text:
Article| Volume 169, ISSUE 6, P1029-1041.e16, June 01, 2017

# Noninvasive Deep Brain Stimulation via Temporally Interfering Electric Fields

• Author Footnotes
16 These authors contributed equally
Open Access

## Highlights

• Noninvasive TI stimulation electrically stimulates neurons at depth selectively
• Neurons are stimulated by interference between multiple electric fields
• Neurons in mouse hippocampus can be stimulated without affecting the overlying cortex

## Summary

We report a noninvasive strategy for electrically stimulating neurons at depth. By delivering to the brain multiple electric fields at frequencies too high to recruit neural firing, but which differ by a frequency within the dynamic range of neural firing, we can electrically stimulate neurons throughout a region where interference between the multiple fields results in a prominent electric field envelope modulated at the difference frequency. We validated this temporal interference (TI) concept via modeling and physics experiments, and verified that neurons in the living mouse brain could follow the electric field envelope. We demonstrate the utility of TI stimulation by stimulating neurons in the hippocampus of living mice without recruiting neurons of the overlying cortex. Finally, we show that by altering the currents delivered to a set of immobile electrodes, we can steerably evoke different motor patterns in living mice.

## Introduction

Physical means of brain stimulation, such as the use of implanted electrodes for deep brain stimulation (DBS), have led to widespread excitement about the possibility of repairing neural dysfunction through direct control of brain circuit dynamics, including multiple FDA-approved therapies for previously intractable brain disorders (
• Greenberg B.D.
• Gabriels L.A.
• Malone Jr., D.A.
• Rezai A.R.
• Friehs G.M.
• Okun M.S.
• Shapira N.A.
• Foote K.D.
• Cosyns P.R.
• Kubu C.S.
• et al.
Deep brain stimulation of the ventral internal capsule/ventral striatum for obsessive-compulsive disorder: worldwide experience.
,
• Kalia S.K.
• Sankar T.
• Lozano A.M.
Deep brain stimulation for Parkinson’s disease and other movement disorders.
). Electrical stimulation via implanted electrodes sparsely activates distributed sets of neurons (
• Histed M.H.
• Bonin V.
• Reid R.C.
Direct activation of sparse, distributed populations of cortical neurons by electrical microstimulation.
), in a fashion different from direct optogenetic control of local cells (
• Mogri M.
• Thompson K.R.
• Henderson J.M.
• Deisseroth K.
Optical deconstruction of parkinsonian neural circuitry.
). The impact of electromagnetic stimulation on brain circuitry is an emergent function of the fields applied, the excitability properties of the neurons themselves, and the configuration of the neural network in which they are embedded (
• Merrill D.R.
• Bikson M.
• Jefferys J.G.R.
Electrical stimulation of excitable tissue: design of efficacious and safe protocols.
). As a result of this complexity, physical means of brain stimulation are often used in a phenomenological way, especially because the excitability properties of neurons vary across different cell types, and thus understanding how a given brain stimulation method impacts a given brain function may require analyzing many factors.
However, some properties of neurons are likely universal—for example, the intrinsic low-pass filtering of electrical signals by the neural membrane (
• Hutcheon B.
• Yarom Y.
Resonance, oscillation and the intrinsic frequency preferences of neurons.
), which prevents neural electrical activity from following very high-frequency oscillating (e.g., ≥ 1 kHz) electric fields. Here, we explore whether the biophysics underpinning such a potentially universal property might support novel strategies for electrical brain stimulation. In particular, if we apply high-frequency oscillating electric fields at multiple sites outside the brain, neurons in the brain will not be able to follow these high-frequency fields directly. However, if two such electric fields are applied at high frequencies that differ by a small amount, which corresponds to a low frequency that neurons can follow, neurons in the brain may be able to demodulate and follow the envelope modulation that results from the temporal interference between these two applied fields, and which oscillates at the difference frequency. If the amplitude of the envelope modulation reaches a maximum at a site deep in the brain, it might be possible to drive deep-lying neurons without recruiting overlying ones. We here test this concept, which we call temporal interference (TI) stimulation, by using computational modeling and phantom measurements, as well as electrophysiological measurements in vivo. We demonstrate the ability of TI stimulation to mediate activation of hippocampal neurons without recruiting overlying cortical neurons and steerably probe motor cortex functionality without physically moving electrodes by altering the current magnitudes delivered to a fixed set of electrodes.

## Results

### TI Stimulation: Concept and Validation of Neural Firing Recruitment

We first set out to examine whether the TI concept could indeed result in well-defined low-frequency envelope modulated electric fields. In the TI concept (Figure 1A), electric currents are applied at high frequencies $f1$ and $f2$ = $f1+Δf$ that fall outside the range of normal neural operation, but which differ by a small amount, $Δf$, that falls within the frequency range that neurons can respond to. The superposition of the two electric fields inside the brain results in an electric field at a frequency of $(f1+f2)/2$, whose envelope is modulated at the frequency $Δf$ (Figure 1B). The amplitude of the envelope modulation at a particular location depends on the vectorial sum of the two applied field vectors at that point and as a result can have a maximum at a point distant from the electrodes, potentially even deep in the brain (Figure 1C). The location of this envelope maximum depends on the electrode configuration, as well as properties of the applied waveforms. For the trapezoidal configuration shown in Figure 1A, the low-frequency envelope oscillates at a frequency of 40 Hz, with waveforms in Figure 1B plotted at the two specific points highlighted by Roman numerals in Figure 1A. For example, Figure 1Bi shows a large envelope modulation amplitude at a location where the two fields are large and aligned, whereas Figure 1Bii shows a small envelope modulation amplitude at a location where the two fields are less aligned.
To assess whether such low-frequency field envelopes could effectively drive neural spiking activity, we applied TI stimulation transcranially to anesthetized living mice, and recorded the responses by using automated whole-cell patch clamp neural recording. Currents were applied via two electrodes on the skull (with a ∼0.5 mm gap between their edges), and recordings were made in the somatosensory cortex. We found that interferential stimulation with two sinusoids at 2.01 kHz and 2 kHz, resulting in a $Δf$ envelope frequency of 10 Hz, was able to recruit neurons to fire at 10 Hz (Figure 1D), as efficaciously as direct 10 Hz stimulation (Figure 1E) that would be expected to broadly affect neural activity (
• Miranda P.C.
• Mekonnen A.
• Ruffini G.
The electric field in the cortex during transcranial current stimulation.
). High-frequency stimulation (with one sinusoid at 2 kHz and no TI) did not result in activity (Figure 1F), beyond a brief transient associated with the beginning of stimulation in some cells (n = 4 out of 6 cells from 2 mice) when 0.25 s sinusoidal ramp-up times were used. When 0.5 s ramp-up times were used, no such transient activity was observed in any cells (n = 5 cells from 2 mice), suggesting that the transient spiking activity observed earlier was due to the speed of the 0.25 s duration ramp-up (Figure 1J). We validated TI stimulation on a population of cortical cells (Figure 1Ii) and found that interferential stimulation with a difference frequency of 10 Hz resulted in spike frequencies of 10.21 ± 0.83 Hz (mean ± SD), for a 1 kHz carrier frequency (n = 6 cells from 2 mice) and 9.68 ± 0.85 Hz for a 2 kHz carrier frequency (n = 7 cells from 3 mice; see Table S1 for full statistics associated with Figure 1Ii).
To validate whether neuronal firing can be manipulated at different depths in tissue, we performed automatic patch clamp recording in the mouse hippocampus. Currents were applied via two electrodes that were located on the skull, with proximal edges 1.5–2 mm apart. We found that interferential stimulation (with two sinusoids at 2.01 kHz and 2 kHz, resulting in a $Δf$ envelope frequency of 10 Hz) was able to recruit neural firing in synchronization with the envelope—with either single spikes (n = 3 cells from 2 mice) or brief bursts of spikes (n = 5 cells from 3 mice; a burst was defined as a $<$ 50 ms spiking event with inter-spike interval $≤$ 15 ms; 1.3 ± 0.37 mean spikes per burst ± SD; 9.07 ± 3.2 ms inter-spike interval) elicited by the TI stimulation (in detail: mouse 1 had one cell with a burst response; mouse 2 had two cells with single spike responses; mouse 3 had two cells with a burst response; mouse 4 had two cells with a burst response and one cell with a single spike response) (Figure 1G). Direct application of high-frequency stimulation (with two sinusoids on the two electrodes, both at 2 kHz) did not result in activity (Figure 1H). No spiking transient was observed because we used the slower, 0.5 s duration ramp-up that we had previously observed to eliminate this transient (Figure 1J; n = 5 cells from 3 mice). We found (Figure 1Iii) that interferential stimulation with a difference frequency of 10 Hz resulted in spike or burst occurrence frequencies of 10.23 ± 0.61 Hz for a 2 kHz carrier frequency (n = 8 cells from 4 mice; see Table S1 for full statistics associated with Figure 1Iii). The timing of the spikes or the first spikes of bursts, relative to the peak of the TI envelope, was −2.8 ± 4.8 ms, i.e., when the envelope amplitude was >97% of its peak amplitude, which was not different from the timing of spikes evoked by 10 Hz stimulation relative to the 10 Hz sinusoid peak (−1.3 ± 2.2 ms; pairwise t test, p = 0.47).
The membrane potential of neurons undergoing TI stimulation repolarized between single spikes, or between brief bursts of spikes, to the baseline membrane potential (cortex, −10.36 ± 27.84 mV, mean difference from baseline ± SD; p = 0.74, pairwise t test; n = 13 cells from 5 mice; hippocampus, 5.5 ± 7.89 mV; p = 0.34; n = 8 cells from 4 mice). The spike frequency during the 20th bout of TI stimulation (tested in three cells in the somatosensory cortex from one mouse; 2 s stimulation followed by 2 s rest) was 9.93 ± 0.2 Hz (mean ± SD), not different from the spike frequency during the 1st bout (p = 0.95; pairwise t test), and the spike amplitude during the 20th bout of TI stimulation was not different from the spike amplitude during the first bout (5.3 ± 3.5 mV, mean amplitude difference ± SD; p = 0.75, pairwise t test); see Figures S1L–S1N for representative traces. The membrane potential of neurons undergoing high-frequency stimulation (with two sinusoids on the two electrodes, both at 2 kHz or at 1 kHz) in both the cortex and the hippocampus was not different from the baseline membrane potential before the stimulation (cortex, 1.67 ± 4.87 mV, mean difference from baseline ± SD, measured 1 s after stimulation onset; p = 0.66, pairwise t test; n = 11 cells from 5 mice; hippocampus, −1.7 ± 5.39 mV; p = 0.91; n = 5 cells from 3 mice).

