The Magnetic Field of an Axon: Ampere versus Biot-Savart

In Homework Problem 14 of Chapter 8 in Intermediate Physics for Medicine and Biology, Russ Hobbie and I ask the reader to calculate the magnetic field produced by the action current in a nerve axon using the law of Biot and Savart, and to compare it with the result found using Ampere’s law. This is a useful exercise, but I’ve always been uncomfortable with one aspect of the calculation. I’ll explain what I mean in today’s post.

In the homework problem you assume the intracellular current is uniform along one section of the axon, and is zero elsewhere (this is a big assumption, but it lets you derive an analytical solution). When you calculate the magnetic field using the law of Biot and Savart, you get a smooth, continuous function valid for any position along the axon. However, when you use Ampere’s law the result seems like it should be discontinuous. For some positions the intracellular current contributes to the current enclosed by the Amperian loop, but for other positions the intracellular current is zero and contributes nothing. How can the magnetic field be smooth and continuous if the intracellular current is discontinuous?

Below I’ll show you an elegant way to resolve this paradox. The bottom line is that the magnetic field you calculate using Ampere’s law is the same continuous function that you’d get using the law of Biot and Savart. I’ll change the details so that you don’t solve the homework problem in the book exactly, but the fundamental idea works for the book’s problem too.

Let the intracellular current be I₀ for −b < x < b, and zero elsewhere, where x is the position along the axon (see the figure above). The axon is surrounded by saline with conductivity σ. The calculation consists of four steps: First calculate the extracellular voltage V_e in the saline, then differentiate V_e to find the x-component of the extracellular current density J_x, next integrate the current density across the area of the Amperian loop to get the return current I_ret (that part of the extracellular current that passes through the loop), and finally determine the net current enclosed by the loop and calculate the magnetic field B.

Case 1: x > b

Begin by calculating the magnetic field at point (x,y) where x > b, so you’re in the region where there is no intracellular current threading the Amperian loop (the green circle in the figure above, having radius r). To determine the extracellular voltage, realize that current crosses the membrane at only two locations: x = b (a positive point source when viewed from the extracellular space) and x = −b (a negative point source). The voltage produced by a point source is inversely proportional to the distance, so

(If you don’t follow how I derived this expression, see Section 7.1 in IPMB.)

To find the x-component of the current density, differentiate V_e with respect to x, multiple by σ, and add a minus sign.

The most difficult part of the calculation is integrating the current density over the area enclosed by the loop to find the return current. This is a two-dimensional integral, with an area element of 2πy dy and limits of the integration from 0 to r (see the figure on the right).

You can look up the needed integral in an integral table, evaluate it at the limits, and fill in any missing steps. The result is

The second term in the brackets is equal to minus one and the fourth term is equal to plus one, which cancel. There is no intracellular current for x > b, so the current enclosed by the loop is just the return current. The magnetic field is

(I switched the order of the two surviving terms and brought the minus sign inside the bracket.) This is exactly the solution you get using the law of Biot and Savart; if you don’t believe me, calculate it yourself.

Case 2: −b < x < b

The calculation for the extracellular potential, current density, and return current in the region −b < x < b is exactly as before

Now comes the interesting part. The second term in the brackets is not equal to minus one as it was earlier. Because x < b the numerator is negative, but the denominator is squared inside the square root so it is positive; the term becomes plus one. Because x > −b the fourth term is also equal to plus one, as before (both the numerator and denominator are positive). These two terms no longer cancel, so the return current becomes

This is different than we found for x > b. Don’t panic; remember that the total current enclosed by the loop is the return current plus the intracellular current. In this case, the intracellular current I₀ exactly cancels the +2 term inside the brackets in the expression for I_ret (remember, there is a minus one half in front of the brackets), so the enclosed current is just what we had for the x > b case, and the magnetic field is again

The equation for the magnetic field is the same for any value of x (you can check the x < −b case yourself; you’ll get the same equation). The “magic” comes from the term (x−b)/√(x−b)² switching from negative to positive, which is exactly what it had to do to cancel the intracellular current. The enclosed current is continuous even though the intracellular and return currents are not. The magnetic field calculated using Ampere’s law is a smooth function for all x, and is equivalent to the result obtained using the law of Biot and Savart (as it must be). Nice!

