Learn · Three
The shape of a spectrum
Alpha power went up. It is one of the most reported findings in the whole of human neuroscience, and on its own it does not say what happened.
There are two ways for the power in a band to increase, and they mean different things about a brain. Telling them apart takes one extra step that is easy to describe and was skipped for decades. This page is that step.
Part one
What a spectrum is made of
Two things, added together, that are usually treated as one. Separating them takes a model, and the model is simpler than the confusion it removes.
The model
A straight line, with bumps on it
Plot the power spectrum of almost any physiological recording on logarithmic axes and the first thing you see is a downward slope. Low frequencies carry more power than high ones, smoothly, across the whole range. That slope is present whether or not anything is rhythmic, and it is the same scale-free structure that turns up in natural images and natural sound.
Sitting on top of it, where a genuine rhythm exists, are bumps. A model of the spectrum is therefore two things added together: an aperiodic background, described by an offset and a slope, and some number of periodic peaks, each with a centre frequency, a height and a width.
Anatomy of a spectrum · a line, plus bumps
The spectrum with the background subtracted
- The spectrum
- Fitted background
- What is left over
In one sentence
The background is not a nuisance to be subtracted before the interesting part. It is one of the two things a spectrum is made of.
Why it matters
Two ways to make a band get bigger
The question it asks
Did the rhythm change, or did the background move underneath it?
Band power is an integral. Add up the power between 8 and 12 Hz and you get a number, and that number rises if the alpha peak grows taller. It also rises if the peak does not move at all and the background tilts. The integral cannot distinguish the two, because it was never asked to.
Same alpha power, two different brains
A · the rhythm grew
Background held still, peak raised.
B · the background tilted
Peak held still, background flattened.
| Measure | A · rhythm grew | B · background tilted | Tells them apart? |
|---|---|---|---|
| Alpha band power | 11.8 | 11.8 | no |
| Aperiodic exponent | 1.49 | 1.34 | yes |
| Peak height above background | 0.54 | 0.35 | yes |
Baseline band power 8.4, both panels raised to 11.8 by searching for the peak height and the background slope that each land on that number.
Blind to · cause. Band power is a good description of how much energy sits inside a window, and it says nothing about where that energy came from.
In one sentence
A result reported as a change in band power is compatible with two different stories, and the analysis that produced it usually cannot say which one it found.
Part two
What the background is
Having separated it, the slope turns out to be worth measuring in its own right, and to move on timescales that make it something you can couple on.
The aperiodic component
The slope is a finding, not a leftover
For a long time the background was treated as the boring part: whatever is left once the peaks have been taken out, a kind of measurement noise to be normalised away. It is not noise. It varies systematically with age, with arousal, with state of consciousness, and it has been argued to track the balance between excitation and inhibition in the underlying population.
Which makes the confusion in the previous section more serious than a methodological quibble. Two studies could report the same change in band power, with one having found a rhythm and the other having found a change in background state, and nothing in either paper would say so.
In one sentence
Everything a spectrum contains that is not a rhythm is still something. Give it a parameter and it stops being a residual and starts being a measurement.
Back to coupling
A slope that moves is a signal
One exponent per recording is a summary. Fit it inside a sliding window and it becomes a trace, one value per moment, of exactly the kind everything else on this site consumes. At that point the question stops being what a person’s exponent is and starts being whether it moves with something in the world.
The exponent as a trace
Something in the environment
The aperiodic exponent, window by window
- Environmental signal
- Exponent, fitted per window
In one sentence
This is the same promotion the complexity measures get: from a number that describes a recording to a number that varies within one. Only the second kind can be coupled to anything.
The fitting here follows the standard approach: fit the background robustly so that peaks do not drag the slope upward, extract peaks from what is left, then refit the background with the peaks removed. It is simplified against a research implementation, which optimises all the peak parameters jointly rather than estimating each from its own neighbourhood. It was checked against known inputs first: with no peaks present the exponent comes back exactly, and with an alpha peak added it comes back within about a hundredth. Still to be written: the knee, which matters whenever the fitting range is wide enough for the slope to bend, and what changes when the same fit is applied across many channels at once.