Problem 30 in Chapter 7 of the 4th edition of

Intermediate Physics for Medicine and Biology is based on a paper by John Clark and Robert Plonsey (

The Extracellular Potential of a Single Active Nerve Fiber in a Volume Conductor, Biophysical Journal, Volume 8, Pages 842-864, 1968). This paper shows how to calculate the extracellular potential from the transmembrane potential, with results shown in our Fig. 7.13.

The calculation involves some mathematical concepts that are slightly advanced for Intermediate Physics for Medicine and Biology. First, the potentials are written in terms for their

Fourier transforms.

Russ Hobbie and I don’t cover Fourier analysis until Chapter 11, so the problem just assumes a sinusoidal spatial dependence. We also introduce

Bessel functions for the first time in the book (to be precise, modified Bessel functions of the first and second kind). Bessel functions arise naturally when solving

Laplace’s equation in

cylindrical coordinates.

I have admired Clark and Plonsey’s paper for years, and was glad to see this problem introduced into the 4th edition of our book.

Robert Plonsey was a professor at

Case Western Reserve University from 1968-1983. He then moved to

Duke University, where he was when I came to know his work while I was a graduate student. I am most familiar with his research on the bidomain model of cardiac tissue, often in collaboration with

Roger Barr (e.g., "

Current Flow Patterns in Two-Dimensional Anisotropic Bisyncytia with Normal and Extreme Conductivities". Biophysical Journal 45: 557-571 and "

Propagation of Excitation in Idealized Anisotropic Two-Dimensional Tissue". Biophysical Journal 45: 1191-1202). Plonsey was elected as a member of the

National Academy of Engineering in 1986 for "the application of electromagnetic field theory to biology, and for distinguished leadership in the emerging profession of biomedical engineering." He retired from Duke in 1996 as the Pfizer Inc./Edmund T. Pratt Jr. University Professor Emeritus of Biomedical Engineering. He has won many awards, such as the 2000 Millennium Medal from the IEEE Engineering in Medicine and Biology Society and the 2004

Ragnar Granit Prize from the Ragnar Granit Foundation.

John Clark is currently a Professor of Electrical and Computer Engineering at

Rice University. He is a Life Fellow in the Institute of Electrical and Electronics Engineers (IEEE) "for contributions to modeling in electrophysiology and cardiopulmonary systems".

One of my earliest papers was an extension of Clark and Plonsey’s model to a strand of cardiac tissue, using the

bidomain model (

A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue. IEEE Transactions of Biomedical Engineering, Volume 33, Pages 467-469, 1986.) The mathematics is almost the same as in their paper--Fourier transforms and Bessel functions--but the difference is that I modeled a multicellular strand of tissue, like a papillary muscle in the heart, that contains of both intracellular and interstitial spaces (the two domains of the “bidomain” model). A comparison of my paper to Clark and Plonsey’s earlier work indicates how influential their research was on my early development as a scientist. They were cited in the first sentence of my paper.