Magnetic Stimulation Using Microcoils: A Skeptical Review

Magnetic Stimulation Using Microcoils:
A Skeptical Review.

Over the last decade I’ve written a lot in this blog about magnetic stimulation of nerves using microcoils, which are tiny current-carrying coils generally from one millimeter down to tens of microns in size (see here, here, here, and here). My latest (and probably last) contribution to this topic was published recently in the BOHR International Journal of Neurology and Neuroscience, titled “Magnetic Stimulation Using Microcoils: A Skeptical Review.” The abstract states

Recently several research groups have used sub-millimeter sized microcoils to perform magnetic stimulation of nerves. This review assesses the magnitude of the electric field induced by these microcoils. In some cases magnetic stimulation is a plausible mechanism for neural excitation, but in other cases the induced electric field is far too small to excite a neuron. These results indicate that microcoil magnetic stimulation may not occur via magnetic stimulation, but by some other mechanism. One alternative mechanism is capacitive coupling.

Basically, this paper reviews the entire field of microcoil stimulation, and finds that in some cases the idea is plausible, but in other cases it is not. One thing I like about the article is that I develop a toy model for calculating the electric field produced by the coil. While the model is an approximation, it should determine the electric field correct to at least an order of magnitude. The strengths of the model are that it provides great insight, it is so simple computationally that anyone can reproduce the calculation, and it supplies a common way to analyze a host of different publications. One goal of Intermediate Physics for Medicine and Biology is to train students in forming and analyzing such toy models. I think that the field of microcoil stimulation illustrates what happens when researchers skip the simple model and go straight to complicated numerical calculations using code that is treated as a black box. These computations often produce numbers but little intuition or understanding. The user cannot tell if something goes wrong. Back when I was teaching, I would urge my students to “think before you calculate!” This review shows why.

The conclusion of the review states

Magnetic stimulation by microcoils is an active and growing field in neural stimulation. It may have promise for the development of neural prostheses. This review, however, suggests there are many unanswered questions in this area of research… Frankly, the field appears prone to errors. One must analyze the articles carefully to separate the wheat from the chaff.

Transcranial magnetic stimulation works by magnetic induction. Stimulation using a millimeter-sized, multi-turn coil placed close to the target neuron and carrying several amps of current may work by magnetic induction, but the electric field it generates is sometimes slightly below the threshold value you expect is required to excite neurons. Nevertheless, magnetic induction in these cases seems plausible. On the other hand, when milliamps of current are passed through a single-turn wire, magnetic induction does not seem to be a plausible mechanism for excitation; the electric field appears to be too weak. One alternative mechanism is capacitive coupling. An important goal of future microcoil magnetic stimulation research is to resolve what exactly is the underlying mechanism.

The article is open access, so anyone can read it online without a subscription. Enjoy!

Note added just before this post was published

After my review article was published earlier this week, I discovered a 2025 PhD dissertation that came to a similar conclusion as my review did. Giovanni Calixte working with his graduate advisor Robert Butera at the Georgia Tech published “An Investigation into the Efficacy and Utility of Micromagnetic Stimulation as a Means of Exciting Neural Tissue.” It’s good work, and you can download it here. I reproduce the summary below. I’ve put the critical sentence near the end of the summary in italics.

Micro-magnetic stimulation (μMS) through micro-scale coils is a rapidly advancing form of neuromodulation that possesses qualities that give it the potential to be used as an alternative to electrical stimulation in cases where it cannot be used. However, reliability of the technique is inconsistent as reported in the peer reviewed literature, suggesting that there is not a strong understanding of the basis of the technique. In this thesis, we provide empirical evidence of the efficacy of μMS in dissociated cortical cultures and a review of μMS’ consistency across early pioneering works. We then simulate the electric fields generated by μMS using COMSOL and couple the fields to neuron models in Python to study their response across a range of parameters in neural stimulation. This is followed by an aim to measure the electric fields of these micro-scale coils to determine whether they surpass stimulus thresholds reported in literature. Our key findings suggest that μMS should not be feasible. Finally, we investigate confounding factors during μMS stimulation that may have effects being mis-attributed to μMS.

I’m delighted that it’s not just my former graduate student Mohammed Alzahrani and me who are skeptical of microcoil magnetic stimulation. We are no longer alone!

Action Current Propagation Across an Electrical Synapse: Magnetic Measurement on a Septated Earthworm Axon

It’s been 40 years since I attended the Biophysical Society Annual Meeting in San Francisco (February 9–13, 1986), coauthoring a poster about earthworms. The lead author was Frans Gielen, a post doc working in John Wikswo’s laboratory at Vanderbilt University when I was a graduate student there. The last author was Peter Brink, a friend of Wikswo’s and an expert on electrical synapses, particularly in an earthworm axon. 

Wikswo and I had been performing experiments on crayfish axons, which are long, straight, and uninterrupted along their length (I’ve written about these experiments before in this blog). We used a wire-wound, ferrite-core toroid to measure the action current along the axon, and compared it to the action potential measured simultaneously with a microelectrode. The interesting thing about the earthworm is that their medial giant axon is divided into segments by septa, which are low resistance electrical synapses also known as gap junctions. At the University of Illinois Brink had studied septa with Lloyd Barr, one of the first researchers to make electrical measurements on a septum (“The Resistance of the Septum of the Median Giant Axon of the Earthworm,” Journal of General Physiology, Volume 69, Pages 517­–536, 1977). Our goal was to see if the gap junctions had high enough resistance to reduce the axial action current at the site of a septum. 

Using a toroid to measure the action current in a nerve axon. 

According to my research notebooks, Brink visited Vanderbilt twice in 1985 for these experiments: first on April 18 and again on July 30. He taught us the dissection—which was easier than the crayfish dissections I had been doing, because the earthworm nerve is robust and not damaged by stretching—and brought the Lucifer Yellow dye needed to visualize the septum. We scanned the toroid along the axon looking for a change in current near the septum. Looking back at the data, it was pretty noisy and inconclusive. But the initial results were enough for a meeting abstract, and we submitted the data to the Biophysical Society meeting.

