouts).
'''Directed vs. non-directed:''' The direction of the interaction between signals (as statistical causation) can be measured with '''directed metrics'''. '''Non-directed metrics '''produce symmetrical connectivity graphs/matrices as connectivity "from Signal <> to Signal <> " is identical to connectivity "from Signal <> to Signal <>".
'''Experimental condition:''' Depending on the neuroscience question, connectivity analyses can be performed on resting-state (spontaneous) or task (e.g., trials) data.
'''Full (NxN) vs. seeded (1xN) connectivity:''' In a '''full connectivity analysis''', the connectivity metric is computed for all the possible node pairs between N time series (noted N×N here). Alternatively, '''seeded connectivity''' (noted 1×N) is performed between one time series of interest (a ''seed, ''e.g., one brain region or a behavioral marker) and N other regions/time series.
'''Time-frequency transformations:''' Some connectivity metrics rely on a [[Tutorials/TimeFrequency|time-frequency representation]] of the signals. These latter are obtained with approaches such as the short-time Fourier transform, Hilbert transform, and Morlet wavelets.
Just as in other areas of electrophysiology studies, connectivity analyses need to be''' guided by mechanistic ''''''hypotheses concerning the expected effects'''.
=== Sensor-level ===
Sensor connectivity analyses present two important limitations:
1. '''Their anatomical interpretation is limited and ambiguous'''.
1. Sensor data is severely corrupted by '''field spread''' and '''volume conduction'''. Hence, activity from one single brain area is detected at multiple, often distant, sensor locations, which may be wrongly interpreted as network connections.
=== Source-level ===
Source connectivity analyses are '''neuroanatomically interpretable''' and can be derived across participants, following spatial normalization and registration.
It is recommended to verify that the outcomes of sensor and source connectivity analyses are compatible with one another ([[https://doi.org/10.1038/s41598-018-30869-w|Lai et al., 2018]]).
'''NxN analyses'''
Whole-brain connectivity analysess at the typical resolution of cortical surfaces in Brainstorm involve '''thousands''' of source locations, making the N×N) derivations impractical. For instance, 15000 cortical vertices would yield a connectivity matrix of 15000x15000x8 bytes = 1.6Gb. If unconstrained cortical sources are used and coherence is computed across 50 frequency bins, the memory allocation increases to 45000X45000x50 = 754 Gb per participant/condition/trial/etc. Reducing the N (via e.g., cortical parcellations, ROIs) is therefore essential.
'''Regions of interest'''
A [[Tutorials/Scouts|scout]] is Brainstorm's name for a ROI. It can be defined based on study priors and other considerations such as the source estimation method, experimental task, and data available ([[https://doi.org/10.1002/hbm.20745|Schhoffen and Gross, 2009]]):
* '''Study priors '''from the literature, working hypotheses,
* '''Association with a non-neuronal signal''' (e.g., cortico-muscular coherence),
* '''Signal strength''', ROIs as regions with strongest activity in an experimental condition,
* '''Whole-brain '''[[https://neuroimage.usc.edu/brainstorm/Tutorials/LabelFreeSurfer#Cortical_parcellations|cortical parcellation]] to reduce N.
{{{#!wiki caution
Whole-brain connectivity estimates may be exposed to the issue of circular analysis ([[https://doi.org/10.1038/nn.2303|Kriegeskorte et al., 2009]]).
}}}
== Requirements ==
Here we skip most of the interface details and focus on the specifics on connectivity analyses with Brainstorm. So please make sure you are familiar with Brainstorm and go through all [[http://neuroimage.usc.edu/brainstorm/Tutorials#Get_started|introduction tutorials]] first.
We first use simulated data to emphasize the theoretical aspects of each connectivity metric with respect to groundtruth outcomes. Real, empirical MEG data are featured later in the tutorial (same auditory oddball dataset as other tutorial sections).
Let's start by creating a new protocol in the Brainstorm database:
* Select the menu File > Create new protocol > type in "'''TutorialConnectivity'''" and select the options:
* '''Yes''', use protocol's default anatomy,
* '''No''', use one channel file per condition.