### Validation of Steerability Using Computational Modeling and Tissue Phantom

To explore the effects of interferential stimulation at a physics level, we modeled the interferential electric field envelope magnitude as in Figure 1A but for a variety of electrode configurations, and we also experimentally assessed these fields by using a tissue phantom comprising a plastic cylinder filled with saline (Figure S2). Current sources were isolated in the circuitry as described in Figure S3. We found that by altering the locations of the electrodes and by setting the currents appropriately, we could enable the interferential envelope modulation to be targeted to specific locations. For example, when the electrodes were arranged in a trapezoidal geometry as in Figure 1A, but with a narrow base, we obtained, both in simulation and in the phantom, a peak of envelope modulation near the surface of the cylinder, at a point in between the two electrodes (Figure 2A), when both electrode pairs were conducting equal currents (1 mA in this example). By moving the electrodes that comprise the narrow base of the trapezoid further and further apart from each other, holding the currents constant, we could steer the location of the peak envelope modulation deeper into the tissue (Figure 2B), approaching the center of the cylinder as the trapezoid converged to a rectangle (Figure 2C). The envelope locus (i.e., distance out to 1/e of the envelope maximum) in Figure 2C was approximately two times larger, and the peak envelope amplitude ten times weaker, than in Figure 2A (see Table 1 for numerical values associated with these three panels). Thus, it is possible to steer the envelope peak to have its maximum at essentially any depth throughout a volume, albeit with a tradeoff between the locus depth and its width and strength. Varying the locations of electrodes causes large changes in steering, with electrode size variation playing a more minor role (Figures S2A–S2C). These analyses were conducted with a cylindrical phantom, but similar field distributions were obtained in simulations with a spherical phantom (Figures S2G–S2L).
Table 1Summary of Envelope Amplitude Attributes in Cylindrical Phantom
Figure 2 PanelElectrodes’ TrapezoidCurrent Ratio (Gray to Black)Envelope Modulation Amplitude $|EAMxˆ|max$Envelope Modulation Amplitude $|EAMyˆ|max$
Bottom BaseTop BasePeak (V/m) (Simulated, Measured)Normalized Peak Location (x,y)Normalized Half Width Half Maximum ($HWHMxˆ,$ $HWHMyˆ$)Peak (V/m) (Simulated, Measured)Normalized Peak Location (x,y)Normalized Half Width Half Maximum ($HWHMxˆ$, $HWHMyˆ$)
A0.391.961:1(3.7, 1)(0, −1)(0.18, 0.1)(2.5, 1.1)(0, −0.77)(0.18, 0.37)
B1.111.961:1(0.9, 0.75)(0, −0.87)(0.33, 0.25)(0.9, 0.7)(0, −0.37)(0.4, 0.52)
C1.961.961:1(0.13, 0.15)(0, −0.59)(0.37, 0.42)(0.21, 0.23)(0, 0)(0.42, 0.49)
D1.961.961:2.5(0.13, 0.14)(−0.2, −0.59)(0.37, 0.42)(0.2, 0.22)(−0.2, 0)(0.38, 0.4)
E1.961.961:4(0.12, 0.13)(−0.35, 0)(0.36, 0.42)(0.19, 0.2)(−0.35, 0)(0.37, 0.34)
Distances were normalized to the phantom’s radius, with (0, 0) at the center of the phantom. $|EAMxˆ|max$ and $|EAMyˆ|max$ are maximum values of the envelope modulation amplitude along $xˆ$ and $yˆ$ directions, respectively. $HWHMxˆ$ and $HWHMyˆ$ are normalized half width at half maximum along $xˆ$ and $yˆ$ directions, respectively. Current ratio is the ratio between the current applied by the gray electrode pair and the current applied by the black electrode pair (the scalar sum was set to 2 mA).
We next explored whether tuning the currents, while holding electrode locations constant, could be used to control the field envelope locus, and in particular to steer the envelope modulation peak away from the centroid of the electrode locations. We started by taking the electrode configuration of Figure 2C, with its rectangular geometry, and adjusting the ratio of currents across the gray versus black electrodes from 1:1 to 1:2.5 (Figure 2D) and 1:4 (Figure 2E). We found that by changing the current ratio between the electrode pairs, while keeping the current sum fixed, the peak envelope modulation became increasingly close to the electrode pair with the lower current, with the peak moving 20% of the radius away from the center in the 1:2.5 case (Figure 2D) and 35% of the radius away from the center in the 1:4 case (Figure 2E). This suggests the possibility of “live steering” of activity from one deep site to another within the brain, without having to physically move the electrodes themselves. By having a larger number of electrodes on the scalp, and tuning the current frequencies and amplitudes appropriately, it may be possible to make the deep targeted stimulation volume smaller, as we computationally model in Figures S2M and S2N.

### Stimulation of Mouse Hippocampus but Not Overlying Cortex

We next aimed to stimulate a deep structure (i.e., mouse hippocampus) while not recruiting overlying structures (i.e., the cortex). We performed simulations like those we did before, but now for the mouse brain, and predicted that 10 Hz transcranial stimulation applied to sites at the surface of the skull (Figure S4A) would broadly recruit neural activity throughout both superficial and deep structures. In contrast, performing TI stimulation with a 10 Hz difference frequency (Figure S4B) would, in our models, result in a peak of 10 Hz envelope modulation at a deep site, with lower envelope modulation amplitudes in more superficial structures. Of course, such models make assumptions about brain electrical parameters and geometry that may vary from brain to brain (
• Peratta C.
• Peratta A.
Modelling the human body exposure to elf electric fields.
), possess limited spatial resolution (our anatomical mouse model had x, y, and z resolutions of 42 μm, 42 μm, and 700 μm, respectively) and do not take into account differences in neural excitability across cell types and brain regions. Thus, to assess whether TI stimulation could indeed recruit activity in deep neural circuits without driving overlying ones, we compared 10 Hz versus TI stimulation in anesthetized mice, using the immediate early gene c-fos as a marker of neural activity, as has been used previously to gauge the focality of brain stimulation, since c-fos functions in the mouse cortex and hippocampus as an indicator of activated neurons (
• Chen R.
• Romero G.
• Christiansen M.G.
• Mohr A.
• Anikeeva P.
Wireless magnetothermal deep brain stimulation.
,
• Dragunow M.
• Robertson H.A.
Kindling stimulation induces c-fos protein(s) in granule cells of the rat dentate gyrus.
,
• Reijmers L.G.
• Perkins B.L.
• Matsuo N.
• Mayford M.
Localization of a stable neural correlate of associative memory.
).
As expected, 10 Hz transcranial stimulation (10 s on then 10 s off, for 20 min) resulted in widespread c-fos expression (measured 90 min after stimulation) in both the cortex and in the hippocampus underlying the electrodes (Figures 3A and 3B), with 13.6% ± 2.2% (mean ± SD used throughout) of cortical cells (as indicated by DAPI-stained nuclei) and 63.9% ± 5.7% of hippocampal cells c-fos-positive underneath the electrodes (Figure 3C). In contrast, there was essentially no c-fos activation on the contralateral side (see Table S2 for complete statistics for Figure 3). Driving the brain with a 2 kHz transcranial current (with the same current magnitude and durations as in the 10 Hz case) resulted in essentially no c-fos positive cells (Figures 3D–3F), in either the cortex or the hippocampus, and on either the electrode-bearing or contralateral side. In contrast, when TI stimulation was applied with frequencies of 2 kHz and 2.01 kHz (with the same current magnitude and duration as in the 10 Hz case), the hippocampus was strongly activated, Figures 3G and 3H, with c-fos in 53.12% ± 14.5% of DAPI-labeled cells (Figure 3I), not significantly different from that recruited by the 10 Hz stimulation (Figure 3C). But, despite the strong hippocampal recruitment, there was essentially zero c-fos in cortical cells—both at a site between the stimulating electrodes where the cortical envelope modulation field would be anticipated to be at its highest value in the cortex, with c-fos in 0.48% ± 0.47% of DAPI-labeled cells, and directly underneath an electrode, with c-fos in 0.32% ± 0.29% of DAPI-labeled cells (Figure 3I). Thus, TI stimulation can recruit neural activation in a deep structure such as the hippocampus without recruiting the overlying cortex. As a control experiment, we separated the electrodes by a larger distance, which would be expected to reduce the envelope modulation field amplitude (Figure S4C), and obtained no activation in the cortex or hippocampus (Figures S4D–S4F).
We did not observe seizures during or after any of these stimulation paradigms (i.e., 10 Hz, 2 kHz, or TI stimulation). Furthermore, c-fos staining was always observed only on the ipsilateral side of stimulation, and not in other analyzed regions, including below the hippocampus or in the contralateral hippocampus, consistent with a local (as opposed to propagating) neural activity profile. The high c-fos expression observed in the dentate gyrus is consistent with c-fos expression patterns observed in rats after strong, unilateral electrical stimulation of the hippocampus via an implanted electrode in animals treated with carbamazepine, which prevents seizures and also prevents bilateral c-fos staining upon unilateral stimulation (
• Dragunow M.
• Robertson H.A.
Kindling stimulation induces c-fos protein(s) in granule cells of the rat dentate gyrus.
).

### Safety Characterization of TI Stimulation

To characterize the safety profile of TI stimulation, we immunohistochemically examined cellular and synaptic molecular profiles in the cortex and the hippocampus after unilateral TI stimulation (2 kHz and 2.01 kHz, 10 s on then 10 s off, for 20 min), as in Figures 3G–3I, but in awake, behaving mice. Mice were sacrificed and transcardially perfused after a 24 hr recovery period to allow for detection of persistent effects (e.g., caspase-3 activation) after a bout of stimulation. Brain sections were fluorescently stained with antibodies for the neuronal marker NeuN (
• Wolf H.K.
• Buslei R.
• Schmidt-Kastner R.
• Schmidt-Kastner P.K.
• Pietsch T.
• Wiestler O.D.
• Blümcke I.
NeuN: a useful neuronal marker for diagnostic histopathology.
), the apoptotic marker cleaved caspase-3 (
• D’Amelio M.
• Sheng M.
• Cecconi F.
Caspase-3 in the central nervous system: beyond apoptosis.
), the DNA damage marker γH2AX (
• Mah L.J.
• El-Osta A.
• Karagiannis T.C.
gammaH2AX: a sensitive molecular marker of DNA damage and repair.
), the microglial marker Iba1 (
• Ito D.
• Imai Y.
• Ohsawa K.
• Nakajima K.
• Fukuuchi Y.
• Kohsaka S.
Microglia-specific localisation of a novel calcium binding protein, Iba1.
), the astrocyte marker GFAP (
• Eng L.F.
• Ghirnikar R.S.
• Lee Y.L.
Glial fibrillary acidic protein: GFAP-thirty-one years (1969-2000).
), and the synaptic protein synaptophysin (Syp) (
• Tarsa L.
• Goda Y.
Synaptophysin regulates activity-dependent synapse formation in cultured hippocampal neurons.
). We compared fluorescence profiles in the brain regions that were stimulated (Stim+) with fluorescence profiles in the contralateral, non-stimulated hemisphere (Stim−), as well as with fluorescence profiles in mice that underwent the same procedure but with current amplitudes set to 0 μA (Sham). We found that TI stimulation did not alter the neuronal density or affect the number of apoptotic cells (Figures 4A and 4B, Figures S5A and S5B, Figure S5J) or induce DNA damage (Figures 4C and 4D, Figures S5C and S5D, Figure S5K), at least as reflected by the stains above, relative to unstimulated or Sham stimulated brains. In addition, TI stimulation did not alter the intensity and density of Iba1 positive cells (Figures 4E and 4F, Figures S5G and S5H, Figure S5M) or GFAP-positive cells (Figures S5N and S5O), suggesting a lack of reactive microglia and astrocytes, respectively, in response to TI stimulation. Finally, TI stimulation did not alter synaptophysin intensity, suggesting no changes to synapse density (Figures 4G and 4H, Figures S5E and S5F, Figure S5L). See Table S3 for full statistics for these experiments, analyzed over cortical and hippocampal regions.
To assess whether high-frequency electric fields heat the brain, we measured brain temperature during stimulation with 2 kHz fields (60 s with 0.5 s ramp-up and ramp-down periods) that were applied via an electrode configuration as in Figures 4A–4H. We measured temperature with a 1-mm-diameter thermocouple probe inserted into the cortex underneath the lateral electrode. We found that the maximal temperature increase at this cortical location during stimulation was 0.069 ± 0.05 C° (mean ± SD increase from baseline temperature; n = 6 mice). This change in brain temperature was not larger than the largest spontaneous deviations from baseline seen during the pre- and post-stimulation periods (Figure 4I; p = 0.81, one-way ANOVA; see Table S3 for full statistics for Figure 4I).