I enjoyed working with Brink, who was about Wikswo’s age, meaning he was several years older than me but still a young professor, fun-loving and irreverent. He went on to a long career at the State University of New York at Stony Brook, rising to become chair of the Department of Physiology and Biophysics. Throughout his career he did a lot of highly cited work on gap junctions, particularly in cardiac tissue.

Below is our abstract to the Biophysical Society Meeting. We never obtained enough good data to write a full research article, so this abstract is my only contribution to earthworm physiology. What I remember most about the meeting was riding a cable car, walking out onto the Golden Gate Bridge, and eating lobster at Fisherman’s Wharf.

Approximate Analytical Solutions of the Bidomain Equations for Electrical Stimulation of Cardiac Tissue With Curving Fibers

Russ Hobbie and I stress qualitative thinking, in addition to quantitative thinking, in Intermediate Physics for Medicine and Biology. What’s the difference? Qualitative thinking is the ability to guess approximately what a solution will look like. As an example, let me share the first part of one of my publications that I always liked. It is from the article “Approximate analytical solutions of the bidomain equations for electrical stimulation of cardiac tissue with curving fibers,” published in Physical Review E (Volume 67, Article number 051925) in 2003. I wrote the paper with my graduate student Debbie Langrill Beaudoin.

In the first paragraph, we wrote “Fig. 1 shows the fiber geometry throughout a sheet of tissue and the direction of the applied electric field. Can you look at Fig. 1 and predict where the tissue will be depolarized and where it will be hyperpolarized?” Figure 1 is shown below. 

Warning: Debbie and I cooked up a way to avoid polarization at the boundaries. Ordinarily you would expect a big hyperpolarization on the left and a depolarization on the right, both restricted to only a few length constants from the edge. Ignore this effect. Just consider the polarization caused by the fiber curvature.

At this point, dear reader, I ask you to stop reading and guess the distribution of polarization. Take a piece of paper, sketch the fiber distribution in Fig. 1, and then mark which areas of the tissue are depolarized and which are hyperpolarized. If you have some colored pencils handy, just color the depolarized region red and the hyperpolarized region blue. Go ahead. I’ll wait... 

 

Okay, let’s see how you did. Below is Fig. 6 of our paper, which gives the result. Depolarization is in red, hyperpolarization is in blue.

Did you get anything like this?

To help explain the polarization distribution, I’ve created two new figures not in the paper. In both, the short gray line segments show the fiber geometry. The purple arrows indicate the direction of the applied electric field. Green shows a component of the intracellular current density. The red D’s and blue H’s indicate depolarization and hyperpolarization. The first of the two figures is shown below, and illustrates what I’ll call “Mechanism 1.” 

Mechanism 1

In the region where the fibers point along the applied electric field (like along the left edge of the tissue), the current is divided approximately equally between the intracellular and extracellular spaces because they have similar conductivities in that direction. So, the green intracellular current density arrows are relatively large there. In the region where the fibers point perpendicular to the applied electric field (like in the center), the intracellular current density is less than the extracellular current density because the intracellular conductivity is smaller than the extracellular conductivity in that direction, so the green intracellular current density arrows are relatively small. Somewhere between these two regions current had to leave the intracellular space and pass out through the membrane, entering the extracellular space. This outward membrane current depolarizes the tissue (red D’s). If the fiber direction then changes back to being parallel to the electric field (like along the right edge of the tissue), some extracellular current must recross the membrane and reenter the cell, which hyperpolarizes the tissue (blue H’s). This behavior is shown in the upper small plot to the left of Fig. 6a.

The next new figure illustrates “Mechanism 2.” Consider what happens where the fibers are oriented at an angle of about 45 degrees to the electric field. In that case, even though the electric field may be horizontal the anisotropy rotates the intracellular current density to be more nearly parallel to the fibers (the direction with the highest conductivity). In other words, the electric field is horizontal, but the intracellular current density rotates counterclockwise. The extracellular current density also rotates, but not as much because the intracellular space is more anisotropic than the extracellular space. Thus, you pick up a component of the intracellular current perpendicular to the applied electric field (shown by the green arrows). If the fibers change direction so they are either parallel or perpendicular to the field, you get no rotation of the current density there, so there is no component of the intracellular current perpendicular to the electric field. At the head of one of those green arrows, the intracellular current density vector ends so the intracellular current must cross the membrane and enter the extracellular space, depolarizing the tissue. At the tail of a green arrow the intracellular current density vector begins so the extracellular current must cross the membrane and enter the intracellular space, hyperpolarizing the tissue. This results in a somewhat complicated pattern of polarization (H’s and D’s), which resembles the pattern shown in the lower small plot to the left of Fig. 6a. 

Mechanism 2

Both of these mechanisms operate simultaneously, so the net polarization is the sum of those two small plots. This results in the Yin-Yang pattern of depolarization and hyperpolarization of Fig. 6a. (Stare at Fig. 6a long enough until you realize this is correct.) Below it, in Fig. 6b, is the result you get if you just mindlessly solve the bidomain equations numerically. (Actually, Debbie solved them, and she did nothing mindlessly, but you know what I mean). The two are qualitatively the same, although there are quantitative differences.

So, how many of you guessed the Yin-Yang pattern? To tell you the truth, I’m not sure I did when Debbie and I first started this analysis. It’s difficult. But at least now I have a way to understand this pretty but nonintuitive pattern. I’ve found that being able to do these hand-waving types of explanations is useful. It lets you understand what is going on, rather than just putting a calculation into a black-box computer program and getting out an answer with no insight. Remember: the purpose of computing is insight, not numbers!

Finally, I really enjoyed starting a research paper off with a puzzle like that in Fig. 1 and ending it with the solution like in Fig. 6. I think you should consider using this trick in your next article.

