1. Introduction

Using magnetoencephalography (MEG) one can make measurements of brain activity with very high temporal resolution.  The localization of the underlying generators in the brain requires solving the “inverse problem” of determining the source distributions that give rise to some set of external electromagnetic measurements.  Numerous inverse algorithms have been used to localize the activity, e.g. [Wood, C.C, 1985, Scherg, M 1992, Mosher, J.C. et Al, 1992, Hamalainen, M., et al, 1993, George, J.S 1995].  Two promising approaches are linear estimation [Hamalainen, M et al, 1993, Liu, A.K et al,1998, Dale, A.M et al 1993, Wang, J.Z et al 1998, Sarvas, J, 1987] and the Spatio-Temporal Fit (SPTF) [Schwartz, D et al 1999].  Neither linear estimation nor SPTF require any a priori information about the number of sources.  Both are user-independent methods that will give the same localization regardless of the initial conditions. 

The linear estimation approach computes an optimal linear inverse operator, which explicitly minimizes the expected error between the estimated and actual activity distribution. The SPTF algorithm applies Principal Component Analysis to the signal over a short moving window of time to build a signal subspace and finally separates simultaneous asynchronous dipolar sources.

The comparison of any two inverse methods is made more complicated by the ill-posed nature of the inverse problem.  Theoretically, there are an infinite number of solutions, which will reproduce the external measurements.  Therefore, with only external experimental measurements, it is not possible to know with full accuracy the position and time course of the actual generators.  Electrophysiological invasive recordings are essential to truly characterize the internal source distributions.  This work essentially compares linear estimation and SPTF using simulated data, allowing for comparisons against the “gold standard” of the known source distributions. 

Our model data was generated using realistic source distributions constrained to the cortical surface with realistic time courses of activation.  The sources were placed in four well-defined areas: frontal, precentral, temporal, and parietal.  We examined single and multiple sources with various amounts of temporal overlap and temporal correlation.

We compared the localization results between the two methods and against the known source distributions.  We quantified the spatial and the temporal accuracy.  The simulated sources always lied inside the gray matter of a realistic head model, from one of our human subjects, with different depths that allowed us to study the effect of various source locations. 

In the first part, we briefly present the linear estimation and SPTF approaches and the generation of the model data. We also describe the original method we used to represent our localization results. Secondly, we present the spatial and temporal accuracy of each method for each simulation dataset.

2. Methods

2.1. Linear Estimation

The linear estimation approach [Hamalainen, M et al, 1993, Liu, A.K et al,1998, Dale, A.M et al 1993, Wang, J.Z et al 1998, Sarvas, J, 1987] computes an optimal linear inverse operator (W), which maps the external electromagnetic field measurements (x) into estimated source activities () within the brain. This inverse operator explicitly minimizes the expected error between the estimated and actual activity distribution (s). The calculation incorporates the forward solution (also called the gain matrix (A)), spatial priors (R) and sensor noise characteristics (C).  For these solutions, we constrained the source locations, but not the source orientations, by the cortical surface (i.e. anatomical constraint).  The expression for the inverse operator is:

                                                                                             (1)

In these modeling studies, we assumed no prior spatial information (R = I) and uncorrelated gaussian noise with a signal-to-noise ratio of 10 (). 

This linear inverse operator provides an estimate of source activity for each time point independently of the other time points.  The estimated source activities are given by the simple expression:

                                                                                                             (2)

2.2. SPTF

The SPTF approach was extensively described and tested in [Schwartz, D et al 1999].  We describe here the method used to introduce anatomical constraints in the SPTF framework. Classically the SPTF searches for an equivalent dipole   to match, through a spatio-temporal analysis, the external electromagnetic field measurements (x) with the computed field   that is generated by the dipole. Let  be the spatio-temporal matrix   containing the signals for   sensors at   latencies. The SPTF performs a Principal Component Analysis (PCA) on the matrix   . The results are   eigenvectors   and   decreasing   eigenvalues. A discontinuity in the order of magnitude of the eigenvalues   indicates the dimension of the signal subspace   of dimension  with .    is then defined as follows:

Figure 1. Source time course used to model the temporal behavior of each source

We computed the MEG signals generated by each sources individually and by 3 combinations of 3 sources with differing amounts of temporal overlap and temporal correlation between the sources (See Table 1). For multiple source configurations, each source had equal source power without equal sensor power. One time point was generated every 10 ms, for 61 time points from 0 to 600 ms. To these signals, we added gaussian noise with a signal-to-noise ratio (peak RMS) of 10.

TABLE 1. Temporal overlap and temporal correlation for the two configurations of three sources used in the simulations: Frontal, Parietal, Precentral and Frontal, Precentral, Temporal.

