Action Potential Simulator

This is a squid, at 6.3 degrees C

Every number on this page comes from Hodgkin and Huxley’s 1952 fit to the giant axon of Loligo, measured at 6.3 degrees C. That matters more than it sounds. The spike here lasts several milliseconds; a mammalian axon at body temperature fires in about one, because the same gates open and shut several times faster when they are warm. The refractory intervals you can measure with the paired stimulus are correspondingly long.

A real neuron also has channels this model does not contain: A type potassium, persistent sodium, calcium channels, calcium activated potassium, and in many cells a hyperpolarisation activated current. Four state variables reproduce the squid impulse remarkably well and are the right thing to learn the mechanism from. They are not a human neuron, and the millisecond figures here should not be quoted as if they were.

Four equations, and nothing else

One equation says the membrane is a capacitor being charged and discharged by whatever current is flowing: C·dV/dt = I − gNa·m³h·(V − ENa) − gK·n⁴·(V − EK) − gL·(V − EL). Three more say that each gating variable relaxes towards a voltage dependent target: dx/dt = αₓ(1 − x) − βₓ·x.

That is the whole model. Threshold is not in it. All-or-none is not in it. Neither refractory period is in it. Every one of those falls out, and watching them appear from four lines of algebra is the reason this simulation is worth more than a labelled diagram of a spike.

The powers are doing real work. m³ means three independent activation gates must all be open, which produces the S shaped delay at the start of the upstroke that a single gate cannot. h is a separate inactivation gate that closes as the membrane depolarises, which is what makes the sodium current transient instead of sustained. That transience is where refractoriness comes from.

Why a threshold exists at all

There is no threshold parameter to set. What there is, is a race. Depolarise the membrane a little and two things start happening at once: sodium activation m begins to rise, which brings in inward current and depolarises further, and potassium activation n begins to rise while leak current grows, both of which pull back towards rest. Sodium is faster. Potassium and leak are stronger at first because they start from an open baseline.

Below some depolarisation the pull back wins and the membrane settles. Above it the sodium feedback wins, and once it wins it accelerates: more depolarisation opens more sodium channels, which depolarises more. Threshold is the crossover point of that race, which is why it is not a property of the sodium channel. Raise the potassium conductance in the controls and the threshold current in the readouts goes up, without sodium having changed at all.

It is also why threshold is not really a voltage. Set the stimulus just under threshold and watch where the trace peaks: that is the dashed orange line. Now shorten the pulse and the same peak voltage no longer fires, because the gates have had less time at that voltage. The threshold current in the readouts is quoted at the pulse duration you have set, and it moves when you change it.

Why it overshoots towards ENa and undershoots towards EK

A membrane potential always sits somewhere between the equilibrium potentials of the ions that can cross it, pulled towards whichever has most conductance at that instant. At rest potassium and leak dominate, so the membrane sits near −65 mV. At the peak of the spike the sodium conductance is briefly an order of magnitude larger than the potassium conductance, so the membrane is dragged most of the way towards ENa = +50 mV.

It never gets there. By the time sodium conductance peaks, h is already closing and n is already opening, so there is never an instant where sodium is the only conductance. The peak here lands around +40 mV, and it is a good check on any implementation that it stops short of ENa rather than touching it.

The undershoot is the same argument in reverse. Potassium conductance is slow to open and slow to close, so it is still raised well after the potential has come back down. For a few milliseconds the membrane is more potassium selective than it is at rest, so it sits closer to EK = −77 mV than to −65 mV. That is the afterhyperpolarisation, and it ends when n finally relaxes.

Two refractory periods, two different causes

This is the part worth spending time on, and it is what the second stimulus is for. Both pulses are identical, and the only thing you change is the gap between them.

• The absolute refractory period. Bring the interval down to about 4 ms and the second stimulus does nothing, at any amplitude the slider offers. The cause is h: sodium inactivation gates closed during the spike and have not reopened, so most sodium channels are simply unavailable. Current cannot substitute for channels that cannot conduct.
• The relative refractory period. Around 8 to 15 ms something different happens. Leave the stimulus at 10 µA/cm² and the second pulse still fails. Raise it to 60 µA/cm² and it succeeds. Here enough sodium channels have recovered, but two things are working against them: potassium conductance is still elevated, and the membrane is hyperpolarised, which means there is further to go to reach threshold. The cell is harder to fire, not impossible to fire, and the threshold has genuinely risen.