The Cardiac Bidomain Model in Twelve Publications

Recently I wrote a review of the bidomain model of cardiac tissue. Russ Hobbie and I discuss the bidomain model in Section 7.9 of Intermediate Physics for Medicine and Biology. It’s a mathematical description of heart muscle that keeps track of the voltages and currents both inside and outside the myocardial cells. What I wrote is not really an academic review article, it’s not a history, and it’s not a memoir. To tell you the truth, I’m not sure what it is. I originally thought I’d try and publish it, but I’m not sure who would accept such an unusual article. So, I decided it would be best to distribute it on my blog. There is little I can do for my dear readers, but I can give them this review.

The format is to describe the bidomain model by considering twelve publications. Below is a list of the articles I chose. Each article is meant to feature one researcher, whose names are listed in bold.

Tung L (1978) A bi-domain model for describing ischemic myocardial dc potentials. PhD Dissertation, Massachusetts Institute of Technology.

Plonsey R, Barr RC (1984) Current flow patterns in two-dimensional anisotrpic bisyncytia with normal and extreme conductivities. Biophys J 45:557–571.

Sepulveda NG, Roth BJ, Wikswo JP, Jr (1989) Current injection into a two-dimensional anisotropic bidomain. Biophys J, 55:987-999. 

Henriquez CS, Plonsey R (1990b) Simulation of propagation along a cylindrical bundle of cardiac tissue. II. Results of the simulation. IEEE Trans Biomed Eng 37:861–875.

Neu JC, Krassowska W (1993) Homogenization of syncytial tissue. Crit Rev Biomed Eng 21:137–199.

Wikswo JP Jr, Lin SF, Abbas RA (1995) Virtual electrodes in cardiac tissue: A common mechanism for anodal and cathodal stimulation. Biophys J 69:2195–2210.

Trayanova N, Skouibine K, Aguel F (1998) The role of cardiac tissue structure in defibrillation. Chaos 8:221–233.

Knisley SB, Trayanova N, Aguel F (1999) Roles of electric field and fiber structure in cardiac electric stimulation. Biophys J 77:1404–1417.

Efimov IR, Cheng Y, van Wagoner DR, Mazgalev T, Tchou PJ (1998) Virtual electrode-induced phase singularity: A basic mechanism of defibrillation failure. Circ Res 82:918–925.

Entcheva E, Eason J, Efimov IR, Cheng Y, Malkin R, Claydon F (1998) Virtual electrode effects in transvenous defibrillation-modulation by structure and interface: Evidence from bidomain simulations and optical mapping. J Cardiovasc Electrophysiol 9:949–961.

Rodriguez B, Li L, Eason JC, Efimov IR, Trayanova NA (2005) Differences between left and right ventricular chamber geometry affect cardiac vulnerability to electric shocks. Circ Res 97:168–175.

Bishop MJ, Boyle PM, Plank G, Welsh DG, Vigmond EJ (2010) Modeling the role of the coronary vasculature during external field stimulation. IEEE Trans Biomed Eng 57:2335–2345.

My biggest worry is that I’ve left too much out. For instance, I could easily have featured other researchers, such as Rick Gray, Jamey Eason, Roger Barr, Marc Lin, Felipe Aguel, David Geselowitz, and others. Also, I suspect there are many researchers who, if they read this review, will be hurt because they are completely ignored. All I can say is, I’m sorry. I tried to relate the story as best as I can remember it, but I may have remembered some things wrong.

You can download my review here. I hope you enjoy reading the article as much as I enjoyed writing it. It was an honor to work on this topic with so many outstanding scientists. As Randy Travis sings, these scientists are my heroes and friends.

 

 

“Heroes and Friends,” by Randy Travis

https://www.youtube.com/watch?v=1JVag-cOSUI

Millikan and the Magnetic Field of a Single Axon

“The Magnetic Field of a Single Axon:
A Comparison of Theory and Experiment.”

Forty years ago this month, I published one of my first scientific papers. “The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment” appeared in the July, 1985 issue of the Biophysical Journal (Volume 48, Pages 93–109). I was a graduate student at Vanderbilt University at the time, and my coauthor was my PhD advisor John Wikswo. When discussing the paper below, I will write “I did this…” and “I thought that…” because I was the one in the lab doing the experiments, but of course it was really Wikswo and I together writing the paper and analyzing the results.

Selected Papers of
Great American Physicists.

In those days I planned to be an experimentalist (like Wikswo). About the time I was writing “The Magnetic Field of a Single Axon,” I read “On the Elementary Electrical Charge and The Avogadro Constant” by Robert Millikan (Physical Review, Volume 11, Pages 109–143, 1913). It had been reprinted in the book Selected Papers of Great American Physicists, published by the American Institute of Physics.

If you are reading this blog, you’re probably are familiar with Millikan’s oil drop experiment. He measured the speed of small droplets of oil suspended in air and placed in gravitational and electric fields, and was able to determine the charge of a single electron. I remember doing this experiment as a undergraduate physics major at the University of Kansas. I was particularly impressed by the way Millikan analyzed his experiment for possible systematic errors: He worried about deviations of the frictional force experienced by the drops from Stokes’ law and corrected for it; he analyzed the possible changes to the density of the oil in small drops; he checked that his 5300 volt battery was calibrated correctly and supplied a constant voltage; and he fussed about convection currents in the air influencing his results. He was especially concerned about his value of the viscosity of air, which he estimated was known to about one part in a thousand. Rooting out systematic errors is a hallmark of a good experimentalist. I wanted to be like Millikan, so I analyzed my magnetic field measurement for a variety of systematic errors.

The first type of error in my experiment was in the parameters used to calculate the magnetic field (so I could compare it to the measured field). I estimated that my largest source of error was in my measurement of the axon radius. This was done using a reticle in the dissecting microscope eyepiece. I only knew the radius to 10% accuracy, in part because I could see that it was not altogether uniform along the axon, and because I could not be sure the axon’s cross section was circular. It was my biggest source of error for calculating the magnitude of the magnetic field, because the field varied as the axon cross-sectional area, which is proportional to the radius squared.

