Neural plasticity · single-cell dynamics

Stochastic Leaky Integrate-and-Fire

τ dV/dt = −(V − V_rest) + RI + σ√(2τ) ξ(t)

Press Start. The cell integrates its input current and fires whenever the membrane reaches threshold.

Cell

τ membrane time constant20 ms How fast the membrane forgets. Sets both the charging speed and how long two inputs can be apart and still add up.

V_rest resting potential −70 mV Where the membrane sits with no input, and where it is put back after a spike.

V_thresh threshold −55 mV Reaching it emits a spike. The model has no spike shape: the upstroke is drawn in, not computed.

R membrane resistance 100 MΩ Converts input current into voltage. A high-resistance cell is easier to drive.

t_ref refractory period 2.0 ms Dead time after a spike. This, and nothing else, is what caps the firing rate at 1/t_ref.

Input

I injected current250 pA Steady current, as in a patch-clamp step.

σ noise amplitude5.0 mV Standard deviation of the membrane fluctuation at rest, standing in for thousands of unresolved synaptic inputs.

Measurement

ISI window5 s How far back the mean ISI and CV look. A short window tracks changes; a long one gives a steadier estimate.

Simulation speed20 ms/frame Milliseconds of model time per animation frame. The physics does not change, only how fast you watch it.

Membrane potential. The dashed line is threshold; the vertical strokes are spikes, drawn in at the moment the threshold is crossed.

membrane V threshold steady state V_rest+RI

The same spikes with the voltage thrown away, which is all a downstream cell gets.

Firing rate, counted in a one-second sliding window.

The f–I curve. The line is the noiseless prediction; the dots are measured, one per second of model time. Move the current slider to fill it in.

predicted measured

spikes0

rate (Hz)0.0

predicted (Hz)49.2

mean ISI (ms)—

CV of ISI—

model time0.0 s

The coefficient of variation of the interval distribution measures how irregular the spiking is. CV near 0 is clock-like; CV near 1 is Poisson-like, which is roughly what cortical cells do in vivo.

What the simulation is doing

The leaky integrate-and-fire neuron is the smallest model that still behaves like a neuron. It throws away the ion channels and keeps two facts: a patch of membrane is a leaky capacitor, and there is a voltage at which the cell emits a spike and starts again. Everything on this page follows from those two sentences.

Between spikes the membrane obeys

τ dV/dt = −(V − V_rest) + R I(t)

The first term is the leak, always pulling V back towards rest at a rate set by τ. The second is the input, pushing it away. With a constant current the two balance at V_∞ = V_rest + RI, drawn on the membrane trace as a horizontal line. If that asymptote is below threshold the noiseless cell never fires, however long you wait; if it is above, the cell fires regularly. The current at which the asymptote exactly touches threshold is the rheobase, and it is the knee in the f–I curve.

Spiking is imposed rather than derived. When V reaches V_thresh the simulation records a spike, resets V to V_rest, and holds it there for the refractory period t_ref. The upstroke drawn on the trace is decoration: the model has no spike shape, and that is a deliberate trade. What it buys is that the interval between spikes can be written down exactly,

T = t_ref + τ ln[ RI / (RI − (V_thresh − V_rest)) ] f = 1000 / T Hz

which is the curve plotted in the f–I panel, and is what the predicted readout reports.

The stochastic part

A real cortical neuron is never given a clean current step. It sits in a barrage of thousands of synaptic inputs that the experiment cannot resolve individually, and their sum looks like noise. This model adds that as a Gaussian term, making the membrane an Ornstein–Uhlenbeck process:

V ← V + (dt/τ)[−(V − V_rest) + RI] + σ √(2dt/τ) N(0,1)

The √(2dt/τ) factor is what makes σ mean something: with it, the membrane at rest has a standard deviation of exactly σ whatever timestep or time constant you choose. Noise changes the behaviour qualitatively, not just cosmetically. Set the current below rheobase, so the prediction is zero, and then raise σ: the cell starts firing anyway, driven entirely by fluctuations that occasionally carry it over threshold. The measured dots lift off the predicted curve near the knee, and the CV climbs towards 1. This is the fluctuation-driven regime, and it is the one cortex appears to operate in.

Parameters

Symbol	Range	What it does
τ	5–50 ms	Membrane time constant, R C. Sets how quickly V approaches its asymptote, and therefore how long ago an input can have arrived and still contribute. Short τ makes a coincidence detector, long τ an integrator. Cortical pyramidal cells sit around 10–30 ms.
V_rest	−80 to −60 mV	Resting potential, set by the balance of leak conductances. Also the reset potential here, so raising it both makes the cell easier to drive and shortens the climb back after each spike.
V_thresh	−65 to −40 mV	Spike threshold. Only the distance V_thresh − V_rest matters for the firing rate, so moving either one has the same effect.
R	20–500 MΩ	Input resistance. Turns picoamps into millivolts: R I in mV is R[MΩ] × I[pA] / 1000. Small cells have high input resistance and are driven by less current.
t_ref	0–10 ms	Absolute refractory period. Sets the ceiling on firing rate, 1/t_ref, which is the flat part at the right of the f–I curve. At t_ref = 0 the curve has no ceiling at all.
I	0–500 pA	Injected current. Rheobase, the current at which firing begins without noise, is (V_thresh − V_rest) × 1000/R and is marked on the f–I panel.
σ	0–15 mV	Standard deviation of the membrane fluctuation at rest. Zero gives the textbook deterministic LIF, where the measured rate should land exactly on the prediction. Raising it lets the cell fire below rheobase and makes the intervals irregular.
ISI window	1–30 s	How much recent history the mean ISI and CV are computed over. It affects the measurement, not the cell. Short windows are noisy estimates; long ones lag behind a parameter change.
speed	2–120 ms/frame	Model milliseconds simulated per animation frame. Purely how fast you watch; the integration timestep is fixed at 0.5 ms.

Things worth doing with it

Turn the noise off. With σ = 0 the spiking becomes perfectly regular, the CV drops to 0, and the measured rate should sit on the predicted one. That agreement is a check on the integration, not a result; press Self-test for the same check done arithmetically.
Find the rheobase. Lower the current until the predicted rate reads zero. The noiseless cell is silent. Now raise σ and watch it fire anyway, irregularly, with a CV near 1. Nothing about the cell changed.
Saturate it. Drive the current to the top. The f–I curve flattens at 1/t_ref, and the CV collapses because a cell spending most of its time refractory has almost no freedom in when to fire.
Sweep the current slowly and let the measured dots accumulate. Below rheobase they sit above the prediction, because noise can only add spikes where the prediction says zero; well above rheobase they converge onto it.

Where the model stops being true

Threshold is a fixed number. In real cells it drifts with recent activity and depends on how fast the membrane is rising, which is why the exponential and adaptive-exponential variants exist.
There is no adaptation. A real pyramidal cell given a long current step fires quickly and then slows down, over hundreds of milliseconds, through calcium-dependent potassium currents. This cell fires at the same rate forever.
The cell is a point. No dendrites, so no distinction between an input at the soma and one two hundred microns out on a branch, and no dendritic spike to back-propagate — which is the signal the STDP rule depends on.
The noise is white and additive. Real synaptic bombardment is correlated in time, and it arrives as conductance rather than current, which lowers the input resistance as it grows. Here R is whatever the slider says, however much noise there is.