A simple non-parametric statistical thresholding for MEG spatial-filter source reconstruction images

Abstract

This paper proposes a simple statistical method for extracting target source activities from spatio-temporal source activities reconstructed from MEG measurements. The method requires measurements in a control condition, which contains only non-target source activities. The method derives, at each pixel location, an empirical probability distribution of the non-target source activity using the time-course reconstruction obtained from the control period. The statistical threshold that can extract the target source activities is derived from the empirical distributions obtained from all pixel locations. Here, the multiple comparison problem is addressed with a two step procedure involving standardizing these empirical distributions and deriving an empirical distribution of the maximum pseudo T value at each pixel location. The results of applying the proposed method to auditory evoked measurements are presented to demonstrate the method’s effectiveness.

1 Introduction

Among the various technologies for noninvasive neural measurement, the major advantage of magnetoencephalography (MEG) is its ability to provide fine temporal resolution, on the order of milliseconds [1]. Neuromagnetic imaging can therefore visualize neural activities with such a fine time resolution, and provide functional information about brain dynamics. One major problem is that the measured MEG signal generally contains not only a magnetic field associated with the signal sources of interest but also contains interference magnetic fields generated from non-target activities. Such non-target activities include spontaneous brain activities or some evoked activities that are not the interest of the current investigation. These non-target activities generally overlap with the target signal activities in the source reconstruction, and they often make interpreting the reconstruction results difficult.

In most studies using positron emission tomography (PET) or functional magnetic resonance imaging (fMRI), the experiments are carefully designed to extract only target activities. A common example of such experimental design contains two kinds of stimuli: a task stimulus and a control stimulus. The task stimulus generally elicits the target cortical activities as well as other activities associated with the target activities. The control stimulus is designed to elicit only the latter activities. Then, by calculating the statistical difference between the images measured with the two kinds of stimuli, the target activities can be revealed. Both parametric [2][3] and non-parametric statistics [4] have been used to calculate such statistical differences.

This paper proposes a simple method of statistical subtraction between task and control measurements. The method is applicable to spatio-temporal source reconstruction from MEG/EEG measurements. It assumes neural activities to be quasi-stochastic, and it uses nonparametric statistics to derive an empirical probability distribution of these activities using the time course reconstruction in the control period. This empirical distribution is then used for deriving an appropriate value for the statistical thresholding. The thresholding can extract the target activities that exist only in the task measurements by eliminating other non-target activities that exist both in the task and control measurements.

In this paper, we present the proposed statistical thresholding method using spatial filter source reconstruction [5]. This is because the formulation of the spatial filter is relatively simple and the spatial filter techniques have been successfully applied to MEG source analysis [6][7][8]. However, the applicability of the proposed method is not limited to the spatial filter formulation and it can be used with any type of source estimation method that can provide the spatio-temporal source reconstruction, i.e., that can reconstruct source time courses at all pixel locations.

2 Spatial filter formulation for MEG source reconstruction

2.1 Definitions

We define the magnetic field measured by the mth detector coil at time t as b_m(t), and a column vector b(t) = [b₁(t), b₂(t), …, b_M(t)]^T as a set of measured data where M is the total number of sensor coils and superscript T indicates the matrix transpose. The spatial location is represented by a three-dimensional vector r: r = (x, y, z). The second-order moment matrix of the measurement is denoted R, i.e., R = ⟨b(t)b^T(t)⟩ where ⟨·⟩ indicates the ensemble average, which is replaced by the time average over a certain time window in practice. (When ⟨b(t)⟩ ≈ 0 holds, R is also equal to the covariance matrix of the measurement.) The magnitude of the source moment is denoted s(r, t). The source orientation is defined as a three-dimensional column vector η(r) = [η_x(r), η_y(r), η_z(r)]^T whose ζ component (where ζ equals x, y, or z) is equal to the cosine of the angle between the direction of the source and the ζ axis.

We define $l_m^{\zeta }\left(\mathit{r}\right)$ as the output of the mth sensor; the output is induced by the unit-magnitude source located at r and pointing in the ζ direction. The column vector l_ζ(r) is defined as ${\mathit{l}}_{\zeta }\left(\mathit{r}\right)={\left[l_1^{\zeta }\right(\mathit{r}),l_2^{\zeta }(\mathit{r}),\cdots ,l_M^{\zeta }(\mathit{r}\left)\right]}^T$. We define the lead field matrix, which represents the sensitivity of the whole sensor array at r, as L(r) = [l_x(r), l_y(r), l_z(r)].

