Source-space connectivity and leakage

Objectives

• Explain leakage in source space
• Apply symmetric orthogonalization
• Parcellate and compute connectivity between parcels
• Interpret source-space connectivity conservatively

Why this matters

The obvious fix for L5.1 is to stop computing connectivity between electrodes and start computing it between sources. It is a real improvement and it does not solve the problem. Source estimation redistributes the mixing rather than removing it, the redistributed version is still instantaneous and still zero-lag, and it now wears anatomical labels — which makes it far more persuasive and no more true.

Concepts

Leakage is the resolution matrix, read off the diagonal

L5.5 defined the resolution matrix Res = W·L, and said that a row of it is a cross-talk function: where the estimate at one location actually draws its signal from. That row is leakage. There is no separate phenomenon to learn here; source leakage is the off-diagonal of an object you already have, and can compute from the inverse operator and the leadfield with no data at all.

The mechanism is unavoidable, not a defect of any particular method. Every linear inverse estimate is ĵ = W·x: each source time course is a weighted sum of all the channels. Two different source locations therefore draw on overlapping sets of channels, and their estimated time courses share signal by construction, however far apart they are in the brain. A single generator in one place produces non-zero estimated activity over a broad region of the source space, with exactly the same structure L5.1 described at the sensors — and for exactly the same reason, since the weights are real numbers.

Three properties follow, and each has a consequence:

• Leakage is instantaneous and linear, so it produces zero-lag relationships. That is what makes the L5.2 toolkit apply unchanged in source space: the imaginary part of coherency, PLI and wPLI, and orthogonalized envelope correlation are the measures that cannot be manufactured by leakage, and coherence, PLV and raw envelope correlation are the ones that can.
• Leakage is spatially structured, not uniform. It is largest between nearby locations, larger for deep locations, and its pattern depends on the montage, the head model, the prior and the regularization. Two studies with different pipelines have different leakage patterns and therefore different connectomes from the same brains.
• Leakage is computable in advance. You can calculate, for any pair of parcels, how much of one’s estimate comes from the other under your own operator, before you look at any data. Doing so and reporting it is the difference between a source connectivity analysis and a picture.

Orthogonalization: removing the zero-lag part

The standard correction is to remove, at each time and frequency, the component of one signal that is exactly in phase with the other — the only part leakage can produce — and to compute the connectivity measure on what remains.

Pairwise orthogonalization does this for one pair at a time: project signal A onto B, subtract, keep the residual, then correlate the two amplitude envelopes. The approach is the one (Hipp et al., 2012) introduced for envelope coupling. It has two awkward properties. It is asymmetric — orthogonalizing A with respect to B does not give the same answer as B with respect to A — which is usually handled by averaging the two directions. And it is inconsistent across a network: correcting each pair separately does not produce a set of signals that are jointly leakage-corrected, so the corrected matrix is not the connectivity of any single set of time courses.

Symmetric multivariate orthogonalization corrects all parcel time courses simultaneously, finding the set of mutually orthogonal signals closest to the originals, so the corrected time courses are a coherent set that can be used for any subsequent analysis. TODO(confirm): the symmetric multivariate method and its properties are not covered by an entry in this site’s reading list and must be attributed to a primary source before publication.

What every version of this costs is the same thing wPLI costs, and it is worth stating in the strongest form: orthogonalization removes all zero-lag coupling, whether it came from leakage or from the brain. Genuine interactions at very short delays, and common drive to two regions, are removed along with the artifact. A leakage-corrected connectome is a map of lagged relationships, and calling it a map of connectivity overstates it.

There is a second, subtler cost. Orthogonalization is applied to estimated time courses that are themselves the output of a smoothing operator, so what is removed is the zero-lag component as the operator represented it. Where leakage is very strong — adjacent parcels, deep parcels — subtracting the in-phase component can remove most of the signal, leaving a residual dominated by noise, and residual-on-residual correlations between two noise-dominated estimates are not informative. Report which parcel pairs had most of their variance removed.

Parcellation, and the sign problem

Working with thousands of source locations is neither tractable nor interpretable, so estimates are aggregated into parcels from an anatomical or functional atlas, and connectivity is computed between parcel time courses. Two choices dominate the result.

Which atlas. Anatomical atlases differ in the number of parcels and where the boundaries sit, and functional ones differ again. Parcel size is the important variable: bigger parcels mean more averaging and less leakage between neighbours, and also less spatial specificity and more chance that one parcel spans two functionally distinct regions. A connectome is not comparable across atlases, and the atlas belongs in the first sentence of the methods.