### Steerable Probing of Motor Functionality without Electrode Movement

We next explored the capability of TI stimulation to activate neurons so as to drive movements. Using ketamine-xylazine anesthetized mice, we applied a current $I1$ via an electrode that was positioned on the skull above the motor cortex region associated with a movement of the contralateral forepaw and a current $I2$ via a second electrode that was positioned on the contralateral skull, above the motor cortex region associated with movement of the whiskers ipsilateral to the $I1$ electrode (Figure 5A) (
• Tennant K.A.
• Donlan N.A.
• Asay A.L.
• Thomas N.
• Kleim J.A.
• Jones T.A.
The organization of the forelimb representation of the C57BL/6 mouse motor cortex as defined by intracortical microstimulation and cytoarchitecture.
).
We first established the motor threshold by systematically increasing the current sum $I1+I2$ in steps of 50 μA while keeping the current ratio $I1:I2$ fixed at $1:4$. We found that if $I1$ and $I2$ were applied with a carrier frequency of 2 kHz and a difference frequency of 10 Hz, the stimulation evoked a 10 Hz periodic movement of the contralateral forepaw with a motor threshold of 916 ± 129 μA (Figure 5B; mean ± SD, n = 6 mice). If $I1$ and $I2$ were applied at the same frequency, no motor movement was observed (n = 9 mice; assessed up to a current sum value of 2 mA). Changing the difference frequency between 1 Hz and 15 Hz changed the motion frequency accordingly, but not the motor threshold (p = 0.88; one-way ANOVA; n = 6 mice; Figure 5B and Movie S1; see Table S4 for full statistics associated with Figure 5B). Increasing the carrier frequency from 1 kHz to 4 kHz linearly increased the motor threshold with a slope of 250 μA/kHz (linear regression, $R2=0.99$; Figure 5C and Movie S2; see Table S4 for full statistics associated with Figure 5C); a 5 kHz carrier resulted in no response at the maximum current sum value tested (2 mA).
We next sought to test whether steering the site of stimulation by changing the current ratio $I1:I2$, as in our physics experiments (Figures 2C–2E), would shift the site of motor cortex activation. We changed the current ratio $I1:I2$, keeping $I1+I2$ fixed, and measured movements evoked in the forepaws, whiskers, and ears (see Movie S3 for representative video). We found that when $I1 (that is, stimulation was steered toward the $I1$ electrode), TI stimulation evoked a movement of the forepaw (Figure 5Di, p = 0.00004, Fisher’s exact test, n = 9 mice) and the whiskers (Figure 5Ei, p = 0.002, n = 9 mice) contralateral to the $I1$ electrode, but no movements ipsilateral to the $I1$ electrode (Figures 5Di–5Fi; see Table S4 for full statistics associated with Figures 5Di–5Fi). In contrast, when $I1>I2$ (stimulation was steered toward the $I2$ electrode), TI stimulation evoked a movement of the whiskers (Figure 5Ei, p = 0.002, n = 9 mice) and ear (Figure 5Fi, p = 0.03, n = 9 mice) ipsilateral to the $I1$ electrode but no movements contralateral to the $I1$ electrode (Figures 5Di–5Fi).
The movement of the forepaw contralateral to the $I1$ electrode was maximal when the current ratio $I1:I2$ was $1:8$, i.e., our condition in which stimulation was maximally steered toward the $I1$ electrode (1.24 ± 0.36 mm, mean movement ± SD used throughout; p = 0.000007, unpaired t test versus null hypothesis of no movement, thresholding at p < 0.0025 Bonferroni corrected for multiple comparisons; n = 9 mice) and gradually decreased as the current ratio increased (Figure 5Dii). (Perhaps because the electrodes were not placed symmetrically, the movement of the forepaw ipsilateral to the $I1$ electrode, though not statistically significant, showed an opposite pattern, with a maximal movement when the current ratio was $8:1$, i.e., when stimulation was maximally steered toward the $I2$ electrode [Figure 5Dii, 0.2 ± 0.6 mm, p = 0.35, n = 9 mice].) The movement of the whiskers contralateral to the $I1$ electrode was maximal when the current ratio $I1:I2$ was $1:4$, i.e., stimulation was partially steered toward the $I1$ electrode (0.4 ± 0.41 mm, p = 0.018, n = 9 mice) and gradually decreased as the current ratio varied from this maximum (Figure 5Eii; the 1:8 condition steers the stimulation more laterally than the 1:4 condition, which perhaps stimulated cortical regions more lateral to the whisker region [
• Tennant K.A.
• Donlan N.A.
• Asay A.L.
• Thomas N.
• Kleim J.A.
• Jones T.A.
The organization of the forelimb representation of the C57BL/6 mouse motor cortex as defined by intracortical microstimulation and cytoarchitecture.
], and thus elicited a lower whisker movement amplitude than that elicited by the 1:4 condition). In contrast, perhaps due to the asymmetrical location of our electrodes, the movement of the whisker ipsilateral to the $I1$ electrode was maximal when the current ratio $I1:I2$ was $8:1$, i.e., stimulation was maximally steered toward the $I2$ electrode (0.96 ± 0.75 mm, p = 0.0016, n = 9 mice) and gradually decreased as the current ratio decreased (Figure 5Eii). A similar trend was seen for the ears (Figure 5Fii; see Table S4 for full statistics associated with Figures 5Dii–5Fii). Thus, TI stimulation can support steering of brain stimulation without physical electrode movement, resulting in tunable elicitation of movements.

## Discussion

In this paper, we present TI stimulation, validating the concept by using modeling as well as both physics and neurophysiology experiments, and demonstrate its utility by performing stimulation of a deep region (mouse hippocampus) without stimulating overlying neurons (cortex), as well as steerable brain stimulation of motor patterns without physical electrode movement. Future studies, perhaps using larger numbers of electrodes and multiple sets of interfering fields, may be able to pinpoint even smaller regions of the brain, or multiple regions of the brain. An open question is how small a focal volume may be achieved. At some point, inhomogeneities in the gray and white matter of the brain may cause difficulty in improving the resolution below that spatial scale, although MRI scans and data-driven sculpting of the electric fields may be able to compensate for this to some degree.
How generalizable might be the effects observed here? There have been reports that strong kHz-frequency electric fields can block the propagation of compound action potentials in peripheral nerves (
• Cuellar J.M.
• Alataris K.
• Walker A.
• Yeomans D.C.
• Antognini J.F.
Effect of high-frequency alternating current on spinal afferent nociceptive transmission.
,
• Kilgore K.L.
Reversible nerve conduction block using kilohertz frequency alternating current.
). Such effects were localized to the immediate vicinity of the electrodes, leaving regions a few millimeters away, perhaps where the magnitude of the fields was lower, unaffected. The magnitude of the fields used in our study to transcranially recruit neural activity in the brain were perhaps one to two orders of magnitude weaker than in these earlier studies, so we anticipate that such effects may have been minimal in our study. This is consistent with our repeatable observation of a lack of physiological effect of 2 kHz electric fields on brain activity. However, future studies might seek to explore how stronger kHz-frequency electric fields affect the brain. Such data might also present an upper limit to the field strengths applicable for TI stimulation. There has been a report of kHz-frequency transcranial electric field stimulation (1–5 kHz) that resulted in neural plasticity similar to that resulting from anodal DC stimulation (
• Chaieb L.
• Antal A.
• Paulus W.
Transcranial alternating current stimulation in the low kHz range increases motor cortex excitability.
), but a later report found that 5 kHz transcranial electric fields, grouped in theta burst patterns, did not result in neural plasticity (
• Kunz P.
• Antal A.
• Hewitt M.
• Neef A.
• Opitz A.
• Paulus W.
5 kHz transcranial alternating current stimulation: lack of cortical excitability changes when grouped in a theta burst pattern.
). In our current dataset, we did not observe effects of kHz-frequency electric fields beyond a brief transient that was observed when short ramp-up times were used (but not with longer ramp-up times), both suggesting that, in studies using kHz-frequency fields, subtle parameters of the stimulation may help determine exactly what effects on neural activity result and presenting an area for future exploration. We found that TI stimulation at amplitudes sufficient to recruit deep brain structures, such as the hippocampus, did not alter the neuronal and synaptic integrity of the underlying tissue 24 hr after stimulation, at least as reflected by the stains we used. Additional time points other than 24 hr post stimulation may provide, in the future, a more detailed picture of the safety of TI stimulation. Furthermore, the safety profile of TI stimulation associated with evoked behavior patterns, such as those utilized here, should be explored in the future.
Given the remarkable therapeutic benefits of DBS for patients with otherwise treatment-resistant movement and affective disorders (
• Kringelbach M.L.
• Jenkinson N.
• Owen S.L.
• Aziz T.Z.
Translational principles of deep brain stimulation.
), the prospects for noninvasive DBS using electricity are potentially exciting. Other methods for noninvasive DBS have been proposed, e.g., using transcranial ultrasound (
• Legon W.
• Sato T.F.
• Opitz A.
• Mueller J.
• Barbour A.
• Williams A.
• Tyler W.J.
Transcranial focused ultrasound modulates the activity of primary somatosensory cortex in humans.
) or using expression of heat-sensitive receptors and injection of thermomagnetic nanoparticles (
• Chen R.
• Romero G.
• Christiansen M.G.
• Mohr A.
• Anikeeva P.
Wireless magnetothermal deep brain stimulation.
), but the unknown mechanism of action (
• Plaksin M.
• Shoham S.
• Kimmel E.
Intramembrane cavitation as a predictive bio-piezoelectric mechanism for ultrasonic brain stimulation.
) and the need to genetically manipulate the brain, respectively, may limit their immediate use in humans. TI stimulation may thus represent a practical strategy for noninvasively stimulating neurons deep in the brain. It uses familiar and well-tested electric fields (
,
International Commission on Non-Ionizing Radiation Protection
Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz).
) and does not require chemical or genetic manipulation of the brain. We anticipate that it might rapidly be deployable into human clinical trials, as well as studies of the human brain.

