How a Neuron Fires

Touch a hot pan and your hand jerks away before you feel the burn. The message that moved it ran from your fingertip to your spinal cord and back out to the muscles of your arm as a string of electrical blips, each lasting about a millisecond. Every nerve cell, or neuron, sends its messages this way. The blips are called action potentials, or spikes.

Much of what we know about them comes from the squid. A squid escapes by squirting water, and the nerve fibers, or axons, that trigger the squirt are often half a millimeter wide or more, wide enough to slide a wire inside. In a series of papers published in 1952, Alan Hodgkin and Andrew Huxley used these giant axons to work out how a spike is made, and summed it up in a handful of equations . Almost every figure on this page runs those equations.

Here is a stretch of giant axon with two electrodes in it. The left one gives the membrane a quick electrical kick that raises its voltage. The right one records the voltage across the membrane, inside relative to outside. Before you try it, a guess:

Give the membrane a kick of and watch the recording. Compare a 5 mV kick with a 7 mV kick, then a 20 mV one:

At rest the inside of the axon sits at about −65 mV, 65 thousandths of a volt below the outside. A small kick nudges it up and it sags straight back. Every kick smaller than about 6.5 mV does that. Go past it, and the answer isn't a bigger nudge but something else entirely: the voltage shoots up to about +40 mV, swings back down below where it started, and creeps back to rest over the next 15 milliseconds. The faint lines are your earlier tries.

That's a spike. A bigger kick makes it start sooner but hardly any bigger: it always has about the same height and shape. A neuron either fires or it doesn't, which is why spikes are called all-or-nothing. These recordings come from an axon kept at 6.3 °C, cold seawater, so the spike is slower than in your own nerves, where it's over in about a millisecond.Every rate in the model triples for each 10 °C of warming, the factor Hodgkin and Huxley measured, so the same equations also describe a warm axon once their rates are scaled. To see where it comes from, we first need to know why there's a voltage at all.

Salt water and a leaky wall

A neuron is a bag of salt water sitting in more salt water. The salts are dissolved as ions, atoms that carry an electric charge, and the two waters have different mixes. Inside the squid axon there are about 400 millimoles per liter of potassium ions, K⁺, and only about 50 of sodium ions, Na⁺. In the squid's blood outside it's the other way round: about 20 of potassium and 440 of sodium. Pumps in the membrane built up those differences, each one trading three sodium ions out for two potassium ions in, at the cost of one molecule of ATP.The sodium–potassium pump. A large share of the brain's energy budget goes to running it, to restore what spikes and synapses let leak.

The membrane is a film of fat about 5 nanometers thick, and ions can't get through it. They can only cross through channels, proteins that form a narrow pore through the film, each kind letting through mostly one kind of ion. In a resting axon, most of the open channels are potassium channels.

Here is a piece of membrane with potassium on both sides and two open potassium channels. The inside also holds large negative molecules, mostly proteins, that can't fit through. The figure starts with no voltage across the membrane. Watch the voltage on the right, then change how much potassium there is outside:

Potassium flows out through the channels for no other reason than that there's more of it inside: more ions bump into a pore from the inside than from the outside. But every K⁺ that leaves carries a positive charge out and leaves an unmatched negative molecule behind. Charge piles up on the two faces of the membrane, positive outside and negative inside, and that voltage pulls positive ions back in. The push from the concentration difference stays the same while the dark pull from the voltage grows, and the flow stops when they balance.

For these concentrations that happens at about −72 mV. In a real axon it takes a few milliseconds; I've slowed it down a few hundred times so you can watch. It's called potassium's equilibrium voltage, \blue{E_\mathrm{K}}, and it follows from how much potassium is on each side, an equation Walther Nernst wrote down in 1889:

The 24 mV is RT/F at 6.3 °C: the gas constant times the temperature, divided by the charge on a mole of ions. Warmer ions push harder. Give the outside as much potassium as the inside and E_K goes to zero, because there's no difference left to push. Take potassium away and it drops further.

The number of ions that has to move is tiny. To charge the membrane to −72 mV takes about 4,500 potassium ions per square micrometer of membrane, while the first micrometer of axon beneath that square holds about 240 million. The concentrations barely change, so the pumps don't have to keep up spike by spike.