One limitation of this calculation is that the action potential has intracellular current only in one direction; a dipole. For an action potential propagating down an axon, the intracellular current first goes in one direction and then in another as the membrane depolarizes and then repolarizes; a quadrupole.

I’ll leave the calculation for this more complicated current distribution as an exercise for you. I suggest you do the calculation using both the Biot-Savart law and Ampere’s law. Enjoy!

This and That

Most of my blog posts are about a single topic related to Intermediate Physics for Medicine and Biology, but today’s post consists of a dozen brief notes. Read to the end for your Christmas gift.

1. Previously in this blog, I’ve mentioned the website medicalphysicsweb.com. That site no longer exists, but was replaced by a page dedicated to medical physics on the Physics World website. Former medicalphysicsweb editor Tami Freeman is still in charge, and the new site is useful for instructors and students using IPMB. I get updates by email.
2. I taught my Biological Physics class (PHY 3250) this fall at Oakland University, and videos of the class meetings are posted on Youtube. The quality is poor; often the blackboard is difficult to read. But if you want to see how I teach the first half of IPMB, take a look.
3. On the Wednesday before Thanksgiving, my class played Trivial Pursuit IPMB. The students had fun and earned extra credit. Earlier this year my wife and I bought two Trivial Pursuit games—complete with game boards and pieces—at a garage sale for a couple dollars, so I was able to accommodate twelve students. You can download the questions at the IPMB homepage. 
4. In 2012 I wrote about the website iBioMagazine. I no longer can find it, but I believe the website iBiology is related to it. I recommend iBiology for physics students trying to improve their knowledge of biology.
5. Today The Rise of Skywalker opens. It’s the final episode in the Star Wars trilogy of trilogies. I remember watching the first Star Wars movie as a teenager in 1977; I can’t wait to see the latest.
6. From the Oakland University campus you can see the Headquarters and Tech Center of Fiat Chrysler Automobiles. This fall the folks at Chrysler introduced a new advertising blitz called the Dodge Horsepower Challenge. Each week they presented a new physics problem about cars, and those who answered correctly were entered in a drawing for a 2019 Dodge Challenger SRT Hellcat Redeye. They needed a physicist to review their problems and solutions, and somehow I got the job. You can find the problems on Youtube, presented by a colorful wrestler named Goldberg.
7. Lately I’ve been republishing these blog posts on medium.com. Oddly, among my most popular stories on Medium is the one about the Fourier series of the cotangent. It has over 160 reads while others that I think are better have just a handful.
8. The Blogger software keeps its own statistics, and claims that my most popular post is about Frank Netter, with over 6000 page views. I think its popularity has to do with Search Engine Optimization.
9. The IPMB Facebook page now has over 200 members. Thanks everyone, and let’s try to finish 2020 with 220.
10. Regular readers know that my two favorite authors are Isaac Asimov and Charles Dickens. Recently I’ve discovered another: P. G. Wodehouse. His books about Bertie Wooster and Jeeves are hilarious, and a joy to read.
11. If you want to know what books I’m reading, you can follow my Goodreads account. Often books in the category Read More Science become subjects of blog posts.
12. Finally, here’s your Christmas present. Last year Oakland University Professor Andrea Eis organized an event—called Encountering the Rare Book—to highlight the OU Kresge Library’s special collections. I was one of the faculty members Andrea asked to select a book from the collection and write a brief essay about it. I chose A Christmas Carol and my essay is below. A Merry Christmas to you all.

I read A Christmas Carol every December, so I was delighted to find a first edition of Charles Dickens’ classic novella in the Rare Book Collection of Kresge Library. I never tire of Dickens’ “ghostly little book.” I love his language, humor, and wonderfully drawn characters.

