Brain Activity Mapping from MEG Data via a Hierarchical Bayesian Algorithm with Automatic Depth Weighting

Abstract

A recently proposed iterated alternating sequential (IAS) MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its ability to correctly identify active brain regions is still missing. We propose novel statistical protocols to quantify the performance of MEG inverse solvers, focusing in particular on how their accuracy and precision at identifying active brain regions. We use these protocols for a systematic study of the performance of the IAS MEG inverse solver, comparing it with three standard inversion methods, wMNE, dSPM, and sLORETA. To avoid the bias of anecdotal tests towards a particular algorithm, the proposed protocols are Monte Carlo sampling based, generating an ensemble of activity patches in each brain region identified in a given atlas. The performance in correctly identifying the active areas is measured by how much, on average, the reconstructed activity is concentrated in the brain region of the simulated active patch. The analysis is based on Bayes factors, interpreting the estimated current activity as data for testing the hypothesis that the active brain region is correctly identified, versus the hypothesis of any erroneous attribution. The methodology allows the presence of a single or several simultaneous activity regions, without assuming that the number of active regions is known. The testing protocols suggest that the IAS solver performs well with both with cortical and subcortical activity estimation.

Appendix

Interpretation of Hyperparameters

In Calvetti et al. (2015), the interpretation of the hyperparameters $\beta \in {\mathbb {R}}$ and $\theta ^*\in {\mathbb {R}}^N$ was discussed: it was shown that $\beta$ allows the user to control the sparsity of the IAS solution, while the empirical Bayesian approach provided a way to relate $\theta ^*$ to the sensitivity scaling. We summarize here the analysis on hyperparameters, developing the discussion of $\theta ^*$ further, so that the parameter tuning can be done easily and semi-automatically.

Parameter $\beta$ and Control of Sparsity

In Calvetti et al. (2015), it was proved (Theorem 2.1) that the sequential minimization that constitutes the core of the IAS algorithm can be interpreted as a fixed point iteration to find a minimizer $\hat{Q} = [\hat{\mathbf {q}}_1,\hat{\mathbf {q}}_2,\ldots ,\hat{\mathbf {q}}_n]^{\mathsf {T}}\in {\mathbb {R}}^{3n}$ of the energy functional (3),

$$\begin{aligned} \hat{Q} = {\mathrm{argmin}}\left\{ {{\mathcal {E}}}(Q,S(Q))\right\} , \quad \hat{\varTheta } = S(\hat{Q}), \end{aligned}$$

where $\hat{\varTheta } = [\hat{\theta }_1;\hat{\theta }_2;\ldots ,\hat{\theta }_n]\in {\mathbb {R}}^n$, and $S:{\mathbb {R}}^{3n} \rightarrow {\mathbb {R}}^n$ is defined componentwise as

$$\begin{aligned} \theta _j = S_j(\mathbf {q}_j) = \theta _j^*\left( \frac{\eta _j}{2} + \sqrt{ \frac{\eta _j^2}{4} + \frac{\Vert \mathbf {q}_j\Vert _{{{\mathsf {C}}}_j}^2}{2\theta _j^*}} \right) ,\quad \eta _j = \beta _j -\frac{5}{2}, \quad 1\le j\le n. \end{aligned}$$

Furthermore, it was shown that if we write $\beta _j = 5/2 + \eta$, $1\le j\le n$, then, as $\eta \rightarrow 0^+$, we have the asymptotic expression

$$\begin{aligned} {{\mathcal {E}}}( Q, S(Q)) = \frac{1}{2} \Vert b -\sum _{j=1}^n {\mathsf {M}}_j\mathbf {q}_j\Vert ^2_{{\mathsf {\Sigma }}} + {\sqrt{2}} \sum _{j=1}^n \frac{ \Vert \mathbf {q}_j\Vert _{{{\mathsf {C}}}_j}}{\sqrt{\theta _j^*}} + {{\mathcal {O}}}(\eta ). \end{aligned}$$