## STAR★Methods

### Key Resources Table

Tabled 1
REAGENT or RESOURCESOURCEIDENTIFIER
Antibodies
c-FosSanta Cruz BiotechnologyCAT#:sc-52
NeuNSynaptic SystemsCAT#:266004
GFAPCell Signaling TechnologyCAT#:12389
Iba1Wako CehmicalsCAT#:019-19741
Synaptophysin, SVP-38Sigma-AldrichCAT#:S5768
Cleaved Caspase-3 (Asp175)Cell Signaling TechnologyCAT#:9664
γH2AX (anti-phospho-histone H2A.X)MilliporeCAT#:05-636
Alexa Fluor 488 secondary antibodyThermoFisher ScientificCAT#:A11008
Alexa Fluor 594 secondary antibodyThermoFisher ScientificCAT#:A11012
Alexa Fluor 647 secondary antibodyThermoFisher ScientificCAT#:A21244
DAPISigma-AldrichCAT#:10236276001
Chemicals, Peptides, and Recombinant Proteins
SignaGel conductive gelParkerLabsCAT#:15-25
ProLong Gold anti-fade reagentInvitrogen, ThermoFischer ScientificCAT#:P36930
Dental cementParkellCAT#:S380
Ten20 conductive pasteWeaver and CompanyCAT#:10-20-4T
Experimental Models: Organisms/Strains
Mouse: C57BL/6Taconic Biosciences; Jackson laboratoryCat#:C57BL/6NTac; 000664
Software and Algorithms
MATLABMathWorksRRID:SCR_001622
LabViewNational InstrumentsRRID:SCR_014325
pClampMolecular DevicesRRID:SCR_011323
Sim4LifeZurich MedTechhttps://www.zurichmedtech.com/sim4life/
Other
Polyimide tubingVention MedicalCAT#:141-0092
2” X 2” Re-Usable TENS/EMS Unit Electrode Pads with GelGurin ProductsCAT#:TE110-2x2WC-10
AutopatcherKodandaramaiah, Suhasa B., et al. Nature protocols 11.4 (2016): 634-654.N/A
Silver wire, diam. 0.25 mmSigma-AldrichCAT#:265578

### Contact for Reagent and Resource Sharing

Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Ed Boyden ( [email protected] ).

### Experimental Model and Subject Details

#### Mouse: C57BL/6

• Sex: Male.
• Age: 8–12 weeks old.
• Source: All animals were purchased from Taconic Biosciences.
• Housing and husbandry: Mice were housed in standard cages in the MIT animal facility with ad libitum food and water in a controlled light-dark cycle environment, with standard monitoring by veterinary staff.
• Allocation of animals to experimental groups: Randomly assigned.
• Committee approval: All animal procedures were approved by the Massachusetts Institute of Technology (MIT) Committee on Animal Care (CAC, Protocol Number: 1115-111-18), and all experiments conformed to the relevant regulatory standards.

#### Mouse: CK-p25

• Sex: Male.
• Age: 4 months old.
• Source: The CK-p25 transgenic mouse was created in the Tsai lab (
• Cruz J.C.
• Tseng H.C.
• Goldman J.A.
• Shih H.
• Tsai L.H.
Aberrant Cdk5 activation by p25 triggers pathological events leading to neurodegeneration and neurofibrillary tangles.
)
• Housing and husbandry: Mice were housed in standard cages in the MIT animal facility with ad libitum food and water in a controlled light-dark cycle environment, with standard monitoring by veterinary staff.
• Allocation of animals to experimental groups: Randomly assigned.
• Committee approval: All animal procedures were approved by the Massachusetts Institute of Technology (MIT) Committee on Animal Care, and all experiments conformed to the relevant regulatory standards.

### Method Details

#### Design and Implementation of the TI Stimulator

The kHz currents were generated using a custom made device consisting of two electrically isolated current sources. To isolate the channels, each waveform was supplied via a balanced pair of current sources that were driven in precisely opposite phase with a ground electrode carrying any imbalance currents (< 1%) from the paired current sources, preventing charging of the body relative to earth ground (Figure S3A). Each current source reliably drove 2 mA of current on 1 kΩ loads up to a frequency of 10 kHz with a resolution of 0.02Hz (Figures S3C and S3D). At load resistances higher than 10 kΩ the current source eventually saturated. The current output had a leakage level < 0.1 μA root mean square (RMS) at 100 kHz bandwidth, measured on a 1 kΩ load resistor with a differential amplifier (7A22, Tektronix). The total harmonic distortion of the current source was < 0.08% at 100 Hz and < 0.4% at 10 kHz (measured with 9 harmonics on a 1 kΩ load resistor). The total harmonic distortion and frequency cross-talk were measured using an FFT spectrum analyzer (SR770, Stanford Research). When the two current sources were applied to a common conductive load, e.g., a resistive bridge (Figure S3B) or a saline bath (Figure S3E), the cross-talk at the terminals of each channel was < 0.1%, allowing almost 100% of the interference to build up inside the load. In comparison, without the anti-phasic drive, approximately 30% cross-talk was measured at both the channel terminals and inside the conductive medium (Figure S3B, Figure S3F).

#### Surgical Procedures

On the day of the experiment, the mice were injected with Meloxicam (1mg/kg) and buprenorphine (0.1mg/kg) and anesthetized with 1%–2% (vol/vol) isoflurane in oxygen. Ophthalmic ointment (Puralube Vet Ointment, Dechra) was applied to the eyes. The scalp and the ventral torso were shaved and sterilized with Betadine and 70% ethanol. Two electrodes made of saline-filled polyimide tubes (Vention Medical Inc) with 1.5 mm outer diameter and 1.4 mm inner diameter or two electrodes made of adhesive electrogel with 1.5 mm diameter (SignaGel, ParkerLabs) were affixed to the skull (polyimide tubes were affixed using dental acrylic). During cortical recording, the positions of the skull electrodes relative to bregma were anteroposterior (AP) −1 mm, mediolateral (ML) −1.5 mm, and AP −1 mm, ML −2 mm; during hippocampus recording, their positions relative to bregma were: AP −2 mm, ML 0.25 mm, and AP −2 mm, ML −2.75 mm.

#### In Vivo Transcranial Stimulation

Transcranial stimulation was applied to anesthetized mice via the two skull electrodes, described above. Each skull electrode was paired with a cloth-base electrode (11 mm diameter conductive area; EL504, BioPac Inc) that was attached to the ventral torso with adhesive electrode gel (for the experiments of Figure 1D-F,Ii; SignaGel, Parker Laboratories Inc; 10-15 mm spacing between the edge of the conductive area of the torso electrodes) or with an adhesive electrode gel based electrode (SignaGel, ParkerLabs) on the cheeks (for the experiments of Figure 1G-H,Iii; approximately 11 mm diameter conductive area). Stimulation was applied for 1.5-2 s periods, with 0.25-0.5 s duration ramp-on periods and 0-0.5 s duration ramp-off periods, with at least a 2 s rest period between consecutive stimulations.

#### Whole-Cell Patch Clamp Recording

In vivo whole cell patching in current clamp mode (i.e., 0 pA holding current) was conducted in the cortex (depth of ∼500 μm below the dura) and CA1 layer of the hippocampus (depth of 1131 ± 157 μm below the dura) of anesthetized mice with an autopatcher (
• Kodandaramaiah S.B.
• Franzesi G.T.
• Chow B.Y.
• Boyden E.S.
• Forest C.R.
Automated whole-cell patch-clamp electrophysiology of neurons in vivo.
). Data were acquired using pClamp software (Molecular Devices) at a 400 kHz sampling rate. Patch electrodes were pulled from thin-walled borosilicate glass capillary tubing using a P-97 puller (Sutter Instruments). Tip electrode resistance was 4.6–7.4 MΩ in artificial cerebrospinal fluid (ACSF), containing 126 mM NaCl, 3 mM KCl, 1.25 mM NaH2PO4, 2 mM CaCl2, 2 mM MgSO4, 24 mM NaHCO3 and 10 mM glucose). The patch electrode solution consisted of (in mM) potassium gluconate 122.5, KCl 12.5, KOH-HEPES 10, KOH-EGTA 0.2, Mg-ATP 2, Na3-GTP 0.3, NaCl 8 (pH 7.35, mOsm 296), with 0.2–0.4 mg/ml biocytin added immediately before use. Capacitance, series resistance and input resistance were frequently measured throughout recording to monitor patch quality and cell health, using 10-pA hyperpolarization/depolarization square current pulses; a 300 pA ramp depolarization over 500 ms was used for AP generation.

#### Data Analysis

Data were analyzed using MATLAB (MathWorks). The recorded traces from each neuron were split into blocks corresponding to each trial within an experiment. Each block consisted of a single stimulation period of 1.5 −2 s duration with 1 s of baseline recorded before and after each stimulation period. To reduce stimulation artifacts for spike identification, traces were filtered using a 5th order Butterworth band-stop filter with cutoff frequencies of 1 kHz and 15 kHz and then with a 3rd order Butterworth high-pass filter with a cutoff frequency of 100 Hz (representative traces from the cortex are shown in the Supplemental Materials without filtering and after filtering with only the band-stop filter; representative traces from the hippocampus are shown after filtering with only the band-stop filter). Single spikes were identified using a running window average that picked out depolarizations of > 40 mV above baseline, which were “peaky” (that is, exhibited amplitudes larger than the average amplitudes of the nearest 3 data points before and after, by > 0.001 mV). Consecutive spikes with inter-spike interval $≤$ 15 ms, which occurred during a period of 50 ms or less, were defined as a spiking burst. Mean spiking frequency during stimulation periods (not including the ramping periods) was computed for each stimulation block and then averaged across neurons for each stimulation condition. Mean spontaneous firing rate was computed by a similar averaging of the firing rates across neurons, but for the 1 s interval before stimulation began. In the case of control 1 kHz or 2 kHz stimulation with no TI, we analyzed data from all complete blocks. Mean membrane potential was computed for a 500 ms period before the onset of 2 kHz or 1 kHz stimulation and was compared with a similar 500 ms period 1 s after stimulation onset, by dividing each period to 10 equally sized epochs and averaging across epochs. Overall, 18 neurons from 8 mice were analyzed with a minimal and maximal number of neurons per mouse of 1 and 4, respectively.

#### Surgical Procedure

On the day of the experiment, to reduce background fos staining, mice were kept undisturbed in the dark, with full access to food and water, for at least an hour after being transferred from the animal facility to the experimental procedure room. Mice were injected with Meloxicam (1mg/kg) and buprenorphine (0.1mg/kg) and anesthetized with 1%–2% (vol/vol) isoflurane in oxygen. The scalp and the ventral torso were shaved and sterilized with Betadine and 70% ethanol. Two cloth-base electrodes with 11 mm diameter conductive area (EL504, BioPac) were attached to the ventral torso with saline electrode gel (SignaGel, Parker Laboratories). The spacing between the edge of the two conductive areas was 10-15 mm. The mice were then placed in a custom stereotax, with ophthalmic ointment (Puralube Vet Ointment, Dechra) applied to the eyes, and with Betadine and 70% ethanol used to sterilize the surgical area. Two polyimide tubes (Vention Medical) with 1.5 mm outer diameter and 1.4 mm inner diameter were affixed to the skull using dental acrylic (C&B Metabond, Parkell). One polyimide tube was located at stereotactic coordinates (relative to bregma) of anteroposterior −2 mm, mediolateral, −0.25 mm. The second polyimide tube was located to the left of the first tube, with a gap between the edges of the electrodes of between 1.5 and 4 mm. Once the dental acrylic set (∼20 min), the mice were removed from the stereotactic apparatus and placed in a custom-built low profile holder. The polyimide tubes were filled with saline solution and a silver wire of 0.25 mm diameter (Sigma-Aldrich) was inserted into each tube for connection to the stimulator.

#### In Vivo Transcranial Stimulation

The cranial tube electrodes and the cloth-base ventral torso electrodes were connected to the stimulator so that each cranial tube electrode was paired with one cloth-base ventral torso electrode. The complex impedance between each pair of electrodes was established by applying short currents of low amplitude (10 Hz and 2 kHz, 10 μA, 0.5 s) and measuring the applied current and resultant voltage waveforms. When the resistance between a pair of electrodes was higher than 1 MΩ, a dental drill was used to thin the skull area inside polyimide tubes (high impedance often resulted from a layer of dental acrylic that leaked into the polyimide tubes and hardened on the skull in that area). 10 s intervals of electrical stimulation, with 0.25 s ramp-on and ramp-off periods, were applied interspersed with 10 s rest intervals over a 20 min experimental time course. Mice were sacrificed after 90 min to allow for c-fos expression.

