EEG and rhythms

The voltage from a few small discs on the scalp is read by how fast it wobbles. Watch the share of power near 10 Hz rise below while the eyes are closed, and the windows that hold a blink get thrown out.

Alpha rises when the eyes close

Put a few small metal discs on someone’s scalp and connect them to a very sensitive amplifier. You see a voltage that wobbles by a few tens of microvolts. It is the summed electrical activity of large groups of brain cells, seen through the skull and scalp. This is the electroencephalogram, or EEG.

Each disc, an electrode, gives one channel, and a clinical recording uses about twenty. On this page I use one channel.

The first thing people ask of an EEG is how fast it wobbles. The answer is sorted into frequency bands, each with a Greek name: delta from 1 to 4 Hz, theta from 4 to 8, alpha from 8 to 13, beta from 13 to 30 and gamma from 30 to 45. These edges are a convention, and books move them by a hertz or two. Gamma reaches higher too, but this page stops at 45 Hz.

The best-known rhythm is alpha, a wave near 10 Hz, strongest at the back of the head. Close your eyes and relax, and it grows. Open them, and it shrinks again. Hans Berger, who recorded the first human EEG, reported this in 1929.

The record

The record lasts 30 s at $f_s=256$ Hz, which is 7680 samples, in microvolts (µV). It adds three parts.

The background is the pink noise of “White, pink, brown” in Power spectral density (24.4). White Gaussian draws from seed 312 go through J. O. Smith’s filter. I run the filter on 2000 extra draws first and drop them, so the record starts settled. Then I scale the rest to an RMS of 10.00 µV.

Pink noise is a common stand-in for the background of an EEG, whose power also falls with frequency. The filter’s coefficients are those of 24.4. At 256 Hz it falls 2.95 dB per octave from 1 to 32 Hz, close to pink’s 3.01.

The alpha rhythm is $A(t)\sin(2\pi\cdot10t+0.4)$, with $t$ in seconds. Its height $A(t)$ is 15 µV from 10 to 20 s, while the eyes are closed, and 3 µV before and after. The change is a straight ramp lasting 0.5 s, centred on 10 s and on 20 s.

Then four blinks. The eye holds a small voltage between its front and back, and a blink changes the field that the electrodes near the eyes pick up: a large, slow bump. Each blink here is

in µV, the bell of “A Gaussian keeps its shape” in Transforms of common signals (8.3), with half-width 0.1 s. Its centre $t_\text{b}$ comes from seed 3121: 3.04, 6.99, 21.74 and 23.11 s, two before the eyes close and two after they open.

Without the blinks, the record stays within ±47.7 µV. With them, it spans −46.3 to 111.5 µV.

Band power, window by window

One Welch estimate of the whole 30 s would mix the eyes-open and eyes-closed parts. “Short window or long window” in Spectrograms & the STFT (15.5) shows the way out: cut the record into short windows, and estimate each window’s PSD.

Each window here is 2 s long, 512 samples, and the next starts 1 s later. That gives 29 windows, centred at 1, 2, …, 29 s.

In each window I use Welch’s method of Averaged periodograms: Bartlett and Welch (25.2): Hann segments of 1 s with half overlap, which gives 3 segments per window. As SciPy’s welch does by default, each segment’s mean is removed first.

The result is the one-sided density $\hat S_x^{\text{1s}}(f)$ of Reading a spectrum: scaling and units (15.4), in µV²/Hz. Its bins are $f_s/256=1$ Hz apart.

A band’s band power is the power of the part of the signal that lies in the band, in µV². As in “Getting a tone’s power from a PSD” of 15.4, I add the density over the band’s bins and multiply by the bin spacing. I write it $P_\text{alpha}$ for alpha, $P_\text{delta}$ for delta, and so on.

Each band takes its bins from its lower edge up to, but not including, its upper edge. For alpha, I add the estimate at 8, 9, 10, 11 and 12 Hz and multiply by 1 Hz. The bin at 13 Hz belongs to beta.

So the five bands share out the bins from 1 to 44 Hz, each bin to exactly one band, with no gaps. Each bin stands for the 1 Hz strip that starts at it, so together the bands cover 1 to 45 Hz.

