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= Tutorial 28: Connectivity =
'''[TUTORIAL UNDER DEVELOPMENT: NOT READY FOR PUBLIC USE] '''
''Authors: Hossein Shahabi, Raymundo Cassani, Takfarinas Medani, François Tadel''
Cognitive and perceptual functions are the result of coordinated activity of functionally specialized regions in the brain. [[http://www.scholarpedia.org/article/Brain_connectivity|Brain connectivity]] investigates how these different regions (or nodes) interact as a network, with the goal of having a better understanding of how the brain processes information. Depending on which connectivity characteristic is studied, a distinction is made between '''structural''' (fiber pathways), '''functional''' (non-directed statistical dependency) and '''effective''' (causal interaction) 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 in Brainstorm, first with simulated data and later with real data.
<>
== General considerations in connectivity analysis ==
Connectivity analyses are commonly performed by computing a bivariate connectivity metric for all the possible pairs of time series or signals. The result of such approach can be presented as a '''connectivity graph''' (left image), where each signal is represented as a node, and the value of the connectivity metric is the value of the edge between the corresponding nodes. This graph representation becomes overwhelming when too many nodes are considered, as such, the connectivity graph can be represented with its '''connectivity matrix''', aka adjacency matrix (right image).
{{attachment:cnx_graph_matrix.png}}
'''Sensors or sources:''' The signals used for the connectivity analysis can be derived from the '''sensor data''' (EEG/MEG signals) or from the reconstructed '''sources''' (voxels or scouts).
'''Directed and non-directed:''' The direction of the interaction between signals (as statistical causation) can be measured with '''directed metrics'''. However, this is not possible with '''non-directed metrics''', as result, the connectivity metric "from Signal <> to Signal <> " is equal to the connectivity metric "from Signal <> to Signal <>".
'''Recording condition:''' Connectivity analyses can be performed on resting-state (spontaneous) and event-related (trials) recordings, the appropriate connectivity method depends on the recording condition.
'''Full network vs point-based connectivity:''' In '''full network''', the connectivity metric is computed for all the possible node pairs in the network (N×N approach), and gives as result a detailed connectivity graph. Alternatively, '''point-based''' analysis is performed solely between one node (aka seed) and the rest of the nodes in the network (1×N approach), this approach is faster to compute and is more useful when you are interested in the connectivity of a specific sensor or source.
'''Temporal resolution:''' Connectivity analyses can be performed in two ways: static and dynamic. Time-varying networks can present the dynamics of brain networks. In contrast, the static graphs illustrate a general perspective of brain connectivity which is helpful in specific conditions. Users need to decide which type of network is more informative for their study.
'''Time-frequency transformation:''' Several connectivity metrics rely on the [[Tutorials/TimeFrequency|time-frequency representation]] of the signals, which is obtained with approaches such as the short-time Fourier transform, Hilbert transform, and Morlet wavelet.
== Requirements ==
Please note that this is an advanced tutorial, it assumes that that you have already followed all the [[http://neuroimage.usc.edu/brainstorm/Tutorials#Get_started|introduction tutorials]]. For readability, most of the interface details are omitted, you must be familiar with the Brainstorm software in order to reproduce the computations illustrated below.
This tutorial is mostly based on simulated data, and independent from the other tutorials. Only one part at the end uses the results of the introduction tutorials (auditory oddball dataset), in order to illustrate how to apply the connectivity processes to real MEG recordings.
Let's start by creating a new protocol in the Brainstorm database:
* Select the menu File > Create new protocol > "'''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 (MVAR model) ==
To compare different connectivity metrics, we use simulated data with known ground truth using a [[https://en.wikipedia.org/wiki/Brain_connectivity_estimators#Multivariate_Autoregressive_Model|multivariate autoregressive (MVAR)]] model. This model consist in three signals in a way that:
* '''Signal 1''': Components at 10 and 25 Hz with a dominant peak at 25 Hz (beta band).
* '''Signal 2''': Same components, but dominant peak is at 10 Hz (alpha band).
* '''Signal 3''': Similar power for both components (10 and 25 Hz).