A tug of war

A real resting membrane isn't quite a pure potassium membrane. A few sodium channels are open too, along with a small leak that lumps together everything else, mostly chloride. Each kind of open channel pulls the voltage toward its own equilibrium: potassium toward −77 mV, the value Hodgkin and Huxley measured, which is a little below the −72 mV above; sodium, with far more of it outside, toward +50 mV; the leak toward −54 mV. The more channels of a kind are open, the harder they pull.

The two sliders set what fraction of the sodium and potassium channels are open, and they start at the resting values. Slide sodium up and watch where the voltage settles:

At rest about one potassium channel in a hundred is open, but only about one sodium channel in 11,000. Between them, the potassium and the leak hold nearly all the rope, and the voltage settles at −65 mV. Open more sodium channels and it slides toward +50. With a quarter of the sodium channels open and 6% of the potassium ones, the mix at the peak of a spike, it reads +40 mV.

The rule behind it: the voltage is an average of the three equilibrium voltages, each weighted by how many of its channels are open, measured as a conductance g:

So a spike is a handover. For about a millisecond the membrane becomes a sodium membrane, and then it goes back to being a potassium one. What hands the rope over are the channels themselves.

Gates that feel the voltage

Hodgkin and Huxley never saw a channel. Nobody would for decades. What they could measure was how easily each kind of ion crossed the membrane, and how that changed with the voltage. The trick was the voltage clamp: an amplifier that feeds in whatever current it takes to hold the membrane at a voltage of the experimenter's choosing, and reports that current. By replacing the sodium in the seawater with an ion that can't get through, they could split the current into its sodium and potassium parts.

Here the clamp holds the membrane at rest and then, 1 ms in, steps it to the voltage set by the slider. The plot shows the sodium and potassium conductances that follow. Try a few steps, then switch off inactivation:

Both kinds of channel respond to the step, in different ways. Sodium conductance shoots up within a fraction of a millisecond and then, although the voltage is held steady, dies away again. Potassium conductance rises later and more slowly, with an S-shaped start, and stays up for as long as the step lasts. The bigger the step, the more channels open and the faster they do it.

With inactivation switched off, sodium stays open too. Inactivation is a second, slower process that shuts sodium channels after they open, and it's what makes the sodium current a brief pulse.

Hodgkin and Huxley described all of this with imaginary gates. Each sodium channel has three fast gates that open when the voltage rises and one slow gate that closes when it rises, and ions get through only when all four are open. Each potassium channel has four slow gates. They fitted the gates' speeds to curves like these, with no idea what a channel looked like. When the structure of a potassium channel was finally worked out, in 1998, it turned out to be built from four identical parts around a central pore , one for each of the four gates they had imagined.

The point of no return

Now the threshold makes sense. Let's go back to the first experiment and watch the two currents in the first few milliseconds after the kick: sodium flowing in, which raises the voltage, against potassium and the leak flowing out, which lowers it. Try a kick of 6.3 mV, just under threshold, then 6.7 mV, just over, and watch which current comes out on top:

Right after the kick, the outward current is ahead: the higher voltage pushes potassium out harder than before. But the kick has also started to open sodium channels, and the sodium current climbs. Where it gets ahead, the shading turns orange and the voltage rises instead of falling. A higher voltage opens more sodium channels, which let in more sodium, which raises the voltage further. This loop is what makes the spike explosive. Once it gets going, the sodium current runs off the chart to several hundred µA/cm², hundreds of times what it was at rest.

Below threshold the loop gets going too, but too slowly. Potassium channels and inactivation also respond to the higher voltage, a little later, and they catch up before sodium runs away. Near 6.5 mV the race is close: sodium creeps ahead, and the spike either fizzles or comes late, more than 4 ms after the kick instead of about 1 ms after a strong one.

One spike, channel by channel

Put the pieces together and we can follow a single spike through a patch of membrane. Here are six sodium channels and six potassium channels in a row, with the front half of the patch cut away so you can see into the pores. Each channel's voltage gate is drawn as a dark trapdoor at the inner end of its pore, and each sodium channel also has an inactivation gate on a short tether. The arrows show ions flowing through open channels. Drag the time slider through the spike, and drag the patch to turn it:

Before the kick every voltage gate is shut, and two of the six sodium channels already have their inactivation gate in place: even at rest, about 40% of sodium channels are plugged. The kick raises the voltage, and one by one the sodium trapdoors swing open. Sodium pours in, the voltage climbs, and more of them open.

