---
title: "Reading Hodgkin and Huxley 1952 with questions in hand"
description: "A reading guide to the 1952 paper: translate its signs, find the equations, constants and method by page, check Tables 4 and 5, then read the authors’ caveats."
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L4 · Chapter 8

# Reading Hodgkin and Huxley 1952 with questions in hand

Sandbox Physics · 2026-10-31

Nearly every equation in the first seven chapters comes from the paper Hodgkin and Huxley published in 1952 in volume 117 of the Journal of Physiology, pages 500 to 544. The scanned pages are free to read on PubMed Central. This chapter does not retell the paper; it offers a way to read it. First translate its signs into the convention of this course, then find the equations, constants and method by page, check them item by item against the recomputations of the earlier chapters, and finally see how the authors themselves limit what their equations mean.

## Before you begin

The membrane equation, gating variables and spike of chapters 1 to 3; chapter 4’s point that these equations average over many channels. Keep the scanned pages of the paper open alongside.

- Translate the paper’s voltage and current signs into a convention with depolarisation and outward current positive.
- Find the membrane equation, the constants of Table 3, the temperature scaling and the numerical method by page.
- Use Tables 4 and 5 and the conduction velocity to check a recomputation, and say where the remaining differences come from.
- Restate how the authors limit the meaning of their equations on p. 541.

## 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.

$$
u=-V,\qquad I_{\mathrm{ours}}=-I_{\mathrm{HH}},\qquad \alpha_m=\frac{0.1\,(V+25)}{e^{(V+25)/10}-1}=\frac{0.1\,(25-u)}{e^{(25-u)/10}-1}
$$

Left: how this course’s depolarisation $u$ and currents relate to the paper’s. Right: the paper’s Eq. (20) for $\alpha_m$ (in $\mathrm{ms^{-1}}$), rewritten in terms of $u$.

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.

$$
K=\frac{2R_2\theta^{2}C_M}{a}\quad\Longrightarrow\quad \theta=\sqrt{\frac{K\,a}{2R_2C_M}},\qquad \phi=3^{(T-6.3)/10}
$$

Left: the relation on p. 524 between $K$ and the conduction velocity $\theta$, with $a$ the axon radius, $R_2$ the resistivity of the axoplasm and $C_M$ the membrane capacity per unit area. Right: the temperature factor.

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

## Try it in the Lab

### [Reproduce the threshold of Fig. 21](https://sandboxphysics.com/labs/hodgkin-huxley-clamp/)

1. Predict first: following Fig. 21 of the paper, what will shocks of 6 and 7 nC/cm² do?
2. Choose the Just below threshold preset and read the peak depolarisation. Then drag the shock charge to 7 nC/cm² and compare the peaks and the times to peak.
3. Explain: the model’s threshold is 6.50 nC/cm², right between the paper’s 6 and 7. The closer the shock is to threshold, the longer the spike takes to reach its peak.

### [Check the 18.5 °C rows of Tables 4 and 5](https://sandboxphysics.com/labs/hodgkin-huxley-clamp/)

1. Predict first: raising the temperature from 6.3 °C to 18.5 °C, will the spike height and the sodium entry per impulse go up or down?
2. Choose the One all-or-none spike preset and read the peak and the sodium and potassium movements. Then drag the temperature to 18.5 °C and read them again.
3. Explain: the spike height falls from about 105 mV to about 97 mV and the sodium entry from about 14.5 to about 4 pmol/cm², close to the 96.8 of Table 4 and the 3.99 of Table 5. At higher temperature sodium inactivation and potassium activation catch up sooner, so the spike is lower and shorter and fewer ions move.

## Exercises with explained answers

1. At some moment the paper has $V=-60\ \mathrm{mV}$ and a positive sodium current. In this course’s convention, what is the depolarisation and which way does the sodium current flow?

   Show the answer

   $u=+60\ \mathrm{mV}$: the membrane is depolarised by 60 mV. Inward current is positive in the paper, so sodium is flowing in; in this course that current is negative.