2.2 Typical non-adaptive and adaptive spatial filters

Spatial filter techniques estimate the source current density by applying a simple linear operation to the measured data, i.e.,

$$\widehat{s}(\mathit{r},t)={\mathit{w}}^T\left(\mathit{r}\right)\mathit{b}\left(t\right)=\sum _{m=1}^M{w}_m\left(r\right)b_m\left(t\right),$$ (1)

where $\widehat{s}(\mathit{r},t)$ is the estimated source magnitude. The column vector

$$\mathit{w}\left(\mathit{r}\right)={\left[w_1\right(\mathit{r}),w_2(\mathit{r}),\dots ,w_M(\mathit{r}\left)\right]}^T$$

represents a set of the filter weights. The filter weights w(r) should only pass the signal from a source with a location r and reject the signals generated at other locations. Since the source is a three-dimensional vector quantity, the weight vector generally depends on the source directions. In this paper, we define w(r) as the weight vector in the optimum direction η_opt(r), which is determined as the direction that gives the maximum spatial-filter outputs at each r.

There are two types of spatial filter techniques. One is a non-adaptive method in which the filter weight is independent of the measurements. The best-known non-adaptive spatial filter is the minimum-norm estimate[9]. The filter weight is expressed as

$$\mathit{w}\left(\mathit{r}\right)={\mathit{G}}^{-1}\mathit{L}\left(\mathit{r}\right){\eta }_{\mathit{opt}}\left(\mathit{r}\right).$$ (2)

The matrix G is often referred to as the gram matrix, which is given by calculating the overlap between the lead fields,

$$\mathit{G}=\int \mathit{L}\left(\mathit{r}\right){\mathit{L}}^T\left(\mathit{r}\right)\mathit{d}\mathit{r}.$$ (3)

The estimated current density is then expressed as

$$\widehat{s}(\mathit{r},t)={\eta }_{\mathit{opt}}^T\left(\mathit{r}\right){\mathit{L}}^T\left(\mathit{r}\right){\mathit{G}}^{-1}\mathit{b}\left(t\right).$$ (4)

The minimum-norm spatial filter in its original form is known to give erroneous source reconstruction results. Therefore, to improve the performance it is often used with some kind of constraint on source distributions [10][11][12].

The other type of spatial filter is an adaptive spatial filter in which the filter weight depends on the measurements. The best-known adaptive spatial filter is the minimum-variance spatial filter, which is customarily referred to as the minimum-variance beamformer [13][14][15]. In this method, the weight vector is given by

$${\mathit{w}}^T\left(\mathit{r}\right)=\frac{{\mathit{l}}^T\left(\mathit{r}\right){\mathit{R}}^{-1}}{{\mathit{l}}^T\left(\mathit{r}\right){\mathit{R}}^{-1}\mathit{l}\left(\mathit{r}\right)},$$ (5)

where l(r) is defined as l(r) = L(r)η_opt(r). (The method for determining η_opt(r) for the adaptive spatial filter is discussed in [16]. Also, in practice, the normalized lead field l(r)/l(r) is used in Eq. (5) to avoid a source location bias caused by the variation of the lead field norm l(r) [17].) The minimum-variance beamformer can be extended to the eigenspace-projection beamformer, which is known to be tolerant of errors in the forward modeling or in the estimation of the data covariance matrix [18]. The extension is attained by projecting the weight vector in Eq. (5) onto the signal subspace of the measurement covariance matrix. That is, redefining the weight vector in Eq. (5) as w_(MV)(r), the weight vector for the eigenspace-projection beamformer is obtained using

$$w\left(r\right)={\mathit{E}}_S{\mathit{E}}_S^T{\mathit{w}}_{\left(\mathrm{MV}\right)}\left(\mathit{r}\right).$$ (6)

In this equation, E_S is a matrix whose columns consist of the signal-level eigenvectors of R, and ${\mathit{E}}_S{\mathit{E}}_S^T$ is the projection matrix that projects a vector onto the signal subspace of R. This eigenspace projection beamformer was used in the experiments described in Sections 5 and 6.

3 Evaluation of statistical significance using parametric statistics

The evaluation of the statistical significance of the spatial filter outputs has typically been performed using parametric statistics[10][13][19][20]. The basic assumption of the parametric method is that the measurement consists of deterministic signal and Gaussian noise, i.e.,

$$\mathit{b}\left(t\right)={\mathit{b}}_I\left(t\right)+\mathit{n}\left(t\right),$$ (7)

where b_I(t) is the signal of interest, i.e., the signal generated from brain sources that are the target of current investigation.