Within a millisecond the slower gates catch up. Inactivation gates swing into the open sodium pores and plug them, so that at the peak most sodium channels are blocked even though their trapdoors stand open. Meanwhile the potassium trapdoors open, potassium flows out, and the voltage falls. The potassium gates are slow to close again, so for a few milliseconds more of them are open than at rest, and the voltage dips below −65 mV, toward −77. The inactivation gates let go more slowly still, over the next 10 ms or so, and only then is the membrane fully ready to fire again.

I've drawn the channels far closer together than they are in a real membrane, and widened their pores so you can see into them. Real voltage gates are not trapdoors either. The voltage sensors are charged parts of the protein that shift when the voltage across the membrane changes, and twist the pore open. Sodium's inactivation gate isn't a ball on a tether either, but the idea holds: a short loop of the protein, on the inside of the cell, swings in against the inner end of the pore and holds it shut .

A pause after each spike

Those slow gates leave the membrane changed for a while after each spike. Many sodium channels are still plugged, and extra potassium channels are still open. To see what that does, give the axon two kicks: a first one that sets off a spike, then a second kick a little later. Drag the gap between them and the size of the second kick:

For the first 6.5 ms or so after the first kick, no second kick sets off a spike, not even one of 50 mV. This stretch is the absolute refractory period, and it lasts as long as most sodium channels are plugged: there's no sodium current to start the loop. After that a second spike becomes possible, but only with a kick well above the usual 6.5 mV, and it comes out smaller. The extra open potassium channels fight the kick, and fewer sodium channels are ready. Over the next 10 ms or so the needed kick falls back to normal: the relative refractory period. Between about 17 and 22 ms it even dips a little below normal, as the voltage rebounds slightly above rest.

In a warm mammalian neuron the absolute refractory period lasts about a millisecond, which caps how fast a neuron can fire at somewhere near a thousand spikes per second; most fire far more slowly. The pause has a second job, which shows up once a spike starts to travel.

Down the axon

So far the whole membrane has done the same thing at once, as if the axon were a single patch. A real axon is long, and a spike starts at one end and travels. The squid's giant axon runs the length of its body; a nerve in your leg has axons a meter long.

Here is a 6 cm stretch of giant axon, seen from the side. Its membrane is colored where current crosses it, orange where sodium flows in and blue where potassium flows out, and the small arrows inside show current flowing along the axon. The plot below it shows the voltage along its length. I've slowed everything down about 1,200 times. Fire it from the left end, then try the other buttons:

The spike moves as a wave. Its front is orange: sodium pouring in. That positive charge doesn't stay put. It spreads along the inside of the axon both ways, the arrows, and ahead of the spike it charges up the next stretch of membrane until that stretch crosses threshold and fires too. Behind the front the membrane turns blue as the potassium channels open and bring the voltage back down. The current also spreads backward, but the membrane there has just fired and is refractory, so the spike can't turn around. Fire from the middle and two spikes run outward; fire from both ends and they meet and wipe each other out, because each runs into the refractory wake of the other.

In a living neuron a spike usually starts at one place, near the cell body, so it only ever travels one way. Here the model runs at 18.5 °C, the temperature of Hodgkin and Huxley's own measurement, and the spike covers the 6 cm at about 19 m/s. They did this calculation in 1952 on a mechanical desk calculator, and got 18.8 m/s against the 21.2 they measured : the equations didn't just describe a spike, they predicted how fast it would travel.Huxley worked the propagating spike out step by step, turning the handle of a mechanical calculator, over a matter of weeks.

Insulation

Nineteen meters per second takes an axon half a millimeter wide. Most of your axons are under 20 µm across, and many under 1 µm. How fast a spike travels depends on how far its current spreads ahead along the inside, and a thinner axon has a narrower, more resistive core. Our nerves solve this a different way: many of their axons are wrapped in myelin, dozens to over a hundred layers of fatty membrane, wound around it by support cells, with a gap about 1 µm wide every millimeter or so. The gaps are called nodes of Ranvier, and the channels are packed into them.

Here are two axons racing over 2 cm. The top one is bare; drag the slider to change how wide it is. The bottom one is 10 µm wide and wrapped in myelin. I've slowed time down about 600 times, and slow motion slows it ten times more:

A bare 10 µm axon carries the spike at under 3 m/s. Making it wider helps, but slowly: four times as wide is only twice as fast, and even at 500 µm, as wide as the squid's, it manages just under 19 m/s. The myelinated axon, 50 times thinner, finishes first at about 39 m/s.