* Right-click on the TutorialConnectivity folder > New subject > Subject01<
><
> {{attachment:protocol_connect.gif}}
== Simulated data ==
To compare connectivity metrics, let's use simulated time series with known ground truth interactions using a [[https://en.wikipedia.org/wiki/Brain_connectivity_estimators#Multivariate_Autoregressive_Model|multivariate autoregressive (MVAR)]] model. The model we'll use consists of the following three signals:
* '''Signal 1''': two oscillatory components. One at 10 Hz (alpha band), and a stronger peak at 25 Hz (beta band).
* '''Signal 2''': Same as Signal 1, with strongest peak at 10 Hz.
* '''Signal 3''': Same as above, with the two components (10 and 25 Hz) of same magnitude.
We will simulate the fact that''' '''the component of''' '''Signal 3 at 25 Hz is driven in part by that of Signal 1 (denoted '''Signal 1>>Signal 3''').
Let's now generate those three time series:
* In the '''''Process1''''' tab, leave the file list empty and click on the button '''''[Run]'''''
* Select process: '''''Simulate > Simulate AR signals'''''.<
><
> {{attachment:sim_process.gif}}
Process options:
* '''Subject name''': Target subject for the simulated signals. Select '''Subject01'''.
* '''Condition name''': Target folder for the simulated signals. Set to '''Simulation'''.
* '''Number of time samples''': Duration of signals, in samples. Set to '''12 000'''.
* '''Sampling frequency''':''' ''' Fs for the simulated signals. Set to '''120 Hz'''.
* '''Interaction specifications:''' Spectral parameters for the signal components and their interactions in the MVAR model: '''From, To / Peak frequencies [Hz] / Peak relative magnitudes [0-1]'''<
> Set to:
{{{
1, 1 / 10, 25 / 0.3, 0.5
2, 2 / 10, 25 / 0.7, 0.3
3, 3 / 10, 25 / 0.2, 0.2
1, 3 / 25 / 0.1
}}}
* '''Display the groundtruth spectral metrics''' of the MVAR-generated time series: transfer function, cross-spectral power density, [[#Coherence|magnitude square coherence]], [[https://en.wikipedia.org/wiki/Brain_connectivity_estimators#Directed_Transfer_Function|directed transfer function (DTF)]] and [[https://en.wikipedia.org/wiki/Brain_connectivity_estimators#Partial_Directed_Coherence|partial directed coherence (PDC)]]. The '''transfer function '''(<>) characterizes the relationships between signals in the frequency domain. It is a non-symmetric representation, which enables the identification of causal dependencies between signal components. The ''auto-transfer functions'' (shown in the graphs along the diagonal below) display the power spectra of each signal. The off-diagonal representations display the interactions between each pair of signals (see Signal 1 >> Signal 3 above). Here, we see the transfer function from signal 1 to signal 3. These transfer functions are our '''ground truth for connectivity values'''.<
><
> {{attachment:sim_ar_spectra_metrics.png||width="600"}}
* '''Get coefficients matrix:''' Shows the coefficients related to the MVAR model. These coefficients can be used in the process '''''Simulate > Simulate AR signals (ARfit)''''' to simulate the same model.<
><
> {{attachment:sim_coef.gif}}
Execution:
* Click '''''Run''' ''to simulate the signals described by the MVAR model. <
><
> {{attachment:sim_db.gif}}
* In the next sections we will compute different connectivity metrics for these simulated signals. As such, place the simulated data in the '''''Process1''''' tab, click on '''''[Run]''''' ( {{https://neuroimage.usc.edu/moin_static198/brainstorm1/img/iconRun.gif}} ) to open the [[Tutorials/PipelineEditor#Selecting_processes|Pipeline editor]], and select the connectivity metric.<
><
> {{attachment:sim_select.gif}}
* This tutorial illustrates only the computation full connectivity graphs ('''''Process 1: NxN'''''). It is however possible to compute the connectivity between 1 signal and the other signals in the same file ('''''Process1: 1xN'''''), or between signals from two different files ('''Process2: AxB'''). These options will be illustrated at the end of the tutorial.
