Introduction to Functional Connectivity

Initialize NeuroAnalyzer 
using NeuroAnalyzer
using Statistics
NeuroAnalyzer.Random.seed!(1)

Brain connectivity describes how brain regions are connected and interact:

  • structural connectivity: anatomical connections (white matter tracts, synapses), e.g. from diffusion MRI
  • functional connectivity: statistical dependencies between the signals of different regions (e.g. correlation, phase synchronization); it is undirected and does not imply a direct anatomical connection or a causal interaction (two regions driven by a common third region are functionally connected)
  • effective connectivity: directed (causal) influences of one region on another (e.g. Granger causality, phase slope index, dynamic causal modelling)

Connectivity in EEG/MEG is usually frequency-specific (e.g. alpha, beta or gamma band) and dynamic (it changes with the cognitive state, the task and over time). It is used to study the organization of brain networks, cognitive processes that rely on the cooperation of distributed regions, and their disruption in neurological and psychiatric disorders (e.g. epilepsy, schizophrenia, dementia).

Measures

Phase-based measures

Phase synchronization (a consistent phase relation between oscillations) is thought to support communication between regions. Phase-based measures use the instantaneous phases of narrowband signals (band-pass filtered and Hilbert-transformed, or from wavelets).

  1. Phase locking value (PLV) and inter-site phase clustering (ISPC): the length of the mean vector of the phase differences, \(|\langle e^{i(\varphi_1 - \varphi_2)} \rangle|\); 0: random phase differences, 1: constant phase difference. ISPC, PLV
  2. Imaginary PLV (iPLV): the imaginary part of the mean vector; zero for a phase difference of 0 or π.
  3. Phase lag index (PLI): the asymmetry of the distribution of phase differences around zero, \(|\langle \mathrm{sign}(\sin(\varphi_1 - \varphi_2)) \rangle|\); insensitive to zero-lag synchronization. PLI
  4. Weighted PLI (wPLI): PLI weighted by the magnitude of the imaginary part of the cross-spectrum, \(|\langle \Im(S_{xy}) \rangle| / \langle |\Im(S_{xy})| \rangle\), calculated across trials (epochs); less sensitive to noise near zero lag than PLI, with a debiased version for small numbers of trials. wPLI
  5. Directed PLI (dPLI) and phase slope index (PSI): which signal leads (directed measures).

Inter-trial phase clustering (ITPC) uses the same formula as ISPC, but across trials at one channel: it measures the consistency of the phase relative to an event (phase locking to the stimulus), not connectivity between regions. ITPC

Spectral measures

Coherence is the frequency-domain correlation between two signals: the normalized cross-spectrum, \(C_{xy}(f) = S_{xy}(f) / \sqrt{S_{xx}(f) S_{yy}(f)}\), estimated over epochs (or segments). It depends on both the phase consistency and the amplitude correlation.

  • Magnitude-squared coherence: \(|C_{xy}|^2\) (0–1).
  • Imaginary coherence: \(\Im(C_{xy})\), zero for zero-lag (instantaneous) interactions, therefore insensitive to volume conduction.

See Coherence.

Amplitude-based measures

Amplitude-based measures capture correlations of power (amplitude envelopes) between regions, which may reflect co-activation even without phase synchronization.

  1. Amplitude envelope correlation (AEC): the correlation of the Hilbert envelopes of band-pass filtered signals; with orthogonalization (removing the zero-lag component of one signal from the other before computing the envelopes) it becomes insensitive to volume conduction. AEC
  2. Envelope-signal correlation and other measures, see Other Methods.

General and non-linear measures

Correlation, covariance, cosine similarity, mutual information and synchronization likelihood (a non-linear measure of generalized synchronization) describe dependencies regardless of frequency, see Other Methods.

Directed measures (effective connectivity)

Granger causality: a signal \(x\) “Granger-causes” \(y\) if the past of \(x\) improves the prediction of \(y\) beyond the past of \(y\) itself (autoregressive models); spectral versions give directed connectivity per frequency. Other measures: the phase slope index, directed PLI, partial directed coherence, transfer entropy.