(7)

In particular, the penalty term in the above expression is a weighted $\ell ^1$-norm for the dipole amplitudes that are measured in the metric defined by the anatomical prior matrices ${{\mathsf {C}}}_j$. This argument demonstrates that at the limit, the IAS algorithm provides an effective algorithm for finding a weighted minimum current estimate (MCE), with the modification given by the anatomical prior (Uutela et al. 1999). In conclusion, we see that the role of the hyperparameter $\beta$ is to control the sparsity of the IAS estimate. In Calvetti et al. (2015), this effect was demonstrated using numerical simulations.

Parameter $\theta ^*$ and Sensitivity

The asymptotic expression (7) is indicative also from the point of view of the interpretation of $\theta _j^*$. It is well-known that MEG algorithms based on penalized minimization of the fidelity to data tend to favor superficial sources, and to compensate this effect, a sensitivity weight is often introduced (see, e.g. Lin et al. 2006a). From the Bayesian point of view, sensitivity weighting is a problematic practice, since traditionally the prior should be independent of the observation model, a condition that the sensitivity weight does not satisfy. However, it is possible to find a satisfactory Bayesian interpretation for $\theta _j^*$ so that it effectively works as a sensitivity weight. The connection between sensitivity and hypermodels is built through the analysis of the signal-to-noise ratio as follows. Consider the linear forward model

$$\begin{aligned} b = {\mathsf {M}}Q + \varepsilon = \sum _{j=1}^n {\mathsf {M}}_j\mathbf {q}_j + \varepsilon = b_0 +\varepsilon ,\quad \varepsilon \sim {{\mathcal {N}}}(0,{{\mathsf {\Sigma }}}). \end{aligned}$$

To estimate the expected power of the noiseless signal appearing in the signal-to-noise ratio defined in (4), observe that from the prior model, conditional on $\varTheta \in {\mathbb {R}}^n$, we have

$$\begin{aligned} {\mathsf {E}}\{\Vert b_0\Vert ^2\mid \varTheta \} = \sum _{j=1}^n \theta _j{\mathrm{trace}}\left( {\mathsf {M}}_j {{\mathsf {C}}}_j {\mathsf {M}}_j^{{\mathsf {T}}}\right) = \sum _{j=1}^n \theta _j \Vert {\mathsf {M}}_j{{\mathsf {C}}}_j^{1/2}\Vert ^2_\mathrm{F}, \end{aligned}$$

where the subscript refers to the Frobenius norm of the matrix. Furthermore, by using the gamma hyperprior model $\theta _j \sim \varGamma (\beta ,\theta _j^*)$ for the vector $\varTheta$, we arrive at

$$\begin{aligned} {\mathsf {E}}\{\Vert b_0\Vert ^2\} = \sum _{j=1}^n {\mathsf {E}}\{\theta _j\} \Vert {\mathsf {M}}_j{{\mathsf {C}}}_j^{1/2}\Vert ^2_\mathrm{F} = \sum _{j=1}^n \beta \theta _j^* \Vert {\mathsf {M}}_j{{\mathsf {C}}}_j^{1/2}\Vert ^2_\mathrm{F}. \end{aligned}$$

The choice of the hyperparameters $\theta _j^*$ must therefore be compatible of what we a priori assume about the distribution of the activity and the resulting SNR. To begin with, assume that we have a reason to believe that only one source is active, but we do not know which one. Then, by the definition (4) of SNR, the active source $j_1$ must satisfy

$$\begin{aligned} \beta \theta _{j_1}^* \Vert {\mathsf {M}}_{j_1}{{\mathsf {C}}}_{j_1}^{1/2}\Vert ^2_\mathrm{F} = ({\mathrm{SNR}}-1)\times {\mathrm{trace}}({\mathsf {\Sigma }}),\quad \text{ or } \quad \theta _{j_1} ^*= \frac{({\mathrm{SNR}}-1)\times {\mathrm{trace}}({\mathsf {\Sigma }})}{\beta \Vert {\mathsf {M}}_{j_1}{{\mathsf {C}}}_{j_1}^{1/2}\Vert ^2_\mathrm{F}}. \end{aligned}$$

