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.

David Cohen: The Father of MEG

David Cohen: The
Father of MEG,
 by Gary Boas.

Gary Boas recently published a short biography of David Cohen, known as the father of magnetoencephalography (MEG). The book begins with Cohen’s childhood in Winnipeg, Canada, including the influence of his uncle who introduced him to electronics and crystal radios. It then describes his college days and his graduate studies at the University of California, Berkeley. He was a professor at the University of Illinois Chicago, where he built his first magnetically shielded room in which he hoped to measure the magnetic fields of the body. Unfortunately, Cohen didn’t get tenure there, mainly for political reasons (and a bias against applied research related to biology and medicine). However, he found a new professorship at the Massachusetts Institute of Technology, where he built an even bigger shielded room. The climax of several years of work came in 1969, when he combined the SQUID magnetometer and his shielded room to make groundbreaking biomagnetic recordings. Boas describes the big event this way:

To address this problem [of noise in his copper-coil based magnetic field detector drowning out the signal], he [David Cohen] turned to James Zimmerman, who had invented a superconducting quantum interference device (SQUID) several years before… The introduction came by way of Ed Edelsack, a U.S. Navy funding officer… In a 2024 retrospective about his biomagnetism work in Boston, David described what happened next.

“Ed put me in touch with Jim, and it was arranged that Jim would bring one of his first SQUIDs to my lab at MIT, to look for biomagnetic signals in the shielded room. Jim arrived near the end of December, complete with SQUID, electronics, and nitrogen-shielded glass dewar. It took a few days to set up his system in the shielded room, and for Jim to tune the SQUID. Finally, we were ready to look at the easiest biomagnetic signal: the signal from the human heart, because it was large and regular. Jim stripped down to his shorts, and it was his heart that we first looked at.”

The results were nothing short of astounding; in terms of the signal measured, they were light years beyond anything David had seen with the copper-coil based detector. By combining the highly sensitive SQUID with the shielded room, which successfully eliminated outside magnetic disturbances, the two researchers were able to produce, for the first time, clear, unambiguous signals showing the magnetic fields produced by various organs of the human body. The implications of this were far reaching, with potential for a wide range of both basic science and clinical applications. David didn’t quite realize this at the time, but he and Zimmerman had just launched a new field of study, biomagnetism…

Having demonstrated the efficacy of the new approach… David switched off the lights in the lab and he and Zimmerman went out to celebrate. It was December 31, 1969. The thrill of possibility hung in the air as they joined other revelers to ring in a new decade—indeed, a new era.

“Biomagnetism: The
First Sixty Years.”

The biography is an interesting read. I always enjoy stories illustrating how physicists become interested in biology and medicine. Russ Hobbie and I discuss the MEG in Chapter 8 of Intermediate Physics for Medicine and Biology.You can also learn more about Cohen's contributions in my review article “Biomagnetism: The First Sixty Years.”

Today Cohen is 97 years old and still active in the field of biomagnetism. The best thing about Boas’s biography is you can read it for free at https://meg.martinos.org/david-cohen-the-father-of-meg. Enjoy! 

The Birth of the MEG: A Brief History
 https://www.youtube.com/watch?v=HxQ8D4cPIHI

 

 

An Alternative to the Linear-Quadratic Model

In Section 16.9 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the linear-quadratic model.

The linear-quadratic model is often used to describe cell survival curves… We use it as a simplified model for DNA damage from ionizing radiation.

Suppose you plate cells in a culture and then expose them to x-rays. In the linear-quadratic model, the probability of cell survival, P, is

P = e^{–αD–βD2}

where D is the dose (in grays) and α and β are constants. At large doses, the quadratic term dominates and P falls as P = e^–βD2. In some experiments, however, at large doses P falls exponentially. It turns out that there is another simple model—called the multi-target single-hit (MTSH) model—describing how P depends on D in survival curves,

P = 1 – (1 –e^–α'D)^N

Let’s compare and contrast these curves. They both have two parameters: α and β for the linear-quadratic model, and α' and N in the MTSH model. Both give P = 1 if D is zero (as they must). They both fall off more slowly at small doses and then faster at large doses. However, while the linear-quadratic model falls off at large dose as e^–βD2, the MTSH model falls off exponentially (linearly in a semilog plot).

If α'D is large, then the exponential is small. We can expand the polynomial using (1 – x)^N = 1 – N x + …, keep only the first two terms, and then use some algebra to shown that at large doses P = N e^–α'D. If you extrapolate this large-dose behavior back to zero dose, you get P = N, which provides a simple way to determine N.

Below is a plot of both curves. The blue curve is the linear-quadratic model with α = 0.1 Gy^-1 and β = 0.1 Gy^-2. The gold curve is the MTSH model with α’=1.2 Gy^-1 and N = 10. The dashed gold line is the extrapolation of the large dose behavior back to zero dose to get N. 

If the survival curve falls off exponentially at large doses use the MTSH model. If it falls off quadratically at large doses use the linear-quadratic model. Sometimes the data doesn’t fit either of these simple toy models. Moreover, often P is difficult to measure when it’s very small, so the large dose behavior is unclear. The two models are based on different assumptions, none of which may apply to your data. Choosing which model to use is not always easy. That’s what makes it so fun.