Beamspace fast fully adaptive brain source localization for limited data sequences

A problem of estimating sources of electrical activity in the brain based on electro-encephalographic (EEG) or magneto-encephalographic (MEG) signals observed on the scalp is known as an important inverse problem in electrophysiology. In both EEG and MEG source localization, the existence of spatially colored and temporally nonstationary noise with low signal-to-noise ratio (SNR) significantly impacts the ability to detect weak sources of interest. To overcome this problem and to improve detection of weak sources, adaptive noise cancellation is required. The most popular adaptive localization approaches are the ones that rely on second order statistics of the measured data such as minimum variance beamforming (MVB) method [1], multiple signal classification (MUSIC) [2–5], FINE [6–8], and maximum likelihood dipole fitting (MLDF) [9] approaches. These methods can provide excellent performance when sufficient duration of measured data is available to accurately evaluate the second-order statistics of interest. With an EEG system with N electrodes (N degrees of freedom (DoFs)), the number of statistically independent data records should be three or more times the number of channels in order to obtain a noise covariance matrix with sufficient accuracy [1]. However, since these data samples are obtained from the measurements, this requirement can rarely be met when the number of observations is small. This is particularly true when measuring evoked potentials (which typically are short in duration), especially when the number of electrodes is large.

To address this issue, researchers have developed techniques with lower complexity and fewer adaptive DoFs. One of these approaches is partial adaptivity, e.g. [10], in which the number of adaptive DoFs is reduced to meet the constraints on the available data. A closely related approach is beamspace processing, e.g. [11], in which the sensor space data is mapped into a lower dimensional space using a linear transformation before applying the desired statistical signal processing algorithm. Reducing the adaptive DoFs yields corresponding reductions in the required sample support and computational load, at the expense of reduced performance and focal brain activity discrimination.

An alternative approach to deal with this limited data case is a multistage approach called the fast fully adaptive (FFA) algorithm that was previously developed by the author [12]. The fully adaptive aspect of this algorithm is that it exploits all available DoFs, while simultaneously reducing the required sample support. The basic idea of the algorithm is to approximate the fully adaptive (and theoretically optimum) scheme using a divide-and-conquer approach.

In this paper, we develop a new form of FFA approach called beamspace FFA for brain source localization. In contrast to the FFA algorithm (named regular FFA in this paper) where the partitioning is applied directly to the measured data in the sensor space, in the beamspace FFA algorithm, first the brain volume is divided into a number of regions of similar volume and then the beamspace transformation is applied to the data without any dimension reduction in order to optimally preserve the sources within each specified region. Then the FFA approach is applied to the beamspaced data. The key advantage of the beamspace FFA algorithm is to reduce the sensitivity to the correlation between the sources in different regions by suppressing the activities (interferences) that are outside each specific region when processing that region. In [12], the regular FFA approach was compared with MVB and beamspace approaches with four different source configuration and showed more accurate localization performance for all types of brain activities. Furthermore, the advantage of the FINE approach in comparison with the classic MUSIC [2] and RAP-MUSIC [3] approaches are presented in [6]. Therefore, in this study the performance of the beamspce FFA approach is evaluated in comparison with regular FFA and widely used FINE approaches using both simulated and experimental data.

The rest of the paper is organized as follows: section 2 presents the system model and briefly reviews the regular FFA approach. Section 3 introduces the beamspace FFA algorithm designed for brain source localization. Section 4 presents several simulation and experimental results used to evaluate the performance of the beamspace FFA and compare that with regular FFA and FINE algorithms. Section 5 concludes the paper.

In this section, we first introduce the system model and then briefly review the regular FFA approach developed in [12]. We consider an EEG recording system with N electrodes. The N electrodes record the brain electrical activity over a short time interval T with sampling frequency f, so the number of time samples is $M=T\times f$. Hence the data can be organized as a $N\times M$ data matrix x.

