Determining the Site of Stimulation During Magnetic Stimulation of a Peripheral Nerve

As December draws to a close and I reflect on all that’s happened over the last twelve months, I conclude that 2012 has been a good year. For me, it has also marked some important anniversaries. Thirty years ago (1982) I graduated from the University of Kansas with a bachelors degree in physics. Twenty-five years ago (1987) I obtained my PhD from Vanderbilt University. And twenty years ago (1992) I was at the National Institutes of Health in Bethesda, Maryland working on magnetic stimulation of nerves.

Today I want to focus on one particular paper published in 1992 that examined magnetic stimulation of a peripheral nerve: “Determining the Site of Stimulation During Magnetic Stimulation of a Peripheral Nerve” (Electroencephalography and Clinical Neurophysiology, Volume 85, Pages 253–264). To understand this article, we must first examine Frank Rattay’s analysis of electrical stimulation. Rattay showed that excitation along a nerve axon occurs where the “activating function” –λ² d²V_e/dx² is largest, with λ the length constant, V_e the extracellular potential produced by a stimulating electrode, and x the distance along the axon. Homework Problem 38 in Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology guides you through Rattay’s derivation. In 1990, Peter Basser and I showed that this result also holds during magnetic stimulation. What is magnetic stimulation? In Chapter 8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Faraday’s law of induction, and then write

Since a changing magnetic field generates an induced electric field, it is possible to stimulate a nerve or muscle cells with out using electrodes…One of the earliest investigations was reported by Barker et al. (1985) who used a solenoid in which the magnetic field changed by 2 T in 110 μs to apply a stimulus to different points on a subject’s arm and skull.

The main difference between Rattay’s analysis of electrical stimulation and our analysis of magnetic stimulation was that Rattay expressed his activation function in terms of the electric potential produced by the stimulus electrode, whereas Basser and I considered the induced electric field along the axon, E_x, and wrote the activating function as λ² dE_x/dx. The most interesting feature of this result is that stimulation does not occur where the electric field is strongest, but instead where its gradient along the axon is greatest. In the early 1990s, this result was surprising (in retrospect, it seems obvious), so we set out to test it experimentally.

Basser and I both worked in NIH’s Biomedical Engineering and Instrumentation Program, and we had neither the expertise nor facilities to perform the needed experiments, but we knew who did. Since arriving at NIH in 1988, I had been working with Mark Hallett and Leo Cohen to develop clinical applications of magnetic stimulation. Also collaborating with Hallett was a delightful couple visiting from Italy, Jan Nilsson and his wife Marcela Panizza. Under Hallett’s overall leadership, with Nilsson and Panizza making the measurements, and with me occasionally making suggestions and cheerleading, we carried out the key experiments that confirmed Basser’s and my prediction about where excitation occurs. These studies were performed on human volunteers (at that time there were people who made their living as paid normal volunteers in clinical studies at NIH) and were carried out in the NIH clinical center. The abstract of our now 20-year old paper said

Magnetic stimulation has not been routinely used for studies of peripheral nerve conduction primarily because of uncertainty about the location of the stimulation site. We performed several experiments to locate the site of nerve stimulation. Uniform latency shifts, similar to those that can be obtained during electrical stimulation, were observed when a magnetic coil was moved along the median nerve in the region of the elbow, thereby ensuring that the properties of the nerve and surrounding volume conductor were uniform. By evoking muscle responses both electrically and magnetically and matching their latencies, amplitudes and shapes, the site of stimulation was determined to be 3.0 ± 0.5 cm from the center of an 8-shaped coil toward the coil handle. When the polarity of the current was reversed by rotating the coil, the latency of the evoked response shifted by 0.65 ± 0.05 msec, which implies that the site of stimulation was displaced 4.1 ± 0.5 cm. Additional evidence of cathode- and anode-like behavior during magnetic stimulation comes from observations of preferential activation of motor responses over H-reflexes with stimulation of a distal site, and of preferential activation of H-reflexes over motor responses with stimulation of a proximal site. Analogous behavior is observed with electrical stimulation. These experiments were motivated by, and are qualitatively consistent with, a mathematical model of magnetic stimulation of an axon.