Finally, assume that we only have a prior idea of how many non-zero sources may be active, and we express this belief in a form of a probability density

$$\begin{aligned} {\mathsf {P}}\{\# \text{ of } \text{ active } \text{ sources } \text{= } k\} =p_k, 1\le k\le n, \end{aligned}$$

where $\sum _k p_k = 1$. If we expect that out of the n dipoles, it is reasonable to expect that $\overline{k} = sn$ are active, a binomial distribution can be used for $p_k$, $k_k\sim {\mathrm{Binom}}(n,s)$, $0<s<1$; in practice the binomial can be approximated by a Poisson distribution, $p_k\sim {\mathrm{Poisson}}(\overline{k})$ with mean $\overline{k}$ provided by the user. Using the previous result, conditioned on k, we arrive at the scaling law

$$\begin{aligned} \theta _{j} ^*= \frac{C}{ \Vert {\mathsf {M}}_j{{\mathsf {C}}}_j^{1/2}\Vert ^2_\mathrm{F}}, \quad 1\le j\le n, \end{aligned}$$

with

$$\begin{aligned} C = ({\mathrm{SNR}}-1)\times {\mathrm{trace}}({\mathsf {\Sigma }}) \, \sum _{k=1}^n\frac{p_k}{k}. \end{aligned}$$

This argument confirms that, in order to match the model with the SNR, the parameters $\theta _j^*$ should indeed be chosen to be inversely proportional to the sensitivity.

Construction of the Activity Patch

To select the vertices in the activity patch ${\mathcal {P}}$, we first pick randomly a seed vertex from the BR of interest, then grow the patch by adding iteratively the neighboring vertices, pruning off at each step those that fall outsize the pertinent BR, and stopping the process as soon as the desired number $N_{\mathcal {P}}$ of vertices have been included. The selected nodes along with the edges of the triangular mesh form a local graph. To generate the activity in the patch, we start by computing a positive graph Laplacian of the patch, which is the matrix ${\mathsf {L}}\in {\mathbb {R}}^{N_{\mathcal {P}}\times N_{\mathcal {P}}}$ with entries

$$\begin{aligned} L_{i,j} = \left\{ \begin{array}{ll} -{\mathrm{deg}}(v_i) &{} \hbox {if} \ i=j, \\ 1 &{} \hbox {if} \ i\ne j \ \hbox {and} \ v_i \ \hbox {is adjacent to} \ v_j, \\ 0 &{} \hbox {otherwise}, \end{array} \right. \end{aligned}$$

where ${\mathrm{deg}}(v_i)$ is the number of the edges that terminate at the vertex $v_i$.

After defining a correlation length $\lambda$, given in units of the number of steps, we draw a random amplitude vector by setting

$$\begin{aligned} Q = ({\mathsf {L}}+ \lambda ^2 {\mathsf {I}}_{N_{\mathcal {P}}})^{-1}W, \end{aligned}$$

where ${\mathsf {I}}_{N_{\mathcal {P}}}\in {\mathbb {R}}^{N_{\mathcal {P}}\times N_{\mathcal {P}}}$ is the unit matrix and $W\in {\mathbb {R}}^{N_{\mathcal {P}}}$ is a standard normal Gaussian random vector, that is, $W\sim {{\mathcal {N}}}(0,{\mathsf {I}}_{N_{\mathcal {P}}})$. The amplitudes are scaled so that the amplitude of the dipole at the seed vertex is one. Finally, we draw the dipole moment directions from the anatomical prior, making sure that adjacent dipoles are not pointing in the opposite sides of the cortex patch.