Editing Magnetoencephalography (section)

== Source localization ==

=== 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>

=== Magnetic source imaging ===
The source locations can be combined with [[magnetic resonance imaging]] (MRI) images to create magnetic source images (MSI).  The two sets of data are combined by measuring the location of a common set of [[Fiduciary marker|fiducial points]] marked during MRI with lipid markers and marked during MEG with electrified coils of wire that give off magnetic fields.  The locations of the fiducial points in each data set are then used to define a common coordinate system so that superimposing the functional MEG data onto the structural MRI data ("[[coregistration]]") is possible.

A criticism of the use of this technique in clinical practice is that it produces colored areas with definite boundaries superimposed upon an MRI scan: the untrained viewer may not realize that the colors do not represent a physiological certainty, not because of the relatively low spatial resolution of MEG, but rather some inherent uncertainty in the probability cloud derived from statistical processes.  However, when the magnetic source image corroborates other data, it can be of clinical utility.

=== Dipole model source localization ===
A widely accepted source-modeling technique for MEG involves calculating a set of equivalent current dipoles (ECDs), which assumes the underlying neuronal sources to be focal. This dipole fitting procedure is non-linear and over-determined, since the number of unknown dipole parameters is smaller than the number of MEG measurements.<ref>{{cite journal | vauthors = Huang MX, Dale AM, Song T, Halgren E, Harrington DL, Podgorny I, Canive JM, Lewis S, Lee RR | title = Vector-based spatial-temporal minimum L1-norm solution for MEG | journal = NeuroImage | volume = 31 | issue = 3 | pages = 1025–37 | date = July 2006 | pmid = 16542857 | doi = 10.1016/j.neuroimage.2006.01.029 | s2cid = 9607000 | url = https://escholarship.org/uc/item/4xr5z4qd }}</ref> Automated multiple dipole model algorithms such as [[multiple signal classification]] (MUSIC) and multi-start spatial and temporal modeling (MSST) are applied to the analysis of MEG responses. The limitations of dipole models for characterizing neuronal responses are (1) difficulties in localizing extended sources with ECDs, (2) problems with accurately estimating the total number of dipoles in advance, and (3) dependency on dipole location, especially depth in the brain.

=== Distributed source models ===
Unlike multiple-dipole modeling, distributed source models divide the source space into a grid containing a large number of dipoles. The inverse problem is to obtain the dipole moments for the grid nodes.<ref>{{cite journal | vauthors = Hämäläinen MS, Ilmoniemi RJ | title = Interpreting magnetic fields of the brain: minimum norm estimates | journal = Medical & Biological Engineering & Computing | volume = 32 | issue = 1 | pages = 35–42 | date = January 1994 | pmid = 8182960 | doi = 10.1007/BF02512476 | s2cid = 6796187 }}</ref> As the number of unknown dipole moments is much greater than the number of MEG sensors, the inverse solution is highly underdetermined, so additional constraints are needed to reduce ambiguity of the solution. The primary advantage of this approach is that no prior specification of the source model is necessary. However, the resulting distributions may be difficult to interpret, because they only reflect a "blurred" (or even distorted) image of the true neuronal source distribution. The matter is complicated by the fact that spatial resolution depends strongly on various parameters such as brain area, depth, orientation, number of sensors etc.<ref>{{cite journal | vauthors = Molins A, Stufflebeam SM, Brown EN, Hämäläinen MS | title = Quantification of the benefit from integrating MEG and EEG data in minimum ℓ<sub>2</sub>-norm estimation | journal = NeuroImage | volume = 42 | issue = 3 | pages = 1069–77 | date = September 2008 | pmid = 18602485 | doi = 10.1016/j.neuroimage.2008.05.064 | s2cid = 6462818 }}</ref>

=== Independent component analysis (ICA) ===
[[Independent component analysis]] (ICA) is another signal processing solution that separates different signals that are statistically independent in time. It is primarily used to remove artifacts such as blinking, eye muscle movement, facial muscle artifacts, cardiac artifacts, etc. from MEG and EEG signals that may be contaminated with outside noise.<ref>{{cite journal | vauthors = Jung TP, Makeig S, Westerfield M, Townsend J, Courchesne E, Sejnowski TJ | title = Removal of eye activity artifacts from visual event-related potentials in normal and clinical subjects | journal = Clinical Neurophysiology | volume = 111 | issue = 10 | pages = 1745–58 | date = October 2000 | pmid = 11018488 | doi = 10.1016/S1388-2457(00)00386-2 | citeseerx = 10.1.1.164.9941 | s2cid = 11044416 }}</ref> However, ICA has poor resolution of highly correlated brain sources.