Credits:
* This process relies on the ARSIM function from the '''ARFit toolbox''': <
>https://github.com/tapios/arfit
* Neumaier A, Schneider T<
>[[http://dx.doi.org/10.1145/382043.382304|Estimation of parameters and eigenmodes of multivariate autoregressive models]]<
>ACM Transactions on Mathematical Software, 2001
* Schneider T, Neumaier A<
>Algorithm 808: [[http://dx.doi.org/10.1145/382043.382316|ARfit – A Matlab package for the estimation of parameters and eigenmodes of multivariate autoregressive models]]<
>ACM Transactions on Mathematical Software, 2001
<
><
>
== Correlation ==
[[https://en.wikipedia.org/wiki/Correlation_and_dependence|Correlation]] is a non-directed connectivity metric that can be used to show similarity, dependence or association among two random variables or signals. While this metric has been widely used in electrophysiology, it should not be considered the best technique to evaluate connectivity. Due to its nature, correlation fails to alleviate the problem of volume conduction and cannot explain the association in different frequency bands. However, it still can provide valuable information in case we deal with a few narrow-banded signals.
=== Process options ===
* Process '''''Connectivity > Correlation NxN''''':<
><
> {{attachment:gui_corr1n.png||width="400"}}
* '''Time window:''' Segment of the signal used for the connectivity analysis. Select: All file.
* '''Compute scalar product:''' If unchecked, the mean of the signals is subtracted before computing the correlation. Uncheck it.
* '''Output options''': Select '''''Save individual results'''''.
=== Result visualization ===
* The results are stored as a N×N connectivity file, icon {{https://neuroimage.usc.edu/moin_static198/brainstorm1/img/iconConnectN.gif}} . Right-click to see its display options:<
><
> {{attachment:corr1n_file.gif}}
* '''Display as graph''': Plots the connectivity graph using a [[https://en.wikipedia.org/wiki/Chord_diagram|chord diagram]] where the color of the edges shows the connectivity metric value. See the [[Tutorials/ConnectivityGraph|connectivity graph tutorial]] for a detailed explanation of the options of this visualization.
* '''Display as image''': Plots the [[https://en.wikipedia.org/wiki/Adjacency_matrix|adjacency matrix]] for the connectivity file.
* '''Display fibers''':Additional option available for source connectivity results when a '''fiber track''' surface is available, as shown [[#Tutorials.2FFiberConnectivity|here]]. <
><
> {{attachment:corr1n_graph.gif}} <
><
> {{attachment:corr1n_image.png}}
Display options:
* '''Diagonal values''': The value of the connectivity metric between a signal and itself is plotted as zero so that it doesn't force scaling the colormap to 1 if the other values are much smaller.
* '''Labels''': Click on the figure to see the signal names and connectivity values as the image legend. In order to see the labels corresponding to each column and row: right-click on the figure > Figure > '''''Show labels'''''. If your signal names are very long, try the option '''''Use short labels'''''.
* '''Colormap''': By default, the NxN colormap is configured to display the absolute values of the connectivity measures. As in the general case, correlation values can be positive or negative, you need to check this option carefully. If you expect to see '''negative values''', make sure to change the colormap configuration: right-click on the figure > Colormap > Uncheck '''''Absolute values'''''. <
><
> {{attachment:corr1n_image_relative.png}}
<
><
>
== Coherence ==
Coherency or complex coherence, <>, is a complex-valued metric that measures the linear relationship of two signals in the frequency domain. Its magnitude square coherence (MSC), <>, often referred to as coherence, measures the covariance of two signals in the frequency domain. For a pair of signals <> and <>, with spectra <> and <>, the MSC is defined as:
. {{{#!latex
\begin{eqnarray*}
C_{xy}(f) &=& \frac{S_{xy}(f)}{\sqrt{ S_{xx}(f)S_{yy}(f)}}\\
|C_{xy}(f)|^2 &=& MSC(f) = \left(\frac{\left |S_{xy}(f) \right |}{\sqrt{ S_{xx}(f)S_{yy}(f) }}\right)^2 = \frac{\left |X(f)Y^*(f) \right |^{2}}{X(f)X^*(f)Y(f)Y^*(f)} \\
S_{xy}(f) &:& \textrm{Cross-spectrum} \\
S_{xx}(f) \quad \textrm{and} \quad S_{yy}(f) &:& \textrm{Auto-spectra or power spectral densities} \\
\end{eqnarray*}
