Russ Hobbie and I discuss magnetic stimulation in Chapter 8 of Intermediate Physics for Medicine and Biology. The technique uses a time-varying magnetic field to induce an electric field in tissue that excites a nerve. I worked on transcranial magnetic stimulation when at the National Institutes of Health in the 1990s. We passed thousands of amps of current through a several centimeter sized, multi-turn coil held above the head. The pulse of current typically rose from zero to its peak in about a tenth of a millisecond. The electric field it induced was large enough to excite neurons in the brain.
Nowadays, researchers such as Seung Woo Lee and his colleagues claim to be able to perform magnetic stimulation using single-turn microcoils, which typically have a size of hundreds of microns and are meant to be implanted in tissue. Lee et al. said they could stimulate neurons using currents of about a twentieth of an amp, and calculated the electric field produced by such a coil for a current as small as one thousandth of an amp. Previously in this blog, I have criticized these studies, saying that the electric field induced in the tissue using such a coil is way too small to excite a nerve. In fact, my graduate student, Mohammed Alzahrani, performed a careful calculation and found that the induced electric field was about 100,000 times smaller than what Lee et al. predicted.
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| “Planar Microcoil Arrays for In Vitro Cellular-Level Micromagnetic Activation of Neurons.” |
Saha et al. passed two amps of current through a coil at a frequency of 2000 Hz. Their coil was square, about 200 microns on each side, and it had five turns. They calculated the electric field 20 microns above the plane of the coil. I will not be reproducing their calculation in detail. Instead, I’ll estimate the induced electric field by examining the equation
where E is the induced electric field, μ0/4π is a constant equal to 10-7 (V/m)/(A/s), N is the number of turns, dI/dt is the rate of change of the coil current, dl is the length of an element of the coil, and R is the distance from that element to where the electric field is calculated. In the past, I have said that the integral is dimensionless and would likely be on the order of one, so the electric field is approximately
For Saha et al.’s coil, this gives
Today, I want to take a closer look at that integral. Let’s calculate the electric field induced a distance z away from a wire that extends from –L/2 to +L/2 along the x axis. The integral, which you can look up in any good integral table, is expressed in terms of the natural logarithm
The integral depends only on the ratio z/L. Getting close to the nerve (small z) and making the coil tiny (small L) do not matter individually; their relative size is what counts. I would like to take L to infinity to find the electric field from a long wire, but in that case the denominator in the argument of the logarithm would go to zero and the logarithm itself would be infinite. So let L, the length of the side of the coil, be 200 microns and let z, the distance from the coil to the measurement point, be 20 microns. Then z/L = 0.1, implying the integral is equal to 4.6. Therefore, the electric field produced by this coil should be about (0.013 V/m)(4.6) = 0.06 V/m. That might be an overestimate, because the width of the five turns is actually about 50 microns, which would spread out the distribution of the electric field and lower its peak strength, and also the other side of the coil might contribute a little. But since I’m estimating, I’ll take the electric field to be 0.06 V/m.
That’s close to the peak electric field calculated by Saha et al. (see their Figure 2a). So unlike Lee et al., Saha et al. seem to calculate the magnitude of the electric field correctly. I do have some reservations about the spatial distribution of the electric field they predict. I would expect it to be large right under the coil and then fall off rapidly away from it. Instead, they find the electric field is large not only under the coil but also a long ways away from it, out near the edge of the 3 mm by 3 mm region where they perform the calculation. Also, their electric field is oddly asymmetric. The electric field in the x direction looks like what I would expect (except for those strange regions near the outer boundary) but their electric field in the y direction is much larger on one side of the coil than on the other, and is spread out over a region much larger than the coil size (see their Figure 2d; I assume there’s a typo in the labels on their color bar, so it actually ranges from +0.02 to –0.05 V/m). Perhaps some of this is due the sealed boundary at the edge of the tissue region, some due to the feed wires that deliver current to the coil, and perhaps some due to an interaction between those two factors. Who knows? But at least they get the magnitude near the coil right, and for me that’s what’s important. Lee et al. had the spatial distribution correct, but their magnitude was way off.
The other concern I have with Saha’s calculation is the response of the neuron. They placed a neuron, which was about 2 mm long, near the coil and used a program called NEURON to calculate its resulting transmembrane voltage. Let’s do this ourselves in an approximate way. The transmembrane voltage should be on the order of the electric field times the length of the neuron. This would be the case if no current entered the cell, so the intracellular space is at a constant voltage, the extracellular voltage varies linearly along the neuron length, and the transmembrane voltage is the intracellular voltage minus the extracellular voltage. That would be 0.06 V/m = 0.06 mV/mm times 2 mm, or 0.12 mV. No neuron is going to fire if the transmembrane potential changes by about a tenth of a millivolt. Yet Saha et al. show an initial response (their stimulus artifact, which is followed immediately by an action potential) that increases by about 70 mV (see their Figure 2b). This I don’t understand. If anything, I’m overestimating the transmembrane potential in my calculation. The neuron is a little less than 2 mm in length, it is off to one side where the electric field has fallen to about a half its peak value, and you could make a good argument that I should use half the length of the neuron rather than its entire length in my estimate (one end is depolarized and the other end hyperpolarized). So, a 0.06 V/m electric field most likely produces less than 0.12 mV of transmembrane voltage, but let’s use 0.12 mV as an upper limit. That’s gotta be subthreshold.
Typical electric field thresholds in the brain are on the order of, and probably somewhat greater than, 10 V/m (see Mohammed’s first paper cited below). Why do Saha et al. find that such a tiny 0.06 V/m electric field fires an action potential in the nerve? I don’t know. NEURON is a black box to me and it is hard to say why its prediction is so odd.
What do I conclude? Saha et al. seem to calculate the induced electric field correctly (at least its peak magnitude, if not its spatial distribution). But their electric field is too small to excite a nerve. Not 100,000 times too small (like for Lee et al.), but perhaps a hundred times too small. I’m not convinced that what they are looking at is magnetic stimulation. What else could it be? My graduate student Mohammed calculated that capacitive coupling might be an alternative mechanism for excitation. Other mechanisms are possible, such as tissue heating or mechanical motion caused by the magnetic force produced by one side of the coil on the other side.
For those who want to look more deeply into this issue, Mohammed’s three papers on this topic are:
Alzahrani, M. and B. J. Roth, 2023, The electric field induced by a microcoil during magnetic stimulation. IEEE Trans. Biomed. Eng., 70:3260–3262.
Alzahrani, M. and B. J. Roth, 2024, The calculation of maximum electric field intensity in brain tissue stimulated by a current pulse through a microcoil via capacitive coupling. Applied Sciences, 14:2994.
Alzahrani, M. and B. J. Roth, 2024, The difference between traditional magnetic stimulation and microcoil stimulation: Threshold and the electric field gradient. Applied Sciences, 14:8349.
The last two are open access, so anyone can read them online. The first one is not open access, but send an email to roth@oakland.edu and I'll be happy to send you the pdf.
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