Friday, September 1, 2017

Anode Break Excitation

Problem 57 in Chapter 6 of Intermediate Physics for Medicine and Biology analyzes anode break excitation.
Problem 57. When a squid nerve axon is hyperpolarized by a stimulus (the transmembrane potential is more negative than resting potential) for a long time and then released, the transmembrane potential drifts back towards resting potential, overshoots vr and becomes more positive than vr, and eventually reaches threshold and fires an action potential. This process is called anode-break excitation: anode because the membrane is hyperpolarized, and break because the excitation does not occur until after the stimulus ends. Modify the program in Figure 6.38 [to solve the Hodgkin-Huxley equations], so that the stimulus lasts 3 ms, and the stimulus strength is −0.15 A m−2. Show that the program predicts anode break stimulation. Determine the mechanism responsible for anode break stimulation. Hint: pay particular attention of the sodium inactivation gate (the h gate). You may want to plot h versus time to see how it behaves.
Anode break is interesting because it is an unexpected, peculiar behavior. I first learned about anode break in Hodgkin and Huxley’s Nobel Prize-winning paper A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve (Journal of Physiology, 117:500-544, 1952). They write:
"Anode break excitation. Our [squid] axons with the long electrode in place often gave anode break responses at the end of a period during which current was made to flow inward through the membrane. The corresponding response of our theoretical model was calculated for the case in which a current sufficient to bring the membrane potential to 30 mV above [Hodgkin and Huxley used an unusual sign convention, in which a transmembrane potential “above” rest means hyperpolarization] the resting potential was suddenly stopped after passing for a time long compared with all the time constants of the membrane. To do this, eqn. (26)
was solved with I = 0 and the initial conditions that V = + 30 mV, and m, n and h [gates opening and closing the sodium and potassium channels] have their steady state values for V = + 30 mV, when t = 0. The calculation was made for a temperature of 6 3° C. A spike resulted, and the time course of membrane potential is plotted in Fig. 22A. A tracing of an experimental anode break response is shown in Fig. 22B; the temperature is 18-50 C, no record near 6° being available. It will be seen that there is good general agreement. (The oscillations after the positive phase in Fig. 22B are exceptionally large; the response of this axon to a small constant current was also unusually oscillatory as shown in Fig. 23.)
The basis of the anode break excitation is that anodal polarization decreases the potassium conductance and removes inactivation [of the sodium channel]. These effects persist for an appreciable time so that the membrane potential reaches its resting value with a reduced outward potassium current and an increased inward sodium current. The total ionic current is therefore inward at V = 0 and the membrane undergoes a depolarization which rapidly becomes regenerative.
Russ Hobbie and I have prepared a solution manual for IPMB that we distribute to instructors. Below is a sample from the solution manual for Problem 57 about anode break excitation. We introduce each homework question by a sentence or two explaining why the problem is important. If you are an instructor—Russ and I will ask you to verify this—and would like a copy of the solution manual, contact us by email.
6.57* Sometimes the true power of a mathematical model becomes evident when it correctly predicts unexpected, odd behavior. In this example, students use numerical computations to show that the Hodgkin-Huxley model predicts anode break excitation.
The plot [below] shows the transmembrane potential as a function of time for anode break stimulation. A stimulus of -0.15 A m−2 lasts from 0.5 to 3.5 ms. The action potential fires about 6 ms after the end of the stimulus.

The mechanism for anode break stimulation can be understood from the plots of the gate variables. During the hyperpolarizing stimulus, the h-gate opens to a value of about 0.8, which is higher than its resting value of about 0.6. After the stimulus ends, the h-gate decreases, but very slowly. Once the transmembrane potential returns to rest (about t = 8 ms), the sodium current is larger than at rest because of the still large value of h. This causes the membrane to further depolarize, until it reaches threshold and fires an action potential. The closing of the n-gate during the hyperpolarizing stimulus also contributes to the anode break mechanism, but because the n-gate is slightly faster than the h-gate, the h-gate provides the main effect. Note that the stimulus must be long enough so the h-gate has time to open. Brief stimuli will not work well.

Hodgkin and Huxley observed anode break excitation in their 1952 paper.
I am not surprised that the Hodgkin-Huxley model correctly describes voltage clamp data from the squid axon; it was designed to do that and the model parameters were fit to the voltage clamp data. Moreover, I am not too surprised that the model correctly predicts the action potential; the purpose of Hodgkin and Huxley's research was to understand nerve excitation and conduction. But I am surprised that the model is so good that it can even reproduce oddball behavior such as anode break excitation. That's impressive!

Finally, anode break excitation in nerves is very different from anode break excitation in cardiac tissue. That is another story.

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