Myelin works by making the membrane between the nodes thick and hard to charge. Under the sheath, the current spreading from one node barely leaks out and barely has to charge the membrane on the way, so it reaches the next node a millimeter away almost at full strength and pushes it past threshold in a few hundredths of a millisecond. In slow motion you can watch the spike skip from node to node, the orange flashing at one gap after another. That's called saltatory conduction, from the Latin for jumping . It saves energy too, because ions only cross the membrane at the nodes.

For this model I gave the nodes channels ten times as dense as along the squid axon, roughly in line with how densely real nodes are packed, and gave the myelin about 200 times less capacitance than bare membrane. Real myelinated axons in a warm body are faster still: the biggest reach about 120 m/s.

When the insulation fails

Myelin is made by other cells, and it can be lost. In multiple sclerosis the body's own immune system strips patches of myelin from axons in the brain and spinal cord; other conditions damage it in the nerves of the limbs. What happens to a spike that reaches a stripped stretch?

Here is a 10 µm myelinated axon 3 cm long, with one internode, the stretch of sheath between two nodes, damaged. Strip away of its myelin and fire the axon from the left. The plot at the bottom shows how much later the spike arrives 6 mm past the damage:

A little damage costs a little time. The spike reaches the last healthy node before the damaged stretch on schedule, but the next jump is harder: the stripped internode has more membrane to charge, and that charge leaks away through it, so the node beyond takes longer to reach threshold. With 15% of the myelin gone the spike arrives about a third of a millisecond late. Lose much more and the current spreading from the last node runs out before the far node fires, and the spike stops dead. In this model that conduction block sets in once a little over half the myelin is gone.

A third of a millisecond sounds harmless, but a nerve carries many axons side by side, and spikes that should arrive together arrive smeared out, or not at all. That is the kind of slowing that clinicians measure when they time how long a flash of light takes to reach the visual part of the brain.A demyelinated axon can sometimes recover conduction by inserting new sodium channels along the bare stretch. This model doesn't, which is why it blocks so sharply. Warmth makes it worse: a hotter axon has a briefer spike, with less charge to spare, which is one reason symptoms of demyelination can come and go with body temperature. This page describes the mechanism; it can't tell anyone what their own symptoms mean.

Firing again and again

Every spike is the same, so a single spike can only say "now". Yet your fingertip can tell a warm pan from a scorching one, and that difference has to travel up the same axons. In 1926 Edgar Adrian and Yngve Zotterman recorded from single sensory fibers and found the answer: a stronger stimulus doesn't make bigger spikes, it makes more of them per second .

So far we've kicked the membrane once and let go. A sensory ending, or a neuron receiving a steady stream of input, is more like a steady current flowing in. Here the squid membrane gets a current switched on at 10 ms and held. Set it to and watch what the membrane does with it:

Four kinds of behavior come out of one model. A weak current lifts the voltage a few millivolts and holds it there. A little more, and the membrane fires one spike and then sits quietly, still depolarized: inactivation and the extra open potassium channels keep it from firing again. Above about 6.3 µA/cm² the spikes keep coming. Each one recovers, the current pushes the membrane back up past threshold, and it fires again, as regular as a clock. Push very hard and the train stops after the first spike: the membrane is held so high that the sodium channels never recover from inactivation.

Plotting the firing rate against the current gives a neuron's signature, its f–I curve. Drag the marker along it:

Two things stand out. The rate climbs with the current, but gently: four times the current of the first steady firing buys less than twice the rate. And the curve doesn't start at zero. The squid axon can't fire slowly: it goes from silent straight to about 55 spikes a second. Alan Hodgkin, sorting nerve fibers by how they fired, called axons that behave this way class 2 . Many neurons in our own cortex are class 1: their rate starts from zero and creeps up, so they can signal a weak input with an occasional spike.

A simpler neuron

The Hodgkin–Huxley model has four variables and a page of rate functions, and a brain has tens of billions of neurons. People who simulate networks often throw almost all of it away and keep one idea: the membrane adds up its input, leaks some of it back out, and fires when it reaches a threshold. That's the leaky integrate-and-fire model, first written down by Louis Lapicque in 1907 , decades before anyone knew about channels.