How the parcel time course is formed. This is where a subtle and common error lives. The orientation of each estimated source within a parcel is arbitrary in sign — a dipole pointing one way with a positive time course and the opposite way with a negative one are the same physical source — and on a folded cortical surface neighbouring vertices genuinely point in opposite directions. Taking a plain mean over the vertices of a parcel therefore lets them cancel, and the cancellation is worst exactly where the folding is most complex. The parcel time course comes out attenuated and dominated by whichever vertices happened not to cancel.

The usual fixes: align the signs to a reference (commonly the sign of the leading eigenvector of the parcel’s data or of its leadfield) before averaging, or take the first principal component of the parcel’s time courses instead of the mean, or use a single representative vertex. They give different answers. Which one was used belongs in the methods, because it changes the amplitude, the spectrum and the connectivity of every parcel.

Interpreting conservatively

A source connectivity result is the product of a long chain — reference, cleaning, head model, coregistration, inverse method, regularization, atlas, parcel aggregation, leakage correction, connectivity measure, statistic — and each link changes the answer. The corresponding discipline:

• Report the leakage you have, not the leakage you assumed away. For every parcel pair you make a claim about, compute the cross-talk under your own operator and report it. A claimed connection between two parcels whose cross-talk is large is not interpretable, corrected or not.
• Use a measure that leakage cannot manufacture, and say what that measure is blind to (zero-lag coupling), so the reader knows which negatives are uninformative.
• Build the null through the whole pipeline. Surrogates computed on sensor data and then passed through the inverse, the parcellation and the correction — the same path the real data took. A null computed on parcel time courses alone omits everything the pipeline did.
• Correct for the number of pairs. 68 parcels give 2278 pairs. This is the same thread as L3.7, L4.7 and L6.1 (pf-uncorrected-timepoint-tests).
• Do not select parcels from the data you then test (pf-post-hoc-windows in its spatial form; L6.4).
• Do not compare connectomes across pipelines — including across papers — without establishing that the leakage structures are comparable. They usually are not.
• Prefer within-study contrasts. A difference between conditions computed through one pipeline shares its leakage structure between the two conditions, so much of the systematic error cancels. This is the strongest thing source connectivity does, and it is much weaker than a connectome.

One more reporting habit, because it is the one that turns leakage correction into information rather than a step. If you report how many parcel pairs survive a threshold, report the count before and after correction — and read the difference, not either count. The difference is how much of the apparent connectivity was zero-lag. It is also a lower bound on the artifact rather than a measurement of it: orthogonalization removes genuine instantaneous coupling as well, so a large drop is consistent with “most of it was leakage” and equally with “much of it was real and instantaneous”, and the analysis cannot separate those two.

What ships on this site, and what does not

Nothing does. A parcel-level analysis needs a cortical surface and an atlas defined on it; the only template anatomy available is ds-fsaverage, whose licence does not clear the site’s gate (L5.4), and ds-mne-sample’s licence is contested and awaits a decision from the author. So there is no shipped parcellation, no shipped leakage matrix, and no shipped connectome anywhere in this level, and the exercise below that asks for a count of surviving parcel pairs carries no answer key for that reason rather than by oversight — it asks instead for the count and the sentence that would have to accompany it.

What can be computed without any anatomy is more than it sounds: leakage is W·L, and W and L can both be built on a sphere model with a volume source space (L5.5). The shape of the leakage — how it falls with distance, how it grows with depth, how it changes with regularization — is available that way, and it is the part of this lesson that generalises.

The data behind this lesson

• ds-lemon eyes-closed resting data is what the notebook analyses. TODO(confirm): ds-lemon’s exact licence terms are recorded as unverified in the site’s catalogue, although both recorded statements are permissive.
• ds-fsaverage is named in this lesson’s frontmatter because the lesson is about why the source-space half cannot be run here. No asset derived from it ships and no notebook downloads it.
• No widget ships for this lesson. The L5.1 simulator demonstrates the sensor-space version of the same mechanism and is the right thing to return to when the source-space version stops feeling inevitable.

Practice

Source-space connectivity and leakage: leakage read off the resolution matrix, a simulation with one known coupling, pairwise and symmetric orthogonalisation, and the same measures on ds-lemon — on geometric regions, because no parcellation ships nb-5-6-source-connectivity

Level 5 ~3 min

notebooks/L5/nb-5-6-source-connectivity.ipynb

Downloads from ds-lemon.

