EEG

Electroencephalogram (EEG) is a passive sensing technique that measures electrophysiological brain activity by recording differences in electric potentials, i.e., voltages, on the scalp.

Theory

From the quasi-static approximations of Maxwell’s equations, we obtain the system of equations:

$$\begin{array}{rl}\nabla \cdot (\sigma \nabla \varphi )& =\nabla \cdot {\mathbf{J}}_p\\ \sigma \nabla \varphi & =\frac{1}{{\mu }_0}(\nabla \times \mathbf{B})-{\mathbf{J}}_p\end{array}$$

where
$\sigma (\mathbf{x},t)$	electrical conductivity, measured in siemens per meter $\frac{\text{S}}{\text{m}}$
$\mathbf{x}$	position vector in three-dimensional space, typically expressed as $\mathbf{x}=(x,y,z)$ and measured in meters $\text{m}$
$t$	time, measured in seconds $\text{s}$
$\varphi (\mathbf{x},t)$	electric potential, measured in volts $\text{V}$
${\mathbf{J}}_p(\mathbf{x},t)$	current density vector field due to primary neuronal activity, measured in amperes per squared meter $\frac{\text{A}}{{\text{m}}^2}$
${\mu }_0$	permeability of free space, measured in henry per meter $\frac{\text{H}}{\text{m}}$
$\mathbf{B}(\mathbf{x},t)$	magnetic field, measured in tesla $\text{T}$

EEG measures the electric potential $\varphi$ at electrode positions ${\mathbf{x}}_{\text{sensors}}$ relative to a reference electric potential at an electrode position $A$, i.e., voltages: $V({\mathbf{x}}_{\text{sensors}},t)=\varphi ({\mathbf{x}}_{\text{sensors}},t)-\varphi (A,t)$

Source reconstruction

We can also use EEG to reconstruct primary current sources from electric potential recordings.

We make explicit the dependence of voltage recordings $V({\mathbf{x}}_{\text{sensors}},t)\in Y$, measured in volts $\text{V}$, at sensor positions ${\mathbf{x}}_{\text{sensors}}$, on the primary current density ${\mathbf{J}}_p(\mathbf{x},t)\in X$, measured in amperes per squared meter $\frac{\text{A}}{{\text{m}}^2}$, due to neuronal activity:

$$\begin{array}{c}V({\mathbf{x}}_{\text{sensors}},t)=\mathcal{F}{\mathbf{J}}_p+\delta V\end{array}$$

where
$X$	Hilbert space of functions representing admissible current densities
$Y$	Hilbert space of functions representing admissible voltage recordings
$\delta V({\mathbf{x}}_{\text{sensors}},t)$	noise component of the $V({\mathbf{x}}_{\text{sensors}},t)$ signal
$\mathcal{F}:X\to Y$	operator that models how ${\mathbf{J}}_p(\mathbf{x},t)$ gives rise to $V({\mathbf{x}}_{\text{sensors}},t)$ in the absence of noise

We can define the action of $\mathcal{F}$ on ${\mathbf{J}}_p$ more precisely as:

$$\begin{array}{c}(\mathcal{F}{\mathbf{J}}_p)(x_{\text{sensors}},t)=\mathcal{R}{\varphi }_{{\mathbf{J}}_p},\end{array}$$

where
${\varphi }_{{\mathbf{J}}_p}(\mathbf{x},t)\in \Phi$	electric potential, measured in volts $\text{V}$, that depends implicitly on ${\mathbf{J}}_p(\mathbf{x},t)$ through the quasi-static approximations of Maxwell’s equations
$\Phi$	Hilbert space of functions that represents all admissible electric potentials
$\mathcal{R}:\Phi \to Y$	operator that extracts ${\varphi }_{{\mathbf{J}}_p}(\mathbf{x},t)$ at sensor positions ${\mathbf{x}}_{\text{sensors}}$ and subtracts the values relative to a reference electric potential ${\varphi }_{{\mathbf{J}}_p}(A,t)$ at a position $A$

Source reconstruction attempts to solve the inverse of equation $(1)$, i.e., determining ${\mathbf{J}}_p(\mathbf{x},t)$ from $V({\mathbf{x}}_{\text{sensors}},t)$. Since $V({\mathbf{x}}_{\text{sensors}},t)$ is unlikely to lie exactly within the range of $\mathcal{F}$ due to the presence of noise and modeling inaccuracies, the problem is reformulated as finding the optimal ${\mathbf{J}}_p^{†}(\mathbf{x},t)$ whose response $\mathcal{F}{\mathbf{J}}_p^{†}$ best matches $V({\mathbf{x}}_{\text{sensors}},t)$. This reformulation involves minimizing the distance between $V({\mathbf{x}}_{\text{sensors}},t)$ and $\mathcal{F}{\mathbf{J}}_p$, defined using a suitable metric and encoded as an objective function, or in the sense of the calculus of variations, a functional $J$:

$$\begin{array}{c}\underset{{\mathbf{J}}_p}{\min }J[{\mathbf{J}}_p].\end{array}$$

A common choice for $J$ is the least-squares misfit:

$$\begin{array}{c}{J}_{\text{LS}}[{\mathbf{J}}_p]=\frac{1}{2}\Vert V({\mathbf{x}}_{\text{sensors}},t)-\mathcal{F}{\mathbf{J}}_p{\Vert }_2^2\end{array}$$

where $||V({\mathbf{x}}_{\text{sensors}},t)|{|}_2^2$ is the $L^2$ norm squared in the space of vectors indexed by ${\mathbf{x}}_{\text{sensors}}$ (discrete) and $t$ (continuous, say).

Conductivity reconstruction

We might also be able to reconstruct for conductivity in the same manner that we reconstruct for the primary current sources. Compared to source reconstruction, the proposed method sets ${\mathbf{J}}_p=0$, simplifying the system of equations, and reconstructs for $\sigma (\mathbf{x},t)$. It is hypothesized that $\sigma (\mathbf{x},t)$ will vary locally in time according to primary current sources, and that the newly formed problem of scalar field reconstruction will have a smaller solution space compared to the vector field reconstruction of ${\mathbf{J}}_p(\mathbf{x},t)$, easing the ill-posedeness of the inverse problem.