in minimum variance adaptive beamforming (MVAB) [6] and. sLORETA [4]. Alternatively, a number of empirical Bayesian approaches have been proposed that ...
ROBUST METHODS FOR RECONSTRUCTING BRAIN ACTIVITY AND FUNCTIONAL CONNECTIVITY BETWEEN BRAIN SOURCES WITH MEG/EEG DATA Julia P. Owen1,2 , David P. Wipf1, Hagai T. Attias3, Kensuke Sekihara4, and Srikantan S. Nagarajan1,2 1
Biomagnetic Imaging Laboratory, Dept. Radiology and Biomedical Imaging, UCSF San Francisco, CA, USA 2
Joint Graduate Group in Bioengineering UCSF/UC Berkeley, San Francisco, CA, USA 3
4
Golden Metallic, Inc. San Francisco, CA, USA
Dept. of Systems Design and Engineering, Tokyo Metropolitan University, Tokyo, Japan
ABSTRACT The synchronous brain activity measured via magentoencephalography (MEG) or electroencephalography (EEG) arises from current dipoles located throughout the cortex. The number, location, time-course, and orientation of these dipoles, called sources, are estimated using a source localization algorithm. Source localization remains a challenging task, one that is significantly compounded by the effects of source correlations and interference from spontaneous brain activity and sensor noise. Likewise, assessing the interactions between the individual sources, known as functional connectivity, is also confounded by noise and correlations in the sensor recordings. In addition, computational complexity has been an obstacle to computing functional connectivity. This paper derives an empirical Bayesian method for performing source localization with MEG and EEG data that includes noise and interference suppression. We demonstrate that this method surpasses standard methods of localization. In addition, we demonstrate that brain source activity inferred from this algorithm is better suited to uncover the interactions between brain areas. Index Terms— Magnetoencephalography, brain, source localization, Bayesian methods, functional connectivity 1. INTRODUCTION Magnetoencephalography (MEG) and related electroencephalography (EEG) are commonly used modalities for non-invasively measuring brain activity using an array of sensors on or near the surface of the scalp to make direct measurements of the magnetic field or voltage potential. The observed field is generated by large ensembles of neurons firing synchronously, which are approximated as current sources. A source localization algorithm is used to solve the inverse problem, i.e. determining the combination of current sources that best explains the field recording. The inverse problem is ill-posed because the number of potential sources is much greater than the number of sensors. Determining the spatial distribution and time courses of these unknown
sources is still an open inverse problem. A variety of algorithms attempt to overcome the ill-posed nature of source localization. These algorithms are heavily dependent on prior assumptions, which in a Bayesian framework, are embedded in the prior distribution on the sources. Such a prior is often considered to be fixed and known, as in minimum variance adaptive beamforming (MVAB) [6] and sLORETA [4]. Alternatively, a number of empirical Bayesian approaches have been proposed that use the data to search for an appropriate prior. While advantageous in many respects, these methods have difficulties estimating complex, correlated source configurations with unknown orientations in the presence of background interference. The first contribution of this paper to is to introduce a novel empirical Bayesian scheme that improves upon existing methods in terms of reconstruction accuracy, robustness, and efficiency. The algorithm derived from this model, which we call Champagne, is designed to estimate the number and location of a small (sparse) set of flexible dipoles that adequately explain the observed sensor data. This method relies on having access to pre- and post-stimulus data, where the pre-stimulus data is thought to contain no stimulus-evoked brain activity. Functional connectivity can be estimated from the output of source localization. Functional connectivity can be described as understanding brain function in terms of the way information is transmitted and integrated across brain networks. In the most complete case, one would like to make inferences from a causal linear model that describes the dependencies among activities across all voxels. However, due to the large number of voxels, solving such a model is computationally expensive and virtually impossible with limited, noisy data. Instead, existing techniques for estimating functional connectivity approximate the full problem in various ways, but there is a tradeoff between reducing the computational complexity and loss of sensitivity. Improving source localization goes hand-in-hand with improving functional connectivity methods. Better estimation of the source locations and time courses, yields a more