2.1. System model

Let x(k) be an $N\times 1$ vector composed of the potentials measured by the electrodes at a given time instant k associated with active dipole sources at locations that are represented by the $3\times 1$ vectors ${{\mathbf{q}}_{i}},\,\,i=1,2,\ldots,L$ and noise at the kth time instant $\mathbf{v}(k)$,

$$\begin{eqnarray}&&\mathbf{x}(k)=\underset{i=1}{\overset{L}{\sum}}\mathbf{H}\left({{\mathbf{q}}_{i}}\right)\mathbf{m}\left({{\mathbf{q}}_{i}},k\right)+\mathbf{v}(k),\end{eqnarray} \tag{ 1 }$$

where

$$\begin{eqnarray}&&\mathbf{m}\left({{\mathbf{q}}_{i}},k\right)={{\alpha}_{{{\mathbf{q}}_{i}}}}(k)\boldsymbol{\mu}\left({{\mathbf{q}}_{i}}\right),\end{eqnarray} \tag{ 2 }$$

${{\alpha}_{{{\mathbf{q}}_{i}}}}(k)$ is a scalar function of time, which describes the variation of the moment amplitude across time, $\boldsymbol{\mu}\left({{\mathbf{q}}_{i}}\right)$ is a $3\times 1$ unit-norm vector whose elements are the x, y and z components of the dipole moment, which assumed to be time invariant, and the columns of the $N\times 3$ lead field matrix $\mathbf{H}\left({{\mathbf{q}}_{i}}\right)$ represent solutions to the forward problem. That is, the first column of $\mathbf{H}\left({{\mathbf{q}}_{i}}\right)$ is the potential at the electrodes due to a dipole source at location ${{\mathbf{q}}_{i}}$ having unity moment in the x direction and zero moment in y and z directions. Similarly, the second and third columns represent the potentials due to sources with unity moment in y and z directions, respectively.

2.2. Regular FFA processing

In this section we illustrate a more interesting variation of FFA named beamspace FFA. In this approach, for each region of interest, we denote the beamspace transformation using the $N\times N$ orthonormal transformation matrix T and obtain the N dimensional beam space data vector at each time instant k as [11]

$$\begin{eqnarray}&&{{\mathbf{x}}_{\text{T}}}(k)={{\mathbf{T}}^{\prime}}\mathbf{x}(k).\end{eqnarray} \tag{ 3 }$$

The rationale behind this transformation is that because the beamspace transformed data is algebraically closest to the transformation matrix for any specific region of interest in the brain, the approach is able to effectively pulling out the source peak in that region.

One method for choosing matrix T in each region of interest r (i.e. ${{\Omega}_{r}}$) could be based on minimizing the mean squared representation error (MSRE) between the original and beamspace representation of sources in that region [11], i.e.

$$\begin{eqnarray}&&\underset{\mathbf{T}}{{\min}}\left({{{\int}^{}}_{{{\Omega}_{r}}}}e_{\mathbf{T}}^{2}(l)\text{d}l\right)\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&e_{\mathbf{T}}^{2}(l)=\|(\mathbf{I}-\mathbf{T}{{\mathbf{T}}^{\prime}})\mathbf{H}(l)m(l)\|_{2}^{2}.\end{eqnarray} \tag{ 5 }$$

Expanding (4) using (5), the minimization problem (4) will be equivalent to the maximization problem

$$\begin{eqnarray}&&\max \,\text{tr}({{\mathbf{T}}^{\prime}}\mathbf{GT})\,\,\text{subject}\,\text{to}~{{\mathbf{T}}^{\prime}}\mathbf{T}=\mathbf{I}\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}&&\mathbf{G}={{{\int}^{}}_{{{\Omega}_{r}}}}\mathbf{H}(l)\mathbf{M}(l){{\mathbf{H}}^{\prime}}(l)\text{d}l,\,\,l\in {{\Omega}_{r}}\end{eqnarray} \tag{ 7 }$$

and $\mathbf{M}(l)=\mathbf{m}(l){{\mathbf{m}}^{\prime}}(l)$, if the moments is known and $\mathbf{M}(l)=\mathbf{I}$ if the moments are assumed unknown. In beamspace beamforming approach presented in [11], the solution to (6) was obtained by choosing the columns of T as the eigenvectors corresponding to the largest eigenvalues of Gin order to reduce the dimension of beamspaced data and so the required sample support. Therefore the corresponding minimum MSRE is given by the sum of the smallest eigenvalues of G. However, the number of eigenvalues to be considered and so the dimension of T is very important to the performance of the beamspace approach. In contrast to the beamspace beamforming approach, here we consider all N eigenvectors of G in matrix T. Therefore all the DoF are used and so the MSRE is equal to zero. Then, the FFA algorithms operate on the beamspaced data ${{\mathbf{x}}_{\text{T}}}$. The beamspace FFA algorithm is identical to its sensor space counterpart (regular FFA), with the exception that the lead field matrix should also be mapped into beamspace as [11]:

$$\begin{eqnarray}&&{{\mathbf{H}}_{\text{T}}}(l)={{\mathbf{T}}^{\prime}}\mathbf{H}(l).\end{eqnarray} \tag{ 8 }$$

