L1-norm vs. L2-norm fitting in optimizing focal multi-channel tES stimulation : linear and semidefinite programming vs. weighted least squares

Background and Objective: This study focuses on Multi-Channel Transcranial Electrical Stimulation, a non-invasive brain method for stimulating neuronal activity under the influence of low-intensity currents. We introduce a mathematical formulation for finding a current pattern that optimizes an L1-norm fit between a given focal target distribution and volumetric current density inside the brain. L1-norm is well-known to favor well-localized or sparse distributions compared to L2-norm (least-squares) fitted estimates. Methods: We present a linear programming approach that performs L1-norm fitting and penalization of the current pattern (L1L1) to control the number of non-zero currents. The optimizer filters a large set of candidate solutions using a two-stage metaheuristic search from a pre-filtered set of candidates. Results: The numerical simulation results obtained with both 8- and 20-channel electrode montages suggest that our hypothesis on the benefits of L1-norm data fitting is valid. Compared to an L1-norm regularized L2-norm fitting (L1L2) via semidefinite programming and weighted Tikhonov least-squares method (TLS), the L1L1 results were overall preferable for maximizing the focused current density at the target position, and the ratio between focused and nuisance current magnitudes. Conclusions: We propose the metaheuristic L1L1 optimization approach as a potential technique to obtain a well-localized stimulus with a controllable magnitude at a given target position. L1L1 finds a current pattern with a steep contrast between the anodal and cathodal electrodes while suppressing the nuisance currents in the brain, hence, providing a potential alternative to modulate the effects of the stimulation, e.g., the sensation experienced by the subject.publishedVersionPeer reviewe

MC-tES in Zeffiro Interface: Sparse Optimized and Regularized Stimulus

This thesis aims to describe the mathematical methodology of multi-channel transcranial electrical stimulation (MC-tES) and its computational implementation in the open Matlab-based Zeffiro Interface (ZI) toolbox. The goal is to extend the current solver capabilities of ZI, and by using the solver implementation, among other things, to enlighten the process of finding a focal optimized, and preferably sparse, current pattern, as well as to provide the necessary codes for further software development. The present implementation covers both forward and inverse MC-tES solver. The former inherits for ZI’s finite element method based forward solver for electroencephalography. Here the mathematical framework of this solver is described and its connection to MC-tES explained. The application of the complete electrode model boundary conditions ensures the high accuracy of the model at the vicinity of the current-injecting electrodes. The inverse problem is approached via L1-regularized optimization and the dual-simplex linear programming algorithm. The performance of the implementation is evaluated in numerical experiments in which the volume current density caused by the stimulus is steered using a synthetic 10 nAm source with a reference extent of 253 mm3 as a target. A rough initial range for the regularization and tolerance parameter is found with somatosensory, visual and auditory cortex as the reference target areas, using a realistic multi-layered head model discretized with 1 mm accuracy

Multi-compartment head modeling in EEG: Unstructured boundary-fitted tetra meshing with subcortical structures.

This paper introduces an automated approach for generating a finite element (FE) discretization of a multi-compartment human head model for electroencephalographic (EEG) source localization. We aim to provide an adaptable FE mesh generation tool for EEG studies. Our technique relies on recursive solid angle labeling of a surface segmentation coupled with smoothing, refinement, inflation, and optimization procedures to enhance the mesh quality. In this study, we performed numerical meshing experiments with the three-layer Ary sphere and a magnetic resonance imaging (MRI)-based multi-compartment head segmentation which incorporates a comprehensive set of subcortical brain structures. These experiments are motivated, on one hand, by the sensitivity of non-invasive subcortical source localization to modeling errors and, on the other hand, by the present lack of open EEG software pipelines to discretize all these structures. Our approach was found to successfully produce an unstructured and boundary-fitted tetrahedral mesh with a sub-one-millimeter fitting error, providing the desired accuracy for the three-dimensional anatomical details, EEG lead field matrix, and source localization. The mesh generator applied in this study has been implemented in the open MATLAB-based Zeffiro Interface toolbox for forward and inverse processing in EEG and it allows for graphics processing unit acceleration

Clustering of simultaneous cortical and subcortical activity.

Clusters obtained via Gaussian mixture modeling (GMM) for the P14/N14 (a-d) and P22/N22 (e-h) components of the experimental SEP dataset [17, 20]. Each cloud shows the ellipsoidal 90% credibility set of the corresponding cluster. Clouds are color-labeled according to their measured intensity levels in cubic/millimeters (mm3).</p

S1 Appendix -

This paper introduces an automated approach for generating a finite element (FE) discretization of a multi-compartment human head model for electroencephalographic (EEG) source localization. We aim to provide an adaptable FE mesh generation tool for EEG studies. Our technique relies on recursive solid angle labeling of a surface segmentation coupled with smoothing, refinement, inflation, and optimization procedures to enhance the mesh quality. In this study, we performed numerical meshing experiments with the three-layer Ary sphere and a magnetic resonance imaging (MRI)-based multi-compartment head segmentation which incorporates a comprehensive set of subcortical brain structures. These experiments are motivated, on one hand, by the sensitivity of non-invasive subcortical source localization to modeling errors and, on the other hand, by the present lack of open EEG software pipelines to discretize all these structures. Our approach was found to successfully produce an unstructured and boundary-fitted tetrahedral mesh with a sub-one-millimeter fitting error, providing the desired accuracy for the three-dimensional anatomical details, EEG lead field matrix, and source localization. The mesh generator applied in this study has been implemented in the open MATLAB-based Zeffiro Interface toolbox for forward and inverse processing in EEG and it allows for graphics processing unit acceleration.</div

Mesh visualization: Ary sphere.

Quadrants of the downsampled surface grids (a-c) and the boundary-fitted tetrahedral mesh (e-g) for 3.0, 2.0 and 1.3 mm (millimeter) mesh sizes, respectively. Surface grids (d) and unfitted tetrahedral mesh (h) for 1.0 mm mesh size are included for comparison. The presented layers are (top-bot): scalp (brown), skull (white), and grey matter.</p

Mind map of the meshing process.

The unfitted mesh is obtained after the first labeling stage, while the boundary-fitting process includes additional processing stages for refinement, re-labeling, smoothing, inflation, and optimization via Delaunay turns. A graphics processing unit (GPU) can be applied to accelerate the solid angle labeling and re-labeling stages as well as the surface extraction stage which finds the compartment boundaries after labeling. The re-labeling process is run recursively as long as one or more compartment labels change their value. A pseudocode of this mind map is provided in data in S1 Appendix.</p

Lead field matrix accuracy.

Accuracy of the lead field matrix evaluated at different eccentricities in Ary sphere, i.e. relative distances from the origin in the brain compartment using relative difference measure (RDM) and magnitude measure (MAG). The horizontal axis corresponds to the eccentricity (relative distance from origin with respect to the grey matter surface) and the vertical one to the difference measure (%) in question.</p

Volume-value of the up-to-three obtained GMM-based clusters (R = Red, G = Green, B = Blue) ordered in descending order with respect to their intensity, measured in cubic millimeters (mm³).

Volume-value of the up-to-three obtained GMM-based clusters (R = Red, G = Green, B = Blue) ordered in descending order with respect to their intensity, measured in cubic millimeters (mm3).</p