Fig 1.
Equivalent circuits for the electrode-gel-skin interface.
(A) Description of the electrode-skin-gel interface as lumped element model. The resistance and capacitance of the skin incorporate spatially varying properties. (B) Distributed circuit model for the spatial extent of the dermal interface. (C) The topology of a single-layer infinite regular grid with edge length ϵ. A node x = (x1, x2) has four direct neighbors (x1±ϵ, x2), (x1, x2±ϵ). (D) The currents in direction x1 and x2 are denoted by I1 and I2, respectively. Note that the injected current J is required to scale with ϵ. (E) The voltages at the nodes are denoted by V.
Fig 2.
The initial voltage distribution in the skin capacitor after the pulse as described by the model.
(A) The initial voltage distribution (shaded red) underneath a recording electrode located at x1 modelled as a box function B with radius b (dashed) convolved with a Gaussian with variance σ1. Darker color means higher density, white means 0. More intense color corresponds to larger voltage amplitude. (B) The initial voltage distribution can have a displacement μ1 w.r.t. its center x1. The reference electrode (shaded blue) also has parameters σ2 and μ2. (C) Some initial voltage distributions can be selected such that their linear combination (green) may approximate the initial voltage distributions at all remaining electrodes.
Fig 3.
The simulated voltage difference between a reference and the recording electrode.
The initial voltage distribution at each electrode is a Gaussian of the form g(σ, x + μ) convolved with the box function B, where g(t, x) is the voltage impulse-response function of our model, and B(x) equals 1/πb2 for |x| ≤ b. The parameters for the reference electrode are fixed to σR = 0.5 and μR = 0. The choice of μR implies that the voltage depends only on the length |μ| of μ and not on its direction. The electrode radius was b = 2. (A, B) Effect of varying σ from 0 to 1 in steps of 0.05 (inset: close-up for σ for half step size). (D, E) Effect of varying |μ| from 0 to 2 in steps of 0.1 (inset: close-up for |μ| for half step size). (C, F) The plot on a log-log scale demonstrates the power law. In comparison to data, we note the asymmetry of positive and negative voltage shapes (compare to Figs 4A, 7A and 11A) and the emergence a local extremum near 0 in some traces (compare to Fig 4A). (G-L) The simulated voltage difference for electrodes approximated by points (b = 0).
Fig 4.
Representative example of TMS artifacts on a human head and assessment of the artifact model.
(A) Example of raw EEG traces displaying TMS-induced artifacts recorded from a human head. Magnetic stimulation was applied at time 0 at electrode Cz on a 64-electrode cap with 32 electrodes on the right hemisphere. The artifacts appear in every electrode trace at different strength or shape. Inset: On the uV scale of physiological brain waves, some traces exhibit an artifact duration of more than 100 ms. (B) The fast initial artifact dynamics related to the magnetic pulse. (C) Averaging out noise using five trials shows the long-lasting artifact decay to baseline. (D) On a log-log scale, the tails in the decay of the artifacts (from C) follow a power-law with an exponent on the order of 2 (red dashed lines). (E) TMS-EEG traces on a human knee. Shown are raw data (gray) of a single recording from 28 electrodes covering the knee following TMS. The artifacts reconstructed with the model are shown in green. (F) Log-log plot. (G) Data after subtraction of the reconstructed artifacts followed by subtraction of average of all traces to remove the common mode. As expected from TMS on a knee, the stimulation does not evoke neuronal activity, such that the artifact-removed traces are flat up to continuation of the typical very slow and small electrode drifts. The area shaded in gray indicates where the artifacts could not be reconstructed. (H) To assess the goodness of fit, we use the χ2-test with significance level α = 5%. Shown is the maximal time span for which the test accepts the fit. Beyond this time, the fit is rejected. Small electrode drifts and noise due to TMS-device recharging can shorten this time, however never below 20 ms. (I-J) Fits of sums of two exponentials. (K) Subtraction of the fits introduces both fast and slow distortions of the data in almost all traces. (L) Correspondingly, the fits are generally not accepted, except mainly in noisy and drifting electrodes. (M) The difference of data reconstructed by the model and by the sum of two exponentials. This equals the difference of the respective reconstructed artifacts. (N) For almost all fits by a sum of two exponentials, the quotients of the decay constants are approximately equal and have the same order of magnitude (a, b, c, d constants, t time).
Fig 5.
TMS-EEG on phantom heads with non-human skin structure.
