Reading Hodgkin and Huxley 1952 with questions in hand

Translate the signs first

Page 503 defines the membrane potential as outside minus inside. Pages 505 and 506 take $V$ as the displacement from rest, negative for a depolarisation, with inward current positive. That is the reverse of today’s habit. This course writes the depolarisation as $u$, with $u=-V$ and outward current positive.

Every potential changes sign in translation. Table 3 gives the sodium equilibrium potential as $V_{\mathrm{Na}}=-115\ \mathrm{mV}$, which is $u_{\mathrm{Na}}=+115$ here; potassium’s $V_{\mathrm K}=+12$ becomes $-12$, and the leak’s $V_l=-10.613$ becomes $+10.613$. The rate constants follow by replacing $V$ with $-u$, as for $\alpha_m$ below.

After translating, check once: the three ionic currents must sum to zero at rest, and the spike must point in the positive direction. If the computed spike points down, a potential or a current has usually missed one sign change.

Source: A. L. Hodgkin and A. F. Huxley · W. Gerstner

Where the equations and constants are

The potassium rate constants are Eqs. (12) and (13), sodium activation Eqs. (20) and (21), and sodium inactivation Eqs. (23) and (24) on p. 516. Eq. (26) on p. 518 combines them with the capacitive current into the full membrane equation. Page 519 gives the units as millivolts, microamperes per square centimetre, millisiemens per square centimetre, microfarads per square centimetre and milliseconds; the rate constants refer to 6.3 °C, other temperatures scale with $Q_{10}=3$, and the other constants do not depend on temperature.

Table 3 on p. 520 rewards a close look: each constant appears with the value chosen, the experimental mean and the range. The membrane capacity was chosen as 1.0 microfarad per square centimetre against a mean of 0.91 and a range of 0.8 to 1.5; the potassium conductance as 36 against a mean of 34 and a range of 26 to 49. The leak potential of −10.613 carries an asterisk: it was chosen to make the total ionic current zero at rest. These constants are a representative set, not measurements on a single axon.

Page 519 also holds a comparison that is easy to miss. Recomputing a voltage-clamp record at 4 °C with the equations (Fig. 11), the theory gives too short a delay near the sodium potential.

Source: A. L. Hodgkin and A. F. Huxley

How they computed a spike without a computer

Page 523 describes the numerical method: step-by-step integration by Hartree’s method, with steps of 0.01 ms at the start, 0.02 ms on the rising phase and up to 1 ms during the oscillations, keeping the voltage to the last microvolt. Other temperatures need no new rate constants: multiplying the capacity by $\phi=3^{(T-6.3)/10}$ and taking $1/\phi$ milliseconds as the time unit is equivalent to computing at 6.3 °C.

Page 526 says the rate constants rise by a factor of 3.84 at 18.5 °C, whereas $3^{(18.5-6.3)/10}=3.82$. Recomputing the membrane action potential at 18.5 °C with 3.82 gives a spike height of 96.92 mV against 96.8 in Table 4, and a maximum rate of rise of 563 V/s against 564. A 0.5% difference is not enough to tell which factor they actually used.

The propagated action potential is much harder, because the equation contains the unknown conduction velocity. On p. 524 they guess a value of $K$, integrate, see whether the solution runs away, and adjust. Page 528 reports $K=10.47\ \mathrm{ms^{-1}}$ at 18.5 °C, a computed velocity of 18.8 m/s and a measured one of 21.2 m/s. Solving the cable equation for the same axon directly by finite differences, three grids extrapolate to $18.73\pm0.006\ \mathrm{m/s}$ with a spike height of 90.58 mV against 90.5 in Table 4.

Source: A. L. Hodgkin and A. F. Huxley

Check Tables 4 and 5 and the figures

Table 4 on p. 527 gives the shape of the spike. For the row at 6.3 °C and 15 nC/cm², our recomputed spike height, amplitude of the positive phase, peak conductance, rising phase, falling phase, duration of the positive phase, interval from peak potential to peak conductance and maximum rate of rise are 105.41, 11.18, 37.02, 0.593, 2.207, 14.22, 0.148 and 310.9; the table prints 105.4, 11.2, 37.0, 0.59, 2.21, 14.15, 0.15 and 311. The largest difference is in the positive-phase duration, 0.07 ms or about 0.5%.

