No spike, no matter how long you wait

Push current into a neuron and its voltage climbs toward a ceiling set by how much current arrives. If that ceiling sits below the threshold that triggers a spike, more time never helps — the voltage gets closer and closer, forever, without ever quite touching it. Cross the ceiling above threshold, though, and firing turns on and keeps going, like clockwork.

What you're seeing

Current trickles into the neuron and voltage rises — but the neuron also leaks, bleeding voltage away at a rate proportional to how high it already is. Those two effects settle into a tug-of-war with a fixed outcome: voltage climbs toward a steady ceiling Vss = R·I and, left alone, never overshoots it. If that ceiling sits at or below the firing threshold, the neuron spends forever approaching a wall it can never quite reach — this is the whole point: below a sharp line on the current dial, no amount of patience produces a spike. Not "a very long wait." Never.

Nudge the current a hair above that line — the exact boundary is I = Vth/R, the rheobase — and the ceiling now sits above threshold. Voltage reaches it, fires, resets instantly to zero, and starts the climb again: a perfectly regular rhythm. Right near the boundary that rhythm is glacially slow — the interval between spikes grows without any ceiling of its own as you creep toward rheobase from above — but firing itself was never a matter of degree. It's a hard yes or no, decided entirely by whether the ceiling clears the bar.

The rule, exactly. Voltage obeys τ dV⁄dt = −V + RI, solved exactly by V(t) = Vss(1−e^−t/τ), Vss = RI, which never crosses Vth if Vss ≤ Vth. The rheobase I_rh = Vth ⁄ R is the exact firing boundary, and above it the inter-spike interval is T = τ ln( Vss ⁄ (Vss − Vth) ). Verified in node (improve/verify/65-neuron.js): the closed form solves the ODE exactly, checked against an independent finite-difference derivative across four (τ, Vss) pairs; subthreshold current produces exactly zero spikes over a 50τ horizon across four parameter sets, with voltage settling at or under threshold; suprathreshold simulated spike times match the closed-form interval to within 0.1% across four parameter sets and many consecutive spikes; the rheobase boundary is exact — current 3% below never fires over 80τ, 3% above always does; doubling the membrane resistance halves the rheobase to within 1e-12, across three cases. Negative control: the firing rate is exactly zero at or below rheobase and strictly increasing above it, and as current approaches rheobase from ever closer — down to one part in a million — the interval between spikes keeps climbing with no ceiling of its own, confirmed numerically across six shrinking gaps rather than merely asserted.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 65.