Connectivity

Authors: Hossein Shahabi, Raymundo Cassani, Takfarinas Medani, François Tadel, Sylvain Baillet

Brain functions (e.g., in cognition, behavior and perception) stem from the coordinated activity of multiple regions. 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 learn more about the specific aspects of electrophysiology for studying human connectomics.

Contents

  1. Introduction
  2. Requirements
  3. Simulated data
  4. Correlation
  5. Coherence
  6. Granger causality
  7. Spectral Granger causality
  8. Envelope correlation
  9. Phase locking value
  10. Phase transfer entropy
  11. Method selection and comparison
  12. Real data: Auditory dataset [TODO]
  13. On the hard drive
  14. Additional documentation
  15. TODO

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).

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 $$x$$ to Signal $$y$$ " is identical to connectivity "from Signal $$y$$ to Signal $$x$$".

Experimental condition: Depending on the neuroscience question, connectivity analyses can be performed on resting-state (spontaneous) or task (e.g., trials) data.

Full vs. ROI connectivity: In a full connectivity analysis, the connectivity metric is computed for all the possible node pairs at all N brain locations (noted N×N here). Alternatively, ROI connectivity is performed between one node of interest (a.k.a. a seed, e.g., one brain region or a behavioral marker) and N other regions (1×N approach).

Time-frequency transformations: Some connectivity metrics rely on a 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.

  2. 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 (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

Scouts are Brainstorm ROIs can be defined in multiple ways depending on study priors of all sorts, and other considerations such as the source estimation method, experimental task, and data available (Schhoffen and Gross, 2009). ROIs can be defined via:

Whole-brain connectivity analyses may be exposed to the issue of circular analysis (Kriegeskorte et al., 2009).

The optimal selection of ROIs to perform source connectivity analysis is still an open question.

Requirements

Please note that this is an advanced tutorial, it assumes that that you have already followed all the 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:

Simulated data

To compare different connectivity metrics, we use simulated data with known ground truth using a multivariate autoregressive (MVAR) model. This model consist in three signals in a way that:

Simulation process:

Process options:

Execution:

Credits:



Correlation

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

Result visualization

Display options:



Coherence

Coherency or complex coherence, $$C_{xy}(f)$$, is a complex-valued metric that measures the linear relationship of two signals in the frequency domain. Its magnitude square coherence (MSC), $$|C_{xy}(f)|^2$$, often referred to as coherence, measures the covariance of two signals in the frequency domain. For a pair of signals $$x(t)$$ and $$y(t)$$, with spectra $$X(f)$$ and $$Y(f)$$, the MSC is defined as:

Two related measures, which alleviate the problem of volume conduction, are imaginary coherence (Nolte et al., 2004), $IC_{xy}(f)$, and the lagged coherence (Pascual-Maqui, 2007), $LC_{xy}(f)$, which are defined as:

where $$\mathrm{Im()}$$ and $$\mathrm{Re()}$$ describe the imaginary and real parts of a complex number.

To calculate coherence values in Brainstorm, select the process.

Process options

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:

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.

res_cohere1n_a.png

res_cohere1n_a2.png

res_cohere1n_b.png

res_cohere1n_c.png

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.

res_cohere1n_d.png

res_cohere1n_e.png



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 $$x(t)$$ (using a linear autoregressive model) is improved by adding signal $$y(t)$$ (also using a linear autoregressive model). If this is true, signal $$y(t)$$ has a Granger causal effect on the first signal. In other words, independent information of the past of signal $$y(t)$$ improves the prediction of signal $$x(t)$$ obtained with the past of signal $$x(t)$$ 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 $$x(t)$$ and $$y(t)$$, as well as the addition of a multiple of $$x(t)$$ to <<latex($$y(t)$$).
See Granger causality - mathematical background for a complete formulation of the method.

Despite the name, Granger causality indicates directionality but not true causality.
For example, if a variable $$w$$ is causing both $$x$$ and $$y$$, but with a smaller delay for $$y$$ than for $$x$$, then the GC measure between $$x$$ and $$y$$ would show a non-zero GC for $$y$$ --> $$x$$, even though $$y$$ is not truly causing $$x$$ (Bressler and Seth, 2011).

Process options

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.

res_granger1n_a.png

res_granger1n_b.png



Spectral Granger causality

GC lacks of resolution in the frequency domain, as such, the spectral Granger causality was developed (Dhamala et al., 2008).

