Basics of the EEG

Abstract

In this chapter, we introduce the essentials of the generation of the EEG. We discuss current dipole sources to model the ionic currents and associated potentials generated by pyramidal cortical cells. We explain why the EEG mainly reflects synchronous activity from large assemblies of these pyramidal cells. In the second part of the chapter, we give an introduction to clinical EEG recordings and its role in ischaemia, epilepsy and coma.

We see in the electroencephalogram a concomitant phenomenon of the continuous nerve processes which take place in the brain, exactly as the electrocardiogram represents a concomitant phenomenon of the contractions of the individual segments of the heart  

— Hans Berger

Appendix: Derivation of the Surface Laplacian

Recall that the divergence of the gradient of the voltage V is defined as the Laplacian of V. If there are no sources or sinks within a particular region were V is measured, this is expressed in Laplace’s equation as

$$\begin{aligned} \nabla ^2 V=0. \end{aligned}$$

(6.15)

that in Cartesian coordinates can be written as

$$\begin{aligned} \frac{\partial ^2V}{\partial x^2}+\frac{\partial ^2V}{\partial y^2}+\frac{\partial ^2V}{\partial z^2}=0. \end{aligned}$$

(6.16)

By taking a coordinate system such that the scalp is on the x-y-plane, and using $E=-\nabla V$, we can rewrite (6.16) as

$$\begin{aligned} \frac{\partial ^2V}{\partial x^2}+\frac{\partial ^2V}{\partial y^2}=\frac{\partial E}{\partial z}. \end{aligned}$$

(6.17)

As in a conducting medium it holds, using Ohm’s law, that $\mathbf{E}=\rho \mathbf{j}$ with $\rho =1/\sigma$ the resistivity (or inverse of the conductivity $\sigma$) and $\mathbf{j}$ the current density it follows that

$$\begin{aligned} \frac{\partial ^2V}{\partial x^2}+\frac{\partial ^2V}{\partial y^2}=\rho \frac{\partial j_z}{\partial z}. \end{aligned}$$

(6.18)

The left hand side of (6.18) is now defined as the surface Laplacian of V

$$\begin{aligned} \text{ Lap}_S(V)=\frac{\partial ^2V}{\partial x^2}+\frac{\partial ^2V}{\partial y^2}. \end{aligned}$$

(6.19)

If Lap$_S(V)$ is nonzero, and assuming no sources are present at the recording position, current lines are diverging below the scalp, which implies the presence of a current source inside the skull.

Assume a univariate function V(x), and take the Taylor series around a point $x=a$. This results in

$$\begin{aligned} V(a+h)=V(a)+V'(a)h + \frac{1}{2!}V''(a)h^2 + \frac{1}{3!}V'''(a)h^3 + \cdots . \end{aligned}$$

(6.20)

Similar for h replaced with $-h$

$$\begin{aligned} V(a-h)=V(a)-V'(a)h + \frac{1}{2!}V''(a)h^2 - \frac{1}{3!}V'''(a)h^3 + \cdots . \end{aligned}$$

(6.21)

Adding these two expressions, we obtain

$$\begin{aligned} V(a+h) + V(a-h)=2V(a)+V''(a)h^2 +\frac{1}{12}V''''(a)h^4+ \cdots \end{aligned}$$

(6.22)

Rewriting results in

$$\begin{aligned} V''(a)=\frac{V(a+h)+V(a-h)=2V(a)}{h^2} -\frac{1}{12}V''''(a)h^2 - \cdots \end{aligned}$$

(6.23)

If we may assume that h is sufficiently small, we arrive at

$$\begin{aligned} V''(a) \approx \frac{V(a+h)+V(a-h)-2V(a)}{h^2}. \end{aligned}$$

(6.24)

Many other possibilities exist to estimate the surface Laplacian of the EEG, including analytical differentiation after first building continuous functions from the recorded data.⁹