L1-norm vs. L2-norm fitting in optimizing focal multi-channel tES stimulation: linear and semidefinite programming vs. weighted least squares - Galaz Prieto, Fernando; Rezaei, Atena; Samavaki, Maryam; Pursiainen, Sampsa (Computer methods and programs in biomedicine, 2022)
•Comparative results using L1-norm regularized L1-norm fitting (L1L1) against L1-norm regularized L2-norm fitting (L1L2) and Tikhonov’s regularized least-squares (TLS) methods for the calculations.•Examination of state-of-the-art 8 and 20 active electrode channel montage for Multi-Channel Transcranial Electrical Stimulation (MC-TES) exercise, i.e., applying more than two electrodes during brain stimulation session.•Analysis of reconstructions, electric fields, current distribution and montage using different number of active electrodes available and applied through a two-stage metaheuristic-based constraints on different regions of the brain. 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.