I enjoy the Ghosts of Christmas Past and Present best; the Ghost of Christmas Yet to Come frightens me. One of my favorite scenes is when Scrooge’s nephew Fred and the Ghost of Christmas Present collude with Topper to catch Fred’s sister-in-law (the plump one with the lace tucker) during a game of blind man’s bluff. I’m a cheapskate focused on my work, so I have a certain sympathy for Ebenezer. I read the book each year as a reminder to not become a “tight-fisted hand at the grindstone.”

All of us in higher education ought to recall the words of the Ghost of Christmas Present at the end of Stave 3, as he revealed two wretched children hidden in his robes: “This boy is Ignorance. This girl is Want. Beware them both…but most of all beware this boy.”

I sometimes wonder if I should have been born a Victorian. I love their physics—Faraday, Maxwell, and Kelvin are my heroes—as well as their literature. A Christmas Carol was published in 1843, the same year that James Joule measured the mechanical equivalent of heat, George Stokes analyzed incompressible fluids, and Ada Lovelace wrote the first computer program. Holding a first edition in your hands connects you to that time; as if Dickens, like Marley’s Ghost, “sat invisible beside you.” The library’s copy has lovely illustrations, which at that time had to be painstakingly hand-colored.

I intend to continue reading A Christmas Carol each year, with the hope that I, like Scrooge, can “become as good a friend, as good a master, and as good a man, as the good old city knew.”

Encountering the Rare Book, an exhibition celebrating
the Special Collections in Kresge Library at
Oakland University, organized by Andrea Eis.

How Russ Hobbie Came to Write Intermediate Physics for Medicine and Biology

In the preface of Intermediate Physics for Medicine and Biology, Russ Hobbie writes

Between 1971 and 1973 I audited all the courses medical students take in their first 2 years at the University of Minnesota.

I was amazed at the amount of physics I found in these courses and how little of it is discussed in the general physics course. I found a great discrepancy between the physics in some papers in the biological research literature and what I knew to be the level of understanding of most biology majors or premed students who have taken a year of physics. It was clear that an intermediate level physics course would help these students. It would provide the physics they need and would relate it directly to the biological problems where it is useful.

This book is the result of my having taught such a course since 1973…

Want to hear more about how Russ came to write IPMB? You can! Russ was interviewed for the University of Minnesota Oral History Project. Below is an excerpt about the origin of the book.

Interview with Russell Hobbie

Interviewed by Professor Clarke A. Chambers. University of Minnesota

Interviewed on September 29, 1994. University of Minnesota Campus

…I wrote Al Sullivan who was the assistant dean of the Medical School asking if it was possible to snoop around over there. Al asked me to have lunch with him one day—it was in October—and said, “What you really ought to do is to attend Medical School.” I said, “I can’t. I’m director of undergraduate studies in Physics. I’m teaching a full load, which is a course each quarter. There’s just no time to do that.” He said, “You could just audit things and skip the labs.” So, for two years, I did that….

I sat through the remainder of the year in embryology, and biochemistry, and anatomy, and pathology, and physiology, and then, in the second year, the organ systems, the neuro psych, the cardiovascular, the pulmonary, the renal, the dermatology, the bones, the GI [gastrointestinal] ….

I really got a fairly good knowledge there and found that there was just too much physics ever to fit it into the pre-med physics course. I also found that there was a tremendous gap between what we teach the pre-meds, who will take one year of physics and that’s it, and what you found in the physiology and biophysics research literature. I convinced the Physics Department that I ought to try teaching a course to try to fill that gap, a 5000 level course that has a year of general physics and a year of calculus as a prereq[uisite] that would appeal to the physiologists and so on. Probably around 1972 or 1973, I started teaching that course, developing it as I went. That turned into a book [Intermediate Physics for Medicine and Biology] that was published by Wiley in 1978 with a second edition about 1988. I’m trying, without much success, to do a third edition right now….