	Temporal overlap and temporal correlation	Time courses
1	Temporal correlation = 0.0
2	Temporal overlap = 50 % to 100 % Temporal correlation = 1.0
3	Temporal correlation = 0.75 to 1.0

 

2.4. Rendering

All results were represented on the inflated cortical surface, which allows the visualization of activity hidden in sulci. The localization of each dataset results in a spatio-temporal movie.  To give a more concise representation of the data, we computed 1) the peak activation and 2) the peak latency over the entire cortical surface.

1. Peak Activation: Maximum value (linear estimation  or SPTF correlation) found in each point of the cortical surface between 0 ms and 600 ms. For both linear estimation and SPTF we used a threshold determined automatically to represent only the significant values.

2. Peak Latency: Latency of the peak activation at each point of the cortical surface.  The peak latency was only represented when the corresponding peak activation was greater than the automatically determined threshold.  We chose to represent peak latencies instead of onset latencies because the peak latencies are insensitive to thresholding effects.  We used a rainbow color map (red, orange, yellow, green, light blue, dark blue) to represent the peak latency.

2.5. Comparison of spatial and temporal accuracy

We compared both quantitatively and qualitatively the localization results from linear estimation and SPTF.  For each the single source activation we evaluated the localization accuracy by comparing the center of mass of the actual source and center of mass of the localized source.  With multiple sources, it becomes problematic to objectively define “sources.”  Therefore, for the multiple sources activation we evaluated the number of sources effectively retrieved versus the actual number of sources. Additionally we qualitatively compared the size of the retrieved sources with the size of the actual sources to evaluate the ability of each algorithm to separate these sources. To evaluate the temporal accuracy, we computed the mean and the standard deviation of the peak latency at each actual source location for each data set.

3. Results

3.1. Single source activation

3.1.1. Spatial accuracy

TABLE 2. Distance between the Center of Mass of the localized source and the Center of Mass of the actual source.

The distances from the estimated center of mass to the actual center of mass for the four different sources are shown in Table 2.  SPTF provides slightly better localization than linear estimation, with an average improvement of 2.4 mm.

3.1.2. Temporal accuracy

TABLE 3. Peak Latencies for Single Source.

	Linear EstimationPeak Latency (ms)	SPTFPeak Latency (ms)	Actual Peak Latency (ms
Frontal	201 ± 7	224 ± 35	210
Temporal	200 ± 0	233 ± 12	210
Precentral	207 ± 4	198 ± 43	210
Parietal	201 ± 3	149 ± 43	210

The peak latencies for the four different source locations are shown in Table 3. The actual peak latency is 210 ms. Linear estimation provides an accurate and consistent estimate of the peak latency. In contrast, SPTF has later peak latencies (frontal and temporal) and earlier peak latencies (precentral and parietal). In addition the variance of the peak latency for SPTF is much larger.

3.1.3. Spatial extent versus Threshold

Figure 2. Spatial extent versus threshold. Peak activation for three different thresholds. We used a heat color map with transparent low values, the mid values are represented in red, and the higher values tend toward yellow. For the linear estimation the threshold is in red (50%), full red and yellow correspond to 80% and 90% of the absolute peak activation, respectively.  For SPTF, threshold is in red (85%) and full red and yellow correspond to 92% and 100% of the absolute peak activation, respectively.

As expected (see Figure 2), raising the threshold results in a more focal localization for both localization approaches. Linear estimation is more sensitive to the threshold selected.  The parietal source is no longer visible when using a threshold that results in sufficiently focal activity of the precentral and frontal source.  For SPTF, one can see that a single threshold yields visible activity in all three sources. 

3.2. Multiple Sources

3.2.1. Spatio-temporal accuracy

Figure 3. Frontal, Parietal and Precentral Source Localization (FPPR 3). Peak activation for linear estimation (top) and SPTF (bottom) are shown on the left.  Peak latency for linear estimation (top) and SPTF (bottom) are shown on the right. The MEG signals are shown on the bottom.

Shown in Figure 3 are the localization results for the Frontal, Parietal and Precentral sources for the third temporal configuration.  Consistent with the single source results, the spatial localization of SPTF is more focal than linear estimation. The frontal and precentral activities are retrieved by both techniques.  However, the parietal activity is only recovered by SPTF.  The peak latencies are shown in Table 4. The temporal agreement between the two methods is not as good as the spatial agreement.  The linear estimation peak latency, for those sources that are well localized, is more accurate and more consistent than the SPTF peak latencies.

 

3.2.2. Linear estimation

 

Figure 4. Frontal, Precentral and Temporal Extended Source Localization (FPRT 2). Peak activation for linear estimation (top) and SPTF (bottom) are shown on the left.  Peak latency for linear estimation (top) and SPTF (bottom) are shown on the right. The MEG signals are shown on the bottom.

 

Shown in Figure 4 are the localization results for Frontal, Precentral and Temporal extended sources with different amount of temporal overlap (FPPRT 1, FPRT 2 and FPRT 3).  These results are similar to those of FPPR.  The spatial localization of SPTF is more focal than the one using the linear estimation.  For both techniques, the frontal and precentral sources are well localized, however the temporal source is poorly localized. The peak latencies for both techniques are shown in Table 5.

