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MEG and EEG detect the induced magnetic field and scalp electrical potential out side of head. In studying of brain functionality by recording of MEG and EEG, the two major difficulties are the ill-posed inverse problem and low Signal-Noise-Ratio (SNR).
First, the forward model, including the lead field vector representing distribution of brain activities to sensor array, is involved in the inverse problem in order to reconstruct brain activities. Therefore, based on the forward model, several groups of methods were proposed to solve the inverse problem or to map sources to cortical surface. An inverse algorithm aims for estimation of real source location and orientation, such as dipole fitting.
An imaging method tries to estimate the statistical map of MEG/EEG signals that indicates the cortical source distribution in probability. The higher probability it is, the more possible stronger activation it has. Otherwise, according to the approach of imaging method or inverse algorithm, it can be separated into two categories such as scanning approach and imaging approach.
For low SNR, it results from the much smaller scale of electrophysiological signal than of environmental noises. In addition, MEG and EEG signals are often corrupted with the background brain activities and artifacts, such as the heartbeat, eye-blink, and environmen-tal noises. Thus, data preprocessing, including bandpass filter and baseline correction, and Independent Component Analysis (ICA) are some widely used techniques for increasing the SNR or rejecting artifacts.
ICA was originally proposed for the purpose of blind source separation to find com-ponents that are mutually statistical independent [21]. Recently, it has been proved a use-ful tool in neurological brain research and is widely-used for analyzing MEG/EEG sig-nals [18, 19], especially for removing artifacts or finding features based on its indepen-dence assumption [4, 8, 17, 22]. Moreover, it may be probable to see the event-related fields (ERFs) after removing noises. However, ICA has limitations that the number of indepen-dent components is less than or equal to the number of sensors and it provides the sensor space distribution of components but no cortical distribution. It is insufficient for studying brain activites.
Electromagnetic Spatiotemporal Independent Component Analysis (EMSICA) was pro-posed by Arthur C. Tsai et al. in 2006 [33] for estimating spatiotemporal independent EEG components and the corresponding cortical source distribution simultaneously. Thus, it
2.1 MEG/EEG Forward Model with Spherical Head Model 7
has the same ability of mapping sources to cortical surface as imaging methods for solving inverse problem. It also has the same ability of separating independent components as ICA.
However, it may be harder to have a best solution for EMSICA than for standard ICA since EMSICA has much more unknown parameters that results from the cortical surface constraints.
We propose an imaging method for mapping independent components to cortical sur-face with less unknown parameters by standard ICA. But it can achieve the same work like EMSICA.
2.1 MEG/EEG Forward Model with Spherical Head Model
Inverse algorithms have to involve forward solution for estimating the properties of brain sources when given a set of MEG/EEG signals measured by an array of external sensors. Therefore, the forward model, describes the distribution of magnetic field outside of head when given a theoretical brain source, should be constructed at first.
The most commonly adopted head model, the spherical model, assumes that it is com-prised by a set of nested concentric sphere shells representing brain, skull and scalp [2, 25].
Each sphere has homogeneous and isotropic conductivity. Under this assumption, consider the simple case of a unit current dipole with the parameter θ = {r, q}, located at r ∈ R3 with orientation q ∈ R3. The lead field vector lθ ∈ RN, a column vector, indicates how this current dipole distributes to the MEG sensor array and can be illustrated as a single topography like Figure 2.1.
lθ = Gr∗ q (2.1)
The gain matrix G ∈ RN ∗3describes the sensibility of N MEG/EEG sensors to the current dipole in the three dimensional Cartesian coordinate system.
Furthermore, concentrate on the general case in volume domain, the MEG/EEG mea-surement m(t) ∈ RN recorded at time t is composed of many time-varying current
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(www.neurevolution.net)
(ibru.vghtpe.gov.tw/chinese/meg.htm)
Brain activities MEG measurment
Figure 2.1: MEG Forward Model.
(The picture of MEG is excerpted from http://ibru.vghtpe.gov.tw/chinese/meg.htm.) (The picture of sensor is excerpted from http://www.neurevolution.net.)
While an oriented current dipole generated by an activated neuron, the MEG instrument ac-quires induced magnetic field by sensor array and output the time course, or measurement.
Each lead field vector with respect to a current dipole can be regarded as a topography or source distribution to MEG sensors.
2.2 Inverse Solution 9
ties with their respective lead field vector illustrated by the following equation
m(t) =X s(t) = [s1(t) s2(t) . . .]T denotes the time-varying current densities. Then, the equation can rewritten as
m(t) = Ls(t) + n(t). (2.3)
Figure 2.1 can be used for explaining the forward model. While a neuron is activated, the induced magnetic field or scalp electric potential is detected by sensors of MEG or EEG. The topography represents the distribution of the magnetic field or electric potential produced by a unit dipole and is so called a lead field vector. Unit dipole with different location and orientation will result in different lead field vector. Thus, linear combined sources with the respective lead field vectors and amplitude plus the additive noises are included in measurement.