Because the temporal waveform of the spatially fixed sources in beamspace FFA approach are influenced by the noise in the measured data, the source time series at a given location may therefore have some fluctuation specially when the SNR is low. One way to remove the temporal fluctuation due to the noise is to assume that the sources are spatially fixed during a short time window and, thus, the linear temporal information can be factored out [13]. The steps of the bemspace FFA algorithm developed for source localization are outlined as follows:

• (1)  
  Divide the brain volume into a reasonable number of smaller regions with approximately similar volume ${{\Omega}_{r}}$, r  =  1, 2, ..., R [6].
• (2)  
  For the rth region ${{\Omega}_{r}}$, transform the data and the corresponding lead field matrices H into beamspace using (3) and (8) respectively, where the beamspace transformation matrix T is the matrix of the eigenvectors of matrix G for that region.
• (3)  
  Given the beamspaced data ${{\mathbf{x}}_{\text{T}}}$ and the transformed lead field matrix ${{\mathbf{H}}_{\text{T}}}$, choose ${{N}^{\prime}}$, the maximum number of adaptive DoFs that can be processed.
• (4)  
  Apply the FFA approach to the beamspaced data ${{\mathbf{x}}_{\text{T}}}$ using the transformed lead field matrix ${{\mathbf{H}}_{\text{T}}}$.
• (5)  
  Repeat steps 2–4 for each location l in each brain region ${{\Omega}_{r}}$.
• (6)  
  Divide the time samples of the FFA output into S short time windows.
• (7)  
  For each time window ${{W}_{s}}$, s  =  1, 2, ..., S, sum up all of the beamspace FFA outputs $\mathbf{y}(k)$ in that time window as:

  $$\begin{eqnarray}&&{{\mathbf{J}}_{s}}=\underset{k\in {{W}_{s}}}{\sum}\mathbf{y}(k)\end{eqnarray} \tag{ 9 }$$

  Here, the window size is 5 samples or 20 ms, which is short enough to fulfill the assumption of spatially fixed brain source and is long enough to sum out the linear temporal information.
• (8)  
  Re-define the solution space by eliminating the nodes with approximately zero value.
• (9)  
  For each time sample in window ${{W}_{s}}$, re-calculate the inverse solution using the weighted minimum norm (WMN) method as

  $$\begin{eqnarray}&&\mathbf{\overset{\scriptscriptstyle\frown}{y}}(k)={{\mathbf{\overline{J}}}_{s}}\mathbf{\overline{J}}_{s}^{\prime}{{\mathbf{\overline{H}}}^{\prime}}{{\left(\mathbf{\overline{H}}{{\mathbf{\overline{J}}}_{s}}\mathbf{\overline{J}}_{s}^{\prime}{{\mathbf{\overline{H}}}^{\prime}}\right)}^{-1}}{{\mathbf{x}}_{\text{T}}}(k),k\in {{W}_{s}},\end{eqnarray} \tag{ 10 }$$

  where the weighting matrix ${{\mathbf{\overline{J}}}_{s}}$ and lead field matrix $\mathbf{\overline{H}}$ are derived from ${{\mathbf{J}}_{s}}$ and H by eliminating the nodes with zero value.

In step 9, since for each time sample, the weight matrix ${{\mathbf{\overline{J}}}_{s}}$ and the source space are identical, thus, the algorithm is linear. It is worth noting that despite the fact that the solution in (10) is calculated using a low-resolution WMN method, it still has very high spatial resolution because the solution space has been confined into a small area.