One entry in the 18.5 °C row does not match: the interval from peak potential to peak conductance is 0.012 ms in the table and 0.004 ms in our run. Both peaks are broad and the interval is only a few microseconds, so the result is very sensitive to time resolution, and the original steps were far coarser.

Table 5 on pp. 532 and 533 gives the sodium and potassium moved per impulse. Integrating over the whole window, the net sodium entry at 6.3 °C is 14.78 pmol/cm², more than the 14.46 in the table. Subtracting the flux that flows at rest anyway over the same window gives 14.459 and 14.328, matching the table’s 14.46 and 14.32 to two decimals. We therefore infer that Table 5 lists the extra movement above the resting flux; this is read back from the numbers, and the paper does not say so in these words.

Figs. 12 and 21 show that at 6 °C a 6 mV depolarisation does not fire and 7 mV does; our threshold is 6.50 nC/cm². Fig. 20 shows refractoriness, Fig. 22 the anode break, and p. 537 discusses why a slowly rising current does not excite.

Source: A. L. Hodgkin and A. F. Huxley

The authors’ own caveats

On p. 541 the authors write: “The agreement must not be taken as evidence that our equations are anything more than an empirical description” of the time course of the permeability changes to sodium and potassium. They also stress that the equations deal only with short-term responses, and only with the isolated squid giant axon.

Pages 542 and 543 list known shortcomings: the membrane is treated as a perfect capacitor, and the delay before the potassium conductance rises is too short. On p. 533 they compare Table 5 with the measurements of Keynes and co-workers and find that the theory gives too little sodium exchange and too much potassium exchange.

Later channel models gave these empirical equations a microscopic reading. In the Markov-chain model of chapter 4 every channel opens and closes independently, and Goldwyn and Shea-Brown explain that it returns to the Hodgkin–Huxley equations as the number of channels goes to infinity. That is later work, not a conclusion of this paper.

Source: A. L. Hodgkin and A. F. Huxley · J. H. Goldwyn and E. Shea-Brown

Worked example: the conduction velocity from K

Problem. Page 528 gives $K=10.47\ \mathrm{ms^{-1}}$ at 18.5 °C, an axon radius $a=238\ \mu\mathrm{m}$, an axoplasm resistivity $R_2=35.4\ \Omega\,\mathrm{cm}$ and a membrane capacity $C_M=1\ \mu\mathrm{F\,cm^{-2}}$. Compute the conduction velocity and compare it with the paper’s 18.8 m/s, the measured 21.2 m/s and our cable-equation 18.73 m/s.

Model. Solve the formula above for $\theta$. Convert everything to centimetres, seconds, ohms and farads: $K=1.047\times10^{4}\ \mathrm{s^{-1}}$, $a=0.0238\ \mathrm{cm}$, $C_M=10^{-6}\ \mathrm{F\,cm^{-2}}$.

Numbers. $\theta^{2}=\frac{1.047\times10^{4}\times0.0238}{2\times35.4\times10^{-6}}=3.52\times10^{6}\ \mathrm{cm^{2}\,s^{-2}}$, so $\theta=1876\ \mathrm{cm/s}=18.76\ \mathrm{m/s}$, which rounds to the paper’s 18.8. The measured 21.2 m/s is 11.5% higher.

Explanation. The reported K and velocity agree with each other, and our independent cable solution of 18.73 m/s is within 0.2% of 18.76 m/s, so the paper’s trial-and-error computation was accurate. The gap to the measurement comes from the equations and constants themselves, not from numerical error.

Check. Units: $\mathrm{s^{-1}\,cm}/(\Omega\,\mathrm{cm}\cdot\mathrm{F\,cm^{-2}})=\mathrm{cm^{2}}/(\mathrm{s}\cdot\Omega\,\mathrm{F})$, and $\Omega\,\mathrm F=\mathrm s$, so the result is in $\mathrm{cm^2\,s^{-2}}$ and its square root is a speed.

Source: A. L. Hodgkin and A. F. Huxley