}}}
Two related measures, which alleviate the problem of volume conduction, are '''imaginary coherence''' ([[https://doi.org/10.1016/j.clinph.2004.04.029|Nolte et al., 2004]]), <>, and the '''lagged coherence''' ([[https://arxiv.org/pdf/0706.1776|Pascual-Maqui, 2007]]), <>, which are defined as:
. {{{#!latex
\begin{eqnarray*}
IC_{xy}(f) &=& \mathrm{Im} \left (C_{xy}(f) \right ) = \frac{\mathrm{Im} \left (S_{xy}(f) \right )}{\sqrt{ S_{xx}(f)S_{yy}(f) }} \\
LC_{xy}(f) &=& \frac{\mathrm{Im} \left (C_{xy}(f) \right )}{\sqrt{ 1 - \left [ \mathrm{Re}\left ( C_{xy}(f) \right ) \right ]^{2} }} = \frac{\mathrm{Im} \left (S_{xy}(f) \right )}{\sqrt{ S_{xx}(f)S_{yy}(f) - \left [ \mathrm{Re}\left ( S_{xy}(f) \right ) \right ]^{2} }} \\
\end{eqnarray*}
}}}
where <> and <> describe the imaginary and real parts of a complex number.
To calculate coherence values in Brainstorm, select the process.
=== Process options ===
* Process '''''Connectivity > Coherence NxN''''':<
><
> {{attachment:gui_cohere1n.png||width="400"}}
* '''Time window:''' Segment of the signal used for the connectivity analysis. Select '''All file'''.
* '''Remove evoked response''': If checked, removes the average of all the files in input (the "trials") from each file, before computing the connectivity measure. Meaningful only in the context of ERP/ERF analyses. Uncheck it.
* '''Process options:''' Different measures. Select: Magnitude squared coherence.
* '''Window length:''' Duration in seconds for the [[https://neuroimage.usc.edu/brainstorm/Tutorials/ArtifactsFilter#Evaluation_of_the_noise_level|spectrum estimation]]. Set to: 1s.
* '''Overlap:''' Percentage of overlap between consecutive windows. Set to: 50%.
* '''Highest frequency''': After the computation, removes all the frequencies above this threshold, mostly for visualization purposes. It should be <= Fs/2. Set to: 60 Hz.
* '''Output options''': Select: Save individual results.
=== Result visualization ===
Coherence is a function of frequency, as such, for each frequency point there is a connectivity graph and a connectivity matrix. Right-click on the coherence result file to see its display options:
* '''Display as graph''': Plot the connectivity graph at a given frequency point.
* '''Display as image''': Plot the connectivity matrix at a given frequency point.
* '''Power spectrum''': Plot coherence as a function of frequency for all the possible node pairs.
Open the 3 representations. These representations are linked such as by clicking on the spectral representation of the coherence, we change the frequency that is displayed in the connectivity graph and matrix. This frequency can be also changed in the Time panel.
|| {{attachment:res_cohere1n_a.png||width="350"}} || || {{attachment:res_cohere1n_a2.png||width="200"}} ||
|| {{attachment:res_cohere1n_b.png||width="350"}} || || {{attachment:res_cohere1n_c.png||width="350"}} ||
In the same way, we can compute the other types of coherence. The figure below presents the spectra for the imaginary coherence (left) and the lagged coherence (right). Both, imaginary and lagged coherence aim to address the volume conduction problem, although they present small differences.
|| {{attachment:res_cohere1n_d.png||width="350"}} || || {{attachment:res_cohere1n_e.png||width="350"}} ||
<
><
>
== Granger causality ==
Granger causality (GC) is a method of directed functional connectivity, which is base on the Wiener-Granger causality methodology. GC is a measure of linear dependence, which tests whether the prediction of signal <> (using a linear autoregressive model) is improved by adding signal <> (also using a linear autoregressive model). If this is true, signal <> has a Granger causal effect on the first signal. In other words, '''independent information''' of the past of signal <> improves the prediction of signal <> obtained with the past of signal <> alone. GC is nonnegative, and zero when there is no Granger causality. As only the past of the signals is considered, the GC metric is directional. The term '''independent''' is emphasized because it creates some interesting properties for GC, such as, that it's invariant under rescaling of <> and <>, as well as the addition of a multiple of <> to <>See [[GrangerCausality|Granger causality - mathematical background]] for a complete formulation of the method.