Volume Conduction

The main problem of sensor-level EEG/MEG connectivity is volume conduction (and field spread): the activity of a single source is recorded by many electrodes at the same time, without delay, so neighbouring channels are correlated and phase-synchronized even without any interaction between brain regions. The common reference adds further shared signal to all channels.

Because volume conduction is instantaneous, it produces phase differences of 0 (or π). Measures based on the imaginary part of the cross-spectrum or on non-zero phase lags (PLI, wPLI, iPLV, imaginary coherence, orthogonalized AEC) ignore it; the price is that true zero-lag interactions are ignored as well.

Simulated example: two channels recording a 10 Hz oscillation in 30 two-second epochs, with independent noise, band-pass filtered at 8–12 Hz:

  • zero lag: channel B records the same source as channel A (scaled), like two electrodes recording one source through volume conduction
  • lag π/4: channel B follows channel A with a phase delay of π/4 (25 ms at 10 Hz), like a true interaction with a conduction delay
fs = 256
n = 2 * fs
n_ep = 30
t = (0:(n - 1)) ./ fs

function simulate(lag)
    a = zeros(n, n_ep)
    b = zeros(n, n_ep)
    for ep in 1:n_ep
        φ = 2π * rand()                     # random phase in each epoch
        src = sin.(2π * 10 .* t .+ φ)
        a[:, ep] = src .+ 0.5 .* randn(n)
        b[:, ep] = (lag == 0 ? 0.8 .* src : sin.(2π * 10 .* t .+ φ .- lag)) .+
                   0.5 .* randn(n)
    end
    obj = create_object(; data_type = "eeg")
    obj.header.recording[:sampling_rate] = fs
    add_channel!(obj;
                 data = permutedims(cat(a, b; dims = 3), (3, 1, 2)),
                 label = ["A", "B"],
                 type = ["eeg", "eeg"],
                 unit = ["μV", "μV"])
    create_time!(obj; fs = fs)
    NeuroAnalyzer.filter!(obj;
                          ch = "all",
                          fprototype = :fir,
                          ftype = :bp,
                          cutoff = (8, 12),
                          bw = 2)
    return obj
end
for (name, lag) in (("zero lag", 0), ("lag π/4", π / 4))
    obj = simulate(lag)
    pv = mean(plv(obj, obj; ch1 = "A", ch2 = "B").pv)
    pl = mean(pli(obj, obj; ch1 = "A", ch2 = "B").pv)
    wp = wpli(obj, obj; ch1 = "A", ch2 = "B", across_epochs = true).pv[1]
    c = coherence(obj, obj; ch1 = "A", ch2 = "B")
    i10 = argmin(abs.(c.f .- 10))
    println(rpad(name, 9),
            " PLV: ", round(pv; digits = 2),
            "  PLI: ", round(pl; digits = 2),
            "  wPLI: ", round(wp; digits = 2),
            "  coherence: ", round(c.msc[1, i10]; digits = 2),
            "  |imag. coherence|: ", round(abs(c.imcoh[1, i10]); digits = 2))
end
zero lag  PLV: 0.99  PLI: 0.3  wPLI: 0.07  coherence: 0.97  |imag. coherence|: 0.0
lag π/4   PLV: 0.97  PLI: 0.98  wPLI: 1.0  coherence: 0.97  |imag. coherence|: 0.7

PLV and coherence are high in both cases and cannot distinguish volume conduction from a true interaction. PLI, wPLI and imaginary coherence are high only for the lagged coupling.

PLI is calculated within each epoch (across its time points) and averaged over epochs; for the zero-lag case it is not exactly 0: within a short epoch of a narrowband signal, noise shifts the phase difference to one side of zero for long stretches. wPLI and imaginary coherence are calculated across epochs (wpli(...; across_epochs = true)), where such random shifts cancel out; debiased = true further removes the positive bias of wPLI caused by a limited number of epochs.