In Eq. (7), n(t) is the noise vector and each element of n(t) is assumed to follow $\mathcal{N}(0,{\sigma }_0^2)$, which indicates the Gaussian distribution with zero mean and variance ${\sigma }_0^2$. The spatial filter outputs $\widehat{s}(\mathit{r},t)$ are expressed as

$$\widehat{s}(\mathit{r},t)={\mathit{w}}^T\left(\mathit{r}\right)\mathit{b}\left(t\right)={\mathit{w}}^T\left(\mathit{r}\right){\mathit{b}}_I\left(t\right)+{\mathit{w}}^T\left(\mathit{r}\right)\mathit{n}\left(t\right).$$ (8)

Therefore, since these Gaussian processes are assumed to be uncorrelated between different sensor recordings, the outputs $\widehat{s}(\mathit{r},t)$ follows $\mathcal{N}\left({\mathit{w}}^T\right(\mathit{r}\left){\mathit{b}}_I\right(t),{\sigma }_0^2{\mid \mid \mathit{w}\left(\mathit{r}\right)\mid \mid }^2)$, which is a Gaussian distribution with a mean of w^T(r)b_I(t) and a variance of ${\sigma }_0^2{\mid \mid \mathit{w}\left(\mathit{r}\right)\mid \mid }^2$. Actually, since ${\sigma }_0^2$ must be estimated from the measured data, the distribution of $\widehat{s}(\mathit{r},t)$ is not exactly represented by the Gaussian distribution but by the t distribution.

The statistical evaluation can be performed by testing the null hypothesis at each pixel location. (The null hypothesis is that there is no signal source activity.) That is, the z score under the null hypothesis, $\widehat{s}(\mathit{r},t)∕({\sigma }_0\mid \mid \mathit{w}(\mathit{r})\mid \mid )$ is calculated and compared to z_α/2, which is the two-tailed z score corresponding to the α level of significance, which is equal to the probability of Type I error. This procedure is performed at each pixel location, and if the calculated z score is higher than z_α/2, the estimated source activity $\widehat{s}(\mathit{r},t)$ is considered to be statistically significant. This procedure can be extended to incorporate the multiple comparison problem [20].

4 Proposed nonparametric statistical significance evaluation

The signal and noise model expressed in Eq. (7) is, in general, insufficient to express real-world measurements, and the measured data should be expressed as

$$\mathit{b}\left(t\right)={\mathit{b}}_I\left(t\right)+{\mathit{b}}_{\xi }\left(t\right)+\mathit{n}\left(t\right),$$ (9)

where b_ξ(t) is the magnetic field generated from sources other than the signal sources, such as spontaneous brain activities or some evoked activities that are not the target of the current investigation. This b_ξ(t) is often referred to as the brain noise. Here, we propose a simple nonparametric method that can take such brain noise into consideration. The spatial filter outputs obtained from b(t) is expressed as

$$\widehat{s}(\mathit{r},t)={\mathit{w}}^T\left(\mathit{r}\right){\mathit{b}}_I\left(t\right)+{\mathit{w}}^T\left(\mathit{r}\right){\mathit{b}}_{\xi }\left(t\right)+{\mathit{w}}^T\left(\mathit{r}\right)\mathit{n}\left(t\right)={\widehat{s}}_I(\mathit{r},t)+{\widehat{s}}_c(\mathit{r},t),$$ (10)

where

$${\widehat{s}}_I(\mathit{r},t)={\mathit{w}}^T\left(\mathit{r}\right){\mathit{b}}_I\left(t\right)\text{and}{\widehat{s}}_c(\mathit{r},t)={\mathit{w}}^T\left(\mathit{r}\right)\left({\mathit{b}}_{\xi }\right(t)+\mathit{n}(t\left)\right).$$ (11)

Here, ${\widehat{s}}_I(\mathit{r},t)$ is the estimated source activity of interest, and ${\widehat{s}}_c(\mathit{r},t)$ is the estimated background interference plus noise. The problem with the parametric modeling described in the preceding section is that it cannot efficiently take the background interference into account, because the Gaussianity assumption may not hold for ${\widehat{s}}_c(\mathit{r},t)$. The key assumption in the proposed method is that the control measurement that can provide b_c(t) = b_ξ(t) + n(t) is available. In general, this assumption is approximately fulfilled, because the pre-stimulus measurement can be considered as a control in many cases. Using this control measurement, the proposed method first derives an empirical distribution of ${\widehat{s}}_c(\mathit{r},t)$, and with this empirical distribution, the method determines the statistical threshold. The procedures are as follows:

