EEG

The primary principle behind Electroencephalography (EEG) is the measurement of extracellular electric potentials generated by synchronous postsynaptic currents in cortical pyramidal neurons. These currents form current dipoles whose fields propagate through the brain, skull, and scalp—the volume conductor—and are detected non-invasively by surface electrodes. Typical scalp potentials range from $10$ to $100~\mu\text{V}$ and are recorded using differential amplifiers with high input impedance $(>1~\text{G}\Omega)$ to minimize loading effects and preserve signal fidelity.

The physics of volume conduction in EEG is governed by the Poisson equation:

$$\nabla \cdot (\sigma \nabla V) = \nabla \cdot \mathbf{J}_s$$

where $V$ is the electric potential, $\sigma$ is the tissue conductivity $(\approx 0.33~\text{S/m})$ in brain tissue, and $\mathbf{J}_s$ represents primary current sources. For an ideal current dipole of moment $\mathbf{p}$ in a homogeneous medium, the potential at distance $r$ is given by:

$$V(r) = \frac{\mathbf{p} \cdot \mathbf{r}}{4\pi \sigma r^3}$$

With dipole moments on the order of $10~\text{nA m}$ and source–sensor distances of $\sim1~\text{cm}$, one predicts scalp potentials of $\sim10~\mu\text{V}$, in line with empirical observations.

In practice, EEG employs standardized electrode montages such as the 10–20 system, ranging from $19$ up to $256$ channels in high-density arrays. Signals are band-pass filtered between $0.1$ and $100~\text{Hz}$ and sampled at $500\text{–}5000~\text{Hz}$, yielding temporal resolution $<1~\text{ms}$. Spatial resolution is inherently limited by volume conduction and skull attenuation, typically $\sim5\text{–}10~\text{cm}$ for conventional setups and improved to $\sim2\text{–}5~\text{cm}$ with high-density recordings. Typical electrode impedances are maintained $<5\text{–}20~\text{k}\Omega$ to optimize signal-to-noise ratios.