#### Histology, Immunohistochemistry, and Imaging

Mice were deeply anesthetized with ketamine/xylazine and sacrificed, then transcardially perfused with cold phosphate-buffered saline (PBS) followed by cold 4% paraformaldehyde (PFA) in 1 x PBS. The brains were dissected and post-fixed in 4% PFA in PBS at 4°C overnight. Free-floating vibratome coronal sections (35 μm) were cut and incubated in a blocking solution containing 10% normal donkey serum, 0.2% Triton X-100, 3% bovine serum albumin, and 0.02% sodium azide in 1 x PBS for 2 hr at room temperature (RT). Sections were labeled with primary anti-c-Fos antibody (1:400; sc-52; Santa Cruz Biotechnology, USA) in the blocking solution at 4°C overnight, followed by the Alexa488-conjugated (1:1000; Invitrogen, ThermoFisher Scientific) secondary antibody for 1 hr at RT. Slices were incubated for 20min in 1 x DAPI dye (Invitrogen, ThermoFisher Scientific) in PBS at RT to label cell nuclei. Samples were then washed 4 × 15 min in PBS with 0.1% Triton X-100. Immunolabeled brain sections were mounted onto glass slides using ProLong Gold anti-fade reagent (Invitrogen, ThermoFisher Scientific, USA) and stored at −20°C. Images of the cortical and hippocampal areas from the stimulated and unstimulated sides of the brain were acquired using a high-resolution multi-channel (sequential) scanning confocal microscope (LSM 510, Zeiss, Germany), using a 10x air objective (NA 0.45).

#### c-fos Quantification

Greyscale images were analyzed using MATLAB (MathWorks). Each greyscale image was processed with contrast-limited adaptive histogram equalization (128 tiles per image) followed by Wiener adaptive noise-removal lowpass filtering (using 5x5 neighboring pixels to estimate the local image mean and SD). DAPI and GFP greyscale images were converted to black and white (BW) masks with global image thresholds established using Otsu’s method. BW masks were smoothed using a morphological disc kernel with a radius of 2 pixels. Masks were visually inspected and Otsu’s threshold was adjusted when required. Cells were quantified from the masked DAPI images. A cell was defined as a region with more than 20 and less than 100 connected pixels. Regions with less than 20 connected pixels were ignored. The number of cells in regions with more than 100 connected pixels was estimated by dividing the number of connected pixels by 100 - the maximal number of connected pixels defined per cell. DAPI cells expressing GFP were quantified from the corresponding GFP BW mask. A cell expressing GFP was defined as a DAPI cell region with connected GFP pixels. The percentage of c-fos expressing cells (a DAPI cell region with connected GFP pixels) was computed in 512 μm x 512 μm regions of interest in the cortex underneath the lateral electrode, the cortex between the electrodes, and the dentate gyrus region of the hippocampus, as well as the corresponding contralateral cortical and hippocampal regions.

#### Brain Slice Montages

GFP greyscale images were converted to RGB images. Images from a single brain slice were visually rearranged and overlapped to form a montage of the imaged slice.

#### Surgical Procedure And Animal Habituation

On the day of the surgery, the mice were injected with Meloxicam (1mg/kg) and buprenorphine (0.1mg/kg) and were anesthetized with 1%–2% (vol/vol) isoflurane in oxygen and placed in a stereotactic frame. The scalp was shaved, ophthalmic ointment (Puralube Vet Ointment, Dechra) was applied to the eyes, and Betadine and 70% ethanol were used to sterilize the surgical area. The scalp was opened and a custom stainless steel headplate was affixed to the skull using dental acrylic (C&B Metabond, Parkell), and the mice were then recovered. Headfixed awake mice were then habituated to restraint for three consecutive days for 15, 30 and 45 min respectively. During this time the animals were placed in a cylinder-shaped tube. Animals were rewarded with sweetened condensed milk (diluted 1:2 in water) every 5-10 min during habituation.

#### In Vivo Transcranial Stimulation

On the day of stimulation, part of the dental cement was removed to enable cranial electrode placement. The stimulation procedure was as described in the ‘In-vivo stimulation followed by c-Fos staining’ section, but electrodes were made of conductive paste (Ten20, Weaver and Company) instead of polyimide tubes filled with saline. The center of one cranial electrode was placed at a point at the midline and −1.5 mm anteroposterior from bregma, and the center of the second cranial electrode was located laterally at +2 mm mediolateral from bregma and at the same anteroposterior coordinate. Each cranial electrode was paired with a larger (approximately 8 mm diameter) electrode that was made of the same conductive paste and was located on the ipsilateral cheek. The complex impedance between each pair of electrodes was established by applying short currents of low amplitude (10 Hz and 2 kHz, 10 μA, 0.5 s) and measuring the applied current and observed voltage waveforms. The stimulation protocol comprised 10 s intervals of electrical stimulation with 0.25 s ramp-on and ramp-off periods, with 10 s rest intervals in between, over a 20 min period. The lateral electrode was driven at a frequency of 2 kHz and current amplitude of 125 μA and the medial electrode was driven at a frequency of 2.01 kHz and a current amplitude of 125 μA. In the case of Sham stimulation, the amplitude of both currents was set to 0 μA; the rest of the procedure was identical.

#### Histology, immunohistochemistry, and imaging

Mice were sacrificed and perfused (cold 4% paraformaldehyde in 1 x PBS) 24 hr later to assess cellular and synaptic integrity by labeling for the neuronal marker NeuN (1:1000, Synaptic Systems, #266004), astrocyte marker GFAP (1:500, Cell Signaling Technology, #12389), microglia marker Iba1 (1:500, Wako Cehmicals, #019-19741), presynaptic marker synaptophysin (1:500, SVP-38, Sigma, #S5768), apoptosis marker cleaved caspase-3 (1:250, Cell Signaling Technology, #9664) and the DNA damage marker γH2AX (1:500. anti-phospho-histone H2A.X, Millipore, #05-636). For immunostaining, 40 μm sections were incubated with blocking buffer (5% normal donkey serum and 0.3% Triton X-100 in PBS) for 1 hr. Primary antibodies were diluted in blocking buffer and incubated with the sections overnight at 4°C. Primary antibodies were visualized using the appropriate secondary antibody conjugates (Alexa Fluor 488, Alexa Fluor 594 and Alexa Fluor 647, ThermoFisher Scientific). We used the CK-p25 transgenic mouse, an established mouse model of neurodegeneration, which exhibits neuronal atrophy, reduced synaptic density and pronounced DNA damage (
• Cruz J.C.
• Tseng H.C.
• Goldman J.A.
• Shih H.
• Tsai L.H.
Aberrant Cdk5 activation by p25 triggers pathological events leading to neurodegeneration and neurofibrillary tangles.
,
• Dobbin M.M.
• Pan L.
• Chen Y.
• Kim D.
• Gao J.
• Ahanonu B.
• Pao P.C.
• Qiu Y.
• Zhao Y.
• Tsai L.H.
SIRT1 collaborates with ATM and HDAC1 to maintain genomic stability in neurons.
,
• Kim D.
• Frank C.L.
• Dobbin M.M.
• Tsunemoto R.K.
• Tu W.
• Peng P.L.
• Guan J.S.
• Lee B.H.
• Moy L.Y.
• Giusti P.
• et al.
Deregulation of HDAC1 by p25/Cdk5 in neurotoxicity.
), as a positive staining control for yH2AX antibody staining. The samples were then washed, stained with DAPI (Sigma, #10236276001) and mounted onto glass slides. Images were acquired on a Zeiss LSM710 laser-scanning confocal microscope using 20x and 40x air objectives, and subsequently analyzed in ImageJ.

#### Surgical Procedure

Surgical procedures were as described in the ‘In-vivo stimulation followed by c-Fos staining’ section.

#### In Vivo Transcranial Stimulation

Stimulation currents ($I1$, 2 kHz, 500 μA; $I2$, 2 kHz, 500 μA) were simultaneously applied with 0.5 ramp-up and ramp-down periods via cranial electrodes that were configured on the skull as described in the ‘In-vivo stimulation in awake mice followed by staining for cell and synapse markers’ section.

#### Intracranial Temperature Measurement

A 1 mm diameter thermocouple (type “K” dual 36 Gauge with Teflon insulation. 36 TT-K-36, OMEGA Engineering) was inserted to the brain underneath the lateral electrode via a 2 mm diameter craniotomy, and continuous temperature measurements were obtained with a temperature logger (NI USB TC01, National Instruments) during 60 s of stimulation as described above, as well as during 30 s periods before and after stimulation.

#### Surgical Procedure and In Vivo Transcranial Stimulation

On the day of the experiment, mice were anesthetized with 100 mg/kg ketamine and 10 mg/kg xylazine. Ophthalmic ointment (Puralube Vet Ointment, Dechra) was applied to the eyes. The scalp was shaved and sterilized with Betadine and 70% ethanol and the mice were headfixed. Two electrically isolated currents $I1$ and $I2$ were applied transcranially via electrodes that were made of conductive paste (1 mm diameter; Ten20, Weaver and Company) and connected to the stimulator via thin silver wires. Current $I1$ was applied via the skull electrode that was located at coordinates relative to bregma AP −1.5 mm, ML +2 mm (n = 5 mice) or ML −2 mm (n = 4 mice). Current $I2$ was applied via the skull electrode that was located 2.5 mm laterally to the $I1$ electrode (distance between centers of electrodes). Both skull electrodes were paired with a 5-8 mm diameter electrode, made of adhesive electrode gel (SignaGel, ParkerLabs), that was attached to the ipsilateral cheek. Stimulation blocks comprised a 0.5 s ramp-up period, a 6 s stimulation period and a 0.5 s ramp-down period. There was approximately 10 s interval between consecutive stimulation blocks.

#### Recording of Motor Activity

Motor activities were recorded using a video camera (Fujinon, YV10x5B-2, 1:1 3/5-50mm 1/3” CS), over a period lasting from 1.5 s pre-stimulation until 1.5 s post-stimulation. The period of stimulation was indicated to the camera with a green LED that was positioned behind a post to avoid a direct illumination of the animal eye.

#### Data Analysis

Motor activities were analyzed offline using MATLAB (MathWorks). Movements of the forepaws and whiskers were measured with the help of an in-frame ruler. In the case of whisker movement, we analyzed movements of the whisker that showed the maximal periodic movement amplitude during stimulation. Movements of the ears were scored by three independent researchers who were blind to the stimulation conditions, per the following scale: 0, no movement; 1, weak movement; 2, strong movement; 3, very strong movement. Prior to scoring movements, the researchers were shown one example video with a weak movement and one example video with a very strong movement, to help calibrate their numerical judgments (these training videos were not included in the analysis).

#### Phantom Construction

A phantom was constructed from a 50 mm diameter petri dish. We mounted sixteen 1 mm diameter silver wire electrodes with equal spacing along the circumference of the phantom (i.e., an interelectrode spacing of 9.8 mm). The electrodes were connected to a 24-channel adaptor box that was connected to the TI stimulator. The phantom was filled with sodium chloride solution. The salt concentration was adjusted until an inter-electrode impedance of 3 kOhm was measured between two opposite electrodes.

