---
title: "Voltage clamp separates the sodium and potassium currents"
description: "Fix the membrane voltage and the capacitive current vanishes, leaving the ionic current. Three gating variables then give the sodium peak after a step to 60 mV."
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language: "en"
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alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/neural-statistical-physics/voltage-clamp-and-gates/"
last_modified: "2026-10-31"
---

[Statistical physics of neurons](https://sandboxphysics.com/stories/neural-statistical-physics/)

L2 · Chapter 2

# Voltage clamp separates the sodium and potassium currents

Sandbox Physics · 2026-10-31

The membrane equation of chapter 1 has three ionic currents, and their conductances change with voltage and time. A change in voltage changes the currents, and the currents change the voltage. Hodgkin and Huxley broke this loop with the voltage clamp: a feedback circuit holds the voltage fixed and records how much current it has to inject. This chapter shows what that experiment measures and the three gating variables they used to describe it.

## Before you begin

The membrane equation and reversal potentials of chapter 1; solving a first-order linear differential equation.

- Explain why the current recorded under voltage clamp is the ionic current.
- Find the steady value and time constant of a gating variable from its rate constants and write its response to a step.
- Compute the peak sodium conductance and the steady potassium current after a step to 60 mV, and compare them with the currents read in the Lab.

## Pin the voltage and the currents come apart

A voltage clamp uses a feedback amplifier to compare the membrane voltage with a command voltage $u_c$ and injects whatever current brings it back. After the step the voltage no longer changes, $du/dt=0$, and the capacitive current is zero. The current the amplifier injects is exactly the ionic current through the channels.

After a step from rest to 60 mV, the recorded current first dips inward and a few milliseconds later turns into a steady outward current. In the earlier papers of the same series, Hodgkin and Huxley had identified the inward part as the sodium current and the late outward part as the potassium current; the 1952 paper summarises those results at its start. Steps of different heights change the size and speed of both currents, and that is all the information needed about how the conductances depend on voltage and time.

The sodium current vanishes at $u_c=E_{\mathrm{Na}}=115\ \mathrm{mV}$ and reverses beyond it. That gives a second way to tell the currents apart, and you can try it in the Lab.

$$
\frac{du}{dt}=0\ \Rightarrow\ I_{\mathrm{ext}}=g_{\mathrm{Na}}\,m^{3}h\,(u_c-E_{\mathrm{Na}})+g_{\mathrm{K}}\,n^{4}\,(u_c-E_{\mathrm{K}})+g_L\,(u_c-E_L)
$$

Under clamp the injected current is the sum of the three ionic currents. $m$, $h$ and $n$ change with time after the step, and so does the current.

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

## Gating variables: each one relaxes at first order

Hodgkin and Huxley wrote the potassium conductance as $g_{\mathrm{K}}n^4$ and the sodium conductance as $g_{\mathrm{Na}}m^3h$, with maxima of 36 and 120 mS/cm². $n$ and $m$ grow with depolarisation and are called activation variables; $h$ falls with depolarisation and is the inactivation variable. Each lies between 0 and 1 and obeys the same first-order equation.

At fixed voltage the solution relaxes exponentially from the old steady value to the new one, at a speed set by the time constant (Eqs. (2.6) and (2.7) of Gerstner and co-workers). The rate constants were fitted at 6.3 °C and are in units of one per millisecond. Every 10 °C of warming multiplies all of them by 3.

Why a cube and a fourth power? After a depolarising step the potassium conductance rises slowly at first and then faster, an S-shaped curve; on repolarisation it falls at once, exponentially. The fourth power of a first-order variable has both shapes: while $n$ climbs exponentially from a small value, $n^4$ at first barely moves. The same holds for $m^3$ in the sodium conductance, and multiplying by the slowly closing $h$ turns it into a current that rises and then falls.

$$
\frac{dx}{dt}=\phi\,\bigl[\alpha_x(u)(1-x)-\beta_x(u)\,x\bigr],\qquad x_\infty=\frac{\alpha_x}{\alpha_x+\beta_x},\quad \tau_x=\frac{1}{\phi\,(\alpha_x+\beta_x)},\quad \phi=3^{(T-6.3\,^{\circ}\mathrm{C})/10\,^{\circ}\mathrm{C}}
$$

Here $x$ stands for $m$, $h$ or $n$, and $\phi$ is the temperature factor, equal to 1 at 6.3 °C.

$$
\begin{aligned}\alpha_m&=\frac{0.1\,(25-u)}{e^{(25-u)/10}-1}, & \beta_m&=4\,e^{-u/18},\\ \alpha_h&=0.07\,e^{-u/20}, & \beta_h&=\frac{1}{e^{(30-u)/10}+1},\\ \alpha_n&=\frac{0.01\,(10-u)}{e^{(10-u)/10}-1}, & \beta_n&=0.125\,e^{-u/80}\end{aligned}
$$

The Hodgkin–Huxley rate constants, with $u$ in mV and depolarisation positive, and rates in $\mathrm{ms^{-1}}$. Eqs. (12), (13), (20), (21), (23) and (24) of the paper use the opposite sign of voltage; substituting $V=-u$ gives the form shown here.