Rather than describe this experiment in detail, I will let you analyze it yourself in a new homework problem (your three-days-late Christmas present). It is similar to a problem from an exam I gave to my biological physics (PHY 325) students.

Section 8.7

Problem 26 ½ (a) Rederive the cable equation for the transmembrane potential v (Eq. 6.55) using one crucial modification: generalize Eq. 6.48 to account for part of the intracellular electric field that arises from Faraday induction and therefore cannot be written as the gradient of a potential,

 Assume you measure v relative to the resting potential so Eq. 6.53 becomes j_m = g_m v, and let the extracellular potential be small so v_i = v. Identify the new source term in the cable equation (the “activating function” for magnetic stimulation), analogous to v_r in Eq. 6.55.

(b) Let

Calculate the activating function and plot both the electric field and the activating function versus x.

(c) Suppose you stimulate a nerve using this activating function, first with one polarity of the current pulse and then the other. What additional delay in the response of the nerve (as measured by the arrival time of the action potential at the far end) will changing polarity cause because of the extra distance the action potential must travel? Assume a = 4 cm and the conduction speed is 60 m/s.

At about the same time as we were doing this study, Paul Maccabee and his colleagues at the SUNY Health Science Center in Brooklyn were carrying out similar experiments using an in-vitro pig nerve model (a nerve in a dish), and came to similar conclusions (“Magnetic Coil Stimulation of Straight and Bent Amphibian and Mammalian Peripheral Nerve In Vitro: Locus of Excitation,” Journal of Physiology, Volume 460, Pages 201–219, 1993). Our paper was published first (Yes!!!) but their results were cleaner and more elegant, in part because they didn’t have the complication of the nerve being surrounded by irregularly shaped muscles and bones. Our paper has been fairly influential (53 citations to date in the Web of Science), but theirs has had an even greater impact (147 citations). A year later Maccabee and I together published a study of a new magnetic stimulation coil design.

What has happened to this cast of characters in the last 20 years? Hallett and Cohen remain at NIH, still doing great work. Nilsson is a biomedical engineer and Panizza is a neurophysiologist in Italy. Basser is at NIH, but is now with the Eunice Kennedy Shriver National Institute of Child Health and Human Development, where he works on MRI diffusion tensor imaging. Paul Maccabee is a neurologist and the Director of the EMG Laboratory at SUNY Brooklyn. I left NIH in 1995, and am now at Oakland University, where I teach, do research, and write a blog so I can wish readers of Intermediate Physics for Medicine and Biology a Happy New Year!

Royal Institution Christmas Lectures

With Christmas approaching, my attention naturally turns to the Royal Institution Christmas Lectures. The Royal Institution (Ri) website states

The Ri is an independent charity dedicated to connecting people with the world of science. We’re about discovery, innovation, inspiration and imagination. You can explore over 200 years of history making science in our Faraday Museum as well as engage with the latest research, ideas and debates in our public science events.

We run science programmes for young people at our Young Scientist Centre, present exciting, demonstration-packed events for schools and run mathematics masterclasses across the UK.

We are most famous for our Christmas Lectures which were started by Michael Faraday in 1825. Check out the 2011 Lectures here and don’t miss them this Christmas on BBC Four.

Anyone can join the Ri. If you’re interested in how the world works, or how to make it work better through science, the Ri is the place for you.

The 2012 Christmas Lectures, “The Modern Alchemist,” will be broadcast on BBC Four on December 26, 27, and 28 at 8pm. Don’t get BBC Four? Neither do I. But that’s OK, because you can watch the Christmas Lectures at the Ri website. In fact, you can watch the Christmas Lectures from past years too. You will have to open an account, which means you will need to give them your email address and other information, but you don’t need to pay anything; it’s free. Kind of like a Christmas present.

