Editing Magnetoencephalography (section)

=== The inverse problem ===
{{main|Inverse problem}}

The challenge posed by MEG is to determine the location of electric activity within the brain from the induced magnetic fields outside the head. Problems such as this, where model parameters (the location of the activity) have to be estimated from measured data (the SQUID signals) are referred to as ''inverse problems'' (in contrast to ''forward problems''<ref>{{cite thesis | url = http://lib.tkk.fi/Diss/2006/isbn9512280914/ | vauthors = Tanzer IO | year = 2006 | title = Numerical Modeling in Electro- and Magnetoencephalography | degree = Ph.D. | publisher = Helsinki University of Technology | location = Finland }}</ref> where the model parameters (e.g. source location) are known and the data (e.g. the field at a given distance) is to be estimated.) The primary difficulty is that the inverse problem does not have a unique solution (i.e., there are infinite possible "correct" answers), and the problem of defining the "best" solution is itself the subject of intensive research<!--

-->.<ref name="HaukWakemanHenson">{{cite journal | vauthors = Hauk O, Wakeman DG, Henson R | title = Comparison of noise-normalized minimum norm estimates for MEG analysis using multiple resolution metrics | journal = NeuroImage | volume = 54 | issue = 3 | pages = 1966–74 | date = February 2011 | pmid = 20884360 | pmc = 3018574 | doi = 10.1016/j.neuroimage.2010.09.053 }}</ref> Possible solutions can be derived using models involving prior knowledge of brain activity.

The source models can be either over-determined or under-determined. An over-determined model may consist of a few point-like sources ("equivalent dipoles"), whose locations are then estimated from the data. Under-determined models may be used in cases where many different distributed areas are activated ("distributed source solutions"): there are infinitely many possible current distributions explaining the measurement results, but the most likely is selected. Localization algorithms make use of given source and head models to find a likely location for an underlying focal field generator.

One type of localization algorithm for overdetermined models operates by [[Expectation-maximization algorithm|expectation-maximization]]: the system is initialized with a first guess.  A loop is started, in which a forward model is used to simulate the magnetic field that would result from the current guess. The guess is adjusted to reduce the discrepancy between the simulated field and the measured field.  This process is iterated until convergence.

Another common technique is [[beamforming]], wherein a theoretical model of the magnetic field produced by a given current dipole is used as a prior, along with second-order statistics of the data in the form of a [[covariance matrix]], to calculate a linear weighting of the [[sensor array]] (the beamformer) via the [[Backus–Gilbert method|Backus-Gilbert inverse]]. This is also known as a linearly constrained minimum variance (LCMV) beamformer. When the beamformer is applied to the data, it produces an estimate of the power in a "virtual channel" at the source location.

The extent to which the constraint-free MEG inverse problem is ill-posed cannot be overemphasized. If one's goal is to estimate the current density within the human brain with say a 5mm resolution then it is well established that the vast majority of the information needed to perform a unique inversion must come not from the magnetic field measurement but rather from the constraints applied to the problem. Furthermore, even when a unique inversion is possible in the presence of such constraints said inversion can be unstable. These conclusions are easily deduced from published works.<ref>{{cite journal | vauthors = Sheltraw D, Coutsias E| journal=Journal of Applied Physics |volume=94|number=8|year=2003 | url = http://www.math.unm.edu/~vageli/papers/JApplPhys_94_5307.pdf  | doi = 10.1063/1.1611262 | title=Invertibility of current density from near-field electromagnetic data|pages=5307–5315| bibcode=2003JAP....94.5307S }}</ref>