The tuning curve
A neuron in visual cortex watches a bar of light tilted at angle $\theta$. Its favorite tilt is $\theta = 90$ deg. For which stimuli does it fire?
At $\theta = 45$ the rate reads $12.99$ Hz. Climb: $24.26$ Hz at $60$, $35.30$ Hz at $75$, and a peak of exactly $40.00$ Hz at $90$. Past the peak the numbers retrace themselves — $24.26$ Hz again at $120$, only $5.41$ Hz by $150$. And the spike row never falls fully silent: even at $\theta = 0$ the model leaves a $0.44$ Hz trickle.
The instrument is drawing one function, the neuron's tuning curve:
a Gaussian bump with preferred angle $90$ deg, width $\sigma = 30$ deg, peak $40$ Hz. Check it by hand at $\theta = 45$: the exponent is $-(45-90)^2/1800 = -1.125$, and $40\,e^{-1.125} = 12.99 \approx 13.0$ Hz — the number on the readout. So this neuron is not a $90$-degree detector. It is a graded dial: angles within roughly $\pm 60$ deg of preferred all produce some response, falling off smoothly, never in steps. That breadth has a price. The curve is symmetric, so a rate of $24.26$ Hz could mean $60$ deg or $120$ deg — one number, two candidate stimuli. A single tuning curve compresses an angle into a rate and loses information doing it. Getting the angle back is the next lesson's job.
About this model
What it represents: the trial-averaged firing rate of an orientation-tuned visual neuron as a smooth Gaussian function of stimulus angle — the standard first picture of how a sensory neuron encodes a stimulus variable. The live spike row plays out that rate as a stream of spikes.
What it leaves out: adaptation — a real neuron's rate sags under a held stimulus; contrast dependence — the curve's height scales with stimulus strength rather than sitting fixed at $40$ Hz; and noise structure — real responses vary from trial to trial in ways this clean curve hides, which lesson 3 confronts directly.
Source: Gerstner, Kistler, Naud and Paninski, Neuronal Dynamics, Ch. 1.5 (EPFL, free online) — rate codes and tuning curves, including the variability this lesson postpones.
In $r(\theta) = 40\,e^{-(\theta-90)^2/(2\cdot 30^2)}$, what does the $30$ control?
Without touching the slider: what should the instrument read at $\theta = 120$?
Transfer: you are told only that this neuron fired at $24.26$ Hz. What can you conclude about the bar?
A tuning curve turns a stimulus into a rate — smoothly, broadly, and with information lost on the way: in this model one neuron's rate narrows the stimulus to two candidate angles and can go no further. The code that can is a population's, and that is lesson 2.