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

TL;DR

This paper introduces an L1-norm based optimization method for multi-channel transcranial electrical stimulation, demonstrating improved focality and control over nuisance currents compared to traditional L2-norm approaches.

Contribution

It presents a novel linear programming approach for L1-norm fitting in brain stimulation, with a metaheuristic search to optimize current patterns for better focality and controllability.

Findings

L1L1 method outperforms L1L2 in focusing current density.

L1L1 reduces nuisance currents and enhances target localization.

Metaheuristic optimization effectively finds optimal stimulation patterns.

Abstract

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 mathematical formulation for finding a current pattern which optimizes a L1-norm fit between a given focal target distribution and volume 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. We present a linear programming approach which 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 in from a pre-filtered set of candidates. The numerical simulation results, obtained with both a 8- and 20-channel electrode montages, suggest that our hypothesis on the benefits of L1-norm data fitting is valid. As compared to L1-norm regularized L2-norm fitting (L1L2) via semidefinite programming and weighted Tikhonov least-squares method, the L1L1 results were overall preferable with respect to maximizing the focused current density at the target position and the ratio between focused and nuisance current magnitudes. 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 meanwhile 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.