Figure 1 from "The Magnetic
Field of a Single Axon."

I measured the magnetic field by threading the axon through a wire-wound ferrite-core toroid (I’ve written about these toroid measurements before in this blog). I assumed the axon was at the center of the toroid, but this was not always the case. I performed calculations assuming the toroid averaged the magnetic field for an off-axis axon, and was able to set an upper limit on this error of about 2%. The magnetic field was not measured at a point but was averaged over the cross-sectional area of the ferrite core. More numerical analysis suggested that I could account for the core area to within about 1%. I was able to show that inductive effects from the toroid were utterly negligible. Finally, I assumed the high permeability ferrite did not affect the magnetic field distribution. This should be true if the axon is concentric with the toroid and aligned properly. I didn’t have a good way to estimate the size of this error.

Figure 2 from "The Magnetic
Field of a Single Axon."

The toroid and axon were suspended in a saline bath (technically, Van Harreveld's solution), and this bath gave rise to other sources of error. I analyzed the magnetic field for different sized baths (the default assumption was an unbounded bath), and for when the bath had a planar insulating boundary. I could do the experiment of measuring the magnetic field as we raised and lowered the volume of fluid in the bath. The effect was negligible. I spent a lot of time worrying about the heterogeneity caused by the axon being embedded in a nerve bundle. I didn’t really know the conductivity of the surrounding nerve bundle, but for reasonable assumptions it didn’t seem to have much effect. Perhaps the biggest heterogeneity in our experiment was the “giant” (~1 mm inner radius, 2 mm outer radius, 1 mm thick) toroid, which was embedded in an insulated epoxy coating. This big chunk of epoxy certainly influenced the current density in the surrounding saline. I had to develop a new way of calculating the extracellular current entirely numerically to estimate this effect. The calculation was so complicated that Wikswo and I didn’t describe it in our paper, but instead cited another paper that we listed as “in preparation” but that in fact never was published. I concluded that the toroid was not a big effect for my nerve axon measurements, although it seemed to be more important when I later studied strands of cardiac tissue.

Figure 3 of "The Magnetic
Field of a Single Axon."

Other miscellaneous potential sources of error include capacitive effects in the saline and an uncertainty in the action potential conduction velocity (measured using a second toroid). I determined the transmembrane potential by taking the difference between the intracellular potential (measured by a glass microelectrode, see more here) and a metal extracellular electrode. However, I could not position the two electrodes too accurately, and the extracellular potential varies considerably over small distances from the axon, so my resulting transmembrane potential certainly had a little bit of error. Measurement of the intracellular potential using the microelectrode was susceptible to capacitive coupling to the surrounding saline bath. I used a “frequency compensator” to supply “negative capacitance” and correct for this coupling, but I could not be sure the correction was accurate enough to avoid introducing any error. One of my goals was to calculate the magnetic field from the transmembrane potential, so any systematic errors in my voltage measurements were concerning. Finally, I worried about cell damage when I pushed the glass microelectrode into the axon. I could check this by putting a second glass microelectrode in nearby and I didn’t see any significant effect, but such things are difficult to be sure about.

All of this analysis of systematic errors, and more, went into our rather long Biophysical Journal paper. It remains one of my favorite publications. I hope Millikan would have been proud. If you want to learn more, see Chapter 8 about Biomagnetism in Intermediate Physics for Medicine and Biology. 

Forty years is a long time, but to this old man it seems like just yesterday.

The First Measurement of the Magnetocardiogram

Biomagnetism: The First Sixty Years.

A couple years ago, I published a review article titled “Biomagnetism: The First Sixty Years.” I wrote about that article before in this blog, but I thought it was time for an update. The paper is popular: according to Google Scholar it has been cited 28 times in two years, which is more citations than any other of my publications in the last decade. I remember working on this paper because it was my Covid project. That year I got Covid for the first—and, so far, only—time. I quarantined myself in our upstairs bedroom, wore a mask, and somehow avoided infecting my wife. I remember having little to do except work on my biomagnetism review.

As a treat, I thought I would reproduce one of the initial sections of the article (references removed) about the first measurement of the magnetocardiogram. Russ Hobbie and I talk about the MCG in Chapter 8 of Intermediate Physics for Medicine and Biology. This excerpt goes into more detail about how MCG measurements began. Enjoy!

2.1. The First Measurement of the Magnetocardiogram

In 1963, Gerhard Baule and Richard McFee first measured the magnetic field generated by the human body. Working in a field in Syracuse, New York, they recorded the magnetic field of the heart: the magnetocardiogram (MCG). To sense the signal, they wound two million turns of wire around a dumbbell-shaped ferrite core that responded to the changing magnetic field by electromagnetic induction. The induced voltage in the pickup coil was detected with a low-noise amplifier.

The ferrite core was about one-third of a meter long, so the magnetic field was not measured at a single point above the chest, but instead was averaged over the entire coil. One question repeatedly examined in this review is spatial resolution. Small detectors are often noisy and large detectors integrate over the area, creating a trade-off between spatial resolution and the signal-to-noise ratio.

The heart’s magnetic field is tiny, on the order of 50–100 pT (Figure 1). A picotesla (pT) is less than a millionth of a millionth as strong as the magnetic field in a magnetic resonance imaging machine. The magnetic field of the earth is about 30,000,000 pT (Figure 1), and the only reason it does not obscure the heart’s field is that the earth’s field is static. That is not strictly true. The earth’s field varies slightly over time, which causes geomagnetic noise that tends to mask the magnetocardiogram (Figure 1). Moreover, even a perfectly static geomagnetic field would influence the MCG if the pickup coil slightly vibrated. A key challenge in biomagnetic recordings, and a major theme in this review, is the battle to lower the noise enough so the signal is detectable

Figure 1. Noise sources in biomagnetism.

