Reading the population
Lesson 1 ended in ambiguity: one broad neuron reading $24.26$ Hz cannot tell $60$ deg from $120$ deg. Now take eight such neurons, preferred angles spread every $22.5$ deg. Can the bank pinpoint the angle?
At $\theta = 45$ the $45$-neuron leads at $40.00$ Hz, its neighbors at $22.5$ and $67.5$ both read $30.19$ Hz, and the decoded needle sits at $45.0$ deg. Slide to $10$ deg, between preferred angles: the $0$-neuron fires at $37.84$ Hz, the $22.5$-neuron at $36.67$ Hz — and the readout still says $10.0$ deg. Wherever you stop, true and decoded agree to within a hundredth of a degree.
The decoder is a population vector. Each neuron contributes a direction — its preferred angle — and its firing rate decides how much that direction counts:
The decoded angle is half the direction of $\vec{p}$. The doubling handles a quirk of orientation: a bar at $0$ deg and a bar at $180$ deg are the same stimulus, so preferred angles live on a $180$-deg circle. Doubling every angle spreads them around a full circle, the sum is taken there, and halving the answer maps it back. What makes the decode work is exactly what made one neuron fail: broad overlap. At $10$ deg two neurons fire hard, the sum interpolates between their directions, and six weaker rates fine-tune the balance. Georgopoulos and colleagues read arm-reach direction out of motor cortex this way in 1982 — a rate-weighted average of preferred directions.
About this model
What it represents: population vector coding — a bank of eight identical Gaussian tuning curves ($\sigma = 30$ deg, peak $40$ Hz, preferred angles every $22.5$ deg, distances wrapped on the $180$-deg orientation circle), decoded by the direction of the rate-weighted sum of doubled preferred-angle vectors.
What it leaves out: noise — these rates are exact, while real decoding degrades with trial-to-trial variability and especially with correlated noise across neurons, which no amount of averaging removes; and optimal decoding — the population vector is simple rather than statistically best, and maximum-likelihood decoders beat it when tuning curves are uneven.
Source: Georgopoulos, Kalaska, Caminiti and Massey (1982), J. Neurosci. 2:1527 — the original population-vector reading of reach direction from motor cortex.
One neuron at $24.26$ Hz was ambiguous between $60$ and $120$ deg. Why does the bank resolve it?
At $\theta = 10$ deg no neuron is centered on the stimulus, yet the decode lands on $10.0$. What produces the in-between answer?
Transfer: a friend deletes the doubling and sums $r_i(\cos\theta_i, \sin\theta_i)$ directly. Where does their decoder break?
Broad tuning is not a defect the code tolerates — it is what makes the population readable. Overlapping curves plus a rate-weighted vector sum turn eight ambiguous numbers into one nearly exact angle. The message lives in the pattern, not in any single cell.