Now the main event: a model of a single real neuron — the Leaky Integrate-and-Fire (LIF) model. The best way to picture it: a leaky bucket you pour electricity into.

The leaky-bucket picture (hold onto this — it carries the whole section):
• The neuron's voltage is the water level in a bucket.
• You pour in current (input) — the water rises. (Integrate.)
• The bucket leaks — water always drains back toward a resting level. (Leaky.)
• If the water reaches the rim, it spills over — the neuron fires a spike — then the bucket empties and refills. (Fire.)

That's literally Leaky-Integrate-and-Fire, in order.

The full equation. It looks like the population equation's more serious sibling:

τ_m · dV/dt = −(V − E_L) + R_m · I

How the membrane voltage V changes over time.

In one sentence, here's the whole equation as a story: you inject current I into the neuron, and it gets scaled by the membrane resistance R_m to push the voltage V up — but some of that push is constantly canceled by the leak term −(V − E_L), which pulls V back toward the resting voltage E_L. Meanwhile the time constant τ_m sets how quickly V changes in response.

Every symbol in plain English:
• V = membrane voltage (the water level).
• E_L = resting potential — the level the bucket leaks back to (≈ −75 mV).
• τ_m (tau) = the time constant — how sluggish the bucket is (big τ = slow to fill/leak, small τ = snappy).
• R_m = membrane resistance (how strongly input current pushes the voltage).
• I = the input current you inject (the tap).

The −(V − E_L) term is the leak (always pulling back to rest); the R_m · I term is the input (pushing up).

🏛️ Fun fact: this model is from Louis Lapicque, 1907 — over a century old, and decades before anyone knew how spikes physically work (Hodgkin–Huxley was 1952). A 115-year-old equation that's still in working neuroscience papers today. Simple ideas age well.