Sparsely optimized multi-electrode transcranial direct current stimulation

Transcranial direct current stimulation (tDCS) has become a widely used technique for non-invasive brain stimulation in the last decade since its successful application to the human cortex [1]. It has shown notable potential for the treatment of neurological and psychiatric disorders, as well as for studying the basics of cognitive and behavioral neuroscience [2]. Currently, tDCS is applied using two rectangular pads placed on the scalp. However, it is well known that the induced electric field in the brain resulting from this configuration is non-focal [3]. Focality in stimulation translates into a better understanding of the elicited effects [4] and it may reduce potential side-effects.

Researchers have recently proposed a multi-electrode stimulation paradigm, where using the superposition of sources, an optimization algorithm calculates the appropriate electrode currents in order to achieve a given intensity and orientation of the induced electric field at a target. The authors show how focality can be improved over the conventional montage [5]. However, to translate this new paradigm into practice one must take into account that there is a limited number of active electrodes which can be used. This poses a hardware constraint which should be included in the optimization techniques.

This work presents a method for realizing optimized multi-electrode tDCS which allows to specify the number of active electrodes while optimizing intensity and orientation of the induced electric field at a given target. For solving the sparsity problem, i.e. to select a subset from the available electrode set, we tested two approaches: (1) a stepwise optimal selection (SOS) algorithm, including electrodes which yield the minimal residual and thus provide the best fit to the predefined electric field and (2) an orthogonal matching pursuit (OMP) [6], a greedy algorithm which finds the most relevant projections of the residual onto the electrodes’ contribution. In both approaches we employ a constrained least-squares optimization approach, taking patient safety into account. Using an anatomically realistic MRI-based finite element model, we calculate the induced electric field in the brain for the different electrode montages by solving the Laplace equation.

We show substantial focality improvement against the conventional setup, namely, 67%, 76% and 78% for four, six and eight active electrodes respectively using SOS (OMP yields a slightly worse performance). Calculating the residual sum of squares and the Bayesian information criterion (BIC), we analyze the trade-off between the number active electrodes and how well the solution matches the predefined electric field. The BIC yields the minimal number of electrodes for an optimal trade-off. We compare our method to optimization approaches with unconstrained number of active electrodes. Most importantly, our method makes optimized multi-electrode tDCS compatible with any stimulator system.