The neuron integrates
Just above threshold, at $I = 1.6$ nA, this cell fires about $18$ Hz. Roughly double the current to $3.0$ nA. The firing rate…
At $1.6$ nA each climb from $-70$ mV to the dashed $-55$ mV line takes $55.5$ ms — the raster ticks at $18.0$ Hz. At $3.0$ nA the climb takes $13.9$ ms: $72.1$ Hz, four times the rate for twice the current. At $1.4$ nA the voltage glides to $-56$ mV and flattens — one millivolt short, forever.
The chart is integrating the equation printed on it, in real time:
This is your accumulation function from calculus, with a leak — a bucket filling at rate $RI$ and draining in proportion to its level. Left alone, $V$ charges toward $V_{rest} + RI$. If that target sits below $-55$ mV — whenever $I < 1.5$ nA, the rheobase — the cell saturates silent. Above it, the time to threshold is $T = \tau \ln\frac{RI}{RI - 15}$, and near rheobase the logarithm blows up: the last millivolt is the slowest, which is why the f–I curve is steepest there.
PREREQUISITEThe climb you are watching is an accumulation function with a leak — the running-total machinery from the integration chapter.About this model
What it represents: the leaky integrate-and-fire neuron — a membrane that integrates input current like an RC circuit, plus one rule stapled on: cross the threshold, count a spike, reset.
What it leaves out: the spike itself — the drawn stroke is decoration, because the model only resets; real spikes come from conductance-based channel dynamics. Also adaptation, which makes real firing rates sag under constant current, and noise, which jitters real interspike intervals. This cell is a metronome; real cells are not.
Source: Gerstner, Kistler, Naud and Paninski, Neuronal Dynamics, Ch. 1.3 (EPFL, free online) on the integrate-and-fire model, and Ch. 1.4 on its limits.
At $I = 1.4$ nA the cell stays silent no matter how long you wait. Why?
Compute the rate at $I = 2.0$ nA: $T = 20 \ln\frac{20}{20-15}$ ms. Roughly what firing rate is that?
Transfer: a slower cell has $\tau = 40$ ms but the same $R$, thresholds, and $I = 2.0$ nA. What happens to its firing rate?
The integrate-and-fire model reduces a neuron to an integrator with a threshold: accumulate, fire, reset, repeat. In this model input strength is coded as firing rate — steepest near rheobase, where the approach to threshold is slowest. Many real neurons are well described by this picture over short stretches; none follow it exactly.