In this section, we use simulated and measured data to evaluate the performance of the beamspace FFA approach. The head model (the model for determining H) in two cases is OpenMEEG BEM head model [14] that was obtained using BrainStorm^® software [15]. This forward model that is developed by the French public research institute INRIA [16] uses a symmetric boundary element method (symmetric BEM). Three realistic layers (scalp, skull, brain) are used to offer better precision than the 3-shell sphere, where the x-, y- and z-axis are oriented along the back–front, left–right, and bottom–top directions, respectively [17]. The conductivities of the 3 layers are set to 1 s/m (scalp), 1/80 s/m (skull) and 1 s/m (brain). In all cases, it is supposed that no prior information regarding the regions of the sources is available. Therefore, for three approaches, the brain was divided into 16 regions with approximately equal volumes to be consistent with the FINE approach, which considered 16 regions as a reasonable number of brain regions as performed in [6]. In particular, the brain as a hemisphere was first divided into four quadrants. Then in each quadrant the brain region is further divided into four horizontal slices.

4.1. Simulation results

Here, we investigate various simulation conditions to estimate the accuracy and resolvability of the proposed beamspace FFA approach in comparison with regular FFA and FINE approaches, relating to the estimation of Event Related Potential (ERP) waveforms. In these simulations, two head models with different number of voxels are used for forward modeling (generating the ERP signal) and inverse solution (localizing the source) to mimic the real situation where the source location is not expected to be exactly at one of the head model voxels. The head model for forward modeling contains 6568 voxels, while the head model for inverse solution contains 3750 voxels that both uniformly distributed throughout the volume of the brain. We named the head model for forward modeling as 'forward head model' and the head model for inverse solution as 'localizing head model'. A 256 channel HydroCel Geodesic Sensor Net (HCGSN) is used in all simulations.

We use the method of [18] to generate the ERP data. An ERP waveform exhibits a peak, which reflects phasic bursts of activity in one or more brain regions that are triggered by experimental events of interest. Specifically, it is assumed that an ERP-like waveform is evoked by each event, but that on any given trial this ERP signal is buried in background EEG noise. Thus, to improve the effective ERP peak-to background EEG signal ration, multiple ERP trials from repeated stimuli are processed to extract the ERP waveform. Typically, this processing involves only simple averaging of the multiple ERP signals. A detailed explanation of the method is available in [12]. The generated ERP data has R  =  100 trials with a sampling frequency of ${{f}_{\text{s}}}=250$ Hz, where each trial has a length of K  =  100 samples, corresponding to a time sequence which may be denoted using Matlab notation as [0: 4: 396] ms. In all simulated data, the ERP signal for each trial consists of a half cycle of a sinusoid with a user-specified frequency $\widetilde{{f}}$ named ERP peak frequency that is centered at the peak location. The peak location of each ERP trial in time is computed by adding a normally distributed random jitter with a standard deviation of ${{T}_{j}}$ to the center location of the ERP peak. In this study, the ERP peak frequency value is $\widetilde{{f}}=10$ Hz and a jitter standard deviation of ${{T}_{j}}=\,\,5\,$ ms is considered around the center of the ERP peak across all trials.

We evaluate the performance of the beamspace FFA method relative to the regular FFA and FINE approaches for the following scenarios. The x, y, and z components of the dipole moment for all scenarios are respectively, $\left[{{\mu}_{x}},{{\mu}_{y}},{{\mu}_{z}}\right]=\left[1,0\text{,}~0\right]$ and the sources are semi-correlated with the correlation factor of $\rho \approx 0.5$, based on the definition [6]:

$$\begin{eqnarray}&&\rho =\frac{\underset{k=1}{\overset{M}{\sum}}{{\alpha}_{1}}(k){{\alpha}_{2}}(k)}{\sqrt{\underset{k=1}{\overset{M}{\sum}}\alpha _{1}^{2}(k)}\sqrt{\underset{k=1}{\overset{M}{\sum}}\alpha _{2}^{2}(k)}},\end{eqnarray} \tag{ 11 }$$

where ${{\alpha}_{1}}(k)$ and ${{\alpha}_{2}}(k)$ denote the source signals at time sample k and M represents the number of time samples.

