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

ABSTRACT

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.