(i) Empirical distribution formation

We calculate ${\widehat{s}}_c(\mathit{r},t_j)$, by applying the spatial filter to the control measurement b_c(t_j) where t_j is the discrete time point in the control measurement. We then calculate $\widehat{F}\left(x\right)$, which is the empirical distribution of the modulus of the time course $\mid {\widehat{s}}_c(\mathit{r},t_j)\mid$, such that $\widehat{F}\left(x\right)=\#\{\mid {\widehat{s}}_c(\mathit{r},t_j)\mid \le x\}∕K_c$ where $\#\{\mid {\widehat{s}}_c(\mathit{r},t_j)\mid \le x\}$ indicates the number of $\mid {\widehat{s}}_c(\mathit{r},t_j)\mid$ which are less than or equal to x, and K_c is the number of total time points in the control measurement. This procedure is repeated and the empirical distribution is calculated at all pixel locations. Since $\widehat{F}\left(x\right)$ is obtained at each pixel location r, $\widehat{F}\left(x\right)$ is rewritten as $\widehat{F}(x\mid \mathit{r})$ below.

(ii) Statistical thresholding without multiple comparisons

Using $\widehat{F}(x\mid \mathit{r})$, we could obtain the statistical threshold at r, Σ(r), such that $\Sigma \left(\mathit{r}\right)={\widehat{F}}^{-1}(1-\alpha \mid \mathit{r})$ where α is a level of significance. In practice, the inverse of the empirical distribution can be calculated by first sorting $\mid {\widehat{s}}_c(\mathit{r},t_j)\mid$ in increasing order:

$$\mid {\widehat{s}}_c(\mathit{r},t_{\left(1\right)})\mid \le \mid {\widehat{s}}_c(\mathit{r},t_{\left(2\right)})\mid \le \cdots \le \mid {\widehat{s}}_c(\mathit{r},t_{\left(K_c\right)})\mid ,$$ (12)

and then by choosing $\mid {\widehat{s}}_c(\mathit{r},t_{\left(q\right)})\mid$ as Σ(r) where q = [(1 − α)K_c] and [·] indicates the maximum integer that does not exceed the value in the parenthesis. However, the statistical threshold obtained in this manner does not take multiple comparisons into consideration, and instead of implementing the above-mentioned procedure, the following procedure is performed.

(iii) Statistical thresholding with multiple comparisons

The proposed method uses maximum statistics [21][22] to address the multiple comparison problem. To utilize maximum statistics, we first standardize the empirical distribution of $\mid {\widehat{s}}_c(\mathit{r},t_j)\mid$ by calculating T (r, t_j) such that

$$T(\mathit{r},t_j)=\frac{\mid {\widehat{s}}_c(\mathit{r},t_j)\mid -{\langle \mid {\widehat{s}}_c(\mathit{r},t_j)\mid \rangle }_c}{\widehat{\sigma }\left(\mathit{r}\right)}.$$ (13)

Here,

$${\widehat{\sigma }}^2\left(\mathit{r}\right)={\langle {\widehat{s}}_c{(\mathit{r},t_j)}^2\rangle }_c-{\langle \mid {\widehat{s}}_c(\mathit{r},t_j)\mid \rangle }_c^2,$$

and ⟨·⟩_c indicates the time average over the control period, i.e.,

$${\langle {\widehat{s}}_c{(\mathit{r},t_j)}^2\rangle }_c=\frac{1}{K_c}\sum _{j=1}^{K_c}{\widehat{s}}_c{(\mathit{r},t_j)}^2\text{and}{\langle \mid {\widehat{s}}_c(\mathit{r},t_j)\mid \rangle }_c=\frac{1}{K_c}\sum _{j=1}^{K_c}\mid {\widehat{s}}_c(\mathit{r},t_j)\mid .$$

