The Forward Problem: Head Models and the Lead Field

Two directions, one of them easy

Source analysis has two directions. The forward problem asks: given a known source configuration and a head model, what will the sensors read? It has a unique, directly computable answer. The inverse problem reverses the arrow — sensors to sources — and, as the next guide shows, has no unique answer at all. The deep point: every inverse method is, at its core, a forward model run many times inside an optimization loop. Master the forward model and the inverse becomes demystified.

The lead field: the linear map you compute once

The single most useful object in source analysis is the lead field. Because the Poisson equation is linear in the source, the mapping from any dipole to the sensors is linear too. Discretize the cortex into N candidate dipole locations, stack all sensor readings into a vector \mathbf{v}, and the forward model collapses to a matrix multiply.

\mathbf{v} = \mathbf{L}\,\mathbf{j} + \boldsymbol{\varepsilon}

The discrete forward model. v ∈ ℝ^M (M sensors), j ∈ ℝ^{3N} (three current components at N sources), L the lead-field (gain) matrix, ε sensor noise.

Once you fix a head model, the map from sources to sensors is just a matrix multiply. Every possible source pattern \mathbf{j} gets turned into sensor readings \mathbf{v} by the fixed lead-field matrix \mathbf{L}, plus noise. Compute \mathbf{L} once, reuse it forever.

\mathbf{v}: The sensor measurements (M of them).
\mathbf{j}: The source currents — 3N unknowns, three directions at N locations.
\mathbf{L}: The lead-field (gain) matrix mapping sources to sensors.
\boldsymbol{\varepsilon}: Sensor noise.

The column interpretation is what to remember. The k-th column of \mathbf{L} is the scalp topography produced by a unit dipole at source k — its signature. The inverse problem then reads: which combination of these known signatures best explains the observed map \mathbf{v}? Once \mathbf{L} is computed for a given head and sensor array, it is reused for every time sample and every inverse method — so the whole cost of physics is paid exactly once.

Head models: from a sphere to a segmented MRI

Computing \mathbf{L} requires a geometric and electrical model of the head. There is a ladder of fidelity, each rung more realistic and more expensive:

Single- or multi-layer spherical models: analytic and instant, but a poor fit to real skull geometry — best for intuition and quick simulation.
Boundary element method (BEM): model the head as nested homogeneous compartments (brain, skull, scalp) with realistic surfaces from MRI, and solve for the potential on those boundaries.
Finite element method (FEM): tessellate the entire volume, allowing anisotropic and inhomogeneous conductivity (e.g. white-matter anisotropy, skull sutures and holes). Most accurate, most expensive.

Crucially, the dominant source of forward error in practice is usually not the numerical method but the conductivity values you plug in — above all the skull-to-brain conductivity ratio, which is uncertain and varies with age and anatomy. A realistic BEM with a wrong skull conductivity can be worse than a sphere with a right one. Fidelity of geometry and fidelity of conductivity are separate battles.

Reciprocity: computing lead fields efficiently

A beautiful shortcut, Helmholtz reciprocity, says the lead field of a sensor can be found by driving unit current into that sensor pair and computing the resulting electric field throughout the brain. This converts one forward solve per source (N solves) into one solve per sensor (M solves), and M ≪ N. It is how practical lead-field calculators stay tractable at thousands of source points.

L_{ik} \;\propto\; \mathbf{E}_i(\mathbf{r}_k)\cdot \hat{\mathbf{e}}_k

Reciprocity: sensor i's sensitivity to a dipole at r_k equals (up to constants) the electric field E_i that same sensor would produce there, projected onto the dipole orientation ê_k.

A clever shortcut: instead of computing what a brain source does at each sensor, imagine driving current backward out of the sensor and seeing what electric field it makes at the source. By reciprocity these are equal, so you can build the whole lead field far more cheaply.

L_{ik}: Sensitivity of sensor i to a dipole at location \mathbf{r}_k.
\mathbf{E}_i(\mathbf{r}_k): The electric field sensor i would create at that location if driven in reverse.
\hat{\mathbf{e}}_k: The dipole's orientation.
\mathbf{E}_i\cdot\hat{\mathbf{e}}_k: The dot product projects that field onto the dipole direction.

The source space: where you allow dipoles to live

The last modeling choice is the source space — the set of candidate dipoles. Two philosophies compete. A focal / equivalent-dipole model places a handful of free-position dipoles and fits their locations, assuming activity comes from a few compact sources. A distributed model instead fixes thousands of dipoles on the cortical surface (often with orientation constrained normal to the sheet) and estimates the amplitude at each. The choice you make here sets up the inverse method of the next guides.