Inverse recovery in EEG
Inverse recovery in EEG: a simple overview
What is inverse recovery in EEG?
- It is a type of inverse problem: we start with measurements of electrical activity on the scalp and try to figure out what happened inside the head, such as where brain signals come from (sources) or how the head’s tissues conduct electricity (conductivity).
The basic idea in plain terms
- The electric potential u inside the head is shaped by how electrical currents flow through tissues with conductivity σ.
- The brain’s activity creates primary currents Jp, while the tissues also produce a return current σE where E is the electric field (E = -∇u).
- The main equation linking these ideas is the elliptic partial differential equation: ∇·(σ ∇u) = S, where S represents the sources inside the brain.
- The goal of inverse recovery is to use measured scalp potentials u to recover either the source term S (the brain activity) or the conductivity σ of head tissues.
Why this problem is hard
- The head has multiple layers (brain, skull, scalp) with different conductivities, which makes the math tricky.
- Because the head is complex and measurements are limited, the problem is ill-posed: many different internal configurations can produce similar scalp signals.
- In simple terms, small measurement errors can lead to big mistakes in estimating the sources or conductivities if we don’t handle the problem carefully.
Two common viewpoints to simplify the problem
1) Constant conductivity (simplified case)
- Some models assume each head layer has a constant conductivity. In this setup, the equations become simpler (like Laplace or Poisson equations) and can sometimes be solved analytically for idealized geometries.
- Researchers use tools like Green’s functions or spherical harmonics to get exact solutions in very simple settings.
- There are benchmark tools (e.g., FindSource3D) that test how well one can locate sources when conductivity is piecewise constant.
2) Non-constant or anisotropic conductivity (more realistic case)
- Real head tissues don’t have uniform conductivity, and skull anisotropy can change how currents travel.
- The inverse problem becomes much harder here. There are mathematical results about when you can uniquely determine σ or S, but they depend on how nicely the head boundary is shaped and other technical assumptions.
- In practice, people rely on approximate or numerical methods, often using perturbation ideas, integral equations, or regularization to get stable solutions.
How people actually solve it (numerical approaches)
- Because exact analytical solutions are only available for very simple cases, simulations and numerical methods are essential.
- Common methods include:
- Finite element method (FEM): flexible for complex head shapes and variable conductivities.
- Boundary element method (BEM): often efficient for problems with layered head geometries.
- To cope with ill-posedness (the tendency to produce unstable results), researchers use regularization:
- Tikhonov regularization, which imposes a smoothness constraint.
- Sparsity constraints, encouraging solutions that are localized in space.
- Bayesian or probabilistic approaches, which incorporate prior information and quantify uncertainty.
- Real-world EEG analysis also models measurement noise and uncertainties in electrode placement to improve robustness.
Why this matters
- Solving the inverse problem helps researchers pinpoint where brain activity starts and how it spreads, which is important in neuroscience and clinical contexts (like epilepsy research or brain-computer interfaces).
- Ongoing work aims to make solutions more accurate and reliable by improving models of head conductivity, using better regularization, and combining EEG with other imaging data.
In short
- Inverse recovery in EEG tries to reconstruct brain sources or tissue conductivities from scalp electrical measurements.
- It relies on solving an elliptic PDE inside the head, often with layered conductivities.
- The problem is mathematically challenging and usually solved with a mix of numerical methods and regularization to obtain stable, useful results.
This page was last edited on 1 February 2026, at 14:44 (CET).