Other ways to reduce volume conduction:

  • Spatial filters: the surface Laplacian or current source density (CSD) reduce the spread of potentials and the effect of the reference.
  • Source space: estimating the activity of brain sources (e.g. beamformers, minimum-norm estimates) before connectivity analysis; source leakage remains and must also be considered.
  • Distance: interpreting connectivity mainly between distant channels.

Graph-Theoretic Measures

Graph theory describes the large-scale organization of brain networks from connectivity matrices:

  • nodes: channels or brain regions
  • edges: connectivity values between nodes (weighted, or binary after thresholding)
Metric Description Use
Degree / strength number (or sum of weights) of the connections of a node identifying hubs
Clustering coefficient how strongly the neighbours of a node are connected with each other local (segregated) processing
Characteristic path length average shortest path between all pairs of nodes global (integrated) processing
Global efficiency average inverse shortest path length integration (robust to disconnected nodes)
Modularity strength of the division of the network into modules (communities) functional modules
Small-worldness high clustering with a short path length, relative to random networks efficient network organization

Graph measures depend strongly on the connectivity measure, the number of nodes, thresholding and normalization; they should be compared with appropriate random (null) networks.

Practical Considerations

Preprocessing:

  • Artifacts: remove artifacts (ICA, rejection); eye movements, muscle activity and line noise produce strong spurious connectivity.
  • Reference: the reference is shared by all channels and affects all sensor-level connectivity measures; the average reference, the Laplacian or source space reduce its influence.
  • Filtering: phase-based measures require narrowband signals (e.g. alpha 8–12 Hz, beta 13–30 Hz); zero-phase filters avoid phase distortion.
  • Epochs: equal numbers and lengths of epochs in the compared conditions.

Biases:

  • PLV, coherence and other measures are biased by the number of epochs (with few epochs, random phases give high values); compare conditions with equal numbers of epochs or use debiased measures (e.g. pairwise phase consistency, debiased wPLI).
  • Power differences change the signal-to-noise ratio and therefore the phase estimates: a difference in connectivity between conditions may be caused by a difference in power.
  • Common input from a third region produces connectivity without a direct interaction.

Statistics:

  • Surrogate data and permutation tests: the observed connectivity is compared with a null distribution (e.g. epochs shuffled between channels, time-shifted or phase-randomized signals).
  • Multiple comparisons: many channel pairs and frequencies are tested; use FDR, Bonferroni or cluster-based permutation tests.
  • Effect sizes (e.g. Cohen’s d) quantify the differences between conditions or groups.

Limitations and Challenges

Challenge Description Approach
Volume conduction one source recorded by many channels gives spurious zero-lag connectivity lag-based measures (PLI, wPLI, imaginary coherence), Laplacian/CSD, source space
Reference the shared reference adds common signal average reference, Laplacian, source space
Common input a third region drives both regions multivariate (partial, conditional) measures
Number of epochs measures are biased with few epochs equal epoch numbers, debiased measures
Power differences connectivity estimates depend on the signal-to-noise ratio control for power, report power
Non-stationarity connectivity changes over time time-resolved (sliding window) connectivity
Artifacts eye, muscle and line-noise artifacts are shared by many channels careful preprocessing
Directionality undirected measures do not show the direction of interactions Granger causality, PSI, dPLI, transfer entropy

Summary

Measure Type Frequency-specific Directed Insensitive to zero-lag volume conduction
PLV / ISPC phase yes no no
iPLV phase yes no yes
PLI phase yes no yes
wPLI phase yes no yes
dPLI phase yes yes yes
PSI phase yes yes yes
Coherence spectral yes no no
Imaginary coherence spectral yes no yes
AEC amplitude yes no no (yes with orthogonalization)
Correlation general no (unless filtered) no no
Mutual information, synchronization likelihood non-linear no (unless filtered) no no
Granger causality directed yes (spectral version) yes partly
ITPC phase consistency across trials (not connectivity) yes – –