FR42 · Neural statistical physics

Hodgkin–Huxley Model: How a Squid Axon Fires an Action Potential

What turns a small shock into an all-or-none spike, and what do the clamp currents reveal? Positive feedback through sodium channels. A shock that depolarizes the membrane past about 6.5 millivolts opens sodium gates faster than potassium gates, so sodium rushes in and the voltage climbs to about 105 millivolts. Slower sodium inactivation and potassium opening then bring it back. Below that threshold the membrane just rings and settles.

Shock, step or voltage-clamp the 1952 Hodgkin–Huxley membrane. Watch the gates open, find the threshold and compare the ion movement with their Table 5.

Peak depolarization $u_{\max}=105\ \mathrm{mV}$

Time from shock to peak $t_{\max}=1.16\ \mathrm{ms}$

Extra sodium entry per impulse $\mathrm{Na^+_{in}}=14.5\ \mathrm{pmol\,cm^{-2}}$

Extra potassium exit per impulse $\mathrm{K^+_{out}}=14.3\ \mathrm{pmol\,cm^{-2}}$

Stimulate the squid axon membrane

Experiment

Shock charge, nC per cm²

15

q

Temperature, °C

6.3

T

Drag the stimulator knob beside the axon to change the stimulus. Pick another protocol to run a different experiment on the same membrane.

Recording, seed and playback

Amplifier noise of the synthetic recording, mV

0.8

\sigma_{\mathrm{amp}}

Model ms shown per second

6

\dot t

Noise seed

42

k

Model, method and sources

$C_M\frac{du}{dt}=-\bar g_{\mathrm{Na}}m^3h\,(u-u_{\mathrm{Na}})-\bar g_{\mathrm K}n^4(u-u_{\mathrm K})-\bar g_l(u-u_l)+I,\quad \frac{dx}{dt}=\phi\,[\alpha_x(1-x)-\beta_x x]$

Rate constants, conductances and reversal potentials are those of the 1952 paper, written for the depolarization $u$ , so $u_{\mathrm{Na}}=115\ \mathrm{mV}$ , $u_{\mathrm K}=-12\ \mathrm{mV}$ and $u_l=10.613\ \mathrm{mV}$ . With the temperature $T$ in degrees Celsius, every rate is multiplied by $\phi=3^{(T-6.3)/10}$ . Because the capacitance is $1\ \mu\mathrm{F\,cm^{-2}}$ , a brief shock of $q\ \mathrm{nC\,cm^{-2}}$ starts the membrane at $u=q\ \mathrm{mV}$ . The equations are integrated with fourth-order Runge–Kutta at 0.002 ms or finer. The synthetic recording adds seeded Gaussian noise to the model trace.

Hodgkin & Huxley (1952), J. Physiol. 117, 500–544 · Goldwyn & Shea-Brown (2011), arXiv:1104.4823v1

Derive the action potential step by step

Model assumptions

The membrane follows the equations of Hodgkin and Huxley (1952) for a space-clamped squid axon, with their rate constants and conductances. The voltage u is the depolarization from rest in mV, positive inside, so it has the opposite sign to the 1952 paper. Currents are per square centimetre and outward positive. The bright trace is the model. The grey trace is a synthetic recording: the model plus amplifier noise drawn from a seeded generator, not measured squid data. The sodium and potassium movement uses the definition of their Table 5, net movement minus the resting flux over the same window.

Synthetic recording over the model truth

Sodium and potassium currents in the model

The same protocol over a range of settings

More numbers and how they are computed

Extreme of the synthetic recording: $0$
Extreme of the model trace: $0$
Amplifier noise added to the recording: $0$
Shock threshold at this temperature: $0$
Rate factor from temperature: $0$
Spikes in the sweep: $0$
Maximum rate of rise: $0$
Window for the ion movement: $0$

Every readout is measured on the model trace. The synthetic recording only shows what an amplifier with this noise would add. The summary curve reruns the model once per point. The exported file holds both traces, the seed and the settings.