• 1.  
  Two cortical sources: For this simulation, two superficial dipole sources were used to generate the measured scalp voltages of the EEG electrodes. The sources were located over superior frontal medial right gyrus, and postcentral right gyrus based on Automated Anatomical Labeling (AAL) digital atlas provided by the Montreal Neurological Institute (MNI) [19], with the MRI coordinates of $\left[x,y,z\right]=\left[95\,\,167\,\,124\right]$ mm and $\left[x,y,z\right]=\left[103\,\,88\,\,151\right]$ mm in forward head model, respectively. The averaged signals over 100 trials from 256 electrodes and the corresponding topography are shown in figures 1(a) and (b), respectively, where the mean of ERP peaks between all electrodes to the mean of maximum background EEG signals or ERP peak-to-background EEG signal ratio is 20log₁₀ (32.15/18.09)  ≈  5 dB (figure 1(a)). The corresponding source localization results for the three methods are shown in figure 2, where the detected locations have been mapped into MRI images. From figure 2 the beamspace FFA approach estimate the locations of the dipoles at [99 169 123] mm and [99 87 148] mm, respectively that were the closest location in the localizing head model to the true locations in the forward head model. The distance between the true and estimated dipole locations are respectively 4.58 mm and 5.10 mm. The regular FFA approach estimates the location of the dipoles at [86 166 121] mm and [106 82 150] mm respectively, where the distance between the true and estimated dipole locations is 9.54 mm and 6.78 mm. The FINE approach fails to detect the source located over superior frontal medial right gyrus and estimates the location of the source in postcentral right gyrus at [99 86 158] mm with the distance of 8.31 mm from the true location. Therefore, the beamspace FFA approach has better accuracy in terms of estimating the dipoles locations. Furthermore, from figure 2, the beamspace FFA has the most focal estimations around the true dipoles locations.
• 2.  
  Two deep sources: Since the voltage fields of the brain activities decrease with the square of distance from skull, therefore the activity from deep sources are more weakly reflected in EEG signals and so more difficult to detect than superficial sources. To compare the performance of the two methods in detecting deep sources, an EEG data set is produced by the activation of two deep dipole sources located at the left hippocampus and left insula regions, where the MRI coordinates of the sources are at $\left[x,y,z\right]=\left[66\,\,89\,\,64\right]$ mm and $\left[x,y,z\right]=\left[55\,\,143\,\,78\right]$ mm in forward head model, respectively. The brain active locations corresponding to three methods are shown in figure 3. From figure 3 the beamspace FFA approach estimates the location of the dipoles at [67 84 66] mm and [54 143 81] mm respectively, where the distance between the true and estimated dipole locations is 5.48 mm and 3.16 mm, while the regular FFA approach estimates the location of the dipoles at [65 92 66] mm and [54 152 78] mm respectively, where the distance between the true and estimated dipole locations are 3.74 mm and 9.05 mm. It is seen that again the FINE approach is unable to recover both sources. Only the source located in the left hippocampus region is detected by the FINE approach, where the estimated location is at [54 94 70] with the distance of 14.32 mm from the true location. Therefore, the beamspace FFA approach has again the best performance.
• 3.  
  One deep and one cortical source: One of the problems in source localization is depth biasing, where the deep sources are mislocalized to superficial source locations, leading to solutions that are closer to the sensors [20]. To compare the performance of methods regarding the problem with depth biasing, an EEG data set is produced by the activation of two sources located at the thalamus left and the superior frontal right gyrus with the MRI coordinates of $\left[x,y,z\right]=\left[73\,\,95\,\,72\right]$ mm and $\left[x,y,z\right]=\left[109\,\,124\,\,148\right]$ mm, respectively. The corresponding MRI mapped localization results are shown in figure 4. The beamspace FFA approach estimates the sources at [76 98 74] mm and [110 123 144] mm with the distances of 4.69 mm and 4.24 mm from the true locations, respectively, while the regular FFA approach estimates the locations at [76 98 74] mm and [109 122 140] mm with the distances of 4.69 mm and 8.25 mm from the true locations and the FINE approach estimates the locations at [75 97 76] mm and [100 119 130] mm with the distances of 4.90 mm and 20.74 mm. Therefore, the beamspace FFA again results in the most accurate localization performance.
• 4.  
  Joint temporal and spatial localization performance: To further compare the performances of the beamspace FFA, regular FFA, and FINE algorithms, we first evaluate their performances when two dipole sources are injected in different locations for the three former scenarios where the distance of the two sources are at least 5 mm. The number of locations for each source and the total number of considered pair-wise locations for the two sources related to all three scenarios are provided in table 1. Tables 2, 4, and 6 demonstrate the percentages of the pair-wise locations for these three scenarios, when (1) at least one source is localized accurately (within 2 cm of the true location and within the 8 ms (2 time samples) of the true time instance) and (2) both sources are localized accurately for three different ERP peak-to-background EEG signal ratios. We then compare the focus around the source location for the two methods by calculating the percentage of pairwise locations where the radii of activity distribution around each source location with the strength of at least 0.7 times of the maximum activity at true location is less than 5 cm. Tables 3, 5, and 7 demonstrate the percentage of concentrated detection for two methods for different ERP peak-to-background EEG signal ratio. According to these tables, the beamspace FFA approach has the best performance in all these scenarios in localizing the two sources with best concentration and resolvability followed by the regular FFA and FINE approaches.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 1. The number of locations for each sources and the total number of considered pair-wise locations for the three scenarios.

