 |
“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.
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.