We then calculate the maximum T value T_max(r) at each pixel location. The maximum T value at the ith pixel location is denoted $T_{max}^i$ where i = 1, …, K_N and K_N indicates the total number of pixels. We next obtain the empirical distribution of $T_{max}^i$, $\widehat{H}\left(x\right)$, such that $\widehat{H}\left(x\right)=\#\{T_{max}^i\le x\}∕K_N$ where $\#\{T_{max}^i\le x\}$ is the number of $T_{max}^i$ values which are less than or equal to x. We can then obtain the threshold of the $T_{max}^i$ value for the α-significance level, $T_{max}^{th}$, such that $T_{max}^{th}={\widehat{H}}^{-1}(1-\alpha )$. The inverse of this empirical distribution can be calculated by first sorting $T_{max}^i$ in increasing order:

$$T_{max}^{\left(1\right)}\le T_{max}^{\left(2\right)}\le \dots \le T_{max}^{\left(K_N\right)},$$ (14)

and choose $T_{max}^{\left(p\right)}$ as $T_{max}^{th}$ where p = [(1−α)K_N]. We finally obtain the statistical threshold for the spatial-filter reconstruction, Σ(r), by using

$$\Sigma \left(\mathit{r}\right)=T_{max}^{\mathit{th}}\widehat{\sigma }\left(\mathit{r}\right)+{\langle \mid {\widehat{s}}_c(\mathit{r},t_j)\mid \rangle }_c.$$ (15)

We evaluate the statistical significance of the spatial filter outputs by comparing the outputs by comparing the outputs $\widehat{s}(\mathit{r},t)$ with Σ(r), and when $\widehat{s}(\mathit{r},t)\ge \Sigma \left(\mathit{r}\right)$, the outputs $\widehat{s}(\mathit{r},t)$ are considered to be statistically significant.

5 Numerical Experiments

We conducted numerical experiments to show the effectiveness of the proposed statistical thresholding. We use a sensor alignment of the 37-sensor array from the Magnes™ (4D Neuroimaging Inc., San Diego) neuromagnetometer. The source-sensor configuration and the coordinate system are illustrated in Fig. 1(a). The coordinate origin is set at the center of the sensor coil located at the center of the array. The three point-like sources, shown by the small filled circles in this figure, are assumed to be located at (0, −1, −6) cm, (0, 1, −6) cm, and (0, −1.6, −7.2) cm on the same plane (x = 0). These locations of the three sources are denoted r₁, r₂, and r₃. The simulated magnetic field is calculated for 400 ms pre-stimulus and 400 ms post-stimulus time windows with a sampling rate of 1 kHz. Here, the nearly orthogonal three time courses shown in Fig. 1(b) are used as the time courses of the three sources.

Fig. 1.

(a) The coordinate system and source-sensor configuration used in the numerical experiments. The coordinate origin was set at the center of the sensor coil located at the center of the array. The three point-like sources, shown by the small filled circles, were assumed to be located at (0, −1, −6) cm, (0, 1, −6) cm, and (0, −1.6, −7.2) cm on the plane of x = 0. The large circle indicates the projection of the sphere used for the forward calculation. (b) The first three panels from top to bottom show the time courses assumed as the time courses of the first, second, and third sources, respectively. The bottom panel shows the simulated magnetic recordings used for the reconstruction experiments.

In these experiments, the pre-stimulus period is considered the control period. The first and second sources are considered the signal sources of interest because they are only active in the post-stimulus (task) period. The third source is considered the control source, because it is active both in the pre- and post-stimulus periods. To simulate brain background activity, (which is so-called brain noise), spontaneous MEG measured from an awake human subject was added to this computer-generated magnetic field to create simulated magnetic recordings. Here, the spontaneous MEG was measured with a sampling rate of 1 kHz using the same sensor array, and averaged over 400 trials. The resulting simulated magnetic recordings are shown in the bottom panel of Fig. 1(b).

The eigenspace-projected adaptive spatial filter [18], mentioned in Section 2.2, was applied to these simulated recordings. The data between 0 and 400 ms was used for calculating the covariance matrix, and the weight vector of the spatial filter was obtained with this covariance matrix. The spatial filter was applied to both the pre- and post-stimulus data, and the reconstructed results are shown in Fig. 2. Figure 2(a) shows the reconstructed time courses at the three source locations, $\widehat{s}({\mathit{r}}_1,t)$, $\widehat{s}({\mathit{r}}_2,t)$, and $\widehat{s}({\mathit{r}}_3,t)$. Three snapshot reconstructions at 220, 265, and 300 ms, $\mid \widehat{s}(\mathit{r},220)\mid$, $\mid \widehat{s}(\mathit{r},265)\mid$, and $\mid \widehat{s}(\mathit{r},300)\mid$, are shown in Fig. 2(b). The reconstruction-averaged over the whole post-stimulus time window between 0 and 400 ms, $\sqrt{{\langle \widehat{s}{(\mathit{r},t)}^2\rangle }_{post}}$ (where ⟨·⟩_post indicates the time average over the whole post-stimulus period)-is shown in Fig. 2(c).