2.2 Inverse Solution
The inverse problem is an ill-posed problem of determining the neuronal sources from MEG/EEG measurement and has no unique solution. Thus, it is impossible to specify distribution of neuronal sources without any further assumptions or anatomical constraints.
According to the revealed MEG/EEG researchs, parametric and imaging methods are the two general approaches for estimating neuronal sources [2, 26].
Parametric or Scanning Methods
The parametric or scanning methods solving the inverse problem under the assump-tion that sources can be represented by a few equivalent current dipoles (ECDs) of un-known locations, orientations and amplitude to be estimated with nonlinear numerical method [2, 13, 16, 26]. Thus, a current dipole is assigned to each tessellation elements,
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numbering in tens of thousands, on the cortical surface orientated to the radial line, or said the local surface normal, during a period and trying to find the best fit of the MEG/EEG measurements.
The most common model used for inverse solution is the least-squares source estimation (Equation 2.4) [2, 7, 11]. It is a brute force approach, but with expensive resource and time cost, of nonlinear search by scanning through all possible set of locations, orientations and amplitude. It attempts to determine the set {θi, si(t)} = {{ri, qi}, si(t)} for i = 1 . . . P that minimizes the square error between recording and the field computed from estimated sources ˜s(t) using forward model ˜L (Equation 2.3).
arg min km(t) − ˜L˜s(t)k (2.4)
Recently, beamforming approaches, performing spatial filter W on data m(t) (Eq. 2.5), are applied in estimating cortical distribution of neuronal sources over least-squares with extra constraints [31,32], such as minimum norm (MNE) [12,15,20,23], minimum variance (LCMV and SAM) [28–30, 34] and maximum contrast (MCB) [5] constraints. [7].
s(t) = WTm(t) (2.5)
However, these methods may result in spatial over-smothing that does not explicitly express anatomical or physiological constraints in the source reconstruction process. It may comprehensively yield unrealistic solution. Moreover, one of the limitations of beam-forming is that coherent sources with the true signal from the scanning location can cause cancellation of the interested signal. Not allowing source estimation throughout the entire event period is another one, thus leaving parts of the event unexplained.
Imaging Approaches
The imaging approaches estimate amplitudes of a set of dipoles with fixed locations within the brain volume. Similar to scanning method mentioned in last section, computing on the volumetric grid is the basic technique for imaging methods. Moreover, since the brain activities are believed to be restricted to the cortex and MEG is most sensitive to cor-tical sources, the imaging method can be constrained to the corcor-tical surface that extracted
2.2 Inverse Solution 11
from an anatomical MR image of the subject. As aforementioned, sources can be placed on each point that forms the triangle mesh with orientation that perpendicular to the surface.
Hence, the inverse problem is simplified to estimate linear parameters only. [6].
Bayesian statistic framework is a widely-used approximation of imaging method [1, 2]
such as FOCUSS [10, 24].
p(S|M) = p(M|S)p(S)
p(M) (2.6)
p(S|M) denotes the conditional probability of an event S assuming M has occurred. That is, applying to the inverse problem, the conditional probability of sources S activated as-suming MEG/EEG measurement M has been recored. In contrast, p(M|S) describes the forward problem that the conditional probability of the measurement M been recorded assuming S has activated and p(S) is the prior. Therefore, the sources are estimated by maximization of the posterior probability.
S = arg max p(M|S)p(S)˜ (2.7)
However, it has been revealed by Hillebrand et al. [14] that small errors in anatomical constraints can incur the large errors in source estimation. Moreover, the higher spatial res-olution it is, the worse effects of errors it has. This may remove or decrease the advantage of estimating sources using imaging approaches and anatomical constraints.
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2.3 Independent Component Analysis (ICA)
Brain activities
Artifacts Measurement m(t)
Topography A Component x(t)
Figure 2.2: Independent Component Analysis. MEG/EEG signals are often corrupted by additive noises including background brain activities, heartbeat, eye-blinking, other elec-trical muscle activities and the environment noises. In general, these interferes occurs inde-pendently to the stimuli or ERFs. Recently, ICA has been proved a useful tool in analyzing MEG/EEG signals, especially for artifacts removal. It attempts to find the unmixing matrix W that makes the output components as independent as possible.
MEG/EEG signals are often corrupted by additive noises including background brain activities, heartbeat, eye-blinking, other electrical muscle activities and the environment noises (Figure 2.2). In general, these interferes occurs independently to the stimuli or event-related fields (ERFs).
