Topographies and scalp maps

Why this matters

A scalp map is the most persuasive figure in an ERP paper and the easiest to over-read. Almost all of its surface is interpolated, its edges are extrapolated from nothing, its colours move when the reference changes, and a colour scale chosen per panel can make two identical distributions look different. Reading maps well means knowing which parts of the picture are measurements and which are the drawing routine.

Concepts

What a topography actually contains

A topography is one time point. At that latency you have one number per electrode — nothing more — and the map is a picture of those numbers positioned on a flattened head. Everything between the electrodes is computed by an interpolator, and everything outside the outermost ring of electrodes is computed by the same interpolator with no data on one side.

The information content of a map is therefore exactly the channel count: a 30-channel recording gives a 30-number map, however smooth it looks. This is worth holding on to because smoothness is the thing that makes a map look like a source. It is not one; source estimation is Level 5.

Spherical-spline interpolation

The standard interpolator for EEG topographies fits a smooth function on a sphere that passes through (or near) the measured values at the electrode positions and minimises a bending-energy penalty elsewhere — the spherical-spline family. Two properties matter in practice.

First, it is linear in the channel values. The interpolated surface is a fixed matrix multiplied by the vector of channel values, so the matrix can be computed once per montage and reused for every time point and every condition. That is precisely what this site does: the matrix is computed in Python from the montage and shipped as an asset, and the browser only multiplies (§4.4). It also means a topography can never show anything the channels did not contain.

Second, it has a smoothing parameter. A regularisation term trades fidelity at the electrodes against smoothness between them. Set it too low and a single noisy channel produces a spike in the map; too high and genuine focal structure is flattened. The same interpolation machinery, with the same trade-off, is what fills in an interpolated bad channel in L2.2 — with the difference that there the output is treated as data, and here it is only a picture.

Extrapolation beyond the electrode hull

Inside the convex hull of the electrodes the interpolator is constrained on all sides. Outside it — between the outermost electrodes and the drawn head outline, which on a typical cap is a band all the way around, and a wide region at the bottom — the surface is extrapolated: continued outward from the nearest data by the smoothness penalty alone. Colour there is a property of the drawing routine’s settings, not of the head.

Three practical consequences:

• A maximum that sits at the edge of the map, beyond the outermost electrodes, is not a measurement. It is what the extrapolator did with the gradient at the rim.
• Different software extrapolates differently — to the head outline, to the electrode hull, or not at all — so the same data plotted in two packages can put the apparent maximum in different places.
• The honest fix is to draw the electrode positions on every map. A reader can then see which parts of the picture are supported.

Shape versus amplitude

Two maps can differ in two quite different ways. They can have the same spatial pattern scaled by a different number — the same generators, more or less active — or they can have different patterns, which requires a different configuration of generators. Only the second is a claim about where the activity came from.

The distinction is invisible on raw maps plotted with a shared colour scale, because a scaled map looks like a weaker version of the same thing only if you happen to notice that the contours are in the same places. The way to decide is to normalise: divide each map by a norm (its maximum absolute value, or its root-mean-square across channels) and plot the result. Maps that differed only in amplitude become identical; maps with different generator configurations stay different.

Two cautions. Normalising makes amplitude differences invisible, so it answers one question by discarding the other — show both. And the comparison is only as good as the noise: at low signal-to-noise, two normalised maps of the same generator differ visibly just from noise, so a shape difference needs a statistical argument, not an eye test. TODO(confirm): formal tests of topographic dissimilarity (for example the global map dissimilarity family) are outside the site’s reading list and this lesson does not cover them.

Reference dependence

Re-referencing subtracts the same value from every channel at each time point (L2.3). On a map that shifts the whole surface up or down by a constant. It therefore cannot change the gradients — the pattern of differences between electrodes is exactly preserved — but it does move where the surface crosses zero, and a colour scale centred on zero turns that into a dramatically different picture: red and blue swap regions while nothing about the underlying distribution has changed.

So the rule for reading maps across papers: the positions of the extrema and the direction of the gradients are comparable; the colours and the zero-crossing are not. And the rule for producing them: state the reference in the caption, keep the scale fixed across the panels being compared, and show a map under a second reference when a claim rests on its appearance. This is pf-reference-changes-everything in its topographic form.

Sequences of maps

One map is a time point; a component is an interval. Plotting a row of maps at evenly spaced latencies — with a single colour scale across the row — shows what a single map cannot:

• whether the distribution is stable across the interval, which is the evidence that one configuration of generators is active throughout, or drifts, which says at least two components overlap;
• when the distribution changes faster than the waveform does, which often marks the handover from one component to the next;
• whether a peak in the waveform corresponds to a change in the map at all.

Choosing the latencies deserves the same discipline as choosing a measurement window (L3.3): evenly spaced latencies across a pre-stated interval, not a hand-picked set that shows the effect at its best.

The data behind this lesson

• ds-erpcore P3, CC BY 4.0, open access, per-subject downloadable; Biosemi ActiveTwo, 30 EEG + 3 EOG channels in a 10-20 placement scheme, 1024 Hz, CMS reference, 60 Hz mains, no software filters, 40 participants per paradigm. TODO(confirm): the author mirrors the ERP CORE entry into the catalogue registry and signs off the dataset page (§10.11 item 8); the shipped asset sidecars also record a licence conflict in the source — the OSF node record says CC BY 4.0, the per-paradigm component’s own LICENSE file says CC BY-SA 4.0 and its dataset_description.json says CC0 — which the author reconciles (§13 item 22).
• The widget draws from grand-average evoked arrays computed under several references, plus a spherical-spline interpolation matrix for the 30-channel montage computed once in Python and shared between the ERP widgets. Points outside the head circle are not drawn.
• The notebook produces topomap sequences for the N170 and the P3 under two references, which is the figure the exercise is keyed to.

Explore

Work it in this order: move the time cursor slowly through the component and watch the map rather than the waveform, then look at the sequence strip to see the same thing laid out; turn on shape-only normalisation and see which pairs of maps collapse onto one another; change the reference with the scale locked and confirm that the gradients hold still while the colours move; finally, look at the outermost electrodes and at the surface beyond them, and notice how much of the coloured area has no electrode behind it.

Practice

The notebook plots topomap sequences for the N170 and the P3 from ds-erpcore under two references, with and without normalisation, drawing the electrode positions on every map. Its final cell identifies the map pairs the exercise asks about.

Exercises