The forward problem is easy; reading it back is not
Given the sources, the potential is just a superposition of 1/r terms — the forward problem is well-posed. But the raw potential is a spatially low-pass, blurred image of the currents, because the 1/r kernel is a smoothing operator. To interpret a laminar recording we want the currents back. That is current source density analysis.
Current source density: the Laplacian of the potential
Rearranging Poisson's equation, the volume current source density is minus the divergence of the current density — proportional to the Laplacian of V.
C(\mathbf{r}) = -\nabla\cdot\mathbf{J} = -\nabla\cdot\big(\sigma \nabla V\big)\;\xrightarrow{\;\sigma=\text{const}\;}\;-\sigma \nabla^{2} VCSD as the (conductivity-weighted) Laplacian of the potential; sinks are negative, sources positive.
Current source density (CSD) tells you where current actually enters (sources) or leaves (sinks) the tissue, instead of just the smeared-out voltage. Mathematically it is the curvature (Laplacian) of the voltage — where the voltage bulges, current is being injected.
- C(\mathbf{r})
- Current source density — current per volume entering or leaving at each point.
- \mathbf{J}
- The current flowing through the tissue.
- \nabla^{2}V
- The Laplacian — a measure of how sharply the voltage curves.
- -\sigma
- Conductivity weighting and sign: sinks come out negative, sources positive.
On a linear laminar probe with contact spacing h, the one-dimensional CSD is approximated by a second spatial difference — a discrete Laplacian along depth.
C_k \approx -\sigma\,\dfrac{V_{k+1} - 2V_k + V_{k-1}}{h^{2}}1-D CSD estimate on a laminar array: a three-point second difference at contact k.
On a linear array of contacts threaded through cortex, you estimate CSD at contact k from just its two neighbours — a discrete stand-in for the second derivative. If the middle contact sticks out from the average of its neighbours, current is flowing there.
- C_k
- The CSD estimate at contact k.
- V_{k-1},V_k,V_{k+1}
- Voltages at the contact and its two immediate neighbours.
- h
- The spacing between neighbouring contacts.
- V_{k+1}-2V_k+V_{k-1}
- The three-point second difference — a discrete measure of curvature.
If both neighbours read 0 but the middle contact reads 1, the second difference is nonzero, flagging a source or sink at k.
# 1-D current-source density from a laminar probe
# V: array (n_depth x n_time), h: contact spacing (m), sigma: S/m
import numpy as np
def csd_1d(V, h, sigma=0.3):
d2V = (V[2:] - 2*V[1:-1] + V[:-2]) / h**2 # second spatial difference
return -sigma * d2V # sinks < 0, sources > 0What actually generates the LFP
The dominant mechanism of LFP genesis is synaptic transmembrane current in aligned pyramidal dendrites, not spikes. An excitatory synapse on apical dendrites makes a sink there and a distributed return source near the soma — a dendritic dipole. Because cortical pyramidal cells are geometrically aligned, these dipoles add rather than cancel (open-field geometry).
Other contributors include dendritic Ca²⁺ spikes, afterhyperpolarizations, and — controversially — spike 'bleed-through' into low frequencies. This mix is why the LFP is powerful but genuinely hard to attribute to a single cause.
Spatial reach and the low-pass illusion
How far does an LFP contact see? Estimates range from a few hundred µm to a couple of mm, and the reason is the correlation structure of the sources: correlated (synchronous) inputs sum coherently and reach far, while uncorrelated sources cancel and stay local. Reach is not a fixed radius — it depends on the synchrony of the underlying activity.
Tissue is also mildly frequency-dependent and anisotropic (extracellular conductivity), which adds some low-pass character, but the biggest 'low-pass in space' effect is geometric (the 1/r kernel), not dielectric.
Why this matters for a decoder
If your features are band power or CSD from an array, remember each channel is a weighted volume average. Two channels can be near-duplicates because they share a recording volume — that inflates apparent dimensionality and fools naïve covariance estimates (a theme you will meet again in the Riemannian and machine-learning tracks). Understanding the forward physics is what tells you when a spatial filter such as the surface Laplacian buys real locality versus just reshaping noise.