Most laboratories contain stray magnetic fields from sources such as electronic equipment, elevators, or passing cars (Figure 1). Baule and McFee avoided much of this noise by performing their experiments at a remote location. Even so, they had to filter out the ubiquitous 60 Hz magnetic field arising from electrical power distribution. A magnetic field changing at 60 Hz is a particular nuisance for biomagnetism because the magnetic field typically exists in a frequency band extending from 1 Hz (1 s between heartbeats) to 1000 Hz (1 ms rise time of a nerve or muscle action potential).

One limitation of a metal pickup coil is the thermal currents in the winding due to the random motion of electrons, creating extraneous magnetic fields caused by the measuring device itself. The ultimate source of noise is thermal currents in the body, but fortunately, their magnetic field is minuscule (Figure 1).

Baule and McFee suppressed background noise by subtracting the output of two pickup coils. A distant source of noise gave the same signal in both coils and did not contribute to their difference. One coil was placed over the heart, and the magnetocardiogram was larger there and did not cancel out. The two coils formed a rudimentary type of gradiometer (Figure 2).

The magnetocardiogram resembled the electrocardiogram (ECG) sensed by electrodes attached to the skin. Baule and McFee speculated that the MCG might contain different information than the ECG, another idea that reappears throughout this review. In a followup article, they theoretically calculated the magnetic field produced by the heart. The interplay between theory and experiments is yet one more subject that frequently arises in this article.

Figure 2. Types of gradiometers.

Extracellular Magnetic Measurements to Determine the Transmembrane Action Potential and the Membrane Conduction Current in a Single Giant Axon

Forty years ago today I was attending my first scientific meeting: The Society for Neuroscience 14^th Annual Meeting, held in Anaheim, California (October 10–15, 1984). As a 24-year-old graduate student in the Department of Physics at Vanderbilt University, I presented a poster based on the abstract shown below: “Extracellular Magnetic Measurements to Determine the Transmembrane Action Potential and the Membrane Conduction Current in a Single Giant Axon.”

I can’t remember much about the meeting. I’m sure I flew to California from Nashville, Tennessee, but I can’t recall if my PhD advisor John Wikswo went with me (his name is not listed on any meeting abstract except the one we presented). I believe the meeting was held at the Anaheim Convention Center. I remember walking along the sidewalk outside of Disneyland, but I didn’t go in (I had visited there with my parents as a child).

Neuroscience Society meetings are huge. This one had over 300 sessions and more than 4000 abstracts submitted. In the Oct. 11, 1984 entry in my research notebook, I wrote “My poster session went OK. Several people were quite enthusiastic.” I took notes from talks I listened to, including James Hudspeth discussing hearing, a Presidential Symposium by Gerald Fischbach, and a talk about synaptic biology and learning by Eric Kandel. I was there when Theodore Bullock and Susumu Hagiwara were awarded the Ralph W. Gerard Prize in Neuroscience.

The research Wikswo and I reported in our abstract was eventually published in my first two peer-reviewed journal articles:

Barach, J. P., B. J. Roth and J. P. Wikswo, Jr., 1985, Magnetic Measurements of Action Currents in a Single Nerve Axon: A Core-Conductor Model. IEEE Transactions on Biomedical Engineering, Volume 32, Pages 136-140.

Roth, B. J. and J. P. Wikswo, Jr., 1985, The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment. Biophysical Journal, Volume 48, Pages 93-109.

Both are cited in Chapter 8 of Intermediate Physics for Medicine and Biology.

This neuroscience abstract was not my first publication. I was listed as a coauthor on an abstract to the 1983 March Meeting of the American Physical Society, based on some research I helped with as an undergraduate physics major at the University of Kansas. But I didn’t attend that meeting. In my CV, I have only one publication listed for 1983 and one again in 1984. Then in 1985, they started coming fast and furious. 

Four decades is a long time, but it seems like yesterday.

The Difference between Traditional Magnetic Stimulation and Microcoil Stimulation: Threshold and the Electric Field Gradient

In Chapter 7 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss electrical stimulation of nerves. In particular, we describe how neural excitation depends on the duration of the stimulus pulse, leading to the strength-duration curve.

The strength-duration curve for current was first described by Lapicque (1909) as

where i is the current required for stimulation, i_R is the rheobase [the minimum current required for a long stimulus pulse], t is the duration of the pulse, and t_c is chronaxie, the duration of the pulse that requires twice the rheobase current.

An axon is difficult to excite using a brief pulse, and you have to apply a strong current. This behavior arises because the axon has its own characteristic time, τ (about 1 ms), which is basically the resistance-capacitance (RC) time constant of the cell membrane. If the stimulus duration is much shorter than this time constant, the stimulus strength must increase.

A nerve axon not only has a time constant τ, but also a space constant λ. Is there a similar spatial behavior when exciting a nerve? This is the question my graduate student Mohammed Alzahrani and I addressed in our recent article “The Difference between Traditional Magnetic Stimulation and Microcoil Stimulation: Threshold and the Electric Field Gradient” (Applied Sciences, Volume 14, Article 8349, 2024). The question becomes important during magnetic stimulation with a microcoil. Magnetic stimulation occurs when a pulse of current is passed through a coil held near the head. The changing magnetic field induces an electric field in the brain, and this electric field excites neurons. Recently, researchers have proposed performing magnetic stimulation using tiny “microcoils” that would be implanted in the brain. (Will such microcoils really work? That’s a long story, see here and here.) If the coil is only 100 microns in size, the induced electric field distribution will be quite localized. In fact, it may exist over a distance that’s short compared to the typical space constant of a nerve axon (about 1 mm). Mohammed and I calculated the response of a nerve to the electric field from a microcoil, and found that for a localized field the stimulus strength required for excitation is large.

Figure 6 of our article, reproduced below, plots the gradient of the induced electric field dE_x/dx (which, in this case, is the stimulus strength) versus the parameter b (which characterizes the spatial width of the electric field distribution). Note that unlike the plot of the strength-duration curve above, Fig. 6 is a log-log plot. 