Fig. 2.

(a) The reconstructed time courses for the first source $\widehat{s}({\mathit{r}}_1,t)$ (upper), the second source $\widehat{s}({\mathit{r}}_2,t)$ (middle), and the third source $\widehat{s}({\mathit{r}}_3,t)$ (bottom). (b) The snapshot reconstruction at 220 ms $\mid \widehat{s}(\mathit{r},220)\mid$ (upper left), and 265 ms $\mid \widehat{s}(\mathit{r},265)\mid$ (upper right), and 300 ms $\mid \widehat{s}(\mathit{r},300)\mid$ (lower left). The time instants at 220, 265, and 300 ms are shown by the three broken vertical lines in Fig. 2(a). (c) The reconstruction averaged over the poststimulus time window, $\sqrt{{\langle \widehat{s}{(\mathit{r},t)}^2\rangle }_{post}}$.

To apply the proposed statistical thresholding, we first calculate the modulus of the reconstructed time courses $\mid \widehat{s}(\mathit{r},t)\mid$ and derive an empirical null distribution from the pre-stimulus portion of $\mid \widehat{s}(\mathit{r},t)\mid$ at each pixel location. The magnitude time courses $\mid \widehat{s}({\mathit{r}}_1,t)\mid$, $\mid \widehat{s}({\mathit{r}}_2,t)\mid$, and $\mid \widehat{s}({\mathit{r}}_3,t)\mid$ are shown in Fig. 3(a). The resultant empirical distributions at r₁, r₂, and r₃, expressed as histograms of the prestimulus values of $\mid \widehat{s}({\mathit{r}}_1,t)\mid$, $\mid \widehat{s}({\mathit{r}}_2,t)\mid$, and $\mid \widehat{s}({\mathit{r}}_3,t)\mid$, are shown in Fig. 3(b). The empirical distributions are then standardized using Eq. (13), and the $T_{max}^i$ values are derived from the standardized distributions at all pixel locations. The distribution of $T_{max}^i$, expressed as a histogram, is shown in Fig. 4(a). Using this distribution, the value of $T_{max}^{th}$ is determined to be 3.40 for a significance level α of 0.05. Finally, using Eq. (15), we derive the threshold Σ(r), and when $\mid \widehat{s}(\mathit{r},t)\mid <\Sigma \left(\mathit{r}\right)$, $\mid \widehat{s}(\mathit{r},t)\mid$ is set equal to zero. The results of this thresholding with a 5% significance level applied to the reconstruction results in Fig. 2(b) are shown in Fig. 4 (b). The thresholded time-averaged reconstruction is shown in Fig. 4(c). The results indicate that the third source, active during both the pre- and post-stimulus periods, is removed from the post-stimulus reconstruction, verifying the effectiveness of the proposed method.

Fig. 3.

(a) Reconstructed magnitude time courses, $\mid \widehat{s}({\mathit{r}}_1,t)\mid$, $\mid \widehat{s}({\mathit{r}}_2,t)\mid$, and $\mid \widehat{s}({\mathit{r}}_3,t)\mid$ are shown from top to bottom, respectively. The horizontal broken lines in the upper two panels show the threshold values at r₁ and r₂. (b) Histograms of the prestimulus values of $\mid \widehat{s}({\mathit{r}}_1,t)\mid$ (upper left), $\mid \widehat{s}({\mathit{r}}_2,t)\mid$ (upper right), and $\mid \widehat{s}({\mathit{r}}_3,t)\mid$ (lower left)

Fig. 4.

(a) Histogram of $T_{max}^i$. The value of $T_{max}^{th}$ is determined to be 3.40 for a significance level α of 0.05 using this distribution. (b) Thresholded results with α = 0.05 for $\mid \widehat{s}(\mathit{r},220)\mid$ (upper left), $\mid \widehat{s}(\mathit{r},265)\mid$ (upper right), and $\mid \widehat{s}(\mathit{r},300)\mid$ (lower left). (c) The thresholded reconstruction for $\sqrt{{\langle \widehat{s}{(\mathit{r},t)}^2\rangle }_{post}}$.

