Poisson: Counting Spikes
⚡ The Poisson distribution counts how many events happen in a fixed window of time or space — like spikes per second. It's the single most-used model of neural firing.
Binomial needed a fixed number of trials n. But how many times could a neuron spike in one second? There's no natural ceiling. Poisson is a point process: it counts discrete events with no upper limit, controlled by a single parameter λ (lambda) — the average number of events per window.
P(x) = (λˣ · e^(−λ)) / x!
The probability of seeing exactly x events when the average is λ. One knob, λ, sets both the center and the shape.
Mantra: "λ is the average count per window." If a neuron fires at an average of 4 spikes/second, then the spikes recorded in any one second follow a Poisson with λ = 4. Some seconds you'll count 2, some 6, occasionally 9 — but 4 on average.
This comes with a big assumption baked in: each spike is independent of the last. Real neurons violate that (after firing they go quiet for a bit — the refractory period), which is exactly the kind of thing modeling reveals. But as a first pass, Poisson is remarkably good.
A Poisson neuron fires at an average rate λ = 4.0 spikes/sec; the measured mean here is 4.06. Drop λ below ~2 and the histogram goes lopsided — you can't have negative spikes, so the pile bunches against 0. More samples just sharpens the shape; it doesn't move it.
The gold line marks λ. Notice: with a small λ (try < 2) the histogram goes lopsided — you can't have negative spikes, so it bunches against 0. Adding samples sharpens the shape but never moves it.
🦌 Ilya says: If you measure the spike counts and want to recover λ, you're doing inference — using data to estimate the hidden parameter. That's the whole next quest. λ isn't just a curve knob; it's often the thing you actually care about.