Figure 6 from Alzahrani and Roth, Appl. Sci., 14:8349, 2024

We wrote

Our strength-spatial extent curve in Figure 6 for magnetic stimulation is analogous to the strength-duration curve for electrical stimulation if we replace the stimulus duration [t] by the spatial extent of the stimulus b and the time constant τ by the [space] constant λ. Our results in Figure 6 have a “spatial rheobase” dE_x/dx value (1853 mV/cm²) for large values of spatial extent b. At small values of b, the value of dE_x/dx rises. If we wanted to define a “spatial chronaxie” (the value of b for which the threshold value of dE_x/dx rises by a factor of two), it would be about half a centimeter.

To learn more about this effect you can read our paper, which was published open access so its available free to everyone. Some researchers have used a value of dE_x/dx found when stimulating with a large coil held outside the head to estimate the threshold stimulus strength using a microcoil. We ended the paper with this warning:

In conclusion, our results show that even in the case of long, straight nerve fibers, you should not use a threshold value of dE_x/dx in a microcoil experiment that was obtained from a traditional magnetic stimulation experiment with a large coil. The threshold value must be scaled to account for the spatial extent of the dE_x/dx distribution. Magnetic stimulation is inherently more difficult for microcoils than for traditional large coils, and for the same reason, electrical stimulation is more difficult for short-duration stimulus pulses than for long-duration pulses. The nerve axon has its own time and space constants, and if the pulse duration or the extent of the dE_x/dx distribution is smaller than these constants, the threshold stimulation will rise. For microcoil stimulation, the increase can be dramatic.

A Comparison of Two Models for Calculating the Electrical Potential in Skeletal Muscle

Roth and Gielen,
Annals of Biomedical Engineering,
Volume 15, Pages 591–602, 1987

Today I want to tell you how Frans Gielen and I wrote the paper “A Comparison of Two Models for Calculating the Electrical Potential in Skeletal Muscle” (Annals of Biomedical Engineering, Volume 15, Pages 591–602, 1987). It’s not one of my more influential works, but it provides insight into the kind of mathematical modeling I do.

The story begins in 1984 when Frans arrived as a post doc in John Wikswo’s Living State Physics Laboratory at Vanderbilt University in Nashville. Tennessee. I had already been working in Wikswo’s lab since 1982 as a graduate student. Frans was from the Netherlands and I called him “that crazy Dutchman.” My girlfriend (now wife) Shirley and I would often go over to Frans and his wife Tiny’s apartment to play bridge. I remember well when they had their first child, Irene. We all became close friends, and would go camping in the Great Smoky Mountains together.

Frans had received his PhD in biophysics from Twente University. In his dissertation he had developed a mathematical model of the electrical conductivity of skeletal muscle. His model was macroscopic, meaning it represented the electrical behavior of the tissue averaged over many cells. It was also anisotropic, so that the conductivity was different if measured parallel or perpendicular to the muscle fiber direction. His PhD dissertation also reported many experiments he performed to test his model. He used the four-electrode method, where two electrodes pass current into the tissue and two others measure the resulting voltage. When the electrodes are placed along the muscle fiber direction, he found that the resulting conductivity depended on the electrode separation. If the current-passing electrodes where very close together then the current was restricted to the extracellular space, resulting in a low conductivity. If, however, the electrodes were farther apart then the current would distribute between the extracellular and intracellular spaces, resulting in a high conductivity.

When Frans arrived at Vanderbilt, he collaborated with Wikswo and me to revise his model. It seemed odd to have the conductivity (a property of the tissue) depend on the electrode separation (a property of the experiment). So we expressed the conductivity using Fourier analysis (a sum of sines and cosines of different frequencies), and let the conductivity depended on the spatial frequency k. Frans’s model already had the conductivity depend on the temporal frequency, ω, because of the muscle fiber’s membrane capacitance. So our revised model had the conductivity σ be a function of both k and ω: σ = σ(k,ω). Our new model had the same behavior as Fran’s original one: for high spatial frequencies the current remained in the extracellular space, but for low spatial frequencies it redistributed between the extracellular and intracellular spaces. The three of us published this result in an article titled “Spatial and Temporal Frequency-Dependent Conductivities in Volume-Conduction Calculations for Skeletal Muscle” (Mathematical Biosciences, Volume 88, Pages 159–189, 1988; the research was done in January 1986, although the paper wasn’t published until April of 1988).

Meanwhile, I was doing experiments using tissue from the heart. My goal was to calculate the magnetic field produced by a strand of cardiac muscle. Current could flow inside the cardiac cells, in the perfusing bath surrounding the strand, or in the extracellular space between the cells. I was stumped about how to incorporate the extracellular space until I read Les Tung’s PhD dissertation, in which he introduced the “bidomain model.” Using this model and Fourier analysis, I was able to derive equations for the magnetic field and test them in a series of experiments. Wikswo and I published these results in the article “A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue” (IEEE Transactions of Biomedical Engineering, Volume 33, Pages 467–469, 1986).

By the summer of 1986 I had two mathematical models for the electrical conductivity of muscle. One was a “monodomain” model (representing an averaging over both the intracellular and extracellular spaces) and one was a “bidomain” model (in which the intracellular and extracellular spaces were each individually averaged over many cells). It was strange to have two models, and I wondered how they were related. One was for skeletal muscle, in which each muscle cell is long and thin but not coupled to its neighbors. The other was for cardiac muscle, which is a syncytium where all the cells are coupled through intercellular junctions. I can remember going into Frans’s office and grumbling that I didn’t know how these two mathematical representations were connected. As I was writing the equations for each model on his chalkboard, it suddenly dawned on me that the main difference between the two models was that for cardiac tissue current could flow perpendicular to the fiber direction by passing through the intercellular junctions, whereas for skeletal muscle there was no intracellular path transverse to the uncoupled fibers. What if I took the bidomain model for cardiac tissue and set the transverse, intracellular conductivity equal to zero? Wouldn’t that, in some way, be equivalent to the skeletal muscle model?