Physics tutorial

A spike is a race between sodium and potassium gates

BackgroundThe membrane of a squid giant axon is a thin insulator with sodium and potassium channels in it. In 1952 Hodgkin and Huxley held its voltage fixed, measured the currents that flowed, and wrote four equations that reproduce the action potential they had not fitted.

Why it mattersThe same equations answer several questions at once: why there is a threshold, why a second shock soon after the first does nothing, why a steady current gives a train of spikes, and why releasing a hyperpolarized membrane can fire it.

Start with the essentials

Focus question: What turns a small shock into an all-or-none spike, and what do the clamp currents reveal?
One-sentence intuition: Depolarization opens the fast sodium activation gates $m$ , sodium flows in and depolarizes the membrane further. The slower inactivation $h$ and potassium activation $n$ then end the spike. In this model the feedback wins only above a shock of about $6.5\ \mathrm{nC\,cm^{-2}}$ at 6.3 °C.

Core mathematical model

Current balance of the membrane

C_M\frac{du}{dt}=-\bar g_{\mathrm{Na}}m^3h\,(u-u_{\mathrm{Na}})-\bar g_{\mathrm K}n^4(u-u_{\mathrm K})-\bar g_l(u-u_l)+I

Here $u$ is the depolarization from rest and outward current is positive. Each ionic current is a conductance times the distance from its reversal potential, $u_{\mathrm{Na}}=115\ \mathrm{mV}$ and $u_{\mathrm K}=-12\ \mathrm{mV}$ .

First-order gates

\frac{dx}{dt}=\phi\,[\alpha_x(u)(1-x)-\beta_x(u)\,x]=\frac{x_\infty(u)-x}{\tau_x(u)},\qquad x\in\{m,h,n\}

Each gate relaxes toward its steady value with a voltage-dependent time constant. Near rest, $\tau_m$ is more than twenty times shorter than $\tau_h$ and $\tau_n$ , which is why sodium wins at first.

A brief shock sets the starting voltage

\Delta u=\frac{q}{C_M},\qquad C_M=1\ \mu\mathrm{F\,cm^{-2}}\ \Rightarrow\ 1\ \mathrm{nC\,cm^{-2}}\mapsto1\ \mathrm{mV}

A shock much shorter than any gate time only charges the capacitor, so its charge per area is the initial depolarization.

Common difficulties

The peak is not the sodium potential

Typical misconceptionAt the top of the spike the membrane reaches the sodium reversal potential.

Better mental modelThe model peaks near $105\ \mathrm{mV}$ , about $10\ \mathrm{mV}$ below $u_{\mathrm{Na}}$ , because potassium and leak conductances are already pulling the other way.

Threshold is not a fixed voltage

Typical misconceptionA spike fires whenever the voltage crosses one fixed value.

Better mental modelThe threshold depends on the state of the gates. Shortly after a spike a larger shock fails, and a release from hyperpolarization fires with no depolarizing shock at all.

Run the experiment

01
Fire one spike
Leave the one-spike preset running and watch the gates beside the axon.
What to observe: The shock starts the membrane at $15\ \mathrm{mV}$ . Sodium gates open first, the voltage peaks near $105\ \mathrm{mV}$ about 1.2 ms later, and potassium brings it back below rest.
02
Step below threshold
Choose Just below threshold.
What to observe: A shock of $6\ \mathrm{nC\,cm^{-2}}$ only rings and decays. The model threshold is $6.50\ \mathrm{nC\,cm^{-2}}$ , consistent with the 1952 report that 7 mV fires and 6 mV does not.
03
Clamp the voltage
Choose Voltage clamp separates the currents.
What to observe: At a command of $60\ \mathrm{mV}$ the current is first inward, about $-1.27\ \mathrm{mA\,cm^{-2}}$ , carried by sodium, then outward, about $1.64\ \mathrm{mA\,cm^{-2}}$ , carried by potassium.

Hodgkin–Huxley Model: How a Squid Axon Fires an Action Potential

A spike is a race between sodium and potassium gates

Core mathematical model

Current balance of the membrane

First-order gates

A brief shock sets the starting voltage

Common difficulties

The peak is not the sodium potential

Threshold is not a fixed voltage

Run the experiment

Fire one spike

Step below threshold

Clamp the voltage

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