Scenario	Number of injected locations for each source	Total number of considered pair-wise locations
Two cortical sources	55	1485
Two deep sources	24	276
One deep and one cortical source	24 for deep sources and 55 for cortical sources	1320 (55  ×  24)

Table 2. Percentage of correct detections for beamspace FFA, regular FFA, and FINE methods using three different ERP peak-to-background EEG signal ratios of 5 dB, 10 dB, and 15 dB when two cortical sources are located in 1485 different locations of the cortex.

Method	Percentage of correct detection for:	ERP-to-background EEG  =  5 dB	ERP-to-background EEG  =  10 dB	ERP-to-background EEG  =  15 dB
Beamspace FFA	At least one source	97.44	99.32	99.86
	Two sources	72.26	76.70	81.41
Regular FFA	At least one source	96.50	97.98	98.45
	Two sources	64.98	69.97	73.47
FINE	At least one source	94.07	95.01	96.63
	Two sources	27.74	44.71	65.32

Table 3. Percentage of focused detections for beamspace FFA, regular FFA, and FINE methods using three different ERP peak-to-background EEG signal ratios of 5 dB, 10 dB, and 15 dB when two cortical sources are located in 1485 different locations of the cortex.

Percentage of focused detections	ERP-to-background EEG  =  5 dB	ERP-to-background EEG  =  10 dB	ERP-to-background EEG  =  15 dB
Beamspace FFA	64.17	74.01	78.92
Regular FFA	53.94	60.94	66.93
FINE	26.53	42.22	60.42

Table 4. Percentage of correct detections for beamspace FFA, regular FFA, and FINE methods using three different ERP peak-to-background EEG signal ratios of 5 dB, 10 dB, and 15 dB when two deep sources are located in 276 pairwise different locations in deep brain regions.

Method	Percentage of correct detection for:	ERP-to-background EEG  =  5 dB	ERP-to-background EEG  =  10 dB	ERP-to-background EEG  =  15 dB
Beamspace FFA	At least one source	88.04	96.74	100
	Two sources	78.62	90.56	100
Regular FFA	At least one source	78.98	87.68	97.82
	Two sources	64.85	76.08	94.20
FINE	At least one source	72.82	84.42	91.67
	Two sources	29.30	51.45	69.93

Table 5. Percentage of focused detections for beamspace FFA, regular FFA, and FINE methods using three different ERP peak-to-background EEG signal ratios of 5 dB, 10 dB, and 15 dB when two deep sources are located in 276 pairwise different locations in deep brain regions.

Percentage of focused detections	ERP-to-background EEG  =  5 dB	ERP-to-background EEG  =  10 dB	ERP-to-background EEG  =  15 dB
Beamspace FFA	56.52	84.06	98.19
Regular FFA	42.75	66.03	81.88
FINE	25.36	48.91	63.77

Table 6. Percentage of correct detections for beamspace FFA, regular FFA, and FINE methods using three different ERP peak-to-background EEG signal ratios of 5 dB, 10 dB, and 15 dB when one deep source and one cortical source are located in 1320 pairwise different locations of the brain.

Method	Percentage of correct detection for:	ERP-to-background EEG  =  5 dB	ERP-to-background EEG  =  10 dB	ERP-to-background EEG  =  15 dB
Beamspace FFA	At least one source	99.39	100	100
	Two sources	76.81	78.18	81.89
Regular FFA	At least one source	93.94	100	100
	Two sources	61.96	75.07	79.39
FINE	At least one source	89.62	95.37	100
	Two sources	14.62	45.68	60.91

Table 7. Percentage of focused detections for beamspace FFA, regular FFA, and FINE methods using three different ERP peak-to-background EEG signal ratios of 5 dB, 10 dB, and 15 dB when one deep source and one cortical source are located in 1320 pairwise different locations of the brain.