I immediately went back to my own office and began to work out the details. This calculation starts on page 85 of my Vanderbilt research notebook #15, dated June 13, 1986. There were several false starts, work scratched out, and a whole page crossed out with a red pen. But by page 92 I had shown that the frequency-dependent conductivity model for skeletal muscle was equivalent to the bidomain model for cardiac muscle if I set the bidomain transverse intracellular conductivity to zero, except for one strange factor that included the membrane impedance, which represented current traveling transverse to the skeletal muscle fibers by shunting across the cell membrane. But this extra factor was important only at high temporal frequencies (when capacitance shorted out the membrane) and otherwise was negligible. I proudly marked the end of my analysis with “QED” (quod erat demonstrandum; Latin for “that which was to be demonstrated,” which often appears at the end of a mathematical proof).

Two pages (85 and 92) from my Research Notebook #15 (June, 1986).

Frans and I published this result in the Annals of Biomedical Engineering, and it is the paper I cite at the top of this blog post. Wikswo was not listed as an author; I think he was traveling that summer, and when he returned to the lab we already had the manuscript prepared, so he let us publish it just under our names. The abstract is given below:

We compare two models for calculating the extracellular electrical potential in skeletal muscle bundles: one a bidomain model, and the other a model using spatial and temporal frequency-dependent conductivities. Under some conditions the two models are nearly identical, However, under other conditions the model using frequency-dependent conductivities provides a more accurate description of the tissue. The bidomain model, having been developed to describe syncytial tissues like cardiac muscle, fails to provide a general description of skeletal muscle bundles due to the non-syncytial nature of skeletal muscle.

Frans left Vanderbilt in December, 1986 and took a job with the Netherlands section of the company Medtronic, famous for making pacemakers and defibrillators. He was instrumental in developing their deep brain stimulation treatment for Parkinson’s disease. I graduated from Vanderbilt in August 1987, stayed for one more year working as a post doc, and then took a job at the National Institutes of Health in Bethesda, Maryland.

Those were fun times working with Frans Gielen. He was a joy to collaborate with. I’ll always remember than June day when—after brainstorming with Frans—I proved how those two models were related.

Short bios of Frans and me published in an article with Wikswo in the IEEE Trans. Biomed. Eng.,
cited on page 237 of Intermediate Physics for Medicine and Biology.

 

The Magnetocardiogram

I recently published a review in the American Institute of Physics journal Biophysics Reviews about the magnetocardiogram (Volume 5, Article 021305, 2024).

The magnetic field produced by the heart’s electrical activity is called the magnetocardiogram (MCG). The first twenty years of MCG research established most of the concepts, instrumentation, and computational algorithms in the field. Additional insights into fundamental mechanisms of biomagnetism were gained by studying isolated hearts or even isolated pieces of cardiac tissue. Much effort has gone into calculating the MCG using computer models, including solving the inverse problem of deducing the bioelectric sources from biomagnetic measurements. Recently, most magnetocardiographic research has focused on clinical applications, driven in part by new technologies to measure weak biomagnetic fields.

This graphical abstract sums the article up. 

Let me highlight one paragraph of the review, about some of my own work on the magnetic field produced by action potential propagation in a slab of cardiac tissue.

The bidomain model led to two views of how an action potential wave front propagating through cardiac muscle produces a magnetic field.⁵⁸ The first view (Fig. 7a) is the traditional one. It shows a depolarization wave front and its associated impressed current propagating to the left (in the x direction) through a slab of tissue. The extracellular current returns through the superfusing saline bath above and below the slab. This geometry generates a magnetic field in the negative y direction, like that for the nerve fiber shown in Fig. 5. This mechanism for producing the magnetic field does not require anisotropy. The second view (Fig. 7b) removes the superfusing bath. If the tissue were isotropic (or anisotropic with equal anisotropy ratios) the intracellular currents would exactly cancel the equal and opposite interstitial currents, producing no net current and no magnetic field. If, however, the tissue has unequal anisotropy ratios and the wave front is propagating at an angle to the fiber axis, the intracellular current will be rotated toward the fiber axis more than the interstitial current, forming a net current flowing in the y direction, perpendicular to the direction of propagation.^59–63 This line of current generates an associated magnetic field. These two views provide different physical pictures of how the magnetic field is produced in cardiac tissue. In one case, the intracellular current forms current dipoles in the direction parallel to propagation, and in the other it forms lines of current in the direction perpendicular to propagation. Holzer et al. recorded the magnetic field created by a wave front in cardiac muscle with no superfusing bath present, and observed a magnetic field distribution consistent with Fig. 7b.⁶⁴ In general, both mechanisms for producing the magnetic field operate simultaneously.

 

FIG. 7. Two mechanisms for how cardiac tissue produces a magnetic field.

This figure is a modified (and colorized) version of an illustration that appeared in our paper in the Journal of Applied Physics.

58. R. A. Murdick and B. J. Roth, “A comparative model of two mechanisms from which a magnetic field arises in the heart,” J. Appl. Phys. 95, 5116–5122 (2004). 

59. B. J. Roth and M. C. Woods, “The magnetic field associated with a plane wave front propa-gating through cardiac tissue,” IEEE Trans. Biomed. Eng. 46, 1288–1292 (1999). 

60. C. R. H. Barbosa, “Simulation of a plane wavefront propagating in cardiac tissue using a cellular automata model,” Phys. Med. Biol. 48, 4151–4164 (2003). 

61. R. Weber dos Santos, F. Dickstein, and D. Marchesin, “Transversal versus longitudinal current propagation on cardiac tissue and its relation to MCG,” Biomed. Tech. 47, 249–252 (2002). 

62. R. Weber dos Santos, O. Kosch, U. Steinhoff, S. Bauer, L. Trahms, and H. Koch, “MCG to ECG source differences: Measurements and a two-dimensional computer model study,” J. Electrocardiol. 37, 123–127 (2004). 

63. R. Weber dos Santos and H. Koch, “Interpreting biomagnetic fields of planar wave fronts in cardiac muscle,” Biophys. J. 88, 3731–3733 (2005). 

