Transcranial Electrical Neuromodulation Based on the Reciprocity Principle

Abstract

A key challenge in multi-electrode transcranial electrical stimulation (TES) or transcranial direct current stimulation (tDCS) is to find a current injection pattern that delivers the necessary current density at a target and minimizes it in the rest of the head, which is mathematically modeled as an optimization problem. Such an optimization with the Least Squares (LS) or Linearly Constrained Minimum Variance (LCMV) algorithms is generally computationally expensive and requires multiple independent current sources. Based on the reciprocity principle in electroencephalography (EEG) and TES, it could be possible to find the optimal TES patterns quickly whenever the solution of the forward EEG problem is available for a brain region of interest. Here, we investigate the reciprocity principle as a guideline for finding optimal current injection patterns in TES that comply with safety constraints. We define four different trial cortical targets in a detailed seven-tissue finite element head model, and analyze the performance of the reciprocity family of TES methods in terms of electrode density, targeting error, focality, intensity, and directionality using the LS and LCMV solutions as the reference standards. It is found that the reciprocity algorithms show good performance comparable to the LCMV and LS solutions. Comparing the 128 and 256 electrode cases, we found that use of greater electrode density improves focality, directionality, and intensity parameters. The results show that reciprocity principle can be used to quickly determine optimal current injection patterns in TES and help to simplify TES protocols that are consistent with hardware and software availability and with safety constraints.

Keywords: high-density electrode arrays; non-invasive neuromodulation; reciprocity principle; transcranial direct current stimulation; transcranial electrical stimulation.

Figures

Figure 1

Head model. (A) Axial slice showing the different tissues of the segmented head and the tetrahedral mesh. (B) Skull, eye balls, and gray matter meshes. (C) Electrodes on the scalp with the denser mesh on the electrode to skin contact surfaces.

Figure 2

Details of the skull FE model. Axial (A), coronal (B), and sagittal (C) slices of the skull obtained from the subject specific CT. The skull base and optic foramina as well as the frontal sinuses and other internal air compartments are clearly seen.

Figure 3

Schematics of the reciprocity targeting. Left: a simulated dipole at position r with moment dr generates a potential distribution Φ with a maximum (point A), a minimum (point B), and a isopotential lines (like the green one). Center: poles A and B can be used for the current injection maximizing the component of the gradient of the potential ∇→ψAB in the direction dr. Right: if any injection pair is on an isopotential line [Φ(A) − Φ(B) = 0], the current density on target is perpendicular to the target orientation.

Figure 4

Schematics of the ROADSS approach. The three columns represent three different targets: radial (perpendicular), oblique, and tangential to cortex orientations at position r. The first row shows the selected sources based on reciprocity (reed dots); the second row shows rm (the red “+” sign) and drmr as a black arrow; and the third row shows rs (blue “–” sign) and the sinks in each case (blue dots). It is clearly seen that in the radial case (first column), the resulting pattern is similar to the “ring” approach and that in the tangential case (third column), the resulting pattern is similar to the “opposite” approach.

Figure 5

Targets. The four trial targets on the cortex, T1, T2, T3, and T4, are shown with the corresponding normal vectors of 5 cm length for better visualization. Normal vectors of targets T1 and T3 are mostly perpendicular and normal vectors of targets T2 and T4 are mostly tangential to the scalp surface. The circles indicate the central position of the electrodes of the 128 EGI sensor net. The normal vector projections to the scalp for T2 and T3 are close to a specific electrode in the 128 montage, while the similar projections for T1 and T4 are in between. The left and right figures are close-up views of the targets with their corresponding normal vectors.

Figure 6

Imprinted electric potential [V] on the scalp for the 128 EGI electrode net. Rows from top to bottom correspond to optimal stimulation of targets T1, T2, T3, and T4; columns correspond to the optimal current injection patterns obtained, from left to right, with the LS, LCMV, “one source,” “opposite,” “ROADSS,” and “ring” methods. The total current injected in each case is 1 mA.

Figure 7

The same as in Figure 6 for the 256 EGI electrode net.

Figure 8

Module of the total delivered current density on the cortex for the 128 EGI electrode net in ampere/square millimeter. Rows from top to bottom correspond to optimal stimulation of targets T1, T2, T3, and T4; columns correspond to the optimal current injection patterns obtained, from left to right, with the LS, LCMV, “one source,” “opposite,” “ROADSS,” and “ring” methods. The total current injected in each case is 1 mA. The cortical ROIs to target are shown in black.

Figure 9

The same as in Figure 8 for the 256 EGI electrode net.

Figure 10

Normal to cortex component of the current density (ampere/square meter) for the 256 EGI electrode net. Rows from top to bottom correspond to optimal stimulation of targets T1, T2, T3, and T4; columns correspond to the optimal current injection patterns obtained, from left to right, with the LS, LCMV, “one source,” “opposite,” “ROADSS,” and “ring” methods. The total current injected in each case is 1 mA. The cortical ROIs to target are shown in black.

Figure 11

Comparison of quantitative metrics of targeting between different methods. (A) Module of the total current density at each target; (B) directional current density on the target as the dot product between the current density and the unitary normal vector at each target surface; and (C) the normalized dot product between each target normal and the resulting current density. The bars depict the results for the 256 electrode net while the “+” marks correspond to the 128 sensor net. The total current injected in each case is 1 mA.

Figure 12

Targeting Error (TE). Distance between the global (A), and local (B) center of gravity and the true central position of the target, for the 256 EGI electrode net (bars) and for the 128 EGI electrode net (“+” marks). Global TEs larger than 5 cm were removed for clarity.

Figure 13

Focality. (A) Focality defined with the half-width at half-maximum radius approach for the whole gray matter domain (the lower the better), i.e., a global focality metric; and (B), the local focality (the larger the better) as defined in Eq. 9. The bars depict the results for the 256 electrode net while the “+” marks correspond to the 128 sensor net.