2. Why does the leak potential of −10.613 mV in Table 3 have five significant figures when the other constants have two or three?

   Show the answer

   It is not a measurement but a value computed from the other constants: it is chosen to make the sodium, potassium and leak currents sum exactly to zero at rest. The extra digits only make that equality hold in the computation.

3. Compare Table 5’s net sodium entry of 14.46 pmol/cm² at 6.3 °C with the measured net entry of 3.5 pmol/cm² in squid at 22 °C quoted on p. 533. Why can you not simply say the theory overestimates fourfold?

   Show the answer

   The temperatures differ. The rate constants grow with temperature with a Q10 of 3, so the spike is shorter when warm and fewer ions move. In the same table the membrane action potential at 18.5 °C gives only 3.99 and the propagated one a net entry of 4.33, already of the same size as 3.5. Match the temperature and the type of stimulus before comparing.

4. Use the sentence on p. 541 to explain why agreement between the equations and experiment does not show that the gating variables correspond to real molecular structures.

   Show the answer

   The authors state that the agreement only shows the equations to be an empirical description of the time course of the permeability changes. Many different microscopic mechanisms can give the same macroscopic conductance time course; settling the mechanism needs experiments that see single channels, which came later.

[Previous: Why a power law cannot prove the brain is critical](https://sandboxphysics.com/stories/neural-statistical-physics/when-a-power-law-misleads/)

## Sources and what we recomputed

1. [A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology 117, 500–544 (1952), PMC1392413](https://pmc.ncbi.nlm.nih.gov/articles/PMC1392413/)

   We read every page, pp. 500 to 544, from the scanned pages in PubMed Central. Sign convention on pp. 505 and 506: V is the displacement from rest with depolarisation negative, and inward current is positive. Rate constants: Eqs. (12), (13), (20), (21), (23) and (24). Eq. (26) and Table 3 on pp. 518 to 520: the full membrane equation and its constants at 6.3 °C, with the leak potential chosen so that the current is zero at rest. Numerical method on p. 523. Figs. 12, 13 and 21: threshold between 6 and 7 mV. Table 4 on p. 527: spike shape. Table 5 on pp. 532 and 533: sodium and potassium movements. Fig. 20: refractory period. Fig. 22: anode break. P. 541: their warning that the equations are an empirical description. We recomputed Tables 4 and 5 for the membrane action potentials, not the propagated one.

2. [W. Gerstner, W. M. Kistler, R. Naud and L. Paninski, Neuronal Dynamics, Cambridge University Press (2014), online edition, Sections 2.1 and 2.2](https://neuronaldynamics.epfl.ch/online/Ch2.S1.html)

   Section 2.1: the Boltzmann factor, Eq. (2.1), and the Nernst potential, Eq. (2.2), with the sodium example near +67 mV and the potassium example near −83 mV. Section 2.2 (Ch2.S2.html): charge conservation on the membrane, Eqs. (2.3) and (2.4), the three ionic currents, Eq. (2.5), first-order gating, Eq. (2.6), and the response to a voltage step, Eq. (2.7). This book writes u as the membrane potential with depolarisation positive, the convention our course uses.

3. [J. H. Goldwyn and E. Shea-Brown, The what and where of adding channel noise to the Hodgkin–Huxley equations, arXiv:1104.4823v1; PLoS Computational Biology 7, e1002247 (2011)](https://arxiv.org/abs/1104.4823v1)

   Version 1, 25 April 2011, read in full. Eq. (1): the Hodgkin–Huxley equation with depolarisation positive. The Markov-chain description of independent channels and its deterministic limit. Eqs. (10) to (13): for a fixed voltage history, the number of open channels is binomial with mean and variance set by the gating variables. Figs. 1 to 3: 100 µm² holds 6000 sodium and 1800 potassium channels, 10 µm² holds 600 and 180, with a spontaneous spike from channel noise alone. Their comparison of current, subunit and conductance noise. The label of Eq. (13) in v1 says sodium where the formula is the potassium variance.