64. J. R. Holzer, L. E. Fong, V. Y. Sidorov, J. P. Wikswo, and F. Baudenbacher, “High resolution magnetic images of planar wave fronts reveal bidomain properties of cardiac tissue,” Biophys. J. 87, 4326–4332 (2004).

The first author is Ryan Murdick, an Oakland University graduate student who analyzed the mechanism of magnetic field production in the heart for his masters degree. He then went to Michigan State University for a PhD in physics and now works for Renaissance Scientific in Boulder, Colorado. I’ve always thought Ryan’s thesis topic about the two mechanisms is underappreciated, and I’m glad I had the opportunity to reintroduce it to the biomagnetism community in my review. It’s hard to believe it has been twenty years since we published that paper. It seems like yesterday.

Biomagnetism: The First Sixty Years

Roth, B. J., 2023,
Biomagnetism: The first sixty years.
Sensors, 23:4218.

The last two blog posts have dealt with biomagnetism: the magnetic fields produced by our bodies. Some of you might have noticed hints about how these posts originated in “another publication.” That other publication is now published! This week, my review article “Biomagnetism: The First Sixty Years” appeared in the journal Sensors. The abstract is given below.

Biomagnetism is the measurement of the weak magnetic fields produced by nerves and muscle. The magnetic field of the heart—the magnetocardiogram (MCG)—is the largest biomagnetic signal generated by the body and was the first measured. Magnetic fields have been detected from isolated tissue, such as a peripheral nerve or cardiac muscle, and these studies have provided insights into the fundamental properties of biomagnetism. The magnetic field of the brain—the magnetoencephalogram (MEG)—has generated much interest and has potential clinical applications to epilepsy, migraine, and psychiatric disorders. The biomagnetic inverse problem, calculating the electrical sources inside the brain from magnetic field recordings made outside the head, is difficult, but several techniques have been introduced to solve it. Traditionally biomagnetic fields are recorded using superconducting quantum interference device (SQUID) magnetometers, but recently new sensors have been developed that allow magnetic measurements without the cryogenic technology required for SQUIDs.

The “First Sixty Years” refers to this year (2023) being six decades since the original biomagnetism publication in 1963, when Baule and McFee first measured the magnetocardiogram. 

My article completes a series of six reviews I’ve published in the last few years. 

• Roth, B. J., 2021, Bidomain modeling of electrical and mechanical properties of cardiac tissue. Biophysics Reviews, 2:041301. 
• Roth, B. J., 2022, The development of transcranial magnetic stimulation. BOHR International Journal of Neurology and Neuroscience, 1:8–20. 
• Roth, B. J., 2023, Can MRI be used as a sensor to record neural activity? Sensors, 23:1337. 
• Roth, B. J., 2023, A mathematical model of mechanotransduction. Academia Biology, 1. 
• Roth, B. J., 2023, Magneto-acoustic imaging in biology. Applied Sciences, 13:3877. 
• Roth, B. J., 2023, Biomagnetism: The first sixty years. Sensors, 23:4218.

Get the whole set! All are open access except the first. If you need a copy of that one, just email me at roth@oakland.edu and I’ll send you a pdf.

I’m not preparing any other reviews, so this will probably be the last one. But, you never know. 

You can learn more about biomagnetism in Chapter 8 of Intermediate Physics for Medicine and Biology.

Enjoy! 

 

The Magnetic Field Associated with a Plane Wave Front Propagating Through Cardiac Tissue

When I was on the faculty at Vanderbilt University, my student Marcella Woods and I examined the magnetic field produced by electrical activity in a sheet of cardiac muscle. I really like this analysis, because it provides a different view of the mechanism producing the magnetic field compared to that used by other researchers studying the magnetocardiogram. In another publication, here is how I describe our research. I hope you find it useful.

Roth and Marcella Woods examined an action potential propagating through a two-dimensional sheet of cardiac muscle [58]. In Fig. 6, a wave front is propagating to the right, so the myocardium on the left is fully depolarized and on the right is at rest. Cardiac muscle is anisotropic, meaning it has a different electrical conductivity parallel to the myocardial fibers than perpendicular to them. In Fig. 6, the fibers are oriented at an angle to the direction of propagation. The intracellular voltage gradient is in the propagation direction (horizontal in Fig. 6), but the anisotropy rotates the intracellular current toward the fiber axis. The same thing happens to the extracellular current, except that in cardiac muscle the intracellular conductivity is more anisotropic than the extracellular conductivity, so the extracellular current is not rotated as far. Continuity requires that the components of the intra- and extracellular current densities in the propagation direction are equal and opposite. Their sum therefore points perpendicular to the direction of propagation, creating a magnetic field that comes out of the plane of the tissue on the left and into the plane on the right (Fig. 6) [58–60].

Figure 6. The current and magnetic field produced by a planar wave front propagating in a two-dimensional sheet of cardiac muscle. The muscle is anisotropic with a higher conductivity along the myocardial fibers.

This perspective of the current and magnetic field in cardiac muscle is unlike that ordinarily adopted when analyzing the magnetocardiogram, where the impressed current is typically taken as in the same direction as propagation. Nonetheless, experiments by Jenny Holzer in Wikswo’s lab confirmed the behavior shown in Fig. 6 [61].

The main references are:

58. Roth, B.J.; Woods, M.C. The magnetic field associated with a plane wave front propagating through cardiac tissue. IEEE Trans. Biomed. Eng. 1999, 46, 1288–1292.

61. Holzer, J.R.; Fong, L.E.; Sidorov, V.Y.; Wikswo, J.P.; Baudenbacher, F. High resolution magnetic images of planar wave fronts reveal bidomain properties of cardiac tissue. Biophys. J. 2004, 87, 4326–4332. 

You can learn more about how magnetic fields are generated by cardiac muscle by reading about what happens at the apex of the heart. Or, solve homework problem 19 in Chapter 8 of Intermediate Physics for Medicine and Biology.