My favorite lecture is from 2010. Mark Miodownik stars in “Why Elephants Can’t Dance but Hamsters Can Skydive.” He talks about an issue discussed in Homework Problem 28 in Chapter 2 of the Fourth edition of Intermediate Physics for Medicine and Biology. Russ Hobbie and I ask the reader to analyze how fast animals of different sizes fall. In “Why Elephants Can’t Dance,” Miodownik performs a brilliant demonstration using two spherical animals—one about the size of a hamster, and the other about the size of a dog—made of some sort of jello-like gel. Suffice to say, the hamster-sized blob of gel does just fine when it hits the ground after a fall, but the dog-sized blob has some problems. The audience for the lecture is mostly children, but as Dickens wrote “it is good to be children sometimes, and never better than at Christmas.” The entire lecture is about why size matters in the animal kingdom.

Miodownik then talks about another topic in animal scaling that Russ and I don’t mention in our book, although I often bring it up when I teach Biological Physics at Oakland University. In two animals with the same shape but different sizes, their weight increases as the cube of their linear size, but the cross-sectional area of their legs increases as the square of the size. Therefore, a large animal has a harder time supporting its weight than a small animal does. Miodownik demonstrates this with two rubber pig-like spheres with rubber legs attached. The small sphere easily stands on its legs, while the large sphere just collapses. As the video says, size really does matter. Of course, elephants solve this problem by making their legs thick, which is why they can’t dance.

I recommend watching “Why Elephants Can’t Dance” while reading Chapter 2 of Intermediate Physics for Medicine and Biology. It will help you understand animal scaling.

Enjoy the Royal Institution Christmas Lectures, and have a Merry Christmas.

Dynamics: The Geometry of Behavior

Dynamics: The Geometry of Behavior,
by Ralph Abraham and Christopher Shaw.

When I was working at the National Institutes of Health in the 1990s, I ran across a wonderful series of books from the Visual Mathematics Library that had a big impact on the way I thought about math. Dynamics: The Geometry of Behavior, by Ralph Abraham and Christopher Shaw, was published in four volumes: 1 Periodic Behavior, 2 Chaotic Behavior, 3 Global Behavior, and 4 Bifurcation Behavior. The fascinating feature of these books was that they contained almost no equations; everything was explained in pictures. At first glance, they look like comic books, but on closer inspection you realize that the math is presented in a very accurate and rigorous way. There are lots of plots of phase planes, and drawings of experimental apparatus that are being modeled by the math. There is hardly a page without pictures, and 90% of many pages are filled with illustrations. I highly recommend these books for anyone interested in developing an intuitive feeling for nonlinear dynamics (which should be everyone).

Their foreword begins

During the Renaissance, algebra was resumed from Near Eastern sources, and geometry from the Greek. Scholars of the time became familiar with classical mathematics. When calculus was born in 1665, the new ideas spread quickly through the intellectual circles of Europe. Our history shows the importance of the diffusion of these mathematical ideas, and their effects upon the subsequent development of the sciences and technology.

Today, there is a cultural resistance to mathematical ideas. Due to the widespread impression that mathematics is difficult to understand, or to a structural flaw in our educational system, or perhaps to other mechanisms, mathematics has become an esoteric subject. Intellectuals of all sorts now carry on their discourse in nearly total ignorance of mathematical ideas. We cannot help thinking that this is a critical situation, as we hold the view that mathematical ideas are essential for the future evolution of our society.

The absence of visual representations in the curriculum may be part of the problem, contributing to mathematical illiteracy, and to the math-avoidance reflex. This series is based on the idea that mathematical concepts may be communicated easily in a format which combines visual, verbal, and symbolic representations in tight coordination. It aims to attack math ignorance with an abundance of visual representations.