accurate estimation of connectivity. Paradoxically, inferences about connectivity can be made from the correlations between source time-courses, but many common localization algorithms (such as MVAB) have significant trouble reconstructing correlated brain activity. Consequently, there is a fundamental problem applying many existing localization methods to functional connectivity estimation. The second contribution of this paper is the application of Champagne to the problem of functional connectivity. The sparseness of Champagne’s solution makes it ideal for this purpose. Not only is Champagne robust to highly correlated dipoles, but it also circumvents the issues of computational complexity by vastly pruning the number of active voxels. We present results from simulated and real MEG data showing Champagne’s efficacy in both reconstructing brain activity and estimating functional connectivity. 2. THE MODEL Source localization can be posed as follows: The measured electromagnetic signal is B ∈ Rdb ×dt , where db equals the number of sensors and dt is the number of time points at which measurements are made. Each unknown source Si ∈ Rdc ×dt is a dc -dimensional neural current dipole , at dt time points, projecting from the i-th (discretized) voxel. B and each Si are related by the likelihood model B=
ds X
Li Si + E,
(1)
i=1
where ds is the number of voxels under consideration, Li ∈ Rdb ×dc is the so-called lead-field matrix for the i-th voxel. The k-th column of Li represents the signal vector that would be observed at the scalp given a unit current source/dipole at the i-th vertex with a fixed orientation in the k-th direction. It is common to assume dc = 2 (for MEG) or dc = 3 (for EEG). Finally, E is a noise-plus-interference term where we assume, for simplicity, that columns are drawn independently from N (0, Σǫ ). We can fully define the assumed likelihood
2
ds
X 1
, p(B|S) ∝ exp − B − L i Si (2)
−1 2 i=1
Σǫ
from the noisepmodel (1), where kXkW denotes the weighted matrix norm trace[X T W X] and S , [S1 , . . . , Sds ]T . The unknown noise covariance Σǫ will be estimated from the data using a variational Bayesian factor analysis (VBFA) model as discussed in Section 4 below; for now we will consider that it is fixed and known. We assume the following for the source prior on S: #! "d s X 1 T −1 . (3) Si Γi Si p (S|Γ) ∝ exp − trace 2 i=1 This is equivalent to applying independently, at each time point, a zero-mean Gaussian distribution with covariance Γi
to each source Si . We define Γ to be the ds dc × ds dc blockdiagonal matrix formed by ordering each Γi along the diagonal of an otherwise zero-valued matrix. If Γ were somehow known, then the conditional distribution p(S|B, Γ) ∝ p(B|S)p(S|Γ) is a fully specified Gaussian distribution. However, since Γ is actually not known, a suitˆ ≈ Γ must first be found. One principled able approximation Γ way to accomplish this is to integrate out the sources S and then minimize the cost function + log |Σb , | , (4) L(Γ) , −2 log p(B|Γ) ≡ trace Cb Σ−1 b T where Cb , d−1 t BB is the empirical covariance and Σb , T Σǫ + LΓL , where L , [L1 , . . . , Lds ].
3. LEARNING THE HYPERPARAMETERS Γ The hyperparameters, Γ, are estimated from post-stimulus data. Minimizing the cost function (4) with respect to Γ can be done in a variety of ways, including the expectationmaximization (EM) algorithm, but this and other generic methods are exceedingly slow when ds is large. Consequently, here we derive an alternative optimization procedure that handles arbitrary/unknown dipole orientations, and converges quickly. This procedure is fully described in [8]; this methods relies on two alterations. The first is replacing B with a matrix e ∈ Rdb ×rank(B) such that B eB e T = Cb , thus removing any B per-iteration dependency on dt . The second is constructing auxiliary functions using sets of upper-bounding hyperplanes and auxillary variables,X and Z. This leads to the modified cost function:
2
ds
e X e L i Xi L(Γ) ≤ L(Γ, X, Z) = B − +
−1
i=1
Σǫ
ds h X i kXi k2Γ−1 + trace ZiT Γi − h∗ (Z),
(5)
i
i=1
where h∗ (Z) is the concave conjugate of log |Σb | [8] and by construction L(Γ) = minX minZ L(Γ, X, Z). We then iteratively optimize L(Γ, X, Z) via coordinate descent over Γ, X, and Z resulting in the update rules: h i e Γ = Γi LTi Σ−1 B e (6) Xinew → E Si |B, b Zinew → ▽Γi log |Σb | = LTi Σ−1 b Li .