{{{#!wiki note
Despite the name, '''Granger causality indicates directionality but not true causality'''. <
> For example, if a variable <> is causing both <> and <>, but with a smaller delay for <> than for <>, then the GC measure between <> and <> would show a non-zero GC for <> --> <>, even though <> is not truly causing <> ([[https://doi.org/10.1016/j.neuroimage.2010.02.059|Bressler and Seth, 2011]]).
}}}
=== Process options ===
* Process '''''Connectivity > Bivariate Granger causality NxN'''''<
><
> {{attachment:gui_granger1n.png||width="400"}}
* '''Time window:''' Segment of the signal used for the connectivity analysis. Select '''All file'''.
* '''Remove evoked response''': If checked, removes the average of all the files in input (the "trials") from each file, before computing the connectivity measure. Meaningful only in the context of ERP/ERF analyses. It is recommended by some authors as it meets the zero-mean stationarity requirement (improves stationarity of the system). However, the problem with this approach is that it does not account for trial-to-trial variability. For a discussion see ([[https://doi.org/10.1016/j.neuroimage.2008.03.025|Wang et al., 2008]]). '''Uncheck''' it.
* '''Maximum Granger model order:''' The most common criteria used to define the order of the model are the [[https://en.wikipedia.org/wiki/Akaike_information_criterion|Akaike’s information]] criterion, the [[https://en.wikipedia.org/wiki/Bayesian_information_criterion|Bayesian-Schwartz’s criterion]], and the [[https://en.wikipedia.org/wiki/Hannan–Quinn_information_criterion|Hannan-Quinn criterion]]. Too low orders may lack the necessary details, while too big orders tend to create spurious values of connectivity. While our simulated signals were created with a model of 4, here we used as model order of '''6''' for a decent connectivity result.
* '''Output options''': Select '''Save individual results'''. <
>
=== Result visualization ===
In the connectivity graph (left) the directionality is shown with an arrow head at the center for the arc connecting nodes. As GC metric is not symmetric, the connectivity matrix (right) is not symmetric. The upper right element of this matrix shows there is a signal flow from signal 1 to signal 3.
|| {{attachment:res_granger1n_a.png||width="350"}} || || {{attachment:res_granger1n_b.png||width="350"}} ||
<
><
>
== Spectral Granger causality ==
GC lacks of resolution in the frequency domain, as such, the spectral Granger causality was developed ([[https://doi.org/10.1103/PhysRevLett.100.018701|Dhamala et al., 2008]]).
=== Process options ===
* Process: '''''Connectivity > Bivariate Granger causality NxN'''''. <
><
> {{attachment:gui_spgranger1n.png||width="400"}} <
><
>With respect to [[#Granger_causality|GC]], spectral GC presents two extra parameters:
* '''Maximum frequency resolution''': Width of frequency bins in PSD estimation. Set to '''1 Hz'''.
* '''Highest frequency''': After the computation, removes all the frequencies above this threshold, mostly for visualization purposes. It should be <= Fs/2. Set to: 60 Hz.
=== Result visualization ===
As with coherence, spectral GC can be plotted as a function of frequency. The plot below clearly shows a peak around 25 Hz for the interaction from signal 1 to signal 3, as expected.
{{attachment:res_spgranger1n.png||width="300"}} <
><
>
== Envelope correlation ==
In the [[Tutorials/TimeFrequency|time-frequency tutorial]] the Morlet wavelets and Hilbert transform were introduced as methods to decompose signals in the time-frequency (TF) domain. The result of this TF transformation can be seen as a set of narrowband complex signals, which are analytic signals.
