Source localization of epileptic spikes using Multiple Sparse Priors

Objective: To evaluate epileptic source estimation using multiple sparse priors (MSP) inverse method and high-resolution, individual electrical head models. Methods: Accurate source localization is dependent on accurate electrical head models and appropriate inverse solvers. Using high-resolution, individual electrical head models in fifteen epilepsy patients, with surgical resection and clinical outcome as criteria for accuracy, performance of MSP method was compared against standardized low-resolution brain electromagnetic tomography (sLORETA) and coherent maximum entropy on the mean (cMEM) methods. Results: The MSP method performed similarly to the sLORETA method and slightly better than the cMEM method in terms of success rate. The MSP and cMEM methods were more focal than sLORETA with the advantage of not requiring an arbitrary selection of a hyperparameter or thresholding of reconstructed current density values to determine focus. MSP and cMEM methods were better than sLORETA in terms of spatial dispersion. Conclusions: Results suggest that the three methods are complementary and could be used together. In practice, the MSP method will be easier to use and interpret compared to sLORETA, and slightly more accurate and faster than the cMEM method. Significance: Source localization of interictal spikes from dense-array electroencephalography data has been shown to be a reliable marker of epileptic foci and useful for pre-surgical planning. The advantages of MSP make it a useful complement to other inverse solvers in clinical practice.


Appendix A: hexahedral FEM validation
In this work we used a hexahedral finite element model (HexaFEM) to compute the forward problem of EEG. In fact, we computed the transcranial electrical stimulation (TES) forward problem which consists of applying an electric current on the scalp. Then, the EEG lead field matrix is obtained from the TES solutions by using the reciprocity principle. In this appendix, we show the validation against the analytic solutions in a spherical model and against the finite difference method (FDM) solver of Geo Source 3 Philips system in a realistic head model. The analytic formulation can be found elsewhere (Fernández-Corazza et al., 2011).

Comparison on a three-layer sphere
We built a three-layer sphere with radii of 9.2, 8.5 and 7.5 centimeters and conductivities 0.33, 0.01 and 0.33 S/m respectively. The electrode layout used is the spherical 256-channel Geodesic sensor-net, with electrode 257 being the reference Cz. A unitary electrical current source was applied to each electrode pair with Cz being fixed. Approximately 2500 dipoles were placed at the internal compartment (separated approximately 10mm from the layer) with normal orientation pointing outwards. quantify the differences. The NRDM is a metric that compares two vectors of the same size based only on the shape of them and not on their strength: it is zero if vectors are colinear, the squared root of 2 if the vectors are orthogonal, and 2 if the vectors are colinear but opposite. The MAG metric compares two vectors of the same size based only on their magnitude: it is 1 if both vectors have equal ℓ2-norm, < 1 if the ℓ2-norm of the first vector is lower than the ℓ2-norm of the second vector, and >1 vice versa.

Appendix B: Details of MSP
The typical signal model for electrodes, dipoles and a time window with samples is: where is the lead field matrix, are the sources, ⋅ accounts for "fixed effects" or "confounds" and is the noise matrix. and are modeled as matrix normal random variables: where is the temporal covariance matrix and is the spatial covariance matrix. Note that the sample covariance estimators are: ̂≈ , and ̂≈ .
The first step is to project the data to eliminate the ⋅ term (model reduction). This is typically done assuming one confound that accounts for the constant shift of the potential reference: ×1 = ( , 1). Thus, the spatial projector is: There is also a spatial projection matrix that is built as described in , where are different empirical source covariance priors with unknown hyperparameters . They are typically "sparse" priors such as some small subnetworks, or patches. As is in the source space, there is no projection needed. Lastly, = models uncorrelated sensor noise.
Both noise and source models are merged into one stochastic model by performing a projection to the signal space: The second step is the estimation of the hyperparameters ( ) from data (or "evidence"). For this purpose, Friston et al. (2008) propose two methods: "greedy search" and automatic relevance determination (ARD). The latter is sparse in nature and it starts with a maximum number of non-zero hyperparameters and it attempts to reduce them. The ARD approach is the one used in both Friston et al. (2008) and here. The hyperparameters are estimated with an expectationmaximization method, where the M-step estimates the hyperparameters and the E-step computes the expected value of the estimates.
Once the hyperparameters corresponding to each prior are computed using the expectationmaximization method, the spatial covariance matrix in the source space is built as: And then projected to the signal space by doing: Electrode space ( × ) = ℎ(1) + ∑ ℎ( + 1)̃ = 1 ( . 8)