Computing Sciences Unit, Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland.
Computing Sciences Unit, Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland.
Comput Methods Programs Biomed. 2022 Nov;226:107084. doi: 10.1016/j.cmpb.2022.107084. Epub 2022 Aug 27.
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.
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.
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.
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.
本研究聚焦于多通道经颅电刺激，这是一种非侵入性的大脑方法，可在低强度电流的影响下刺激神经元活动。我们引入了一种数学公式，用于找到一种电流模式，该模式可以使给定的焦点目标分布与大脑内体积电流密度之间的 L1 范数拟合达到最佳。与 L2 范数（最小二乘）拟合估计相比，L1 范数更有利于良好的局部化或稀疏分布。
我们提出了一种线性规划方法，该方法执行 L1 范数拟合和电流模式的惩罚（L1L1），以控制非零电流的数量。优化器使用两级元启发式搜索从预过滤的候选者集中筛选大量候选解决方案。
使用 8 通道和 20 通道电极排列获得的数值模拟结果表明，我们关于 L1 范数数据拟合优势的假设是有效的。与通过半定规划和加权 Tikhonov 最小二乘法（TLS）进行的 L1 范数正则化 L2 范数拟合（L1L2）相比，L1L1 结果总体上更有利于在目标位置最大化聚焦电流密度，以及聚焦电流与杂散电流幅度之比。
我们提出了元启发式 L1L1 优化方法作为一种潜在的技术，可在给定目标位置获得具有可控幅度的良好局部刺激。L1L1 在抑制大脑中的杂散电流的同时，在阳极和阴极电极之间形成陡峭的对比，从而提供了一种潜在的替代方法来调节刺激的效果，例如，受试者所经历的感觉。