The [[https://en.wikipedia.org/wiki/Analytic_signal|analytic signal]], <>, is a complex signal uniquely associated to a real signal, <>, that has been useful in signal processing due to its characteristics, more specifically, its '''module''' <>, and '''phase''' <>, correspond to the '''instantaneous amplitude''' (or envelope) and '''instantaneous phase''' of the associated real signal <>. The real part of <> is its associated real signal <>, and the imaginary part is the Hilbert transform of the same real signal <>.
. {{{#!latex
\begin{eqnarray*}
\tilde{x}(t)= x(t) + j\mathcal{H}\left\{ x(t) \right\} = a_{\tilde{x}}(t)e^{j\phi_{\tilde{x}}(t)} \\
\end{eqnarray*}
}}}
The instantaneous amplitude (or envelope) of these band analytic signals can be used to carry out pairwise connectivity analysis with metrics such as correlation and coherence (including lagged coherence).
In computing the envelope correlation, an optional step is to orthogonalize the envelopes by removing their real part of coherence before the correlation ([[https://psycnet.apa.org/doi/10.1038/nn.3101|Hipp et al., 2012]]). This orthogonalization process alleviates the effect of volume conduction in MEG/EEG signals.
=== Process options ===
* Process: '''''Connectivity > Envelope Correlation N×N [2020]''''' <
><
> {{attachment:gui_henv1n_ha.png||width="400"}}
* '''Time window:''' Segment of the signal used for the connectivity analysis. Select: All file.
* '''Remove evoked response''': If checked, removes the average of all the files in input (the "trials") from each file, before computing the connectivity measure. Meaningful only in the context of ERP/ERF analyses. Uncheck it.
* '''Time-frequency transformation method:''' Either '''Hilbert transform''' or '''Morlet wavelets'''. Each of this methods requires additional parameters that are found in an external panel that opens by clicking on '''''Edit'''''. See the [[Tutorials/TimeFrequency|time-frequency tutorial]]. In this example, the sampling frequency of the signals being 120Hz, make sure you remove the "gamma2" band.<
><
> {{attachment:gui_henv1n_hb.png||width="227",height="276"}} {{attachment:gui_henv1n_wb.png||width="385",height="401"}}
* '''Signal splitting:''' This process has the capability of splitting the input data into several blocks for performing time-frequency transformation, and then merging them to build a single file. This feature helps to save a huge amount of memory and, at the same time, avoids breaking a long-time recording to short-time signals, which makes inconsistency in dynamic network representation of spontaneous data. The maximum number of blocks which can be specified is 20. Set to '''1'''.
* '''Connectivity measure:''' This is the connectivity metric that will be used with the envelopes. Select '''Envelope correlation (orthogonalized)'''.
* '''Time resolution:''' If Dynamic is selected, connectivity is saved for each window of analysis. If Static is selected, the connectivity results from all the windows are averaged. Select '''Dynamic'''.
* '''Estimation window length''': Duration in milliseconds to compute the connectivity measure. Set to '''5000 ms'''
* '''Sliding window overlap:''' Percentage of overlap between consecutive windows to compute the connectivity measure. Set to '''50%'''.
* '''Use the parallel processing toolbox:''' Enables the use of the parallel processing toolbox in Matlab to accelerate the computational procedure. '''Uncheck''' it.
* '''Output configuration:''' Generally, the above calculation results in a 4-D matrix, where dimensions represent channels (1st and 2nd dimensions), time points (3rd dimension), and frequency (4th dimension). In the case that we analyze event-related data, we have also several files (trials). However, due to the poor signal-to-noise ratio of a single trial, an individual realization of connectivity matrices for each of them is not in our interests. Consequently, we need to average connectivity matrices '''among all trials''' of a specific event. The second option of this part performs this averaging.
=== Result visualization ===
Similar to the results from coherence and spectral Granger causality, the envelope correlation can be plotted as a function of frequency, and as a function of time if the '''Time resolution''' option is set to '''Dynamic'''. Below, the results obtained with the Hilbert transform (left) and with Morlet wavelet (right) for the first 5-s window (top) and the 5-s last window (bottom).
|| {{attachment:res_henv1n_h.png||width="350"}} ||