The relative band power of alpha is $P_\text{alpha}$ divided by the same sum taken over all bins from 1 to 44 Hz, in percent. That sum is also the sum of the five band powers, so the five shares add up to 100 %. A share does not change when the whole recording gets louder or quieter.

Blinks fake delta

A blink is a slow bump. By 8.3, its spectrum is a bell too, with half-width $1/0.1=10$ rad/s, which is 1.6 Hz. So its energy sits at low frequencies: of the part between 1 and 45 Hz, 99.9 % lies in delta.

A window that holds a blink therefore shows a large delta share that has nothing to do with the brain. A part of a recording that does not come from the brain is an artefact.

The simplest defence is to throw such windows away, which is called rejection. A blink here reaches about 100 µV, and the rest of the record stays within ±47.7 µV. So I reject a window if any of its samples goes beyond ±75 µV.

Watching the shares

The picture at the top of the page works like a sound-level meter set to one octave band, watching one instrument of an orchestra enter and leave. It draws the record and then each window’s band shares.

Notice the alpha readout: 72 % in the first closed-eye windows, 22 % after the eyes open again. Over all nine kept windows with the eyes closed, centred 11 to 19 s, alpha’s share averages 73 %.

The windows centred at 10 and 20 s straddle a change of the eyes. Their alpha shares, 58 % and 61 %, lie in between, and no span counts them.

Now look at the crosses. The ten rejected windows are exactly those that hold a part of a blink above the limit. On average 75 % of their power sits in delta, against 19 % in the windows kept. The kept windows hold no visible part of a blink: the largest blink value inside one is about $10^{-10}$ µV.

Average the trials, keep the response

Some brain responses are tied to an event: a sound, a flash, a letter on a screen. The voltage that follows an event and is caused by it is an event-related potential, or ERP.

A well-known one is the P300, a positive wave some 300 ms or more after an event the person was waiting for, such as a rare tone among common ones.

The trouble is size. The response on this page peaks at 5 µV, while the background is 10 µV RMS, so one trial hides it. The cure is to repeat the event many times, cut out the stretch of record after each one, and average.

Epochs and their average

Each stretch is an epoch, or trial: here 1 s, 256 samples, starting at the event. The index $n$ counts samples after the event.

I write trial $r$ as $x^{(r)}[n]$, the notation for a realisation in Random processes (24.2). Each trial is one realisation, and averaging trials sample by sample is the average across the ensemble of “Across the ensemble, along a realisation”. The average of the first $N$ trials is

The hat means “estimated”, as in 24.1: $\hat\mu_x[n]$ estimates the ensemble mean at each $n$.

Each trial is the response plus that trial’s background. The response is the same every time, so the average keeps it whole. The backgrounds differ, and if they are unrelated to each other, “A longer average: less noise, more delay” of Simple smoothing filters (18.2) applies: averaging $N$ of them divides their RMS by $\sqrt N$.

With 10 µV of background, 100 trials should leave about $10/\sqrt{100}=1$ µV.

The synthetic trials

The response, with $t$ in seconds after the event and the result in µV, is

It is a bell of 5 µV at 300 ms, shaped like a P300, and an earlier dip of 3 µV at 100 ms. On the 256 Hz grid its peak is 5.00 µV, at 301 ms.

The background is a fresh pink stream from seed 3120, made as before: the same filter, 2000 dropped draws, scaled to 10.00 µV RMS over its 25 600 samples. I cut it into 100 epochs, one after another, as if the events came once a second.

One second of pink noise holds more or less of its slow parts, so the epochs differ in size: their RMS runs from 6.92 to 15.09 µV.

Averaging works like stacking many photographs of the same face, taken with a shaky camera: the face sharpens, and the shake blurs away.

Watch the solid average close in on the dashed response as the trials pile up. Its noise left is the RMS of the average minus the response, and 18.2 predicts $10/\sqrt N$ µV for it.

Notice the last frame. After 100 trials the noise left is 1.1 µV, against 1.0 µV for $10/\sqrt{100}$. The peak of the average is 6.09 µV at 293 ms, so the noise left still lifts it by 1.09 µV and moves it 8 ms early.

Why the noise left is not 1.0

The $\sqrt N$ rule assumes that the trials’ backgrounds are unrelated. These are cut one after another from one stream, and pink noise’s slowest parts last longer than a second. A sample and the one a second later have a correlation coefficient of 0.11.