In sum, the purpose of this series is to encourage the diffusion of mathematical ideas, by presenting them visually.

Dynamics: The Geometry of Behavior,
by Ralph Abraham and Christopher Shaw.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I do not suppress mathematical expressions. I suspect many of our readers would claim we have too many, rather than too few, equations. Nevertheless, we try to convey our subject in figures as well as math, visually as well as symbolically. We discuss nonlinear dynamics in Chapter 10, and we have some state space figures that are similar to those found in Abraham and Shaw (although they use 4-color figures—green, red, blue, and black—while we use the less attractive black and white). I believe all the illustrations in Abraham and Shaw are hand-drawn, giving them a charm that often is lacking in this age of computer-generated drawings. Unfortunately, Russ and I never cited Abraham and Shaw. One reason I write this blog is to alert our readers to books and articles that don’t appear in the pages of Intermediate Physics for Medicine and Biology.

Dynamics: The Geometry of Behavior is one of those rare gems that you should become familiar with, both for what it can teach and also for its beauty. To learn more about The Visual Mathematics Library, see Ralph Abraham’s webpage.

Lord Rayleigh, Biological Physicist

Theory of Sound,
by Lord Rayleigh.

I am a big fan of Victorian physicists. Among my heroes are Faraday, Maxwell, and Kelvin. Another leading Victorian was John William Strutt, also known as Lord Rayleigh (1842–1919). Russ Hobbie and I mention Rayleigh in the 4th edition of Intermediate Physics for Medicine and Biology, in the context of Rayleigh Scattering. In Chapter 15 on the interaction of x-rays with matter, we write

A photon can also scatter elastically from an atom, with none of the electrons leaving their energy levels. This (γ, γ) process is called coherent scattering (sometimes called Rayleigh scattering), and its cross section is σ_coh. The entire atom recoils; if one substitutes the atomic mass in Eqs. 15.15 and 15.16, one finds that the atomic recoil kinetic energy is negligible.

In Rayleigh scattering, the oscillating electric field in an electromagnetic wave exerts a force on electrons. These electrons are displaced by this force, and therefore oscillate at the same frequency as the wave. An oscillating charge emits electromagnetic radiation. The net result is scattering of the incident wave. If the electrons are free, this is known as Thomson scattering. If the electrons are bound to an atom, and the frequency of the light is less than the natural frequency of oscillation of the bound electrons, then it is known as Rayleigh scattering. Light scattering is complicated when the wavelength is similar to or smaller than the size of the scatterer, because light scattered from different regions within the particle interfere. However, Rayleigh scattering assumes that the wavelength is large compared to the size of the scatterer, so interference is not important.

Rayleigh scattering not only plays a role in the scattering of x-rays, but also is responsible for the scattering of visible light. The Rayleigh scattering cross section varies as the 4th power of the frequency, or inversely with the 4th power of the wavelength. When we look at the sky, we see the scattered light. Since the short wavelength blue light is scattered much more than the long wavelength red light, the sky appears blue.

Lord Rayleigh made other important contributions to physics. For example, he wrote an influential book on the Theory of Sound, and he won the Nobel Prize in 1904 for his discovery of the element argon. He succeeded Maxwell as the Cavendish Professor of Physics (see this video: https://www.youtube.com/watch?v=tkwLavjqsBI to learn more).

Was Rayleigh a biological physicist? Yes! Rayleigh was one of the first to explain how we localize sound. His Duplex Theory suggests that we can determine the direction a sound came by sensing the arrival time difference at each of our two ears for low frequencies, and sensing the intensity difference between the ears for high frequencies.

Lord Rayleigh was born 170 years ago this fall (November 12, 1842). J. J. Thomson studied under Rayleigh, and Ernest Rutherford studied under Thomson. Previously in this blog, I described how I am descended, academically speaking, from Rutherford. This means Lord Rayleigh is, again academically speaking, my great-great-great-great-great-great grandfather.