6 Experiments

We applied the proposed method to auditory-evoked MEG data to test its effectiveness. The auditory-evoked fields were measured using the 275-channel Omega-275 ™ (CTF Systems Inc., Port Coquitlam) whole-cortex biomagnetometer installed at the Biomagnetic Imaging Laboratory, University of California, San Francisco. The auditory stimulus (1-kHz pure tone) was presented to the subject’s right ear. The average inter-stimulus interval was 2 s, with the interval randomly varied between 1.75 s and 2.25 s. The sampling frequency was set at 4 kHz, and an on-line filter with a bandwidth from 1 to 2 kHz was used. A total of 400 epochs were measured and these 400 epochs were averaged to produce the auditory-evoked recordings shown in Fig. 5. Here, although clear P50m and N100m peaks can be observed, we can see that these averaged recordings contain a considerable amount of periodic background activity.

Fig. 5.

(a) The 400-epoch-averaged auditory-evoked fields measured using the 275-channel sensor array. Among the 275 sensor recordings, the recordings from 132 sensors covering the subject’s left hemisphere are displayed.

The eigenspace-projected adaptive beamformer [18] was applied to these averaged recordings. The data between 0 to 200 ms was used to calculate the covariance matrix R, and the weight vector was obtained with this covariance matrix. Two latencies, 44 ms and 86 ms, are selected. One is near the peak of P50m and the other is near the peak of N100m. These time points are shown in the two vertical broken lines in Fig. 5. The snapshots of the source reconstruction results at these latencies are shown in Fig. 6(a) and (b). Both sets of the results contain a clear localized source in the left temporal lobe probably near the primary auditory area. However, the reconstruction results at 44 ms also contain other diffuse activity. The time-averaged reconstruction obtained from the whole prestimulus period (−400 − 0 ms), $\sqrt{{\langle \widehat{s}{(\mathit{r},t)}^2\rangle }_{pre}}$, is shown in Fig. 6(c) (where ⟨·⟩_pre indicates the time average over this prestimulus period.) These results also contain a diffuse source similar to that found in the snapshot at 44 ms, suggesting that these diffuse sources were caused by the periodic background activities observed in the waveforms in Fig. 5.

Fig. 6.

The maximum-intensity projections of the source reconstruction obtained using the eigenspace-projected adaptive spatial filter. (a) Snapshot at 44 ms of latency, (b) snapshot at 86 ms of latency, and (c) time averaged reconstruction obtained from the whole prestimulus period (−400 and 0 ms). The left column shows the maximum intensity projections of the three-dimensional reconstruction onto the axial plane. The middle column shows those onto the coronal plane. The left column shows those onto the sagittal plane. The upper case letters L and R show the left and the right hemispheres, respectively.

To apply the proposed statistical thresholding, we use the measurements taken during the prestimulus period as a control. We reconstruct the time course $\mid {\widehat{s}}_c(\mathit{r},t_j)\mid$ of the prestimulus period and the empirical distribution of $\mid {\widehat{s}}_c(\mathit{r},t_j)\mid$ is calculated at each pixel location. The empirical distributions are then standardized using Eq. (13) to determine $T_{max}^i$ for the ith pixel, and the same procedure is repeated at all pixel locations to derive the distribution of $T_{max}^i$. The empirical distribution of $T_{max}^i$ is shown as a histogram in Fig. 7(a). Here, the number of pixels in the three-dimensional reconstruction grid was 6800. The value of $T_{max}^{th}$ was determined to be 3.34 using this empirical distribution. Then, using Eq. (15), we derive the threshold Σ(r). When $\mid \widehat{s}(\mathit{r},t)\mid <\Sigma \left(\mathit{r}\right)$, $\mid \widehat{s}(\mathit{r},t)\mid$ is set to zero. The results of this thresholding applied to the snapshot images in Fig. 6(a) and (b) are shown in Figs. 7 (b) and (c), respectively. Here, the significance level α was set at 0.05. The diffuse source activities contained in the results in Fig. 6(a) have been removed in Fig. 7(b). These results demonstrate the effectiveness of the proposed statistical thresholding in removing the influence of background source activities in the reconstruction results.

Fig. 7.

(a) Histogram of $T_{max}^i$ from all pixel locations. (b) The results of the proposed statistical thresholding applied to the snapshot shown in Fig. 6(a). (c) The results of the proposed statistical thresholding applied to the snapshot shown in Fig. 6(b).

