Connectivity and Spatial Analysis

Level 5 starts from one number. Put a single generator in a head, take the two electrodes furthest apart on the scalp, and their coherence is exactly 1 — not high, not approximately, at every frequency and any recording length. That is not a simulation result; it is what falls out of the arithmetic when every sensor carries a real scalar multiple of one waveform. The seven lessons are what you can still say afterwards: the measures that instantaneous mixing cannot manufacture and what each of them becomes blind to in exchange; the surface Laplacian, which is reference-free and sharpens the map but removes deep and broad generators along the way; the forward model, which makes the mixing explicit and turns out to be the only well-posed part of source analysis; the inverse problem, which is not; leakage, which is volume conduction reappearing in source space wearing anatomical labels; and common spatial patterns, where the same mixing becomes an advantage for prediction and a trap for interpretation.

Two threads run through it. The first is that robustness is always bought with blindness, in a place you can name. Imaginary coherence and wPLI cannot be produced by zero-lag mixing and cannot see a genuine zero-lag interaction; CSD is reference-free and cannot see a deep generator; orthogonalization removes leakage and removes real instantaneous coupling with it; a beamformer suppresses interference and cancels correlated sources. In every case the honest report names the trade, so that a reader knows which negative results are uninformative. The second is the gap between what a picture shows and what it establishes, which is why pf-deep-source-claims exists: a deep dipole and a broad superficial patch produce the same scalp map to within a few percent at every depth tested, so no method operating on those measurements can separate them, however sharp the colours. L5.2 is the fourth stop on the statistics thread from L3.7 and L4.7 to L6.1 and L6.4 — a connectivity matrix is a multiple-comparisons problem with a floor that is not zero, and a surrogate null has to be built through the whole pipeline.

One constraint shapes what this level ships, and it is teaching material rather than an apology. No realistic head model is available, because the two template anatomies that would supply one do not clear the site’s licensing policy: ds-fsaverage is governed by the FreeSurfer licence, which permits derivative works only by propagating the whole agreement onto every copy, and ds-mne-sample’s licence is genuinely contested and remains an open decision for this site’s author. So the forward-model comparison ships as two concentric spheres, one shell against four, which asks the same question of the same physics and is answerable from what ships — and the lessons say plainly what that comparison cannot show, and why nothing better is here. L5.4’s “template versus individual MRI” objective is precisely about the fact that a template anatomy is somebody’s data with somebody’s terms attached.