…after I started teaching the course, I can remember Professor Jack Johnson from Physiology wanted to come and sit it in; and I was quite nervous about this because I was afraid I might get some of the physiology wrong. He reminded me, in no uncertain terms, that I’d been sitting through his course and turnabout really was fair play…

I think that, as I look back at my own career, the thing that I think that was most important, that has certainly given me the greatest intellectual satisfaction is the book.

If you can’t get enough of Russ, watch him in this video about his computer program MacDose.

Russell Hobbie demonstrates MacDose, part 1.

https://www.youtube.com/watch?v=eZGfHBYMuHg&t=8s

Russell Hobbie demonstrates MacDose, part 2.

https://www.youtube.com/watch?v=Rwp8eeKFycQ

Russell Hobbie demonstrates MacDose, part 3.

https://www.youtube.com/watch?v=hB4U7T0gH6M

The Dimensionality of Color Vision in Carriers of Anomalous Trichromacy

Russ Hobbie and I discuss color vision in Chapter 14 of Intermediate Physics for Medicine and Biology.

14.15 Color Vision

The eye can detect color because there are three types of cones in the retina, each of which responds to a different wavelength of light (trichromate vision): red, green, and blue, the primary colors...

From Photon to Neuron:
Light, Imaging, Vision,
by Philip Nelson.

Imagine my shock when I read about possible tetrachromate vision in Philip Nelson’s book From Photon to Neuron. I downloaded the article Phil cited—“The Dimensionality of Color Vision in Carriers of Anomalous Trichromacy,” by Gabriele Jordan, Samir Deeb, Jenny Bosten, and John Mollon, Journal of Vision, Volume 10, doi:10.1167/10.8.12, 2010—and quote the abstract below.

Some 12% of women are carriers of the mild, X-linked forms of color vision deficiencies called “anomalous trichromacy.” Owing to random X chromosome inactivation, their retinae must contain four classes of cone rather than the normal three; and it has previously been speculated that these female carriers might be tetrachromatic, capable of discriminating spectral stimuli that are indistinguishable to the normal trichromat. However, the existing evidence is sparse and inconclusive. Here, we address the question using (a) a forced-choice version of the Rayleigh test, (b) a test using multidimensional scaling to reveal directly the dimensionality of the participants' color space, and (c) molecular genetic analyses to estimate the X-linked cone peak sensitivities of a selected sample of strong candidates for tetrachromacy. Our results suggest that most carriers of color anomaly do not exhibit four-dimensional color vision, and so we believe that anomalous trichromacy is unlikely to be maintained by an advantage to the carriers in discriminating colors. However, 1 of 24 obligate carriers of deuteranomaly exhibited tetrachromatic behavior on all our tests; this participant has three well-separated cone photopigments in the long-wave spectral region in addition to her short-wave cone. We assess the likelihood that behavioral tetrachromacy exists in the human population.

Flatland: A Romance of Many Dimensions,
by Edwin A. Abbott. If you haven’t
read Flatland, ask Santa for a copy
this Christmas (or click on this link).

Wow! IPMB claims that “other animals...[can] have more than three types [of cones]” but offers no hint that people can. How cool is that? This is like finding someone who lives in a four-dimensional world. Would a tetrachromat explaining color to me (a trichromat) be like Square in Flatland describing a sphere to the Triangles? (Square was thrown in prison for that!) What would life be like with four color receptors (red, green, blue, and orange) instead of three? Would you perceive a fundamentally different world, or would any difference be subtle? Could we use CRISPR or some other gene editing tool to expand our color vision? (I’ll take a dozen different cones, please.) Is it fair that only women can be tetrachromats? (No, but let’s not go there.) Is tetrachromacy a superpower?

San Diego woman Concetta Antico diagnosed with “super vision.”

I don’t know how accurate this news story is, but it’s interesting.

https://www.youtube.com/watch?v=QjocalycuiQ