Brain signals, montages, and frequency bands

"They asked me to read a mind. I got a millivolt smear of a billion neurons shouting through a skull, and a solemn argument about which electrode to subtract from which."

An Overpromised AI Agent

What is actually on the wire: postsynaptic potentials and volume conduction

An EEG electrode does not hear neurons fire. Action potentials are too brief and too spatially incoherent to sum at the scalp. What EEG measures is the slower, spatially aligned postsynaptic potentials of cortical pyramidal neurons, whose long apical dendrites are stacked in parallel, all pointing perpendicular to the cortical surface. When thousands of these dendrites are excited together, they act like a field of tiny aligned batteries, and their currents add into a measurable extracellular field. A single neuron contributes nothing detectable; only the synchronized activity of a patch of cortex roughly a few square centimeters across produces a scalp signal you can see.

That field then travels to the electrode by volume conduction: the head is a passive conductor, so current spreads and blurs through the tissue layers before reaching the scalp. Two consequences dominate everything that follows. First, the amplitude is tiny. Scalp EEG is on the order of 10 to 100 microvolts, roughly a thousand times smaller than the ECG of Chapter 29, which is exactly why electrode impedance and amplifier noise, introduced in Chapter 28, are make-or-break here. Second, the signal is spatially smeared: the skull is a poor conductor and acts as a spatial low-pass filter, so a compact cortical source appears at the scalp as a broad blob spanning several electrodes. EEG therefore trades away spatial resolution (centimeters, not millimeters) to buy temporal resolution (sub-millisecond). That trade is the defining property of the modality and the reason the invasive arrays of Section 31.5 exist at all.

The 10-20 system and the montage question

To make recordings comparable across people and labs, electrodes are placed by the international 10-20 system: positions are defined as fractions (10 percent and 20 percent) of the distances between skull landmarks (nasion to inion, and ear to ear), so the same label lands over the same brain region regardless of head size. Names encode region and hemisphere: F for frontal, C central, P parietal, O occipital, T temporal; odd numbers sit on the left, even on the right, and z marks the midline (Cz, Pz). Denser caps (the 10-10 and 10-5 extensions) interpolate more sites for 64, 128, or 256 channels. A consumer headset might expose just Fp1, Fp2, and a couple of others; a clinical or research cap covers the whole scalp.

A montage is the recipe that turns raw electrode potentials into the channels you analyze, and it is entirely a choice about the reference. Four families matter. A referential montage subtracts one shared electrode (an earlobe, the mastoid, or the vertex) from every other; it is simple but inherits any noise on that single reference. A bipolar montage subtracts adjacent electrodes in chains, which emphasizes local gradients and is the traditional reading montage for clinical epilepsy review. A common average reference (CAR) subtracts the mean of all electrodes from each, which is the default for most machine-learning pipelines because it is unbiased toward any one site, though it assumes good coverage and clean channels. A surface Laplacian subtracts a local neighborhood average, sharpening focal sources by undoing some of the volume-conduction blur. There is no universally correct montage; each is a spatial filter that trades sensitivity to focal versus distributed sources, and the right one depends on what you are trying to decode.

Frequency bands: the rhythms that carry information

EEG is analyzed in frequency bands because cortical populations tend to synchronize at characteristic rates, and those rates map, imperfectly but usefully, onto brain states. By long convention the bands are delta (roughly 0.5 to 4 Hz, deep sleep and large slow waves), theta (4 to 8 Hz, drowsiness, memory, and mid-frontal effort), alpha (8 to 12 Hz, relaxed wakefulness, strongest over the occiput with eyes closed), beta (12 to 30 Hz, active concentration and motor control), and gamma (above 30 Hz, fast local processing, but faint and easily confused with muscle artifact at the scalp). These boundaries are conventions, not laws of physics; different sub-fields shift them by a hertz or two, so a band definition is a parameter you set and document, not a constant you inherit.

The workhorse feature is band power: the fraction of signal energy inside a band, computed from the power spectral density (PSD) of a short window. The PSD is the same object built in Chapter 7, and for a band \([f_1, f_2]\) the band power is simply the area under the PSD there,

where \(S(f)\) is the one-sided PSD. Two practical wrinkles decide whether this feature is trustworthy. First, EEG spectra are broadly \(1/f\): power falls off steeply with frequency, so raw band powers are dominated by the slow bands and are usually compared as relative band power (each band divided by total power) or in decibels. Second, the sampling rules of Chapter 3 bite hard: line noise at 50 or 60 Hz sits right inside the gamma band, and any muscle tension adds broadband high-frequency power, so a gamma feature is only as honest as the artifact handling that precedes it, which is the whole subject of Section 31.2.

The program below synthesizes a short EEG-like epoch as a \(1/f\) background plus an injected alpha oscillation, re-references two channels to their common average, and computes relative band power with Welch's method. It makes the reference choice and the band-power feature concrete before we clean real recordings in the next section.

import numpy as np
from scipy import signal

fs = 256.0                                   # samples per second
t = np.arange(0, 4.0, 1 / fs)               # a 4-second epoch
rng = np.random.default_rng(0)

# 1/f-ish background: cumulative sum of noise, plus an 10 Hz alpha burst
def channel(alpha_uv):
    pink = np.cumsum(rng.standard_normal(t.size))      # brown/pink-ish drift
    pink = 20 * (pink - pink.mean()) / pink.std()
    alpha = alpha_uv * np.sin(2 * np.pi * 10 * t)      # posterior alpha
    return pink + alpha

Oz = channel(alpha_uv=15)                    # strong occipital alpha
Fz = channel(alpha_uv=2)                     # weak frontal alpha
car = 0.5 * (Oz + Fz)                        # common average reference
Oz_car = Oz - car                            # re-referenced channel

f, psd = signal.welch(Oz_car, fs=fs, nperseg=int(fs))   # PSD of the epoch
bands = {"theta": (4, 8), "alpha": (8, 12), "beta": (12, 30)}
total = np.trapz(psd, f)
for name, (lo, hi) in bands.items():
    m = (f >= lo) & (f < hi)
    rel = np.trapz(psd[m], f[m]) / total
    print(f"{name:>5}: relative power = {100 * rel:5.1f} %")

As Listing 31.1 shows, band power is a handful of lines once the reference and window are fixed, and the alpha injection dominates the relative spectrum exactly as posterior alpha does in a relaxed subject. The same PSD machinery, per channel, is the feature substrate under the classical motor-imagery and SSVEP decoders of Section 31.3, and it is what the EEG foundation models of Section 31.4 aim to replace with learned representations.

import mne
raw = mne.io.read_raw_edf("subject.edf", preload=True)   # channels + montage
raw.set_eeg_reference("average")                          # common average ref
raw.filter(1., 40.)                                       # band-pass in place
psd = raw.compute_psd(fmin=1, fmax=40, method="welch")   # per-channel PSD
alpha = psd.get_data(fmin=8, fmax=12).sum(axis=-1)       # alpha band power