Those are two mechanisms, not one mechanism with two names, but in this model no single control moves only one of them. Lower the sodium conductance far enough and both periods lengthen, because there is less margin while h recovers: at 40 mS/cm² even 60 µA/cm² fails until the gap is about 8 ms rather than 7 ms. Raise the potassium conductance and both lengthen too, because the extra outward current raises the threshold the recovering sodium channels have to beat.

One further detail you may notice: at intervals around 20 ms the second spike can be very slightly taller than the first. That is not an artefact. During the undershoot the membrane is hyperpolarised, so the steady state of h is higher than at rest, and h transiently overshoots its resting value on the way back. Briefly, more sodium channels are available than usual.

What all-or-none actually claims

The claim is not that the spike is digital, or that the model contains a switch. It is that the amplitude is set by the equilibrium potentials and the conductance ratio at the peak, and those barely depend on how hard the membrane was pushed. Try it: at 10 µA/cm² the peak is near +39 mV, and at 40 µA/cm² it is near +41 mV. Four times the stimulus buys about two millivolts.

This is the whole reason neural coding works the way it does. A spike cannot carry the strength of a stimulus in its height, because its height is not available to be modulated. What does change with stimulus strength is the latency, which you can see shorten as you raise the current, and, in a cell driven continuously, the firing rate. Intensity is in the rate. It is also why a spike can propagate down a metre of axon without fading: each patch of membrane regenerates a full sized spike from its own conductances rather than passing along an attenuating signal.

What is not fixed is the amplitude across different membranes. Halve the sodium conductance and the spike gets smaller and eventually fails altogether. All-or-none holds for one membrane across a range of stimuli, which is a much narrower claim than it is usually given credit for.

Reading the conductance panel

The lower panel is the explanation for the upper one. Sodium conductance rises first, steeply, and then falls while the potential is still high, which is inactivation rather than the potential coming back down. Potassium conductance rises later and more slowly and is still falling when the sodium curve has returned to nothing.

Look at the ordering rather than the sizes. If potassium rose as fast as sodium there would be no spike at all, and you can demonstrate that by pushing the potassium conductance up towards 100 mS/cm²: the threshold current climbs, and the spike, when you finally get one, is shorter and smaller.

What this model leaves out

Space. This is one patch of membrane, so nothing propagates. There is no conduction velocity here, no myelin, and no saltatory conduction, all of which need the cable equation and an axon with length.

Temperature. The rate constants are fixed at 6.3 degrees C. Real work applies a Q₁₀ factor of about 3 to the gating rates, which is what turns a several millisecond squid spike into a one millisecond mammalian one.

Ion accumulation and the pump. ENa and EK are held constant, as if the concentrations either side never change. In reality each spike moves a small amount of sodium in and potassium out, and the Na/K ATPase restores it. Over one spike the shift is negligible, which is why holding them fixed is fair; over a long high frequency train in a fine axon it is not.

Other channels, and single channel behaviour. No calcium, no A type potassium, no persistent sodium, no synaptic conductances. The gating variables are also deterministic fractions of a large population; a real patch with a handful of channels is stochastic, and can fire spontaneously from channel noise alone, which nothing here can do.

Common mistakes

• Treating threshold as a fixed voltage. It depends on how fast you arrive, on the pulse duration, and on both conductances. A slow depolarisation can pass the voltage a fast one fired at and never fire, because sodium inactivates on the way.
• Calling the refractory period one thing. Absolute is sodium inactivation. Relative is raised potassium conductance plus the hyperpolarisation. Different causes, different remedies, and only the second one yields to a stronger stimulus.
• Reading these millisecond figures as human values. Squid, 6.3 degrees C. A mammalian spike is about 1 ms wide.
• Saying the sodium channel closes to end the spike. It inactivates, which is a separate gate and a separate state. An inactivated channel cannot be reopened by depolarisation; it has to be repolarised first. That distinction is exactly why there is a refractory period.
• Thinking a bigger stimulus gives a bigger spike. It gives an earlier spike. The height is set by ENa, EK and the conductance ratio.
• Believing the potential reaches ENa. It never does, in this model or in a real axon, because potassium and inactivation are already under way at the peak.
• Assuming the concentration gradients change during a spike. They barely move. The potential swings by 100 mV on a charge transfer far too small to shift the concentrations, which is why the equilibrium potentials can be treated as constants.
• Expecting the undershoot to pass EK. It cannot. Potassium conductance pulls towards EK and no further, and leak is still pulling the other way.