Inverse problem · Grey Matter

The inverse problem is the task of working out which currents inside the head produced the voltages or magnetic fields measured outside it, and it is the step that turns an EEG or MEG recording into a map of brain activity.

Inverse problem. The inverse problem is the task of working out which currents inside the head produced the voltages or magnetic fields measured outside it, and it is the step that turns an EEG or MEG recording into a map of brain activity.

The forward direction is well defined: given a current source and a model of the head (scalp, skull, fluid and brain, each with its conductivity), physics predicts the field at every sensor. The inverse direction is not. Helmholtz showed in 1853 that the currents inside a conductor cannot be recovered uniquely from the fields outside it: some arrangements of currents produce no external field at all, and any of them can be added to a solution without changing the measurement. With a few dozen or a few hundred sensors and many thousands of possible source locations, infinitely many answers fit the data.

Every source map is the answer that a chosen assumption picks out. One or a few point sources (dipole fits), the smallest total current that explains the data (minimum-norm methods), smoothness across the cortex, or filters that scan the brain one location at a time are the common choices, and they can disagree.

Distance makes it worse. The potential of a current dipole falls with the square of distance, roughly as 1/r21/r^21/r2, so deep sources reach the sensors weak and smeared, and a small deep source can look like a broad superficial one.

The head model matters. Errors in the assumed skull conductivity or in the shape of the head shift the solution, which is why clinical source localisation uses the subject's own MRI.

Prior knowledge is what makes it work in practice. Constraining sources to the cortical sheet, using the MRI anatomy, and combining EEG with MEG (which see different orientations) narrows the answer considerably.

Reading a brain from outside is underdetermined by physics.

More sensors and less noise sharpen the answer, but no sensor array outside the head removes the need for an assumption about the sources.

Questions: Why can a full helmet of MEG sensors still not say for certain where a signal came from? The answer is not unique. Helmholtz proved in 1853 that the currents inside a conductor cannot be recovered from the fields outside it, since some current patterns produce no external field and can be added to any solution unseen. A few hundred sensors facing many thousands of possible source locations leave infinitely many current maps that fit the data equally well. MEG source maps therefore come from adding assumptions (a few point sources, the smallest total current, sources confined to the cortical sheet of the subject's MRI), and different assumptions can place the same signal differently, especially for deep sources.