The leaky integrator with a threshold and a reset — the simplest neuron that spikes. Sum your inputs; the membrane charges toward V_rest + R·I with time constant tau; the instant it touches V_th it fires an all-or-none spike and snaps back to V_reset. Below a rheobase current it never fires at all; above it, the train is periodic and the firing rate rises with current — the f–I curve. Down the center: current goes in, the engine integrates, the spike train comes out. The blue team builds it; the red team breaks it.
source Lapicque, Recherches quantitatives sur l'excitation electrique des nerfs traitee comme une polarisation (1907), J. Physiol. Pathol. Gen. 9:620–635; Eng. tr. Brunel & van Rossum, Biol. Cybern. (2007) — pubmed.ncbi.nlm.nih.gov/17968583. AMBER: no stable open link to the 1907 French original; params illustrative (typical cortical values), no medical advice. Rendered, not quoted.
One line of biophysics, an RC membrane driven by a current:
tau dV/dt = −(V − V_rest) + R·I ; if V ≥ V_th → spike, V ← V_reset
Fixed parameters (illustrative AMBER): tau=10 ms, R=100 MΩ, V_rest=−65, V_th=−50, V_reset=−70 mV. For the current input, live state:
| quantity | value |
|---|
Lapicque (1907) modeled excitation as the polarization of an RC circuit: charge the membrane until a threshold, then it discharges. That is the whole idea — sum, cross, fire, reset.
Strip everything else from a spike and this is what survives. the-hodgkin-huxley resolves the spike into four gated ionic currents and derives the all-or-none event; the integrate-and-fire throws the spike's shape away and keeps only threshold + reset. The full neuron is this sphere's premise; the caricature is what computational neuroscience runs by the million.
The blue team's live check: re-integrate the membrane and confirm the model still fires periodically above rheobase and its ISI matches the closed form. If red removes the reset, this badge is where it shows.
The neuron takes one driving input current I (nA) across a membrane with fixed passive parameters. Everything downstream is determined by whether the steady value V_rest + R·I clears threshold:
| symbol | meaning | value |
|---|---|---|
| I | input current | you set it ↓ |
| tau | membrane time constant | 10 ms |
| R | membrane resistance | 100 MΩ |
| V_rest | resting potential | −65 mV |
| V_th | spike threshold | −50 mV |
| V_reset | post-spike reset | −70 mV |
The threshold gap is V_th − V_rest = 15 mV, so the rheobase current is 15 mV / 100 MΩ = 0.15 nA. Feed I into the panel below.
Drag the current. Below 0.15 nA the membrane charges and stalls; above it, a periodic train appears and packs tighter as I rises.
The trace is a live forward integration of the ODE — charging, threshold crossings (spikes), and resets — never a lookup. Dashed line = V_th; the closed-form ISI is checked against it.
What the machine produces, proven: a rheobase at 0.15 nA (below it, zero spikes forever); above it a periodic train whose interval is the closed form T = tau·ln((V_inf−V_reset)/(V_inf−V_th)) with V_inf = V_rest + R·I; and an f–I curve that rises monotonically with current (62, 118, 221 Hz at I = 0.20, 0.30, 0.50 nA). Every number self-checks against numeric integration.
The blue team's witness (left) re-integrates these live; the red team (right) tries to make the train lie.
And without a refractory floor the f–I curve is unbounded — as I→∞ the ISI → 0. Real neurons cap at f_max = 1/t_ref. That ceiling is deliberately omitted here AMBER so the pure leaky-integrator law is visible; it is a known limit, not a bug hidden in the engine.
"Lapicque wrote down the integrate-and-fire neuron." Cut. He modeled excitation as RC polarization and gave the strength–duration law; the explicit leaky-integrate-and-fire formalism and name came later (Brunel & van Rossum trace the lineage).
"The threshold is a real voltage the membrane reaches." Cut. In this model the threshold is an imposed rule, not an emergent property — deriving it from ion channels is HH's job, not this one's.
"A stronger current always means the neuron fires." Kept, corrected. Only above rheobase (0.15 nA). Below it the membrane charges to a steady value under threshold and sits there — forever silent.
The red team's move: remove the reset. Let V keep rising after it crosses threshold instead of snapping back to V_reset. The neuron fires once, then sits stuck above threshold — the periodic train is gone.
Remove the reset and the train collapses to a single spike. The witness (window 7) re-integrates, finds no periodic firing, disagrees with the closed-form rate, and turns red. Nothing is faked; the attack is real and it is caught.