Think of the membrane as a bucket with a hole in it. The input is a tap, the water level is the voltage, and the hole lets it drain back toward rest. When the water reaches the line, the neuron fires and the bucket empties. Turn the input up to and try a slower or faster leak:

With the input just under threshold the water rises until the leak through the hole matches the flow from the tap, a little below the line, and stays there forever. Just over, and the bucket fills slowly to the line, fires and starts again: about 20 spikes a second, and fewer still the closer the input sits to threshold. The leak time sets how fast the bucket forgets: a slow leak remembers its input for a long time and fills smoothly, a fast one forgets in a few milliseconds.

This model can fire as slowly as you like, which makes it a class 1 neuron. Put it on the same footing as the squid model, measuring each input as a multiple of the smallest one that makes it fire at all, and compare their curves:

The integrate-and-fire neuron starts at zero and rises smoothly; the squid model is silent until nearly three times its spiking threshold and then jumps. What the simple model keeps is the essential bookkeeping: inputs add up, they leak away, and crossing a threshold produces a spike. What it loses is everything that came from the channels: the shape of the spike, its height, the refractory pause (which it fakes with a fixed dead time of 2 ms), the jump to fast firing, the block at high currents. It also can't travel; it has no axon. For many questions about networks those losses don't matter, and the simple model runs thousands of times faster. For questions about the spike itself, they are the whole story.

Across the gap

An axon ends in hundreds or thousands of small swellings, each pressed against another cell: a muscle fiber, a gland, most often another neuron. The spike can't jump the gap. At almost every one of these contacts, a synapse, the electrical signal is turned into a chemical one and back again.

Here is one synapse between two neurons, with the gap drawn far wider than its real 20 nanometers. Send a spike down into the terminal and follow what happens, in very slow motion:

The spike's arrival opens calcium channels in the terminal, and calcium floods in. Within a fraction of a millisecond the calcium makes a few of the vesicles waiting at the membrane fuse with it and spill their contents into the gap. The molecules of neurotransmitter cross 20 nanometers in microseconds and lock onto receptors on the next cell, which are channels too: binding transmitter opens them. Then the transmitter is pumped away or broken down, the receptors close, and it's over. From spike to receptor takes about half a millisecond, the synaptic delay.

What the next cell feels depends on which channels open. Glutamate, the brain's main excitatory transmitter, opens channels that let sodium in, and the next cell's voltage rises: an excitatory postsynaptic potential, or EPSP. Switch to GABA, the main inhibitory one, and the channels let chloride in instead, and the voltage dips a little: an IPSP. Either way the change is small, typically well under a millivolt at a neuron's cell body, and far from the 15 mV or so it would take to reach threshold. One synapse is a whisper.Some synapses are loud. Where a nerve meets a muscle fiber, one spike releases hundreds of vesicles and the muscle's voltage jumps by tens of millivolts, enough to fire it every time.

Bernard Katz and his colleagues worked out how release happens by recording from a frog's nerve–muscle junction with too little calcium in the bath, so that each spike released only a few vesicles, or none . The responses came in steps. Here is a synapse with six release sites, each releasing one vesicle with some probability when a spike arrives. Lower the release probability to and let the trials pile up:

The responses aren't spread out evenly. They pile up at zero, at about 0.4 mV, at twice that, three times that: whole numbers of a basic step. Each step is one vesicle, which holds a few thousand molecules of transmitter. Katz called the step a quantum. Release is a game of chance played at every site with every spike, and the dashed curve, which assumes nothing but that each site releases independently with the probability you set, matches the pile-ups. With little calcium, most spikes release nothing at all; with plenty, most release four or five vesicles.

That makes calcium a volume knob. It's also one of the places the brain changes itself: strengthening or weakening a synapse, by changing how likely it is to release or how many receptors wait on the other side, is a large part of how learning is thought to be stored.

Adding up

A typical neuron in your cortex receives thousands of synapses . Each nudges its voltage by a fraction of a millivolt, up or down, and the cell has to turn that crowd of whispers into a decision: fire or don't. It does it by adding them up. The sum is taken at the start of the axon, the axon's initial segment, where sodium channels are packed densely enough to start a spike.

For the rest of this page the receiving neuron is the simple leaky integrate-and-fire kind, set up like a cortical cell: it rests at −65 mV, fires at −50, and forgets its input over about 15 ms. To keep the figures readable, I've made each synapse about ten times stronger than a typical one, 6 mV, so a few are enough.Real cortical neurons usually need tens to hundreds of inputs arriving close together to fire.

