Inject a steady current into a neuron and it fires at some rate. The function mapping input current → output firing rate is the neuron's input-output transfer function (or I/O curve). It behaves like a dimmer switch — and now we'll make it precise.

The input is the injected current, I, measured in arbitrary units (au) — a stand-in for stimulus strength, from weak to strong.

The output is the firing rate, r — spikes per second, here normalized from 0 (silent) to 1 (maximum).

For most neurons the curve is a sigmoid: near-silent at low current, a steep rise through a middle band, then saturating (flattening) at high current. So the neuron ignores weak input, responds sharply over a middle range, and tops out when driven hard.

r(I) = 1 / ( 1 + e^(−a·(I − θ)) )

The sigmoid transfer function. (NMA's full version subtracts a small constant so r(0)≈0 and adds a noise term η — but a and θ set the shape.)

The curve has tunable parameters, and each reshapes it predictably:

• θ (theta) — the threshold. The current at which the neuron is half-activated — the midpoint of the rise. It lives on the x-axis (it's a current value, in au). Raise θ → the whole curve shifts right (the neuron needs more current before it responds).

• a — the gain parameter. How steep the rise is. Raise a → a sharper, more switch-like transition; lower a → a gradual, gentle rise.

So what is "gain," exactly? Gain is the slope of the transfer function — the derivative d(r)/dI. It answers one question: if I nudge the input current up by a tiny bit, how much does the firing rate change?

• High gain → a small change in current produces a large change in firing rate → the neuron is very sensitive to input there.
• Low gain → changing the current barely moves the firing rate → insensitive (when silent or saturated).

"Gain" and "sensitivity" are the same thing: how strongly the output reacts to the input.

Because the sigmoid is steepest at the threshold θ and flat at both ends, the gain is a bump centered on θ — peak sensitivity right at the wake-up point, near-zero when the neuron is silent or saturated. The widget shows both at once: the transfer function on top, its slope (the gain) below.

Tune it. Slide θ (threshold) and a (gain parameter) and watch the transfer function and its gain respond. Predict first: at which current will the neuron be most sensitive?

What to take away:
• Raise θ → transfer curve and gain bump both shift to higher current (more drive needed).
• Raise a → transfer curve steepens; the gain bump grows taller and narrower (higher peak sensitivity, over a narrower current range).

This input-output view — and the gain as its derivative — is exactly how computational neuroscientists characterize a single neuron. You'll meet this same equation again in Week 3.