A neuron is a leaky bucket. Current pours charge in; the membrane leaks it out; and when the level crosses a threshold the neuron spikes and empties. Push a weak current and nothing happens — the leak wins. Push harder and it fires, faster and faster. Slide the current and find the tipping point.
The leaky integrate-and-fire model: the membrane voltage obeys τ·dV/dt = −V + R·I; when V reaches the threshold the neuron emits a spike and resets to zero. Below a critical current (the rheobase) the leak balances the input and V never reaches threshold — no spikes. Above it, the rate rises with current. The instrument integrates the real ODE. A fail-loud self-check throws unless a sub-rheobase current gives ZERO spikes while a strong current fires faster than a moderate one — the threshold nonlinearity that makes a neuron a decision, not an amplifier.
A point-neuron with instantaneous reset and no refractory period or ion channels (Hodgkin-Huxley adds those); the leak-integrate-threshold-reset dynamics and the rheobase are exact for this model (Lapicque 1907).