Open in Colab Download Read it here

The notebook computes the leakage structure itself — the resolution matrix of an inverse operator built on a sphere model, and the cross-talk between regions as a function of distance, depth and regularization — and then computes wPLI and orthogonalized envelope correlation on ds-lemon with and without leakage correction, so that the size of the correction is a measured quantity rather than a step. The parcel-level analysis on a cortical atlas that §6 asks for cannot be run from this site’s notebooks while the template anatomy fails the licence gate, and the notebook says so where that cell would have been.

Exercises

Exercise ex-5-6-orthogonalization

Multiple select

Which statements about leakage correction by orthogonalization are true? Select all that apply.

Options (select all that apply)

It removes the component of one signal that is exactly in phase with the other, which is the only component leakage can produceIt removes genuine zero-lag interactions between the two regions along with the leakageAfter correction, a non-zero connectivity value can no longer be an artifact of the inverse operatorPairwise orthogonalization is asymmetric, so correcting A against B is not the same as correcting B against AWhere leakage is very strong, most of a parcel time course can be removed, leaving a noise-dominated residual

Check

Hint

Exercise ex-5-6-parcel-sign

Multiple choice

You form a parcel time course by taking the plain mean of the estimated source time courses at every vertex in the parcel. The result is much weaker than the individual vertex estimates. What happened?

Options

The mean is a low-pass filter over vertices, so it attenuated the high frequenciesSource orientation has an arbitrary sign, and on a folded surface neighbouring vertices point in opposite directions, so their time courses cancelled in the averageLeakage from neighbouring parcels added an out-of-phase component that partly cancelled the parcel signalThe inverse operator underestimates amplitude, and averaging compounds the underestimate

Check

Hint

Exercise ex-5-6-surviving-pairs

Numeric

In a parcel-level analysis of eyes-closed alpha connectivity, how many parcel pairs exceed the surrogate threshold before leakage correction, and how many after it?

Your answer parcel pairs

No answer key yet — work it out and compare with the notebook.

Check

Hint

Exercise ex-5-6-conservative-claim

Free response

A source-space analysis reports increased alpha-band connectivity between left and right parietal parcels in one condition. Write the sentence you would accept in a results section, and list what must be reported for a reader to evaluate it.

Your answer

Reveal model answer

Pitfalls

Pitfall

Zero-lag connectivity from one source

Symptom

All sensors coherent; connectivity maps mirror power maps.

Cause

Volume conduction is instantaneous, linear mixing: every electrode sees a fixed weighted sum of every source, with real weights and no delay (L5.1). With a single generator, each sensor’s signal is xi(t) = ai · s(t) — one waveform times a real number. Take the Fourier transform of a segment and Xi = ai · S, so the coherence estimate is

Detect

• Plot the independence floor on the same colour scale as the matrix. 1/K where K is the number of averaged segments. If the floor is visible next to the data, most of this pitfall becomes self-diagnosing. - Compare the connectivity map with the power map. If the “network” is wherever the band is strong, the null hypothesis is one generator, and it has not been tested. - Compute the imaginary par…

Fix

• Report a measure that instantaneous mixing cannot manufacture: imaginary coherence, PLI or wPLI for phase, orthogonalized amplitude-envelope correlation for envelopes. State in the same sentence that these are blind to genuine zero-lag coupling, so that a null result is reported as uninformative about instantaneous interaction rather than as absence of connection. - Sharpen before measuring. Th…

Full entry with example →

In other tools

In other toolsEEGLAB · FieldTrip — names only

The equivalents of what this lesson does, for a reader who works in another toolbox. Function names only: their own documentation is the place to learn how to call them.

EEGLAB

• pop_roi_activityROIconnect plugin (install separately)
• pop_roi_connectROIconnect plugin (install separately)

FieldTrip

• ft_connectivityanalysisFieldTrip
• ft_sourceparcellateFieldTrip
• ft_networkanalysisFieldTrip

Names checked 2026-09-18 against EEGLAB 2026.0.0 (plugins at the versions in EEGLAB’s own plugin list) and FieldTrip 20251218.

Reading

1. Hipp et al. (2012). Orthogonalized envelopes. unverified
2. Bastos & Schoffelen (2016). Connectivity tutorial review. unverified