(A, B) Log-log plots of TMS artifacts on a watermelon and on a muskmelon show artifact decay is very different from a human head in that it does not follow a power law of order 1 (dotted line) or 2 (dashed). (C) The artifact traces on the phantom qualitatively resemble the artifact on a human head even though they are much smaller and generally shorter-lasting. (D) Example of an artifact which contains an additional recharging artifact consisting of two waves at an interval of 20 ms, where the first wave appears within around 20 ms after the pulse. This artifact can appear when the TMS device is operated with two boosters. It is visible in around half of all trials at different intensities and is not affected by acquisition rate. Inset: Subtraction of the common average from the traces turns the recharging artifact into two ‘blips’.
Fig 6.
The TMS-induced artifacts before and after skin preparation by puncturing and exfoliation.
(A, B) Sample of raw EEG traces from a 64-electrodes cap after TMS application at CPz at time 0. Acquisition rate was 8 kHz to resolve the fast initial artifact dynamics. The recording before (red) and after skin puncture underneath the EEG electrodes (blue) shows no difference in the dynamics of the pulse artifact (A) but a reduction of the artifact decay (B). (C, D) Shown is the envelope (shaded area) of the two distributions of all artifacts before and after puncture. These distributions of artifacts were obtained by combining all sets of 64-electrodes cap traces from 2 subjects, each stimulated at both CPz and CP3 in a total of 71 TMS pulses. The pulse artifact is not changed (within an accuracy of one time step) by skin puncturing (C). The amplitude of the decay artifact (D) is reduced as shown by the shaded area, corresponding to the 5%-to-95% percentile of the distribution of all artifacts. (E) We compare two physical models of decay, (shifted) power laws a/(t + b)2 and exponentials cexp(−dt) (t time, a, b, c, d constants). Both models are least-squares fitted to all traces which do not change sign and have amplitude larger than 1.5 mV (dashed line). All fits are done to 25 ms starting from the point of reaching 1.5 mV. Evaluation of the fits by R2 shows the power law is better than the exponential with and without skin treatment. Specifically, skin puncturing does not decrease the difference of R2 by median (solid lines). (F-J) Same as (A-E) with skin exfoliation (green) instead of skin puncturing compared to control (red). Stimulation site was Cz and FP2, sampling rate 8 kHz.
Fig 7.
Effect of placing the CMS and DRL electrodes at different locations relative to the TMS coil.
(A, B, C) Recordings from electrodes F4 and C6 (triangles) for TMS application at C2 (cross) are shown for different placements of the CMS (circles) and DRL (squares) electrodes. Different colors correspond to different placements of CMS-DRL (red = standard positions assigned on the Biosemi cap, green = P3-P7, blue = CP3-TP7, pink = C3-T7, cyan = FC3-FT7, yellow = F3-F7). (D) EEG traces after TMS stimulation at time 0 with the CMS and DRL electrodes placed on the index finger of the left hand. The corresponding electrode wires had a minimal distance of 50 cm from the TMS coil. The points of stimulation consisted of Cz, CPz, and POz. Shown are 13 trials with 13 electrode traces each acquired at 16 kHz. The red traces correspond to the raw, unmodified electrode traces. The green traces correspond to the same data where in addition, in every trial, the average of all 13 electrodes was subtracted from the data. Because the raw traces exhibit strong 50 Hz noise due to the unusual ground electrode placement, the grand average of the red traces over all trials is also shown (blue trace). (E) Same data on a longer time scale. Note the similarity of the variety of the green traces to the artifacts in Fig 4A. (F) On a log-log scale, the red follow a power-law with an exponent on the order of 1 (black dotted line). The green traces follow a power-law upon late-stage decay to baseline with an exponent on the order of 2 (black dashed lines).
Fig 8.
Effect of changing the sampling rate.
(A, B) Decreasing acquisition rate (16384 Hz, 8192 Hz, 4096 Hz, 2048 Hz, 1024 Hz) lead to a progressive time shift of the TMS artifacts and a decrease of TMS pulse artifact amplitude. The shifting time can be found by time-shifting the traces for each sampling rate backwards until they coincide with the 16 kHz traces. The optimal time is found when the sum of distances between these traces, evaluated directly after the pulse artifact, becomes minimal. The optimal times coincide for the watermelon and the human head (C). Note that time shifting will not change power law decay tails as can also be seen in (D).
