Every textbook gives it one, usually about −55 mV. It is not a property of the cell. What is actually there is a boundary in state space, and seeing that means giving up the idea of a neuron as a switch.
The standard account goes like this. A neuron sits at about −70 mV. Inputs nudge the voltage up. At roughly −55 mV it fires. Below that nothing, above it a spike — hence all-or-none.
It is a useful account and it is how I carried the idea around for years. It is also wrong in a specific way worth chasing, because chasing it forces you to stop thinking of voltage as a number on a dial and start thinking about where the cell is.
Two observations the standard account cannot hold. First, the threshold moves. Inject current fast and a cell fires near −55 mV; inject the same charge slowly as a ramp and it fires higher, or not at all however far you push. Physiologists have known this since the 1930s and call it accommodation. A property that changes with how you approach it is not a property of the cell.
Second, for some neurons it is not well defined at all. There is a class of cells for which no amount of careful measurement yields a clean threshold voltage, because the mathematics underneath does not contain one.
Take everything you would need to predict what a cell does next and write it as a list of numbers. For a simplified neuron the list is two long: membrane voltage, and a second quantity tracking how much the cell has already reacted.
Two numbers means a plane, and the cell's condition at any instant is one point on it. Not a level on a dial — a position. As time passes the point moves, and the rule that moves it is just ions crossing a membrane: read where you are, compute how fast to go, step.
Start with one dimension, because the two ideas that matter are visible there with nothing in the way. The lower line in the figure is the state space: every position a state could occupy, with an arrow showing which way it moves from there. The curve above it is not part of that space — it is the rule that generates the arrows.
Some positions are resting places — the motion vanishes. More usefully, they come in two kinds. At a green one the arrows converge, so a state knocked slightly off returns. At a red one they diverge, so the slightest disturbance sends it away, which is why nothing is ever found sitting there.
That distinction is why so much of this hinges on what happens to tiny disturbances. It decides whether a state is something you can observe at all.
Now the second dimension. Horizontal is voltage, vertical is the recovery variable. Two curves are drawn: on the purple one the voltage holds still, on the amber one the recovery holds still, and the neuron rests where they cross. That crossing is what −70 mV is. Not a setting, but an attracting position the cell returns to when synaptic noise shoves it around.
Watch a small disturbance in the figure. It is pushed off, and it comes back. Resting potential is measurable precisely because it is stable; a resting state that repelled would be as unobservable as a marble balanced on a hilltop.
Here is where the textbook story breaks, and it breaks immediately.
The figure now holds two states at exactly the same voltage. They differ only in how much recovery is already underway — one has more braking applied than the other. Both are released.
One fires. One does not. They had the same voltage.
The dividing line is a curve in the plane, not a value on the voltage axis. Asking for the threshold voltage is like asking for the latitude of a coastline.
Which explains accommodation, incidentally. Injecting current slowly lets the recovery variable climb alongside the voltage, so the state creeps parallel to the dividing curve and can travel a long way up the voltage axis without ever crossing it. Inject fast and recovery lags, the state moves horizontally, and it cuts straight through. What matters is the crossing, not the height.
In the model on the right that boundary is extraordinarily steep but not perfectly sharp; in others it is anchored on a saddle point and genuinely is a curve. Either way it is a place, not a number.
Look at the path a firing state takes. It does not rise and stop. It swings far out, loops, and returns — and it takes the same wide loop no matter how hard it was pushed, so long as it was pushed past the boundary.
The figure releases several states with quite different shoves. Their excursions are indistinguishable.
That is where all-or-none comes from, and the usual phrasing hides it. A spike's size is not set by the stimulus; it is set by the geometry of the loop, which belongs to the cell. Push harder and the state crosses the boundary sooner — the excursion afterwards is the same shape. The amplitude of the input has been discarded.
Under sustained current the loop closes into a cycle the state runs indefinitely, one spike per lap. Which is also why repetitive firing needs two dimensions: on a line you cannot come back around.
So if there is no threshold voltage, what does turning up the current do?
It takes the resting place away.
Drag the slider. The curve rises, and the green resting point slides along it toward the region where the cell is unstable. Past a critical current there is no attracting rest state left at all — nothing for the cell to settle into — and the only thing remaining is the loop. The cell circles because it has nowhere else to be, and circling is firing.
In this model the resting point does not vanish; it loses stability while staying put, and a cycle appears around it. In other cells it is destroyed outright, colliding with the saddle that anchored the boundary and annihilating with it. Different mechanisms, same consequence: nowhere to settle, so the state circles.
Which of those two routes a cell takes is not a detail, and the figure shows the fingerprint.
When the resting point is destroyed by collision, the state afterwards has to crawl through the region where the two points just vanished, where the speed is still nearly zero. So the first spikes are enormously far apart, and the firing rate climbs smoothly from near nothing. These cells are called Type I, and they can fire arbitrarily slowly.
When the resting point merely destabilises, there is no such bottleneck and no saddle — which means no boundary curve at all. These cells have no threshold in even the generalised sense, and they cannot fire slowly: the cycle is born already turning at a definite rate, so the firing rate jumps from zero to something like forty per second with nothing in between. These are Type II, and they are the cells I promised earlier.
Both transitions have names — a saddle-node bifurcation on an invariant circle, and an Andronov–Hopf bifurcation. The choice between them determines whether the cell acts as an integrator, summing its inputs and reporting the total as a rate, or as a resonator, answering inputs at its preferred rhythm and ignoring the rest. Two different computations out of the same equations under different parameters.
Not that the textbook is useless. For fast synaptic input, in a Type I cell, at a fixed recent history, there is a voltage above which firing becomes reliable, and treating it as a threshold is a good approximation that every working neuroscientist uses.
What changes is the object you are reasoning about. A switch has a setting. A dynamical system has a geometry — resting places, the boundaries between what they capture, the loops that appear when the resting places are taken away, and the parameters that make all of it appear and vanish. With the second picture, accommodation stops being an anomaly, all-or-none stops being a postulate, and Type I versus Type II stops being a taxonomy and becomes a consequence.
I am here because I want to know how much of that transfers. The same apparatus — where are the fixed points, what happens to small perturbations, will this settle or circle or run away — is what decides whether a recurrent network forgets its input or explodes into NaN. Whether that resemblance is deep or merely notational is what I am trying to find out, and it is what most of this site will be about.