4. Discussion

We found little variation in the spatial localization as a function of source extent for both techniques.  This essentially reflects the poor spatial resolution of the forward operator.  That is, nearby locations in the brain generate very similar forward solutions.  Subsequently, those locations are difficult to separate.  If one has a priori information that suggests the sources are indeed focal, single and multiple dipole approaches may be more appropriate than either linear estimation or SPTF. 

The center of mass of activation was slightly better for SPTF than linear estimation.  Comparison of the spatial extent of activation between the two techniques shows that SPTF is consistently more focal than linear estimation.  Since SPTF performs the localization, not on the actual signal, but on a reconstructed signal subspace generated from a moving time window, those localizations become more robust in presence of noise.  This allows a more aggressive thresholding of the estimated probability of activation.  In addition, small amplitude sources are better reconstructed using SPTF than linear estimation because the probability may still remain high even while the absolute activation is small compared to other sources.

The entire issue of thresholding is, however, a difficult one, and altering the threshold in any localization technique will alter the spatial extent of the localization.  We addressed this issue using an automatic thresholding approach for both methods.  This gives us an objective way to compare the results from linear estimation and SPTF.  For a single source, it is always possible to generate a more focal solution by increasing the threshold. When there are multiple sources with equal source power but not necessarily equal sensor power, altering the threshold can result in variable spatial localizations where entire sources may be lost at higher and more focal thresholds. In physiological measurements, one would expect comparable source power rather than sensor power. The sensor power for a given source is determined by the local cortical morphology.

In comparison to the spatial localization, the temporal information given by linear estimation is more consistent and, in some cases, more accurate than SPTF.  Since the linear estimation approach works directly with the measurement data, no temporal dynamics of well-reconstructed sources are lost.  For SPTF, the estimated probability does not necessarily correlate to the actual intensity of the source.  Subsequently, the maximum of probability does not always occur at the actual peak activation.

In the multiple source conditions, the parietal and temporal sources were difficult to localize, while the frontal and precentral were always well reconstructed.  Alone, both the parietal and temporal sources were well retrieved.  This variability in localization accuracy demonstrates the effect of the sensor sensitivity to different locations in the brain.  For a given activity, the sensor power (i.e. norm of the gain vector) is smaller for the parietal and temporal sources than for the frontal and precentral.  As a result, when several sources are active, the frontal and precentral sources will dominate the measurements.  This discrepancy in sensor power will also result in lower signal to noise ratios for the parietal and temporal sources, resulting in poor localization accuracy in these anatomic regions.  This effect is likely the cause of the poor localizations in the temporal lobe as reported by [25].  The effect of sensor power is greater in the linear estimation approach, since a low threshold results in additional spurious activity for our parietal and temporal sources.  Hence, with real data, it is not possible to differentiate “noise activity” from weak sources.

Linear estimation gives a more consistent temporal localization than SPTF. The temporal discrepancies between SPTF and linear estimation are higher than the ones seen for single source activation. For multiple sources activation the estimated probability computed by SPTF is higher when the source is activated alone and decreases when additional sources appear in the signal. Thus the estimated peak latency usually occurs earlier than the actual peak. This is especially obvious for the SPTF estimate of the frontal source whose peak latency occurs at 50 ms, well before the actual peak at 160 ms.

One of the theoretical drawbacks of SPTF is its sensitivity to correlated sources. SPTF looks for single dipolar activity in the signal subspace and thus is not able to separate multiple correlated sources. We expected the SPTF results to be worst for FPPR 3 and FPRT 2, which had the greatest correlation between sources. In addition, we expected the linear estimation results to be insensitive to correlation. Not surprisingly, the linear estimation localizations were essentially the same for all levels of correlation. However, the SPTF spatial localizations remained very accurate even for FPPR 3 and FPRT 2. This somewhat unexpected result is due to the use of the anatomic constraint. When two sources are highly correlated the SPTF maximum probabilities would be in some other location beside the actual sources. This maximum typically would not be on the cortical surface while secondary peaks will still be found at the actual source locations.

5. Conclusion

Our results demonstrate the complementary strengths of linear estimation and SPTF. While both linear estimation and SPTF provide comparably accurate spatial localization of single and multiple extended sources, SPTF gives more focal solutions. Surprisingly, SPTF is not very sensitive, in terms of spatial accuracy, to the temporal correlation between the sources. Linear estimation provides more accurate temporal information especially when several sources are simultaneously activated. We can exploit the superior spatial accuracy of SPTF by incorporating the SPTF spatial localizations as a prior in the linear estimation approach. In future modeling and experimental work we will explore and utilize the greater spatio-temporal accuracy of this combined linear estimation – SPTF technique.

Acknowledgement

We thank Dr. Anders Dale at the MGH-NMR Center, Mass. General Hospital for helpful discussion and support.

6. References