#### Electric Field Measurement

The electric field was measured using two orthogonal 3.6 mm-spaced dipole electrodes constructed from medical stainless steel needle electrodes. The location of the probe was adjusted across an 36 mm × 36 mm matrix with 6 mm steps using two large range motorized linear stages (Compumotor NEMA 23 types Model S57-51-MO, Parker Hannifin Corporation). The signal from each dipole electrode was fed into two separate custom built ultra-high input impedance differential electrometer type buffer amplifiers and then the outputs of these amplifiers were differentially fed into lock-in amplifiers (SR830, Stanford Research Systems) before readout with a digital multimeter (3457A, Hewlett Packard). The movements of the probe and the readouts of the measurements from the digital multimeter were controlled by a Labview script. The measurements at each location were averaged several times to reduce noise.

#### Post-processing

For 2D electric field maps, measurement points were linearly interpolated (with interpolation factor 2) using MATLAB’s interp2 function.

#### Electromagnetic Field Computation

Electromagnetic simulations were performed on the Sim4Life platform (Zurich MedTech AG) using the ‘stationary current’ solver - a real valued quasi-electrostatic finite element method (FEM) solver for the ohmic current dominated regime. The simulation solved the equation $∇σ∇ϕ=0$, where $σ$ is the local electrical conductivity and $ϕ$ is the electric potential from which the electric field and the current density can be obtained as $E→=−∇ϕ$ and $j→=σ⋅E→$ respectively. The solver is suitable for the frequencies used in this paper, as displacement currents can be neglected compared to ohmic ones. The solver discretizes the model using adaptive, rectilinear meshes. Tissue properties have been assigned according to the IT’IS Foundation database of tissue properties (

Hasgall, P.A., Di Gennaro, F., Baumgartner, C., Neufeld, E., Gosselin, M.C., Payne, D., Klingenböck, A., and Kuster, N. (2015). IT’IS database for thermal and electromagnetic parameters of biological tissues, version 3.0. doi: 10.13099/VIP21000-03-0, https://www.itis.ethz.ch/virtual-population/tissue-properties/overview/.

). The same conductivity values were used at all frequencies. The stimulating currents were normalized by integrating the normal current density over a surface surrounding an electrode. The simulations were performed with Dirichlet boundary conditions at active electrodes. The spatial distribution of the envelope modulation amplitude caused by temporal interference was computed from the fields of both electrode pairs using $|E→AM(n,→r→)|=||(E→1(r→)+E→2(r→))⋅n→|−|(E→1(r→)−E→2(r→))⋅n→||$ where $E→1(r→)$ and $E→2(r→)$ are the fields generated by the first and second electrode pair, respectively, at the location $r→(x,y,z)$ and $n→$ is an unit vector along the direction of interest (e.g., normal to the surface). The maximal envelope modulation amplitude along any orientation which results from the vector fields $E→1(r→)$ and $E→2(r→)$ at location $r→(x,y,z)$ was computed. Assuming without loss of generality (as the numbering of the channels can be swapped and the sign of $E→2$ can be inverted) that $|E→1|>|E→2|$ and that the angle $α$ (angle between $E→1$ and $E→a$) is smaller than $(π/2)$, the maximal modulation amplitude is obtained using:

#### Phantom Model

The homogeneous phantom model consisted of a saline medium with a conductivity of $σ=0.333S/m$. The inhomogeneous (‘4-layer’) model consisted of scalp ($d=0.05R$, $σ=0.333S/m$), skull ($d=0.085R$, $σ=0.0083S/m$), cerebrospinal fluid ($d=0.023R$, $σ=1.79S/m$) and brain ($d=0.83R$, $σ=0.333S/m$) layers, where $d$ is the layer thickness normalized to the overall sphere’s radius $R$.

#### Mouse Model

A computational mouse model (IT’IS Foundation, Male OF1 Mouse) was developed by segmentation of a male OF1 Mouse, weighing 35.5 g, according to the method described in (
• Kainz W.
• Nikoloski N.
• Oesch W.
• Berdiñas-Torres V.
• Fröhlich J.
• Neubauer G.
• Kuster N.
Development of novel whole-body exposure setups for rats providing high efficiency, National Toxicology Program (NTP) compatibility and well-characterized exposure.
). The resolution of the model in the x, y, z directions was 42 μm, 42 μm and 700 μm, respectively. The model did not include CSF. The model was fitted with two small cranial electrodes with outer radius of 0.5 mm and inner radius of 0.17 mm filled with saline $(σ=0.333S/m)$ and two large surface electrodes of radius 2.2 mm with conductive gel $(σ=1.79S/m)$ on the ventral torso. The grid resolution in the electrode vicinity was high (0.04 mm) to properly resolve field gradients, while the coarsest grid step in the exposed area was in the order of 0.12 mm.

### Quantification and Statistical Analysis

The reasoning behind our sample sizes is not based upon a power analysis, since the goal was to create a new technology. As noted in (
• Dell R.B.
• Holleran S.
• Ramakrishnan R.
Sample size determination.
), “In experiments based on the success or failure of a desired goal, the number of animals required is difficult to estimate...” As noted in the aforementioned paper, “The number of animals required is usually estimated by experience instead of by any formal statistical calculation, although the procedures will be terminated [when the goal is achieved].” These numbers reflect our past experience in developing neurotechnologies.

#### Definition of center and dispersion

Spike firing rate (Figure 1Ii), spike firing or bursting rate (Figure 1Iii): shown values are mean ± SD.

#### Definition of n

Number of cells.

#### Statistical test and definition of significance

Significance (p < 0.05) was characterized by one-way ANOVA followed by post hoc test with Bonferroni correction for multiple comparisons.

#### Randomization strategy

The order of stimulation conditions (i.e., 10 Hz stimulation, 2 kHz stimulation, and TI stimulation) was randomized.

#### Inclusion/exclusion of data

We included all cells that responded to a control 10 Hz stimulation apart from one neuron that was excluded from the hippocampus analysis due to an unphysiologically high rest potential of ∼−37 mV.
Statistical details can be found in the legends of Figure 1Ii and Figure 1Iii, the Results section TI Stimulation: Concept and Validation of Neural Firing Recruitment, and Table S1.

#### Definition of Center and Dispersion

Percentage of c-fos expressing cells (a DAPI cell region with connected GFP pixels) was computed in 512 μm x 512 μm regions of interest: shown values are mean values ± SD.

#### Definition of n

Number of animals.

#### Statistical test and definition of significance

Significance (p < 0.05) was characterized by one-way ANOVA followed by post hoc test with Bonferroni correction for multiple comparisons.

#### Randomization Strategy

Each animal was exposed to a single stimulation condition (i.e., 10 Hz stimulation, 2 kHz stimulation, TI stimulation, or TI stimulation with a large inter-electrode distance).

#### Inclusion/Exclusion of Data

We included all animals that underwent the stimulation procedure.
Statistical details can be found in the legends of Figure 3 and Figure S4, the Results section Stimulation of Mouse Hippocampus but Not Overlying Cortex, and Table S2.

#### Definition of Center and Dispersion

Immunohistochemical: shown values are mean ± SEM.

#### Definition of n

Number of brain sections.

#### Statistical Test and Definition of Significance

Significance (p < 0.05) characterized by one-way ANOVA.

#### Randomization Strategy

Each animal was exposed to a single stimulation condition (i.e., TI stimulation or Sham).

#### Inclusion/Exclusion of Data

We included all animals that underwent the stimulation procedure.
Statistical details can be found in the legends of Figure 4, Figure S5, the Results section Safety Characterization of TI Stimulation, and in Table S3. In-vivo temperature measurement (related to Figure 4I).

#### Definition of Center and Dispersion

Instantaneous change in brain temperature from baseline: shown values are mean ± SD.

#### Definition of n

Number of animals.

#### Statistical test and definition of significance

Significance (p < 0.05) characterized by one-way ANOVA.

N/A.

#### Inclusion/exclusion of data

We included all animals that underwent the stimulation procedure.
Statistical details can be found in the legend of Figure 4I.

#### Definition of center and dispersion

Motor threshold (Figures 5B and 5C) shown values are mean ± SD; number of responsive animals (Figure 5 panels Di, Ei, and Fi); evoked movements (Figure 5 panels Dii, Eii, and Fii) shown values are mean ± SEM.

#### Definition of n

Number of animals.

#### Statistical test and definition of significance

Significance (p < 0.05) was characterized by one-way ANOVA followed by post hoc test with Bonferroni correction for multiple comparisons.

#### Randomization strategy

The order of stimulation conditions (i.e., TI stimulation at different difference frequencies, Figure 5B; TI stimulation at different carrier frequencies, Figure 5C; TI stimulation at different current ratios, Figure 5 panels Dii, Eii, and Fii) was randomized.

#### Inclusion/exclusion of data

We included all animals that underwent the stimulation procedure.
Statistical details can be found in the legend of Figure 5, the Results section Steerable Probing of Motor Functionality without Electrode Movement, and in Table S4.
All the statistical analyses in this manuscript were performed by using MATLAB statistics toolbox (Mathworks).

## Author Contributions

N.G. designed the TI stimulator device, simulations, and experiments and wrote the paper. D.B. designed and implemented the TI stimulator. N.G., S.B.K. and H.J.S. designed and conducted in-vivo rodent stimulation and electrophysiology experiments. N.G., S.B. K., and A.R. designed and conducted in-vivo rodent stimulation experiments followed by c-fos staining. N.G. and N.D. designed and conducted in-vivo rodent stimulation experiments with evoke motor activities. H.J.S. and N.D. designed and conducted awake in-vivo rodent stimulation experiments, and N.D. followed the experiments with staining and imaging for cell and synapses markers. A.M.C. and E.N designed, developed, and conducted electromagnetic FEM simulations in phantom and mouse models. N.K., L.H.T., and A.P.L. oversaw and designed experiments. E.S.B designed devices and experiments, wrote the paper, and oversaw experiments.

## Acknowledgments

E.S.B. was funded by NIH Director’s Pioneer Award 1DP1NS087724 and NIH Director’s Transformative Research Award 1R01MH103910-01 , the New York Stem Cell Foundation-Robertson Investigator Award , the MIT Center for Brains, Minds, and Machines ( NSF CCF-1231216 ), Jeremy and Joyce Wertheimer , Google , NSF CAREER Award CBET 1053233 , the MIT Synthetic Intelligence Project , the MIT Media Lab , the MIT McGovern Institute , and the MIT Neurotechnology Fund . N.G. was funded by a Wellcome Trust MIT fellowship ( 097443/Z/11/Z ). L.H.-T. was funded by NIH grants 1RF1AG047661 and NS051874 , the Belfer Neurodegeneration Consortium , the Glenn Foundation for Aging Research , the Alana Foundation , Cure Alzheimer's Fund , and the JBP Foundation . A.R. was funded by a NARSAD grant. A.P.L. was partly supported by the  Sidney R. Baer Jr. Foundation , the NIH ( R01MH100186 , R01HD069776 , R01NS073601 , R21 NS082870 , R21 MH099196 , R21 NS085491 , R21 HD07616 ), the Football Players Health Study at Harvard University , and Harvard Catalyst | The Harvard Clinical and Translational Science Center (NCRR and the NCATS NIH, UL1 RR025758 ). We acknowledge, for helpful discussions, A.C. Singer. The content is solely the responsibility of the authors and does not necessarily represent the official views of Harvard Catalyst, Harvard University or its affiliated academic health care centers, MIT, the NIH, or the Sidney R. Baer Jr. Foundation. N.G., E.S.B., and D.B. have applied for a patent on the technology, assigned to MIT. N.G. is a founder of a medical device company that licensed the patent from MIT. A.P.L. serves on the scientific advisory boards for Nexstim, Neuronix, Starlab Neuroscience, Neuroelectrics, Axilum Robotics, Magstim, and Neosync and is listed as an inventor on several issued and pending patents on the real-time integration of transcranial magnetic stimulation with electroencephalography and magnetic resonance imaging.