Independent component analysis (ICA) was originally proposed for the purpose of
2.4 Electromagnetic Spatiotemporal Independent Component Analysis
(EMSICA) 13
blind source separation to find components that are mutually statistical independent [21].
It performs best when raw or unaveraged signals are applied as inputs. Recently, it has been proved a useful tool in neurological brain research and is widely-used for analyzing MEG/EEG signals [18, 19], especially for removing the interferences mentioned above as a preprocessing step based on its independence assumption [4, 8, 17, 22]. Moreover, it may be probable to see the ERFs after removing noises and be helpful for the following source estimation process.
The ICA task is casted as follows:
m(t) = Ax(t) (2.8)
, A ∈ RN ∗K is so called a mixing matrix that compounds the K independent components x(t) ∈ RKinto MEG measurement m(t). Each single column vector aiof mixing matrix A is respected to the ithcomponent xi = [xi(t1) . . . xi(tT)], for i = 1 . . . K, which specifies its distribution to MEG sensors. In contrast, the equation 2.8 can also be written as
WTm(t) = x(t) (2.9)
where W ∈ RN ∗K is the unmixing matrix. Each single column vector wi of unmixing matrix W is a filter extracting the corresponding component xi from MEG measurement.
However, the amount of output independent components is limited to the number of input channels. That is, at most N components will be outputted if we send a N -channel MEG measurement to ICA.
2.4 Electromagnetic Spatiotemporal Independent Compo-nent Analysis (EMSICA)
Electromagnetic Spatiotemporal Independent Component Analysis (EMSICA) was pro-posed by Arthur C. Tsai et al. in 2006 [33] for estimating spatiotemporal independent EEG components and the corresponding cortical source distribution. It is implemented using Bayesian statistical framework for imaging independent brain activities under physiologi-cal source constraints.
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First, it assumes that P brain activities are consisted of the K spatiotemporal indepen-dent components and the matrix B ∈ P ∗ K describes the linear combination.
s(t) = Bx(t) (2.10)
Each column vector bi ,where B = [b1. . . bK] , represents the cortical distribution of component xi(t) for i = 1 . . . P . Thus, the forward solution (Eq. 2.3) becomes
m(t) = Ls(t) = LBx(t). (2.11)
Topography
Cortical distribution Component
Figure 2.3: EMSICA.
(This picture is excerpted from [33])
EMSICA attempts to estimate the cortical distribution B that makes the corresponding output components as independent as possible. Moreover, compare equation 2.8 for ICA and 2.11, distribution of independent components to sensors are obtained by Equation 2.12.
Thus, there are distributions to both cortical surface and sensor space.
Similar to ICA, EMSICA attempts to estimate the cortical distribution B that makes
2.5 Comparison between ICA and EMSICA 15
the corresponding output components as independent as possible and K, the amount of components, must be less than or equal to P or said the number of brain activities.
2.5 Comparison between ICA and EMSICA
EMSICA and ICA both attempt to extract independent components from input signals and to find the respected distribution. According to the Equation 2.8 from ICA and 2.11 from EMSICA, the relation between distribution to sensor space from ICA and to cortical surface from EMSICA can be illustrated by
A = LB. (2.12)
(a) (b)
Figure 2.4: Infomax ICA in sensor space vs. EMSICA on cortical surface.
(These two pictures are excerpted from [33])
(a) Topography with respect to 12 components, accounting for sensorimotor mu, frontal midline theta, central and lateral posterior alpha rhythms, separated using infomax ICA.
(b) Cortical distribution of the 12 components extracted by EMSICA and the corresponding topography by applying lead field matrix to cortical distribution map. Results of EMSICA similar to the ones of ICA demonstrates that EMSICA has the convinced ability to separate components well like what the widely-used ICA has.
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Figure 2.4 displays the results of a two-back visual memory working memory task an-alyzing by standard ICA and EMSICA. The topographies, outputted by ICA, displayed in left-panel and the ones, transformed from cortical distribution by lead field matrix, dis-played in right-panel are similar. This demonstrates that EMSICA has the convinced ability to separate components well like what the widely-used ICA has.
Moreover, EMSICA, within a single procedure, solves not only independent compo-nents but also the imaging of cortical source distribution by involving the forward solution and implemented using Bayesian framework. This makes the difference between ICA who is not an imaging method and just separates components in the same space as input signals.
On the other hand, ICA can split components in cortical space as the second procedure but an imaging or inverse method to measurement required at first [35].
Comparing these two algorithms, EMSICA has the ability to estimate cortical source distribution but also has too many variables, at least 110,000 triangular points consisted of the cortical surface in our case, locations of dipolar sources need to be solved. In contrast, standard ICA cannot estimate cortical source distribution directly when applied measure-ment but not cortical sources. Furthermore, ICA has same accuracy in sensor space but handles the smaller data and fewer variables than EMSICA does.