7 Discussion and conclusion

We have developed a simple and novel non-parametric statistical thresholding procedure for tomographic reconstruction results from MEG data. An alternative method to ours has been recently proposed [21]. That method uses permutation tests to assess the statistical significance of the source reconstruction results. Thus, it not only requires intensive computer time but also requires raw epoch data to be stored. On the contrary, the method proposed in this paper does not use such computer intensive re-sampling methods as the permutation tests or the bootstrap, and can thus be implemented with much less computer time. In addition, our method does not require raw epoch data to be stored but uses rather only the averaged data. Moreover, in the permutation tests, a mixture of the control and task data is used to derive a null distribution, thus the null distribution contains signal information, although it ought to be derived solely based on the property of noise. In contrast, the proposed method derives a null distribution only from the control data, and thus it can achieve greater detectability than that from the permutation tests.

The proposed method uses the maximum statistics to address the multiple comparison problems. Without such a multiple comparison procedure, 100α% of pixels may exhibit false-positive activations in a single snapshot image (where α is the level of significance). The multiple comparison procedure, however, reduces this false positive probability to the level at which only 100α% of snapshot images may contain the false positive activations. However, since the proposed procedure does not perform the multiple comparisons in the temporal dimension, a reconstructed time course may still contain 100α% of time points that exhibit false activations. This fact can be seen in Fig. 3(a) in which the derived threshold values are indicated by the horizontal broken lines in the upper two panels. Particularly in the middle panel, several time points in the pre-stimulus period exceed the threshold value, resulting in the false positive activations at these time points. To avoid such false activations, the multiple comparisons should also be performed in the temporal dimension, and for such space-time multiple comparisons, an empirical distribution should be derived not only at each spatial point but also at each time point. Such a method for space-time multiple comparisons is currently under investigation, with the results to be published in the near future.

Although the proposed method has been described as a method for assessing the statistical significance of the source reconstruction results, the method can also be applied to determine the statistical significance of b_I(t) in the sensor recordings b(t). The null hypothesis here is b_I(t) = 0 in the task measurements. Let us denote the kth component of the vectors b(t), b_c(t) and b_I(t) as b^k(t), $b_c^k\left(t\right)$ and $b_I^k\left(t\right)$, respectively. To determine the statistical significance of $b_I^k\left(t\right)$, the empirical distribution of $\mid b_c^k\left(t_j\right)\mid$ is first standardized by calculating

$$T_c^k\left(t_j\right)=\frac{\mid b_c^k\left(t_j\right)\mid -{\langle \mid b_c^k\left(t_j\right)\mid \rangle }_c}{{\widehat{\sigma }}_b^k},$$ (16)

where

$${\left({\widehat{\sigma }}_b^k\right)}^2={\langle b_c^k{\left(t_j\right)}^2\rangle }_c-{\langle \mid b_c^k\left(t_j\right)\mid \rangle }_c^2,$$

t_j is a discrete time point in the control period, and ⟨·⟩_c again indicates the time average over the control period. We then calculate the maximum T value for the kth-channel recording, $T_{max}^k$, and obtain the empirical distribution of $T_{max}^k$, ${\widehat{H}}_b\left(x\right)$, such that ${\widehat{H}}_b\left(x\right)=\#\{T_{max}^k\le x\}∕M^{\ast }$. We then obtain the threshold of the T_max value for the α-significance level, $T_{max}^{th}$, such that $T_{max}^{th}={\widehat{H}}_b^{-1}(1-\alpha )$, and finally obtain the statistical threshold for the kth sensor recording, ${\Sigma }_b^k$, by using

$${\Sigma }_b^k=T_{max}^{th}{\widehat{\sigma }}_b^k+{\langle \mid b_c^k\left(t_j\right)\mid \rangle }_c.$$ (17)

Therefore, when $b^k\left(t\right)>{\Sigma }_b^k$, the null hypothesis is rejected and we can conclude that a statistically significant b_I(t) exists.

In conclusion, this paper proposes a simple and efficient method of statistical thresholding for MEG spatio-temporal source reconstruction. The method assumes the neural source activities to be quasi stochastic, and it derives an empirical distribution of non-target sources from the time course reconstruction in the control period. It then derives the value of statistical threshold based on the empirical distribution with the multiple comparisons taken into account. In summary, the method provides an effective means of removing source activities not of interest to the current measurements and of extracting the source activity of interest.