Fig 9.
TMS artifacts of a passive EEG system (g.tec) with ring electrodes.
Traces were recorded from nine electrodes (F2, F5, F6, C3, C4, CZ, CP5, P5, POz) at 38,4 kHz on a single subject. TMS was applied at time 0. Stimulation sites were C3 (red traces, 4 trials), CPz (green, 3 trials), and CP2 (blue, 3 trials). (A) The TMS pulse artifact is well-resolved due to the high acquisition rate and has a duration of 0.39 ms. Note the short-time artifact dynamics visible in the green traces, which is different from what can be seen in the Biosemi artifacts. (B, C) The skin-capacitor discharge artifacts for a short time (B) and long time (C). (D) On a log-log scale, the late-stage decay of all artifacts exhibits a power law of order 2 like in the Biosemi EEG system. (E, F, G) Single trials with reconstructed artifact fits overlaid on the artefactual data (gray) for each stimulation site as indicated by color. (H) Log-log plot of the artifact fits for all trials overlaid on the data. The first 15 samples corresponding to 0.39 ms containing the TMS pulse were omitted. For each stimulation site the starting time point for the fit is selected as the first point at which all the artifacts in all electrodes already decay faster than a power law of order 1. This is actually the place where a power law of order 1 (magenta) is tangential to the slowest decaying artifact. These time points are marked by vertical lines that correspond to each stimulation site (for CP2 0.31 ms, C3 at 0.63 ms, CPz at 3.12 ms following the end of the TMS pulse). (I, J, K) Reconstructed data by subtraction of the artifact fits. Note that the noise in the form of spikes (arrows in I, J; also visible in E, F), possibly from the TMS stimulator, is reconstructed without distortion.
Fig 10.
Cranial muscle activation by TMS.
Cranial muscle activation experiments were performed with untreated skin (red traces) and, to assess artifact reconstruction, both punctured and exfoliated skin (blue traces) to downscale the TMS discharge artifacts. TMS stimulation was applied at F1 (A-D, subject 1, 32 electrodes) and C5 (E-H, subject 2, 64 electrodes). Sampling rate was 8 kHz, except for stimulation at F1 for untreated skin (C), where it was 16 kHz. The muscle artifacts have amplitudes of up to 4 mV and a duration of up to a few tens of milliseconds. Their biphasic dynamics fall within the first 7 ms (C, D) and 13 ms (G, H).
Fig 11.
Computationally reconstructed artifacts for given EEG data from a 64-electrodes cap recorded at 1024 Hz.
TMS application at FC1. Two electrodes with strong drifts were removed, all data average-referenced. (A) The artifact fits (colored) and original EEG traces (gray). (B) Log-log plot of A. (C) The same sample, still unfiltered, with the artifact fits subtracted. At this low sampling rate the sample points between time of stimulation (origin) and 7.8 ms afterwards could not be reconstructed and are therefore interpolated with splines (shaded in gray). The traces with strong 50 Hz noise (corresponding to 5 electrodes) are not shown (but see D). Note the two spiking recharging artifacts of the TMS stimulator occurring with an exact timing of 20 ms (bars) also found on the phantom head. (D) The five 50 Hz affected traces (blue, the remaining signals are grey). The subtraction of the artifact fit reconstructs the 50 Hz signal corroborating the performance of the reconstruction. Note that recharging artifacts of the TMS stimulator are aligned with the 50 Hz signal. (E) The full sample on a longer time scale high-pass filtered at 1 Hz. (F) Trial average over 99 trials. All trials were additionally low-pass at 300 Hz and notch-filtered at 50 Hz. The brain response to TMS stimulation manifests as TMS-evoked potentials continuing over several hundred milliseconds.
Fig 12.
TMS-evoked responses due to stimulation of the right motor cortex.
(A) Artifact fits (colored) to data (gray) of a single trial. (B) The fitted artifacts and data on a longer time scale. (C) Log-log plot of the artifact fits. (D) The same data with the artifact fits subtracted. The first 1.6 ms, for which data could not be reconstructed, was marked as gray area. (E-F) Electrodes Cz (red), C2 (green), C4 (blue) of the filtered single trial and of the average over 20 filtered trials (thin and thick lines, respectively). Filtering: 1 Hz high-pass, 50 Hz notch. TMS-evoked potentials as reported in the literature are marked at 15, 30, 45, 60, 100, and 180 ms. All data are average-referenced.