Process options

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.

res_spgranger1n.png

Envelope correlation

In the 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 analytic signal, $$\tilde{x}(t)$$, is a complex signal uniquely associated to a real signal, $$x(t)$$, that has been useful in signal processing due to its characteristics, more specifically, its module $$a_{\tilde{x}}(t)$$, and phase $$\phi_{\tilde{x}}(t)$$, correspond to the instantaneous amplitude (or envelope) and instantaneous phase of the associated real signal $$x(t)$$. The real part of $$\tilde{x}(t)$$ is its associated real signal $$x(t)$$, and the imaginary part is the Hilbert transform of the same real signal $$x(t)$$.

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 (Hipp et al., 2012). This orthogonalization process alleviates the effect of volume conduction in MEG/EEG signals.

Process options

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).

res_henv1n_h.png

First 5-s window

res_henv1n_w.png

res_henv1n_h2.png

Last 5-s window

res_henv1n_w2.png

Phase locking value

An alternative class of connectivity metrics considers only the relative instantaneous phase between the two signals, i.e., phase-locking or synchronization (Tass et al., 1998). Phase locking is a fundamental concept in dynamical systems that has been used in control systems (the phase-locked loop) and in the analysis of nonlinear, chaotic and non-stationary systems. Since the brain is a nonlinear dynamical system, phase locking is an appropriate approach to quantifying connectivity. A more pragmatic argument for its use in studies of LFPs, EEG, and MEG is that it is robust to fluctuations in amplitude that may contain less information about interactions than does the relative phase (Lachaux et al., 1999; Mormann et al., 2000).

The most commonly used phase connectivity metric is the phase-locking value (PLV), which is defined as the length of the average vector of many unit vectors whose phase angle corresponds to the phase difference between two signals (Tass et al., 1998). If the distribution of the phase difference between the two signals is uniform, the length of such an average vector will be zero. Conversely, if the phases of the two signals are strongly coupled, the length of the average vector will approach unity. For event-related studies, we would expect the phase difference across trials to be uniform distributed unless the phase is locked to the stimulus. In that case, we may have nonuniform marginals which could in principle lead to indications of phase locking between two signals. Considering a pair of narrow-band analytic signals $$\tilde{x}(t)$$ and $$\tilde{y}(t)$$, obtained from the TF transformation using the Hilbert transform:

\begin{eqnarray*} \mathrm{PLV} = \left | E\left [ e^{j\Delta \phi (t)} \right ] \right | \\ \end{eqnarray*}

with:

\begin{eqnarray*} \Delta \phi (t) = \phi_{\tilde{x}}(t) - \phi_{\tilde{y}}(t) = arg\left ( \frac{\tilde{x}(t)\tilde{y}^{*}(t)}{\left | \tilde{x}(t) \right |\left | \tilde{y}(t) \right |} \right ) \\ \end{eqnarray*}

plv.png

Process options

Result visualization

PLV is frequency resolved, and it was computed for the delta, theta, alpha, beta and gamma bands. With the simulated data, we expect a higher PLV value in the beta band (15 to 29 Hz), between signal 1 and signal 3. This result is seen as a peak at 22 Hz (center of beta band) shown in PLV as a function of frequency.

res_plv1n.png

Phase transfer entropy

Phase transfer entropy (PTE) is a directed connectivity metric that quantifies the transfer entropy (TE) between two instantaneous phase time series (Lobier et al., 2014). Similar to GC, TE estimates whether including the past of both source and target time-series influences the ability to predict the future of the target time-series. In PTE, if a phase signal $$\phi_{\tilde{x}}(t)$$ causes the signal $$\phi_{\tilde{y}}(t)$$, the mutual information, between $$\phi_{\tilde{y}}(t)$$ and the past of $$\phi_{\tilde{x}}(t)$$ i.e. $$\phi_{\tilde{x}}(t')$$ is larger than the mutual information of $$\phi_{\tilde{y}}(t)$$, the past of $$\phi_{\tilde{y}}(t)$$ i.e. $$\phi_{\tilde{y}}(t')$$ and $$\phi_{\tilde{x}}(t')$$. This relationship can be seen on the Venn diagram below, where $$\mathrm{I()}$$ and $$\mathrm{H()}$$, indicate mutual information and the individual entropies respectively. Lastly, PTE cannot be negative, and its magnitude does not have a meaningful upper bound.

pte.png

Process options

Result visualization

PTE was computed for the delta, theta, alpha, beta and gamma bands. With the simulated data, we expect a higher PTE value in the beta band, From signal 1 To signal 3, as PTE is directed metric. This is confirmed with a peak at 22 Hz (center of beta band) shown in PTE frequency representation.

res_pte1n.png .

Method selection and comparison

The following table list the available connectivity metrics in Brainstorm and their description.