Γnew i → Zi
−1/2
“
1/2
Zi
Xi XiT Zi
1/2
”1/2
−1/2
Zi
(7)
,
(8)
In summary then, to estimate Γ, we need simply iterate (6), (7), and (8), and with each pass we are guaranteed to reduce (or leave unchanged) L(Γ); we refer to the resultant algorithm as Champagne. The convergence rate is orders of magnitude faster than EM-based algorithms. Upon convergence Xi is our source estimate for the i-th voxel.
Cb ≈ Λ + EE T + AAT ,
(9)
db ×de
represents a matrix of de learned interferwhere E ∈ R ence factors, Λ is a diagonal noise matrix, and A ∈ Rdb ×da represents signal factors. Then, we set Σǫ → Λ + EE T and proceed as in Section 3. 5. PERFORMANCE: LOCALIZATION AND TIME-COURSE RECONSTRUCTION We conducted tests using simulated data with realistic source configurations. The brain volume was segmented into 5mm voxels and a two orientation (dc = 2) forward leadfield was calculated using a single spherical-shell model [5]. The data time courses were partitioned into a pre-stimulus period where there is only noise and interfering brain activity and a post-stimulus period where there is the same (statistically) noise and interference factors plus source activity of interest. The pre-stimulus activity consisted of the resting-state sensor recordings collected from a human subject and is presumed to have spontaneous activity (i.e., non-stimulus evoked sources) and sensor noise; this activity was on-going and continued into the post-stimulus period, where damped-sinusoidal sources were seeded and projected to the sensors through the leadfield. We were able to adjust the signal-to-noise-plusinterefence ratio (SNIR), the correlations between the different voxel time-courses (inter-dipole), and the correlations between the two orientations of the dipoles (intra-dipole) to examine the algorithm performance on unknown correlated sources and dipole orientations. We obtained aggregate data on the performance of our method at SNIR levels of -5, 0, 5, 10dB, with 100 randomly (located) seeded sources at each level. The sources in these simulations had an inter- and intra-dipole correlation coefficients of 0.5. We chose to test our algorithm against two standard source localization algorithms: minimum variance adaptive beamforming (MVAB) [6] and a non-adaptive spatial filtering method, sLORETA [4]. We used two features: source localization accuracy (measured with the A′ metric [7]) and estimation of time course (measured with the correlation coefficient of the true time course and the estimated time course). Both the A′ metric and correlation coefficient range from 0 to 1, with 1 implying perfect localization or time course estimation. Figure 1 displays comparative results at different SNIR
1
1
Average Correlation Coefficient
The noise-plus-interference term, E, from (1) is calculated from the pre-stimulus data. The learning procedure desrcibed in the previous section boils down to fitting a structured maximum-likelihood covariance estimate Σb = Σǫ + LΓLT to the data covariance Cb , where LΓLT reflects the brain signals of interest while Σǫ will capture all interfering factors, e.g., spontaneous brain activity, sensor noise, muscle artifacts, etc. Σǫ is estimated from VBFA. While details can be found in [2], the procedure computes the approximation
levels with standard errors. Champagne quite significantly outperforms MVAB and sLORETA. We also ran Champagne, MVAB, and sLORETA on an auditory data-set that is known to have highly correlated dipoles in the right and left auditory cortices. Figure 2 shows that Champagne is able to recover bilateral dipoles, while MVAB and sLORETA find only the source on the right. 0.9 0.8
A Prime Metric
4. LEARNING THE INTERFERENCE E
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0.8
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Fig. 1. Performance evaluation: (a) Aggregate localization results for MVAB, sLORETA, and CHAMP for recovering three correlated sources with unknown orientations. (b) Estimated time-course correlation coefficient results.