## Supplemental Information

• Table S1. Interferential Neural Activation in Intact Mouse Brain, Related to Figure 1
• Table S2. Summary of Group Mean c-fos Expression Fractions: % of DAPI-positive cells, Related to Figure 3
• Table S3. Safety and Temperature Assessment, in Awake Mice, Related to Figure 4
• Table S4. Interferentially Evoked Motor Activity, Related to Figure 5

## References

• Chaieb L.
• Antal A.
• Paulus W.
Transcranial alternating current stimulation in the low kHz range increases motor cortex excitability.
Restor. Neurol. Neurosci. 2011; 29: 167-175
• Chen R.
• Romero G.
• Christiansen M.G.
• Mohr A.
• Anikeeva P.
Wireless magnetothermal deep brain stimulation.
Science. 2015; 347: 1477-1480
• Cruz J.C.
• Tseng H.C.
• Goldman J.A.
• Shih H.
• Tsai L.H.
Aberrant Cdk5 activation by p25 triggers pathological events leading to neurodegeneration and neurofibrillary tangles.
Neuron. 2003; 40: 471-483
• Cuellar J.M.
• Alataris K.
• Walker A.
• Yeomans D.C.
• Antognini J.F.
Effect of high-frequency alternating current on spinal afferent nociceptive transmission.
Neuromodulation. 2013; 16 (discussion 327): 318-327
• D’Amelio M.
• Sheng M.
• Cecconi F.
Caspase-3 in the central nervous system: beyond apoptosis.
Trends Neurosci. 2012; 35: 700-709
• Dell R.B.
• Holleran S.
• Ramakrishnan R.
Sample size determination.
ILAR J. 2002; 43: 207-213
• Dobbin M.M.
• Pan L.
• Chen Y.
• Kim D.
• Gao J.
• Ahanonu B.
• Pao P.C.
• Qiu Y.
• Zhao Y.
• Tsai L.H.
SIRT1 collaborates with ATM and HDAC1 to maintain genomic stability in neurons.
Nat. Neurosci. 2013; 16: 1008-1015
• Dragunow M.
• Robertson H.A.
Kindling stimulation induces c-fos protein(s) in granule cells of the rat dentate gyrus.
Nature. 1987; 329: 441-442
• Eng L.F.
• Ghirnikar R.S.
• Lee Y.L.
Glial fibrillary acidic protein: GFAP-thirty-one years (1969-2000).
Neurochem. Res. 2000; 25: 1439-1451
• Mogri M.
• Thompson K.R.
• Henderson J.M.
• Deisseroth K.
Optical deconstruction of parkinsonian neural circuitry.
Science. 2009; 324: 354-359
• Greenberg B.D.
• Gabriels L.A.
• Malone Jr., D.A.
• Rezai A.R.
• Friehs G.M.
• Okun M.S.
• Shapira N.A.
• Foote K.D.
• Cosyns P.R.
• Kubu C.S.
• et al.
Deep brain stimulation of the ventral internal capsule/ventral striatum for obsessive-compulsive disorder: worldwide experience.
Mol. Psychiatry. 2010; 15: 64-79
1. Hasgall, P.A., Di Gennaro, F., Baumgartner, C., Neufeld, E., Gosselin, M.C., Payne, D., Klingenböck, A., and Kuster, N. (2015). IT’IS database for thermal and electromagnetic parameters of biological tissues, version 3.0. doi: 10.13099/VIP21000-03-0, https://www.itis.ethz.ch/virtual-population/tissue-properties/overview/.

• Histed M.H.
• Bonin V.
• Reid R.C.
Direct activation of sparse, distributed populations of cortical neurons by electrical microstimulation.
Neuron. 2009; 63: 508-522
• Hutcheon B.
• Yarom Y.
Resonance, oscillation and the intrinsic frequency preferences of neurons.
Trends Neurosci. 2000; 23: 216-222
• International Commission on Non-Ionizing Radiation Protection
Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz).
Health Phys. 2010; 99: 818-836
• Ito D.
• Imai Y.
• Ohsawa K.
• Nakajima K.
• Fukuuchi Y.
• Kohsaka S.
Microglia-specific localisation of a novel calcium binding protein, Iba1.
Brain Res. Mol. Brain Res. 1998; 57: 1-9
• Kainz W.
• Nikoloski N.
• Oesch W.
• Berdiñas-Torres V.
• Fröhlich J.
• Neubauer G.
• Kuster N.
Development of novel whole-body exposure setups for rats providing high efficiency, National Toxicology Program (NTP) compatibility and well-characterized exposure.
Phys. Med. Biol. 2006; 51: 5211-5229
• Kalia S.K.
• Sankar T.
• Lozano A.M.
Deep brain stimulation for Parkinson’s disease and other movement disorders.
Curr. Opin. Neurol. 2013; 26: 374-380
• Kilgore K.L.
Reversible nerve conduction block using kilohertz frequency alternating current.
Neuromodulation. 2014; 17 (discussion 254–255): 242-254
• Kim D.
• Frank C.L.
• Dobbin M.M.
• Tsunemoto R.K.
• Tu W.
• Peng P.L.
• Guan J.S.
• Lee B.H.
• Moy L.Y.
• Giusti P.
• et al.
Deregulation of HDAC1 by p25/Cdk5 in neurotoxicity.
Neuron. 2008; 60: 803-817
• Kodandaramaiah S.B.
• Franzesi G.T.
• Chow B.Y.
• Boyden E.S.
• Forest C.R.
Automated whole-cell patch-clamp electrophysiology of neurons in vivo.
Nat. Methods. 2012; 9: 585-587
• Kringelbach M.L.
• Jenkinson N.
• Owen S.L.
• Aziz T.Z.
Translational principles of deep brain stimulation.
Nat. Rev. Neurosci. 2007; 8: 623-635
• Kunz P.
• Antal A.
• Hewitt M.
• Neef A.
• Opitz A.
• Paulus W.
5 kHz transcranial alternating current stimulation: lack of cortical excitability changes when grouped in a theta burst pattern.
Front. Hum. Neurosci. 2017; 10
• Legon W.
• Sato T.F.
• Opitz A.
• Mueller J.
• Barbour A.
• Williams A.
• Tyler W.J.
Transcranial focused ultrasound modulates the activity of primary somatosensory cortex in humans.
Nat. Neurosci. 2014; 17: 322-329
• Mah L.J.
• El-Osta A.
• Karagiannis T.C.
gammaH2AX: a sensitive molecular marker of DNA damage and repair.
Leukemia. 2010; 24: 679-686
• Merrill D.R.
• Bikson M.
• Jefferys J.G.R.
Electrical stimulation of excitable tissue: design of efficacious and safe protocols.
J. Neurosci. Methods. 2005; 141: 171-198
• Miranda P.C.
• Mekonnen A.
• Ruffini G.
The electric field in the cortex during transcranial current stimulation.
Neuroimage. 2013; 70: 48-58
• Peratta C.
• Peratta A.
Modelling the human body exposure to elf electric fields.
WIT Press, 2010
• Plaksin M.
• Shoham S.
• Kimmel E.
Intramembrane cavitation as a predictive bio-piezoelectric mechanism for ultrasonic brain stimulation.
Phys. Rev. X. 2014; 4
• Reijmers L.G.
• Perkins B.L.
• Matsuo N.
• Mayford M.
Localization of a stable neural correlate of associative memory.
Science. 2007; 317: 1230-1233
2. Stavroulakis P. Biological Effects of Electromagnetic Fields Mechanisms, Modeling, Biological Effects, Therapeutic Effects, International Standards, Exposure Criteria. Springer Berlin Heidelberg, 2014
• Tarsa L.
• Goda Y.
Synaptophysin regulates activity-dependent synapse formation in cultured hippocampal neurons.
Proc. Natl. Acad. Sci. USA. 2002; 99: 1012-1016
• Tennant K.A.
• Donlan N.A.
• Asay A.L.
• Thomas N.
• Kleim J.A.
• Jones T.A.
The organization of the forelimb representation of the C57BL/6 mouse motor cortex as defined by intracortical microstimulation and cytoarchitecture.
Cereb. Cortex. 2011; 21: 865-876
• Wolf H.K.
• Buslei R.
• Schmidt-Kastner R.
• Schmidt-Kastner P.K.
• Pietsch T.
• Wiestler O.D.
• Blümcke I.
NeuN: a useful neuronal marker for diagnostic histopathology.
J. Histochem. Cytochem. 1996; 44: 1167-1171


text:
# How can ideal gases be “elastic” and “ideal” at the same time?

The elastic properties of any substance is because of the restoring intermolecular forces operating in it.

And I read that the isothermal elasticity of an ideal gas is given by

$$E_{isothermal} = P$$

And also we can define elasticity of the ideal gas in an adiabatic process.

But I don't know how can we define such quantities for ideal gases too. In the definition of ideal gases, I read that there is no intermolecular forces in it.

So my question is that How do we define Elasticity for an ideal gas ?

If elasticity can be defined in terms of properties other than intermolecular forces then please explain it since I have not heard of other factors.

• Don't you think this property should more closely be observed as an effect of their nature to expand rather than due to intermolecular forces? – SteelCubes Jan 16 at 14:42
• Where did you read this? Does $E_{isothermal}$ mean isothermal elasticity? And what is $P$? – Bob D Jan 16 at 14:46
• @Bob D yes it means the same . I read it in my book . I can add a pic of it if you want .. – Ankit Jan 16 at 14:48
• @BobD - Kindly take a look here. – SteelCubes Jan 16 at 14:49
• @Bob D the derivation linked by SteelCubes is the one in my book.. – Ankit Jan 16 at 14:51

"Elasticity" can have slightly different meanings in different fields. In the thermodynamics (and I mean thermodynamics, not statistical mechanics) of spatially uniform systems, a system or a material (a class of systems) is called elastic if its state does not depend on rates of change of physical quantities. The state of an ideal gas at time $$t$$ is taken to be determined by the instantaneous temperature and volume (or density): $$\bigl(T(t),V(t)\bigr)$$, and is therefore elastic. Another example of elastic material, with the same state variables, is rubber.

The state of a solid or fluid (gases included) at time $$t$$ during a spatially uniform process may depend not only on temperature and volume, but for example also on the rate of change of either or both of them: $$\bigl(T(t), V(t), \dot{T}(t), \dot{V}(t)\bigr)$$, where $$\dot{T} := \frac{\mathrm{d}T}{\mathrm{d}t}$$ etc., or even higher-order derivatives. In this case the material is not elastic.

For example, suppose we keep a real gas at constant temperature $$T$$, and measure its pressure during compression when the instantaneous value of its volume is $$V$$; and then measure it during expansion when the instantaneous value of its volume is again $$V$$. We will find that the pressure measured in the two cases is different: it's higher during compression than during expansion. This happens because the pressure of a real gas is not only a function of $$(T,V)$$, but at the very least of $$(T,V,\dot{V})$$ (and possibly other rates of change), and $$\dot{V}$$ is different during compression ($$\dot{V}<0$$) and during expansion ($$\dot{V}>0$$). Real gases are therefore not elastic.

