The Magnetic Field of a Single Axon

When I was in graduate school, working in John Wikswo’s lab at Vanderbilt University, I calculated and measured the magnetic field of a single nerve axon. The calculation makes a nice, although slightly advanced, homework problem for Chapter 8 of the 4th edition of Intermediate Physics for Medicine and Biology.

Section 8.2

Problem 14.5 Use Ampere’s law to calculate the magnetic field produced by a nerve axon.

(a) First, solve Problem 30 of Chapter 7 to obtain the electrical potential inside (V_i) and outside (V_o) an axon. The solution will be in terms of the modified Bessel functions I₀(kr) and K₀(kr), where k is a spatial frequency and r is the radial distance from the center of the axon. Assume the axon has a radius a.

(b) Find the axial component of the current density, J, both inside and outside the axon using J_iz = − σ_i dV_i/dz and J_oz = − σ_o dV_o/dz, where σ_i and σ_o are the intracellular and extracellular conductivities (Eqs. 6.16b and 6.26).

(c) Integrate J_iz over the axon cross-section to get the total intracellular current. Then integrate J_oz over an annulus from a to the radius r, to get the “return current.”

(d) Use Ampere’s law (Eq. 8.9) to calculate the magnetic field. Take the line integral of Ampere's law as a closed loop of radius r concentric with the axon (r greater than a). The current enclosed by this loop is simply the sum of the intracellular and return currents calculated in (c).

In part (c), you will need the Bessel function integrals

∫ I₀(x) x dx = x I₁(x)
∫ K₀(x) x dx = - x K₁(x) .

To check your solution, see Eq. A13 of “The Magnetic Field of a Single Axon” (Roth and Wikswo, Biophysical Journal, Volume 48, Pages 93–109, 1985). However, that paper uses complex exponentials whereas Problem 30 of Chapter 7 uses sines and cosines, introducing a slight difference between your expression and that in Eq. A13 of the Roth and Wikswo paper.

Classical Electrodynamics,
by John David Jackson.

I recall the day I derived this expression for the magnetic field. I was puzzled because another graduate student in Wikswo's lab, James Woosley, had derived a different expression for the magnetic field of an axon using the Biot-Savart law (Sec. 8.2.3). How could there be two seemingly different expressions for the magnetic field? Previous discussions with Prof. John Barach had given me a hint. He had derived two expressions for the magnetic field produced by a battery in a bucket of saline, using either Ampere’s law or the Biot-Savart law, and showed that they were the same (he eventually published this in “The Effect of Ohmic Return Current on Biomagnetic Fields”, Journal of Theoretical Biology, Volume 125, Pages 187–191, 1987). I worried about this problem for some time, until one evening (September 22, 1983; Wikswo insisted that I keep careful records in my lab notebook) I was working on my electricity and magnetism homework and found the solution staring at me: Eq. 3.147 in Jackson’s famous textbook Classical Electrodynamics (here I quote the current 3rd Edition, but at the time I was using my now tattered 2nd Edition with the red cover). This equation defines the Wronskian condition for Bessel functions

I₀(x) K₁(x) + K₀(x) I₁(x) = 1/x .

I didn't have all my work at home, so I remember riding my bike (I didn’t yet own a car) back to the lab in the rain so I could check if the Wronskian would resolve the difference between my expression and Woosley’s. It did; the two expressions were equivalent (in my usually dry notebook, I wrote “HA! It works”). You can calculate the magnetic field using either Ampere’s law or the Biot-Savart law, and you get the same result. To see how these two equations predict the same magnetic field in a slightly easier case (like that considered by Barach), solve Problem 13 of Chapter 8 in the 4th edition of Intermediate Physics for Medicine and Biology.

For those of you interested in Woosley’s expression, you can find its derivation in “The Magnetic Field of a Single Axon: A Volume Conductor Model” (Woosley, Roth, and Wikswo, Mathematical Biosciences, Volume 76, Pages 1–36, 1985). In particular, they state on page 13

If we... rearrange terms, and use a relation which can be derived from the Wronskian... we can show that... Equation (45), derived from Ampere’s law, is identical to... Equation (36), derived from the law of Biot and Savart.

Vanderbilt Research Notebook 4, Page 21, September 22, 1983.