(a) Champagne
(b)MVAB
(c)sLORETA
Fig. 2. Performance on a real auditory data-set that has highly correlated, bilateral sources. 6. PERFORMANCE: FUNCTIONAL CONNECTIVITY Simulated data was created as described above with SN IR = 10dB and an intra-dipole correlation of 0.5. The sensor data is shown in Figure 3(a). We decided to simulate a network of 7 nodes (or voxels). The inter-dipole correlations are depicted in the diagram found in Figure 3(b) where the color of the lines between the sources denotes the strength of correlation, with red being high and blue being weak (see colorbar in Figure 3). The line type indicates whether the mixing was instantaneous (dashed) or non-instantaneous (solid). We chose two pair-wise connectivity metrics: coherence and imaginary coherence. Coherence is a traditional metric of connectivity and is the frequency domain representation of cross-correlation. The coherence is a complex-valued quantity; we looked at both the magnitude of the coherence and the imaginary part of the coherence alone. Imaginary coherence is a relatively new metric developed for use with MEG and EEG data[3]. It only reflects the coherence that is noninstantaneously mixed. Functional connectivity methods with MEG are subject to spurious correlations arising from instantaneous correlations at the sensors. While imaginary coherence under-estimates the coupling between two brain areas, it
7. DISCUSSION This paper derives a novel empirical Bayesian algorithm for MEG source reconstruction that readily handles multiple correlated sources with unknown orientations, a situation that commonly arises even with simple imaging tasks. Based on a principled cost function and fast, convergent update rules, this procedure displays significant advantages over many existing methods. We have restricted most of our exposition and analyses to MEG; however, preliminary work with EEG is also promising. The results we have obtained on functional connectivity demonstrate that Champagne is better suited for these types of analyses than beamforming and other algorithms that fail to reconstruct correlated sources. This line of research will be extended to real data sets to evaluate the functional connectivity during task performance. It is also possible to combine the estimation of source location, time course and functional connectivity into one Bayesian framework. We are working to develop such an integrated model. 8. REFERENCES [1] A.G. Guggisberg, et al., “Mapping functional connectivity in patients with brain lesions,” Annals of Neurology. vol. 63 no. 2, pp. 193-203 2007. [2] S.S. Nagarajan, et al., “A probabilistic algorithm for robust interference suppression in bioelectromagnetic sensor data,” Stat Med. vol. 26, no. 21, pp. 3886–910 Sept. 2007.
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has been shown to be more robust than coherence when using MEG source reconstructions. [1] Figure 3(b), (c), and (d) show composite plots for the ground truth, Champagne, and MVAB source reconstructions respectively. These plots combine the reconstruction and functional connectivity results. Champagne is able to reconstruct all the sources, while MVAB fails to reconstruct the sources even at this high SNIR (10dB). The black circles mark the true locations of the sources and the surface plot shows the maximum intensity projection of the power of the source estimate at every voxel, illustrating the inferred location of the sources. The lines between the sources depict the functional connectivity results. We used the coherence measure to reconstruct the correlations (shown by the color of the lines) and imaginary coherence to determine which correlations are instantaneous or non-instantaneous (shown by dashed versus solid lines). Coherence and imaginary coherence contain complimentary information. The similarity of the ground truth and Champagne plots demonstrates that these two quantities can be used in conjunction to uncover the strength and lags (instantaneous vs. non-instantaneous) of interactions in a network of brain areas. MVAB only reconstructs the location of one source accurately, so we do not compute the functional connectivity results for the MVAB results.
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Fig. 3. (a) Simulated sensor data (SNIR=10dB). Red-line depicts pre- and post-stimulus divide. Maximum intensity projection (showing the localization accuracy) and network connections for (b) ground truth, (c) Champagne, and (d) MVAB. For all three, solid line indicates non-zero lag and dashed line indicates zero-lag and the color bar depicts the color scale for the strength of correlation. For (c) the magnitude of the coherence sets the color and imaginary part of the coherence sets the dashed vs. solid aspect of the lines. [3] G. Nolte, et al. Identifying true brain interaction from EEG data using the imaginary part of coherency,” Clinical Neurophysiology. vol. 115, no. 10, pp. 2292-307, 2004. [4] R.D. Pascual-Marqui, “Standardized low resolution brain electromagnetic tomography (sloreta): Technical details,” Methods and Findings in Experimental and Clinical Pharmacology, vol. 24, no. Suppl D, pp. 5–12, 2002. [5] J. Sarvas, “Basic methematical and electromagnetic concepts of the biomagnetic inverse problem,” Phys. Med. Biol., vol. 32, pp. 11–22, 1987. [6] K. Sekihara, M. Sahani, and S.S. Nagarajan, “Localization bias and spatial resolution of adaptive and nonadaptive spatial filters for MEG source reconstruction,” NeuroImage, vol. 25, pp. 1056–1067, 2005. [7] J.G. Snodgrass, and J.Corwin, “Pragmatics of Measuring Recognition Memory: Applications to Dementia and Amnesia,” Journal of Experimental Psychology: General, vol. 17, no. 1 pp. 34-50, 1988. [8] D. Wipf, et al., “Estimating the Location and Orientation of Complex, Correlated Neural Activity using MEG,” Advances in Neural Information Processing Systems 21 2009.