* '''Signal 1>>Signal 3''': Signal 3 is influenced by Signal 1, with a peak at 25 Hz.
Simulation process:
* In the '''''Process1''''' tab, leave the file list empty and click on the button '''''[Run]'''''
* Select process: '''''Simulate > Simulate AR signals (ARfit) with spectra'''''.<
><
> {{attachment:simulate_ar_spectra.png||width="400"}}
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 interaction between signals in the MVAR model, given as: '''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
}}}
* '''View spectral metrics''': Display spectral metrics related to the MVAR process: 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, <>, contains information about the relationships between signals and their spectral characteristics; it is non-symmetric, so it allows for finding causal dependencies. The auto-transfer function (diagonal elements in the image) correspond to the power spectrum of the signals. The off-diagonal terms represent the interactions between different signals. 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_ar_spectra_coeff.png||width="600"}}
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}}
== 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''''' process:<
><
> {{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 on this file 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.<
><
> {{attachment:corr1n_graph.gif}}
* '''Display as image''': Plots the [[https://en.wikipedia.org/wiki/Adjacency_matrix|adjacency matrix]] for the connectivity file. <
>
{{{#!wiki note
The value of the connectivity metric between a signal and itself plotted as '''zero''' so that it doesn't force scaling the colormap to 1 if the other values are much smaller.
}}}
<
><
> {{attachment:corr1n_image.png}}
* '''Display fibers''':Additional option available for source connectivity results when a '''fiber track''' surface is available, as shown [[#Tutorials.2FFiberConnectivity|here]].
|| {{attachment:res_corr1n_a.png||width="350"}} || || {{attachment:res_corr1n_b.png||width="350"}} ||
*
{{{#!wiki note
In '''Display as image''', the value of the connectivity metric between a signal and itself plotted as '''zero''' so that it doesn't force scaling the colormap to 1 if the other values are much smaller.
}}}
== Coherence ==
Coherency or complex coherence, <>, is a complex-valued metric that measures of the linear relationship of two signals in the frequency domain. And, 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 '''''Connectivity > Coherence NxN''''' process.
{{attachment:gui_cohere1n.png||width="400"}} <
>
=== Process options ===
* '''Time window:''' Segment of the signal used for the connectivity analysis. Select '''All file'''.
* '''Remove evoked response''': If checked, removes the averaged evoked from each trial. '''Uncheck''' it.
* '''Process options:''' Different types of coherence. Select '''Magnitude squared coherence'''.
* '''Window length for PSD estimation''' Duration in seconds for the PSD estimation. Set to '''1 s'''.
* '''Overlap for PSD estimation:''' Percentage of overlap between consecutive windows for PSD estimation. Set to '''50%'''.
* '''Highest frequency of interest''': Highest frequency for the analysis, 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]]).
}}}
To compute the Granger causality values in Brainstorm, select the '''''Connectivity > Bivariate Granger causality NxN''''' process.
{{attachment:gui_granger1n.png||width="400"}} <
><
>
=== Process options ===
* '''Time window:''' Segment of the signal used for the connectivity analysis. Select '''All file'''.
* '''Remove evoked response''': If checked, removes the averaged evoked from each trial. 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]]). The process to calculate this metric is found in '''''Connectivity > Bivariate Granger causality NxN'''''.
{{attachment:gui_spgranger1n.png||width="400"}} <
><
>
=== Process options ===
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 of interest''': Highest frequency for the analysis, 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 (2020) ==
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. These connectivity metrics can be computed with the '''''Connectivity > Envelope Correlation N×N [2020]''''' process. B
{{attachment:gui_henv1n_ha.png||width="400"}}
=== Process options ===
* '''Time window:''' Segment of the signal used for the connectivity analysis. Select '''All file'''.
* '''Remove evoked response''': If checked, removes the averaged evoked from each trial. '''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]].
|| {{attachment:gui_henv1n_hb.png||width="250"}} || || {{attachment:gui_henv1n_wb.png||width="400"}} ||
* '''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, avoid 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"}} ||