Percentage of focused detections	ERP-to-background EEG  =  5 dB	ERP-to-background EEG  =  10 dB	ERP-to-background EEG  =  15 dB
Beamspace FFA	55.75	63.86	75.23
Regular FFA	33.94	48.93	64.69
FINE	9.47	41.89	58.48

4.2. Results using real EEG data

In order to evaluate the performance of the algorithm in dealing with real experimental data, we use an EEG data corresponding to the opening and closing of left fist from the EEG Motor Movement/Imagery Dataset recorded using BCI2000 [21] instrumentation system available through Physionet [22]. The EEG activity for this task was recorded on 64 channels placed on their scalp as per the standard international 10–10 system with the sampling frequency of 160 Hz. The recorded EEG waveforms from 64 electrodes of the measured data are shown in figure 5. The length of the data is chosen to be 1.25 s from 625 ms before the maximum of EEG activity to 625 ms after the maximum to include just one task. Figure 6 shows the most active area of the brain at the time of maximum EEG activity (time 0 in figure 5) for the opening and closing of the left fist for the three methods. From figures 6(a) and (b) the beamspace FFA and regular FFA approaches localized the activity in the right pre- and post-central gyrus and right middle frontal gyrus which are the projection of the left hand in the premotor cortex, supplementary motor cortex, and primary somatosensory cortex that are proven to be active regions for this task with fMRI technique [23]. However, regular FFA approach also shows the activity in the right precuneus area, which is not among the active regions for this task. From figure 6(c), the FINE approach shows the activity on both right and left primary motor cortex and could not estimate the active areas accurately.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

This study presents an alternative FFA beamforming named beamspace FFA for brain source localization. In this approach, first the brain volume is divided into a number of regions of similar volume and then the beamspace transformation is applied to the data without any dimension reduction in order to optimally preserve the sources within each specified region. Then, the FFA approach is applied to the beamspaced data. The key advantage of the beamspace FFA algorithm is to reduce the sensitivity to the correlation between the sources in different regions by suppressing the activities (interferences) that are outside each specific region when processing that region. In fact the beamspace technique applied to the FFA approach offers the potential for significant reduction in estimated source location variability especially in scenarios with limited data. The performance of the beamspace FFA scheme is compared by the regular FFA approach and widely used FINE approach with three illustrative examples of two dipole source configurations using simulated data on a three realistic layers head model that was obtained using BrainStorm^® software. The simulation results show that the beamspace FFA method can localize all types of brain activities more accurately than both the regular FFA and FINE approaches. We then compare the performance of the beamspace FFA method with the regular FFA and FINE approaches by computing the percentage of correct estimation of the time and spatial location of at least one source and two sources for the three scenarios of having two cortical, two deep, and one deep and one cortical sources with 1485, 276, and 1320 pair wise locations, respectively (tables 2, 4, and 6). Three different ERP peak-to-background EEG signal ratios of 5 dB, 10 dB and 15 dB were considered for each scenario. We then compared the focus around the source location for the three methods by calculating the percentage of pairwise locations where the radii of activity distribution around each source location with the strength of at least 0.7 times of the activity at true location is less than 5 cm (tables 3, 5, 7). From the tables, the beamspace FFA brain localization results show considerably improved performance in comparison to the regular FFA and FINE approaches in the 'data poor' scenario (small number of time samples and low ERP peak-to-background EEG signal ratios).

As a final experiment, we compared the performance of the beamspace FFA method with the regular FFA and FINE approaches using the EEG data corresponding to the opening and closing of left fist from the EEG Motor Movement/Imagery dataset recorded using BCI2000 instrumentation system. This study showed that the active locations yielded by the beamspace FFA approach were in the regions that are proven to be active regions for this task with fMRI technique, while the regular FFA approach showed activity in the precuneus area in addition to the beamspace FFA active locations, which is not among the proven active regions for this task and the FINE approach could not estimate the active areas accurately. This improved performance of the beamspace FFA in comparison to FINE scheme arises at the cost of increased computational complexity due to multiple, although smaller, fully adaptive processes that must be executed. However, the computational time per brain location could be reduced by a parallel implementation of these processes.