What the sensors actually see (a Vol I recap, sharpened)
In Volume I you learned that scalp EEG reflects the summed activity of many neurons. The graduate version is more specific and more constraining: a sensor does not see spikes or single cells, but the extracellular potential produced by slow post-synaptic currents flowing along the long apical dendrites of spatially aligned cortical pyramidal cells. Only when tens of thousands of these cells share a geometry and are active near-synchronously do their fields add rather than cancel — which is why EEG is exquisitely sensitive to organized cortical rhythms and nearly blind to disorganized deep firing.
The convenient abstraction for a small patch of such coherent activity is the equivalent current dipole — a single vector \mathbf{p} (units A·m) whose direction is the net current flow and whose magnitude scales with the synchronously active population. Everything downstream — forward models, inverse solvers, beamformers — is built on this one idealization, so it pays to hold it precisely: a dipole is a source–sink current pair, not a charge.
The quasi-static approximation and the governing equation
At EEG/MEG frequencies (roughly DC to a few hundred hertz) and head-sized distances, the electromagnetic wavelength is enormous and capacitive/inductive effects in tissue are negligible. This is the quasi-static approximation: Maxwell's equations decouple, propagation delay effectively vanishes, and the electric potential obeys a Poisson-type equation at every instant. It is what lets us treat each time sample independently.
\nabla\cdot\left(\sigma\,\nabla\phi\right) = \nabla\cdot\mathbf{J}^{p}The quasi-static Poisson equation. Ohmic (volume) currents σ∇φ are driven by the divergence of the primary neural current J^p — the source term. Conductivity σ varies sharply across tissues.
The same physics as inside cortex, now for the whole head: neural currents \mathbf{J}^p act as the source, and the head's tissues — with wildly different conductivities — shape how the resulting voltage \phi spreads to the scalp. Skull, brain and scalp each bend the current differently.
- \phi(\mathbf{r})
- The electric potential throughout the head.
- \sigma
- Tissue conductivity, which jumps sharply between brain, skull and scalp.
- \mathbf{J}^{p}
- The primary neural current — the true source of the signal.
- \nabla\cdot
- Divergence; the source term is where this primary current diverges.
Read this as bookkeeping of current. The neural generator injects a primary current \mathbf{J}^p; charge conservation forces return currents to spread through the entire head volume — brain, CSF, skull, scalp — and it is those return currents, arriving at the electrodes, that we measure. The source is local; the fields it creates are global. That single fact is the origin of every difficulty in this track.
One dipole, an infinite medium, and then a real head
In the simplest case — a single dipole \mathbf{p} at position \mathbf{r}_0 inside an infinite, homogeneous medium of conductivity σ — the Poisson equation has a clean closed-form solution.
\phi(\mathbf{r}) = \frac{1}{4\pi\sigma}\,\frac{\mathbf{p}\cdot(\mathbf{r}-\mathbf{r}_0)}{\lVert\mathbf{r}-\mathbf{r}_0\rVert^{3}}Potential of a current dipole in an infinite homogeneous conductor. Note the 1/r² falloff and the cosine dependence on dipole orientation relative to the sensor direction.
One neural dipole in a uniform head produces a voltage that fades as 1/r^2 and depends on orientation — strongest along the dipole's axis, zero broadside. It's the building block from which any source configuration is assembled.
- \mathbf{p}
- The dipole moment — the strength and direction of the source.
- \mathbf{r}-\mathbf{r}_0
- The vector from the source location \mathbf{r}_0 to the observation point.
- \lVert\mathbf{r}-\mathbf{r}_0\rVert
- The distance between source and observation point.
- \mathbf{p}\cdot(\mathbf{r}-\mathbf{r}_0)
- A dot product giving the cosine (orientation) dependence.
The real head is neither infinite nor homogeneous. The field must cross concentric shells of wildly different conductivity: brain and scalp conduct moderately, CSF very well, and the skull perhaps tens of times more resistively than the tissue around it. Every boundary bends and attenuates the field. The dominant effect is the skull acting as a thick insulating shell that spreads current tangentially before it ever reaches the scalp.
Spatial low-pass filtering and the reference problem
The combined effect of volume conduction is a spatial low-pass filter: a point source in cortex becomes a broad, centimeters-wide blob on the scalp. Two nearby cortical patches produce scalp maps that overlap almost completely, which is the root cause of EEG's poor spatial resolution — and precisely why source localization is worth doing at all. Note this is a spatial, not temporal, smoothing: EEG keeps millisecond timing while losing millimeter geography.
A second, often-underappreciated complication: EEG measures potential differences, so every number depends on the choice of reference. There is no true zero of potential on the head; an average reference, a linked-mastoid reference, or a surface Laplacian each define a different signal — and therefore a different topography feeding your inverse solver. This is the reference problem, and it silently reshapes maps before any source estimation runs.
MEG's complementary geometry
Because magnetic fields pass through skull and scalp almost undistorted, MEG largely sidesteps the conductivity-blurring that plagues EEG — but it pays with a blind spot: a purely radial dipole (pointing straight out of a spherical head) produces essentially no external magnetic field. MEG therefore images mostly the tangential sources in sulcal walls, while EEG sees both radial and tangential. Combining the two constrains the inverse problem better than either alone, a fact we will exploit in the last guide. The frontier hardware chasing this — the optically pumped magnetometer — reappears there too.