Latent Addition

The usual way to find the time constant τ of a nerve axon is to use various stimuli S(t) of different durations, and then fit the data to a strength-duration curve (discussed in Chapter 7 of Intermediate Physics for Medicine and Biology). Suppose, however, that you didn’t want to vary the duration. Perhaps you own a stimulator that can only output square stimulus pulses of one duration. Can you still determine the time constant?

“The Time Constants of Motor and  Sensory Peripheral Nerve Fibers Measured  with the Method of Latent Addition.”

You can, using the method of latent addition. When I worked at the National Institutes of Health in the 1990s, I was part of a team that used the method of latent addition to measure the time constants of peripheral nerve fibers (M. Panizza, J. Nilsson, B. J. Roth, J. Rothwell and M. Hallett, 1994, “The time constants of motor and sensory peripheral nerve fibers measured with the method of latent addition,” Electroenceph. clin. Neurophysiol., Volume 93, Pages 147-154). The abstract is below.

The time constants of motor and sensory fibers in the human ulnar, median and tibial nerves were determined using the method of latent addition. Two square-wave stimuli were applied: the first one was subthreshold and the second, at various delays relative to the first, was adjusted to achieve threshold activation. Strength-delay curves were obtained, from which the time constant was determined using a mathematical model. Sensory fibers had time constants that were about 3 times the time constant for motor fibers. The strength-delay curves gave similar time constants as those obtained from strength-duration curves.

As the abstract explains, the idea is that you stimulate first with a conditioning pulse and then with a test pulse. You can measure the threshold strength of the test pulse as a function of the delay between the two pulses, and from that data you can determine the time constant. Rather than explain the details, I’ll present the method as a homework problem and let you work it all out.

Section 6.12

Problem 37½. Consider a space-clamped axon with a membrane time constant τ. Initially (t V is zero (measured as a change from the rest potential). At t = 0, apply a stimulus of strength S₁ for a duration d. We assume this first stimulus is subthreshold. At t = d the stimulus is turned off, and V decays toward rest. At time t = s (where s > d), apply a second stimulus S₂, again having the same duration d. At time t = s + d, the second stimulus turns off and the V again decays back to rest. The voltage obeys the space-clamped cable equation

τ dV/dt + V = S(t) 

(a) Solve for V(t). You should get four expressions, corresponding to different ranges of time.

(b) Set the voltage at the end of the second pulse equal to the threshold voltage, V(t = s + d) = V_threshold. Solve for S₂ (the threshold strength of the second pulse) as a function of S₁, s, d, τ, and V_threshold.

(c) Find an expression for the threshold strength of the first pulse, S_1,threshold, and write the expression you found in part (b) for S₂ as a function of S₁, s, τ, and S_1,threshold.

(d) Divide through by S_1,threshold to get a new relationship for S₂/S_1,threshold that is a function of s, τ, and S₁/S_1,threshold.

(e) Assume that S₁/S_1,threshold = 0.9. What is the expression for S₂/S_1,threshold? Next express S₁/S_1,threshold and S₂/S_1,threshold as percentages rather than fractions of S_1,threshold. What is the final expression for the percentage S₂ as a function of s?

I will let you work out the intermediate steps, but the final expression is S₂ = 100 – 90 e^–s/τ. This equation can be fit to data for various values of s to get τ. The method worked well, and we were able to determine the time constants of motor and sensory fibers. 

Let me add a few limitations. First, nerve fibers are not space clamped. This is an approximation that works best if the electrodes are not too close to the nerve. Second, the model assumes that for a subthreshold pulse the membrane resistance is fixed, but if you stimulate at 90% of threshold some active behavior and ion channel opening or closing might occur. Third, the model was derived for s > d, but it also works for s = 0 (an unexpected bonus). Finally, I recommend using a short duration pulse, d τ. That way, you can use a wider range of values of s, include some fairly short ones, s τ.

Now I’ll tell you a bit about the other researchers in this project. I’ve talked about Mark Hallett before in this blog, who died last year. He worked in the intramural program for the National Institute of Neurological Disorders and Stroke, one of the National Institutes of Health. He was particularly well known for his research on transcranial magnetic stimulation, but he also was interested in purely electrical measurements of nerves. He led a large lab in the NIH Clinical Center on the Bethesda campus.

Marcela Panizza and Jan Nilsson were visiting researchers from Italy, who would often come to NIH to collaborate with Hallett. They were a husband and wife team with Panizza, a native Italian, being a medical doctor and Nilsson, originally from The Netherlands, being a biomedical engineer. I published several papers with them, and they were a joy to work with. They were the ones who performed the experiments and obtained all the data.

John Rothwell is a leading magnetic stimulation researcher from the Queen Square Institute of Neurology in London. I believe I met Rothwell at a meeting (probably the 1989 International Motor Evoked Potential Symposium in Chicago, which had a big influence on my career), but he wasn’t at NIH working with us, and it was Panizza and Nilsson (or maybe Hallett) who corresponded with him. Rothwell was the one who had proposed the method of latent addition, and had used it initially to determine nerve time constants. 

My job was mainly to write the appendix, which was almost the same as the above homework problem. Now, having solved that problem, any of you, dear readers, could write that.

John Rothwell 

https://www.youtube.com/watch?v=ElXresr5w0Q