Objective Transcranial direct current stimulation (tDCS) aims to alter brain function noninvasively via electrodes placed on the scalp. Conventional tDCS uses two relatively large patch electrodes to deliver electrical currents to the brain region of interest (ROI). Recent studies have shown that using dense arrays containing up to 512 smaller electrodes may increase the precision of targeting ROIs. However, this creates a need for methods to determine effective and safe stimulus patterns as the degrees of freedom is much higher with such arrays. Several approaches to this problem have appeared in the literature. In this paper, we describe a new method for calculating optimal electrode stimulus pattern for targeted and directional modulation in dense array tDCS which differs in some important aspects with methods reported to date. Approach We optimize stimulus pattern of dense arrays with fixed electrode placement to maximize the current density in a particular direction in the ROI. We impose a flexible set of safety constraints on the current power in the brain, individual electrode currents, and total injected current, to protect subject safety. The proposed optimization problem is convex and thus efficiently solved using existing optimization software to find unique and globally optimal electrode stimulus patterns. Main results Solutions for four anatomical ROIs based on a realistic head model are shown as exemplary results. To illustrate the differences between our approach and previously introduced methods, we compare our method with two of the other leading methods in the literature. We also report on extensive simulations that show the effect of the values chosen for each proposed safety constraint bound on the optimized stimulus patterns. Significance The proposed optimization approach employs volume based ROIs, easily adapts to different sets of safety constraints, and takes negligible time to compute. In-depth comparison study gives insight into the relationship between different objective criteria and optimized stimulus patterns. In addition, the analysis of the interaction between optimized stimulus patterns and safety constraint bounds suggests that more precise current localization in the ROI, with supposably improved safety criterion, may be achieved by careful selection of the constraint bounds.

Let us consider a case where all of the elements in some continuous slices are missing in tensor data. In this case, the nuclear-norm and total variation regularization methods usually fail to recover the missing elements. The key problem is capturing some delay/shift-invariant structure. In this study, we consider a low-rank model in an embedded space of a tensor. For this purpose, we extend a delay embedding for a time series to a "multi-way delay-embedding transform" for a tensor, which takes a given incomplete tensor as the input and outputs a higher-order incomplete Hankel tensor. The higher-order tensor is then recovered by Tucker-based low-rank tensor factorization. Finally, an estimated tensor can be obtained by using the inverse multiway delay embedding transform of the recovered higherorder tensor. Our experiments showed that the proposed method successfully recovered missing slices for some color images and functional magnetic resonance images.

Dense array transcranial direct current stimulation (tDCS) has become of increasing interest as a noninvasive modality to modulate brain function. To target a particular brain region of interest (ROI), using a dense electrode array placed on the scalp, the current injection pattern can be appropriately optimized. Previous optimization methods have assumed availability of individually controlled current sources for each non-reference electrode. This may be costly and impractical in a clinical setting. However, using fewer current sources than electrodes results in a non-convex combinatorial optimization problem. In this paper, we present a novel use of the branch and bound (BB) algorithm to find sub-optimal stimulus patterns with fewer current sources than electrodes. We present simulation results for both focal and spatially extended cortical ROIs. Our results suggest that only a few (2-3) independently controlled current sources can achieve comparable results to a full set (125 sources) to a tolerance of 5%. BB is computationally 3-5 orders of magnitude less demanding than exhaustive search.

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