To make a connection with some more familiar aspects of elasticity, suppose we make a system undergo the following processes at constant temperature:

1. Between time 0 and $$\Delta t$$ we let the volume change according to some given time dependence $$V(t) = f(t)$$. For many systems the work done by the system during this process will be $$W_\text{I} = \int_0^{\Delta t} p[\dotso]\ \mathrm{d}V(t) \equiv \int_0^{\Delta t} p[\dotso]\ \frac{\mathrm{d}V}{\mathrm{d}t}\ \mathrm{d}t = \int_0^{\Delta t} p[\dotso]\ \dot{f}(t)\ \mathrm{d}t$$ where I have left the state-dependence of the pressure undefined as yet. This formula is valid for an ideal or real gas. It also holds for a piece of rubber if we use its elongation $$L$$ instead of the volume $$V$$, and keep in mind that $$p$$ will have opposite sign – it's a tension rather than a pressure – and a different state dependence.

2. Between time $$\Delta t$$ and $$2\Delta t$$ we let the volume change in the reverse way: $$V(t) = f(\Delta t-t)$$. The work done by the system in this process is $$W_\text{II} = \int_{\Delta t}^{2\Delta t} p[\dotso]\ \mathrm{d}V(t) \equiv \int_{\Delta t}^{2\Delta t} p[\dotso]\ \frac{\mathrm{d}V}{\mathrm{d}t}\ \mathrm{d}t = -\int_{\Delta t}^{2\Delta t} p[\dotso]\ \dot{f}(\Delta t -t)\ \mathrm{d}t\ .$$

If the pressure (or tension) only depends on temperature and volume, and not on the rate of change of the latter, we have $$p(t) = p[T, V(t)] = p[T, f(t)]$$ in the first process, and $$p(t) = p[T, V(t)] = p[T, f(\Delta-t)]$$ in the second. By a simple change of variable we see that the integrals in the two processeses are opposite: $$W_\text{I} = -W_\text{II}$$. That is, the work done/received by the system in the first process is equal to the work received/done in the second: it's fully returned. So we can say that the work done on the system is fully stored there. This is characteristic of elastic materials, and you see why an ideal gas is also an elastic material. Indeed, it's elastic because it's ideal.

If pressure depends on the rate of change (or even the whole history) of the volume, we have $$p(t) = p[T, V(t), \dot{V}(t)] = p[T, f(t), \dot{f}(t)]$$ in the first process, and $$p(t) = p[T, V(t), \dot{V}(t)] = p[T, f(\Delta t-t), -\dot{f}(\Delta t- t)]$$ in the second. In this case a change of variable only leads us to $$W_\text{I} = \int_0^{\Delta t} p[T, f(t), \dot{f}(t)]\ \dot{f}(t)\ \mathrm{d}t$$ and $$W_\text{II} = -\int_0^{\Delta t} p[T, f(t), -\dot{f}(t)]\ \dot{f}(t)\ \mathrm{d}t$$, and there is no specific relation between the two integrals (unless $$p$$ has some special symmetric dependence on $$\dot{V}$$). In fact they could be both positive or both negative. For real gases, if the first process is an expansion ($$\dot{f}(t)>0$$) and the second the reversed compression, then the first integral turns out to be less than the opposite of the second: $$W_\text{I} < -W_\text{II}$$, which is an expression of dissipation and irreversibility. For the relation between elasticity, irreversibility, and the second law, see Astarita's text below.

In the more general thermodynamics of spatially non-uniform systems (continuum thermomechanics), a material is called elastic if its stress tensor (the generalization of pressure) is rate-independent and memoryless (that is, the material is not capable of hysteresis).

The microscopic interpretation can be quite different depending on the material. In fact, elasticity depends not only on the microscopic composition but also on the macroscopic level of description (coarseness in space and time) that we're using.

Here are some references:

An extensive discussion of elasticity for spatially uniform systems, and its relation with reversibility, irreversibility, and entropy can be found in

see especially chapter 2.

The general classification of materials as elastic, plastic, viscoelastic, or viscoplastic is given for example in

see section 6.4 there, and also in

Quoting from section 43 there:

A material is called elastic if it is simple [a technical term, roughly referring to the dependence upon rotations] and if the stress at time $$t$$ depends only on the local configuration at time $$t$$, and not on the entire past history of the motion.

From this definition you see again why an ideal gas is an elastic material. This text has also a brief discussion of the historical evolution of the notion of elasticity.

Regarding the micro-macro relationship and its dependence on the chosen level of description, a great text is

format: plain


text:
# 13.2.5: Interpreting Beyond 2x2 in Graphs

Our graphs so far have focused on the simplest case for factorial designs, the 2x2 design, with two IVs, each with 2 levels. It is worth spending some time looking at a few more complicated designs and how to interpret them.  We talked about more complicated designs in the Factorial Notations and Square Tables section, but here's a more focused approach to interpreting the graphs of these advanced designs.

## 2x3 design

In a 2x3 design there are two IVs. IV1 has two levels, and IV2 has three levels. Typically, there would be one DV. Let’s talk about the main effects and interaction for this design.

First, let’s make the design concrete. Let’s imagine we are running a memory experiment. We give people some words to remember, and then test them to see how many they can correctly remember. Our DV is the proportion (percentage) that participants remembered correctly out of all tries. We know that people forget things over time. Our first IV will be time of test, immediate versus one week later. The time of test IV will produce a forgetting effect. Generally, people will have a higher proportion correct on an immediate test of their memory for things they just saw, compared to testing a week later.

We might be interested in manipulations that reduce the amount of forgetting that happens over the week. Let’s make the second IV the number of time people got to study the items before the memory test, once, twice or three times. We call IV2 the repetition manipulation.

We might expect data that looks like Figure $$\PageIndex{1}$$.  The top line shows the means when there is no delay (Immediate) for the three levels of repetition.  Notice that the proportion correct (y-axis) increases for the Immediate group with each repetition.  The bottom ling shows the One Week Delay group over the three levels of repetition.  Again, more repetition seems to increase the proportion correct.  Let’s talk about this graph in terms of main effects and interaction.

First, the main effect of delay (time of test) is shown by in each differently-colored line, and seems obvious; the red line is on the top, way above the aqua line. This tells us that the proportion correct on the memory test is always higher when the memory test is taken immediately compared to after one week.

Second, the main effect of repetition is presented on the x-axis, and seems to be clearly present. The more times people saw the items in the memory test (once, twice, or three times), the more they remembered, as measured by increasingly higher proportion correct as a function of number of repetitions.

Is there an interaction? Yes, there is. Remember, an interaction occurs when the effect of one IV depends on the levels of an another. The delay IV measures the forgetting effect. Does the size of the forgetting effet change across the levels of the repetition variable? Yes it does. With one repetition the forgetting effect is 0.9 - 0.6 = 0.4. With two repetitions, the forgetting effect is a little bit smaller, and with three, the repetition is even smaller still. So, the size of the forgetting effect changes as a function of the levels of the repetition IV. There is evidence in the means for an interaction. You would have to conduct an inferential test on the interaction term to see if these differences were likely or unlikely to be due to sampling error.

If there was no interaction, and say, no main effect of repetition, we would see something like Figure $$\PageIndex{2}$$.

What would you say about the interaction if you saw something like Figure $$\PageIndex{3}$$?  The Immediate group is high, but repetition doesn't seem to matter.  The One Week Delay group is flat until the third repetition, then increases the proportion correct.

There is evidence in the means for an interaction. Remember, we are measuring the forgetting effect (effect of delay) three times. The forgetting effect is the same for repetition condition 1 and 2, but it is much smaller for repetition condition 3. The size of the forgetting effect depends on the levels of the repetition IV, so here again there is an interaction.

## 2x2x2 designs

Let’s take it up a notch and look at a 2x2x2 design. Here, there are three IVs with 2 levels each. There are three main effects, three two-way (2x2) interactions, and one 3-way (2x2x2) interaction.

We will use the same example as before but add an additional manipualtion of the kind of material that is to be remembered. For example, we could present words during an encoding phase either visually or spoken (auditory) over headphones.

Figure $$\PageIndex{4}$$ shows two pairs of lines, one side (the panel on the left) is for the auditory information to be remembered, and the panel on the right is when the information was presented visually.  The lines still show the delay, and the y-axis still shows the number of repetitions.  In this version of the study, the was only two repetitions levels:  once or twice.   The two lines on the left show auditory IV levels and the two lines on the right show visual information.  The top lines show when there's no delay, and the diagonal lines show when there is a week delay.  There are only two levels of repetition, so there are only two dots representing this IV (1 repetition on the right and 2 repetitions on the left for both auditory and visual information).

You can think of the 2x2x2, as two 2x2s, one for auditory and one for visual. What’s the take home from this example data? We can see that the graphs for auditory and visual are the same. They both show a 2x2 interaction between delay and repetition. People forgot more things across the week when they studied the material once, compared to when they studied the material twice. There is a main effect of delay, there is a main effect of repetition, but there is no main effect of modality (no difference between auditory or visual information), and there is not a three-way interaction.

What is a three-way interaction anyway? That would occur if there was a difference between the 2x2 interactions. For example, consider the next pattern of results (Figure $$\PageIndex{5}$$).

We are looking at a 3-way interaction between modality, repetition and delay in Figure $$\PageIndex{5}$$. What is going on here? These results would be very strange, but here is an interpretation.

For auditory stimuli, we see that there is a small forgetting effect when people studied things once, but the forgetting effect gets bigger if they studies things twice. A pattern like this would generally be very strange, usually people would do better if they got to review the material twice.

The visual stimuli show a different pattern. This different pattern is where we get the three-way interaction.  Here, the forgetting effect is large when studying visual things once, and it get’s smaller when studying visual things twice.

We see that there is an interaction between delay (the forgetting effect) and repetition for the auditory stimuli; BUT, this interaction effect is different from the interaction effect we see for the visual stimuli. The 2x2 interaction for the auditory stimuli is different from the 2x2 interaction for the visual stimuli. In other words, there is an interaction between the two interactions, as a result there is a three-way interaction, called a 2x2x2 interaction.

We will note a general pattern here. Imagine you had a 2x2x2x2 design. That would have a 4-way interaction. What would that mean? It would mean that the pattern of the 2x2x2 interaction changes across the levels of the 4th IV. If two three-way interactions are different, then there is a four-way interaction.


text:
# Integral problem

Hello all,

Can you help me to derive the following integral?

f(t)=sqrt(r^2-(h*cos(t)-a*sin(t))^2)*(a*cos(t)+h*sin(t))*(h*cos(t)-a*sin(t))

Integral (f(t),t)?

Please note that r,h, and a are constant values.

Any help would be really appreciated.

Thank you
Nejla

Last edited:

mathman


text:
# Door 4? No, no, door 7

You find yourself on a game show and there are 10 doors. Behind nine of them are assorted random farm animals including a variety of pigs, goats and geese, but behind one is a brand new shiny gold car made of real gold!

The game proceeds as follows:

You pick a door.

The game show host then opens a door you didn't choose that he knows has only farm animals behind it.

You are then given the option to switch to a different unopened door, after which he opens another door with farm animals behind it.

This process continues until there are only two doors left, one being your current choice. (Every time the host opens a door you are given another option to switch or stick with the one you currently have selected.)

If you play optimally, your chances of winning the car are of the form $$\dfrac ab$$, where $$a$$ and $$b$$ are coprime positive integers. Find $$a+b$$.

Details and Assumptions:

• You know ahead of time that this process will continue until you are down to only two doors.

• Assume that you prefer the car over any of the farm animals!

Photo credit: http://iloboyou.com/

More probability questions:

Check out my other problems.

×