  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.