Metric

Directionality

Domain

1×N

N×N

Time resolved

Process

Info

Correlation

Non-directed

Time

✅

✅

✅

bst_corrn.m

Link

Coherence

Non-directed

Frequency

✅

✅

✅

bst_cohn.m

Link

Granger causality

Directed

Time

✅

✅

❌

bst_granger.m

Link

Spectral Granger causality

Directed

Frequency

✅

✅

❌

bst_granger_spectral.m

Link

Envelope Correlation (2020)

Non-directed

T-F

✅

✅

✅

bst_henv.m

Link

Phase locking value

Non-directed

Phase

✅

✅

❌

bst_connectivity.m

Link

Phase transfer entropy

Directed

Phase

❌

✅

❌

PhaseTE_MF.m

Link

Real data: Auditory dataset [TODO]

TODO: Rewrite this section with coherence on scouts, with single trials.

--

In this section we will use the results obtained in the introduction tutorials. The goal here is only to illustrate the interface of these tools on real data, as the results are not particularly interesting. The corticomuscular coherence tutorial provides more complete and meaninful guidelines for computing connectivity measures on real MEG recordings.

Let's go back to our auditory oddball dataset, available in the protocol TutorialIntroduction if you have followed the introduction tutorials. As the stimulus is played on both ears, we expect to observe highly correlated activity in the primary auditory cortices, in both experimental conditions (standard and deviant beeps), around 0 to 150 ms after the stimulus due to auditory evoked responses at 50 and 100 ms (M50, M100).

In the rest of this section, correlation is computed on the scouts time series of the average response for the standard and deviant conditions. On practice connectivity metrics are computed trial-wise and the results are aggregated. An example of this approach can be seen in the corticomuscular coherence tutorial.

In the Scouts tutorial, we have created 4 scouts for these regions of interest: A1L and A1R for the left and right primary cortices respectively, IFGL for the left inferior frontal gyrus, and M1L for the left primary motor cortex.

scouts_avg_rel.png

Let's compute source-domain correlation for standard and deviant conditions for the Run #1: S01_AEF_20131218_01_600Hz_notch data.

  1. Drag and drop the source files associated to the average standard response (Avg: standard (193 files)) and the average deviant response (Avg: deviant (193 files)) within the Process1 tab, select the option 'source process' ( https://neuroimage.usc.edu/moin_static198/brainstorm1/img/iconResultList.gif ), and click on the [Run] button.

  2. Add a the process, Connectivity > Correlation NxN. Set the time window from 0 to 150 ms. Check the Use scouts option, and select User scouts as atlas in the drop menu. Set the Scout function to Mean, for When to apply the scout function select Before. And select Save individual results (one file per input). Finally click on [Run].

  3. Display the connectivity matrices as Image (top) and as Graph (bottom). Adjust the colormaps to show signed values (no absolute), and set the a custom range from -0.8 to 0.8. Lastly, adjust the Intensity threshold for the graph to the maximum (around 0.7), so only the strongest correlations are kept, mainly, between scouts in the left and right auditory cortices.

Standard

Deviant

corr_0_150_std_intro.png

corr_0_150_dev_intro.png

corrg_0_150_std_intro.png

corrg_0_150_dev_intro.png

In this early response period, we appreciate that the correlation matrices for the standard ann deviant response are quite similar, and the strongest correlation happens between the scouts on the primary auditory cortices; just as expected.

On the hard drive

File name

The connectivity file structure is an extension of the time-frequency structure. The file names start with timefreq_, followed by connect1 (1xN) or connectn (NxN or AxB), the connectivity method and a time stamp. Example: timefreq_connectn_corr_220120_1350.mat.

File structure

Right click on of the first connectivity file computed here > File > View file contents.

The data structure is the same as for the time-frequency files. Only the fields that some specificity related with the connectivity analysis are documented here.

Connectivity matrix encoding

Let's consider the structure TfMat, loaded from a connectivity file, for example by right-clicking on the file > File > Export to Matlab. The connectivity matrix R can be obtained with function GetConnectMatrix. The size of R is [Na x Nb x Ntime x Nfreq]. 

R = bst_memory('GetConnectMatrix', TfMat);

If the matrix is not symmetrical (ie. not compressed): for each time and frequency, the list of values from the first dimension of the TF variable are reshaped into a 2D matrix: [Na x Nb].

If the matrix is symmetrical and compressed: only the lower triangular matrix is saved in the TF variable. The full matrix is first reconstructed with function process_compress_sym>Expand, then reshaped into [Na x Nb].

Saving the connectivity matrix R back in the TF variable is possible by reshaping to [Nx1] (R(:)) and then compressing again the matrix: 

TfMat.TF = process_compress_sym('Compress', R(:));

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Tutorials/Connectivity (last edited 2022-01-26 22:27:26 by SylvainBaillet)