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

## Worked example: a step to 60 mV

Problem. At 6.3 °C the membrane is clamped from rest to $u_c=60\ \mathrm{mV}$. Find the new steady values and time constants of the three gates, the peak sodium conductance and when it occurs, the peak sodium current and the potassium current after 10 ms.

Model. At rest $m=0.0529$, $h=0.596$ and $n=0.318$. After the step each variable relaxes to its new steady value along the exponential below. The sodium conductance is $120\,m^3h$, maximised over time; a current is the conductance times $u_c-E$.

Numbers. The rate formulas give $\alpha_m=3.609$ and $\beta_m=0.143$, so $m_\infty=0.962$ and $\tau_m=0.267\ \mathrm{ms}$; $h_\infty=0.0036$ and $\tau_h=1.05\ \mathrm{ms}$; $n_\infty=0.895$ and $\tau_n=1.78\ \mathrm{ms}$. The sodium conductance peaks at $26.6\ \mathrm{mS\,cm^{-2}}$ at $t=0.67\ \mathrm{ms}$, and the sodium current is $26.6\times(60-115)=-1.46\ \mathrm{mA\,cm^{-2}}$, inward. At 10 ms $g_{\mathrm{K}}=22.9\ \mathrm{mS\,cm^{-2}}$ and the potassium current is $22.9\times(60+12)=1.65\ \mathrm{mA\,cm^{-2}}$, outward.

Explanation. $m$ is four times faster than $h$, so the sodium channels open and are then shut, leaving an inward pulse less than a millisecond wide. $n$ is slower still, so the potassium current takes over only after the sodium current fades. Fast activation, slower inactivation and a still slower potassium current are the three time scales that the next chapter assembles into a spike.

Check. The Lab preset Voltage clamp separates the currents uses this same step. It reads $-1.27\ \mathrm{mA\,cm^{-2}}$ for the total current at the sodium peak: the sodium current of −1.46 plus the potassium current of +0.16 already flowing at that moment and a leak of +0.015. At the end of the step it reads $1.64\ \mathrm{mA\,cm^{-2}}$, which matches the potassium and leak currents above minus the small sodium current that remains.

$$
x(t)=x_\infty(u_c)-\bigl[x_\infty(u_c)-x_\infty(0)\bigr]\,e^{-t/\tau_x(u_c)}
$$

The gating variable after a step. Each of the three gates has its own steady value and time constant.

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

## Try it in the Lab

### [Clamp at the sodium reversal potential](https://sandboxphysics.com/labs/hodgkin-huxley-clamp/)

1. Predict first: if the clamp command is set to 115 mV, will the early inward dip grow, shrink or disappear?
2. Choose the Voltage clamp separates the currents preset and read the early and late currents. Then drag the clamp command to 115 mV and watch the current trace.
3. Explain: 115 mV is the sodium reversal potential, so however far the sodium channels open there is no net sodium current; only the outward potassium and leak currents remain.

## Exercises with explained answers

1. By what factor do the rate constants grow from 6.3 °C to 18.5 °C? Hodgkin and Huxley print 3.84 on p. 526. Is that right?

   Show the answer

   $\phi=3^{1.22}=3.82$. The printed 3.84 differs from the formula by 0.5%. Our model uses the formula and our model notes record the difference.

2. Long after a clamp step to 60 mV, what are the sodium conductance and current?

   Show the answer

   $g_{\mathrm{Na}}=120\times0.962^3\times0.0036=0.39\ \mathrm{mS\,cm^{-2}}$ and the current is $0.39\times(60-115)=-0.021\ \mathrm{mA\,cm^{-2}}$, only 1.5% of the peak. Inactivation shuts almost all of the sodium current off.

3. What is the time constant of $m$ at rest? Why is sodium activation the fastest process in the model?

   Show the answer

   $\alpha_m(0)=2.5/(e^{2.5}-1)=0.224$ and $\beta_m(0)=4$, so $\tau_m=1/4.22=0.237\ \mathrm{ms}$. At the same voltage $\tau_h=8.5\ \mathrm{ms}$ and $\tau_n=5.5\ \mathrm{ms}$, twenty to thirty times slower.

4. Why does the fourth power of n delay the rise of the potassium conductance but not its fall?

   Show the answer

   On the rise, $n$ starts at 0.32 and $n^4$ starts at only 0.01, so it grows noticeably only after $n$ has climbed, giving an S-shaped curve. On the fall, $n$ drops exponentially from near 0.9 and $n^4$ falls at once, roughly as $e^{-4t/\tau_n}$.

[Previous: The membrane stores charge like a capacitor](https://sandboxphysics.com/stories/neural-statistical-physics/the-membrane-as-a-capacitor/) [Next: The action potential from one equation, all or none](https://sandboxphysics.com/stories/neural-statistical-physics/the-action-potential/)

## 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.