So neighbouring trials do not cancel fully. Counting that correlation, the square root of the expected noise power after 100 trials is 1.14 µV, not 1.0.

One record also scatters about that value. For 2000 other random streams made the same way, the noise left after 100 trials fell between 0.7 and 1.8 µV in 94 % of cases, so 1.1 µV is ordinary.

At ten trials this record was lucky. Its 2.2 µV is below the 3.2 µV of $10/\sqrt{10}$, and only 1.6 % of those 2000 streams did better. So $1/\sqrt N$ tells you the trend, not the value for one record.

What the average assumes

The average rests on two assumptions. The response must be the same in every trial, at the same delay after the event. The background must be unrelated to the event, so that it does not repeat from trial to trial.

If the response wanders in time, the average smears it. Suppose the P300’s delay varies from trial to trial, as a Gaussian draw with a spread of 30 ms. Averaging over many trials adds that spread to the bell’s own, and for bells the squares add, as the variances of independent draws do in Random variables for signals (24.1).

The bell’s half-width becomes $\sqrt{50^2+30^2}=58.3$ ms, and its peak falls to $5\times50/58.3=4.29$ µV. Real P300s do wander, so an averaged P300 is lower and wider than the response in each trial.

If something outside the brain repeats with the event, it survives the average, as the response does. A person who blinks after every flash adds a blink to every trial at the same delay, and the average keeps it.

Worked example

1. Averaging gain. One hundred trials divide unrelated noise by $\sqrt{100}=10$: 10 µV becomes 1 µV in theory. This record measured 1.1 µV, and the correlation of neighbouring trials predicted 1.14 µV. To halve the noise left again you need four times as many trials, 400.

2. Band edges in bins. Segments of 1 s at 256 Hz have bins $256/256=1$ Hz apart. Alpha, 8 to 13 Hz, takes the bins at 8, 9, 10, 11 and 12 Hz, so it is 5 bins wide.

Segments of 2 s would give bins 0.5 Hz apart and 10 alpha bins. But a 2 s window then holds only one segment, $K=1$: the raw periodogram, with the full scatter of 25.2.

3. Predicting alpha’s share. In the nine kept windows with the eyes closed, the background’s five band powers add up to about 50 µV² on average, and its alpha band holds about 4 µV². So only about half of its 100 µV² falls in the bands: pink noise keeps much of its power below 1 Hz, 40 % of it.

The rhythm of height 15 µV has power $15^2/2=112.5$ µV². Its Hann lobe covers 9 to 11 Hz, so all of it lands in alpha. Alpha’s share should be about

That is close to the 73 % of all nine closed windows. The measured value averages the windows’ own shares, which is not quite the share of the averages. With the eyes open again, from 20 to 30 s, the 3 µV rhythm adds 4.5 µV². The same sum, with that span’s background, gives 20 % against the readout’s 22 %.

Where you’ll meet this

Sleep laboratories score a night in 30 s epochs, reading the EEG’s rhythms together with eye and muscle signals. Deep sleep is marked by slow, large delta waves. A lighter stage is marked by sleep spindles: bursts of 11 to 16 Hz that last half a second or more.

Neurofeedback systems show a person the power in one band as it changes, often alpha, and ask them to raise or lower it.

Hearing tests for newborns use averaged responses. Clicks play through a small earphone, and the device averages the EEG after each click to find the response of the brainstem’s hearing pathway. That response is well under 1 µV, so the device averages thousands of clicks.

A brain–computer interface can use the P300 to spell. Letters flash in rows and columns while the user counts the flashes of the wanted letter. Those flashes bring a P300, and averaging over several rounds finds the row and column that did.

The heart’s signal is processed with filters and a detector instead, in ECG processing (31.1). Band power over short windows returns for machines in Vibration and machine monitoring (32.1).

For more, see M. X Cohen, Analyzing Neural Time Series Data (2014), ch. 11 and 18; S. J. Luck, An Introduction to the Event-Related Potential Technique (2nd ed., 2014), ch. 1 and 8; and L. Sörnmo and P. Laguna, Bioelectrical Signal Processing in Cardiac and Neurological Applications (2005), ch. 3 and 4.

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