What Volume I left as a black box
In Vol I you learned the BCI pipeline: acquire, filter, extract features, decode. But two arrows in that chain were black boxes. First, how a neuron's action potential turns into an extracellular voltage an electrode can sense. Second, how the electrode converts that voltage into a number. This track opens both boxes, and everything downstream — every filter, feature and decoder — inherits the assumptions we make here.
The quasi-static approximation
The exact description of the field is Maxwell's equations, but neural signals live below about 10 kHz. At those frequencies the electromagnetic wavelength in tissue is on the order of kilometers, and inductive, propagation and displacement-current terms are negligible. This is the quasi-static approximation: the electric and magnetic parts decouple, and at each instant the extracellular potential obeys a Poisson-type equation.
\nabla \cdot \big(\sigma(\mathbf{r})\, \nabla V(\mathbf{r})\big) = -I_v(\mathbf{r})Poisson equation for the extracellular potential; σ is tissue conductivity and I_v the volume current-source density.
This is the master equation that turns neural currents into the voltages you record. It says: wherever neurons dump current into the tissue (the source I_v), the surrounding voltage V must bend and spread so that current can flow away through the tissue's conductivity \sigma — much like water pressure self-adjusts so that whatever you pour in flows out.
- \sigma(\mathbf{r})
- How easily the tissue at location \mathbf{r} carries current (conductivity); larger \sigma lets current flow more freely.
- V(\mathbf{r})
- The extracellular voltage at each point — exactly what your electrode measures.
- \nabla\cdot
- The divergence: how much current is spreading out from a point.
- I_v(\mathbf{r})
- The current neurons inject per unit volume at that point — the source term.
In a homogeneous isotropic medium (σ constant) this reduces to \nabla^2 V = -I_v/\sigma, and the potential from a single point current source I is the workhorse formula of extracellular biophysics.
V(r) = \dfrac{I}{4\pi \sigma r}Point current source in an infinite homogeneous medium (σ ≈ 0.3 S/m in cortex). The potential falls as 1/r.
The simplest possible case: a single point of current sitting in a uniform sea of tissue. The voltage you measure just fades with distance as 1/r — double the distance and you halve the voltage.
- I
- The current injected at the point source (in amps).
- \sigma
- Tissue conductivity, about 0.3 S/m in cortex.
- r
- Distance from the source to your electrode.
- 1/r
- The falloff law — voltage shrinks in proportion to distance.
At 0.1 mm from the source the voltage is ten times larger than at 1 mm.
A neuron is a distribution of sources and sinks
A neuron injects no net charge — charge is conserved. Where current enters the cell (inward Na+ influx during a spike) the extracellular medium sees a sink; where current returns to the medium (capacitive and leak return) it sees a source. These are the primary and return currents. Seen from a distance, the leading approximation to the field is a current dipole.
V(\mathbf{r}) \approx \dfrac{p\,\cos\theta}{4\pi\sigma r^{2}},\qquad p = I\,dFar field of a current dipole of moment p = I·d; it falls as 1/r² and depends on orientation θ.
A real neuron has current entering in one place and leaving in another — a tiny 'battery' called a dipole. Far away its voltage falls off faster (1/r^2) than a lone source, and it depends on viewing angle: strong off the ends, zero off the side.
- p = I\,d
- The dipole moment — current times the separation d between source and sink; the 'strength' of the battery.
- \theta
- The angle between the dipole's axis and the direction to your electrode.
- \cos\theta
- Makes the signal maximal in line with the dipole and vanish to the side.
- 1/r^{2}
- The faster falloff of a dipole compared with a single source.
The dipole picture is why geometry matters: aligned pyramidal cells in cortex sum into an 'open field' whose potential reaches far, while symmetric cells (e.g. many interneurons) make a 'closed field' that nearly cancels at a distance.
Two signals in one voltage: spikes and LFP
The recorded trace is a superposition. Fast Na+/K+ spike currents create the extracellular action potential — on the order of tens to a couple hundred µV within roughly 50–100 µm, dominant in the ~300 Hz–5 kHz band. Slower synaptic and dendritic currents create the local field potential (below ~300 Hz). Same physics, different current sources and spatial scales.
The spatial reach differs too: a spike is only visible very locally (which is exactly what makes spike sorting possible), whereas the LFP mixes contributions from a much larger volume.
Where this track goes
- This guide: the field a current source makes in tissue (the biophysics you record through).
- Next: reconstruct the source from the field — CSD and LFP genesis.
- Then the electrode itself: the double layer, half-cell potential and Randles circuit.
- Then impedance, thermal noise and electrode materials.
- Finally the write direction: stimulation biophysics and the open problems.