First, inputs that arrive one after another at the same synapse. Set the gap between them to :

Each input alone, the dashed bumps, lifts the voltage by 6 mV and lets it fade. Far apart, each bump has mostly faded before the next arrives, and the voltage never gets near −50. Close together, they land on each other's shoulders and the sum crosses threshold. This is temporal summation, and the window for it is set by how fast the membrane leaks: a neuron that forgets in 15 ms adds up inputs that arrive within about that long of each other.

Inputs also arrive at different places. Most synapses sit on the dendrites, the branches that fan out from the cell body, and a bump that starts far out along a dendrite shrinks as it spreads toward the cell body, leaking out of the dendrite's membrane on the way. Here are three excitatory synapses on three dendrites and one inhibitory synapse on the cell body. All of them fire together at 5 ms. Drag the orange synapses along their dendrites:

Out at the tip of a dendrite, 600 µm from the cell body, an 8 mV input arrives as barely 1 mV. Drag all three synapses in close and their bumps add up past threshold; spread them out and the same three inputs fall short. This is spatial summation. In this model a bump shrinks by a factor of e, to about a third, for every 300 µm it travels, a figure in the range measured for real dendrites, which depends on how thick they are and how leaky their membrane is.

Now switch on the inhibitory synapse. It sits right on the cell body, next to the axon, and its dip cancels much of what arrives from the dendrites. Inhibitory synapses on the cell body are placed like that in real neurons, as a last word on whether the neuron fires. A single neuron is already a small calculator: it weighs each input by where it lands and when it arrives, subtracts the inhibition, and compares the total to a threshold.Real dendrites do more than leak. Many have their own voltage-gated channels and can fire local spikes, which lets a neuron compute things this simple sum can't.

Back to the hot pan

Now we can follow the whole message. Heat opens channels in the endings of sensory neurons in your fingertip, a steady current flows in, and the endings fire: a little heat, a slow train of spikes, a scorching pan, a fast one. The first, sharp sensation travels toward your spinal cord on thin myelinated fibers at several to tens of meters per second, each spike jumping from node to node. In the spinal cord those fibers end in thousands of synapses. Glutamate spills across, and the neurons there add up the excitation and fire in turn, the ones that drive the muscles of your arm among them. The reflex is wired through your spinal cord, so it doesn't wait for the brain. Your hand is moving before the slower, burning pain arrives a second later, on thin bare fibers that carry their spikes at about a meter per second.

References

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 (1952), 500–544. The paper whose equations run most figures here.
2. D. A. Doyle, J. Morais Cabral, R. A. Pfuetzner, A. Kuo, J. M. Gulbis, S. L. Cohen, B. T. Chait and R. MacKinnon, “The structure of the potassium channel: molecular basis of K⁺ conduction and selectivity”, Science 280 (1998), 69–77.
3. Bertil Hille, Ion Channels of Excitable Membranes, 3rd edition (Sinauer, 2001).
4. A. F. Huxley and R. Stämpfli, “Evidence for saltatory conduction in peripheral myelinated nerve fibres”, Journal of Physiology 108 (1949), 315–339.
5. E. D. Adrian and Y. Zotterman, “The impulses produced by sensory nerve-endings. Part 2. The response of a single end-organ”, Journal of Physiology 61 (1926), 151–171.
6. A. L. Hodgkin, “The local electric changes associated with repetitive action in a non-medullated axon”, Journal of Physiology 107 (1948), 165–181.
7. L. Lapicque, “Recherches quantitatives sur l'excitation électrique des nerfs traitée comme une polarisation”, Journal de Physiologie et de Pathologie Générale 9 (1907), 620–635.
8. J. del Castillo and B. Katz, “Quantal components of the end-plate potential”, Journal of Physiology 124 (1954), 560–573.
9. E. R. Kandel, J. H. Schwartz, T. M. Jessell, S. A. Siegelbaum and A. J. Hudspeth, Principles of Neural Science, 5th edition (McGraw-Hill, 2013).

Nearly everything a nervous system does runs on spikes like these: a few thousand ions crossing each square micrometer of membrane, a handful of gates that move with the voltage, and a loop that turns a nudge into an all-or-nothing signal. It took two people with a squid, an amplifier and a desk calculator to write it down, and their equations still describe it well more than seventy years later.