Extended Data Figure 8: Asymptotic and bounded information variability is quenched in large populations.

Panels in the top row show results when tuning curve centers are bounded to lie within the stimulus range, whereas the bottom row shows results when these centers can extend significantly beyond that range. a. Our measure of encoded information, using linear averaging of the population FI. b. Zhang et al.’s measure of encoded information, which we have shown (SI 6) equals ours minus components that depend on the variability of per-neuron encoded information, inversely scaled by the size of the population. Here we measure the encoded information in populations with heterogeneity $\alpha =8\times {10}^{-2}(v=3.5$). At such a setting, the difference in our measures vanishes for populations as small is $N=800$ neurons. c. If tuning curve centers are bounded, encoded information varies across the stimulus range. The resulting variance, which is shown here, additionally reduces Zhang et al.’s information measure. This source of variability is independent of population size, but is here estimated with a population of size $N=51,200$. d. Variation in per-neuron FI that is agnostic to whether tuning curve centers are bounded. e. The total variability (the sum of variabilities depicted in panels c. and d. divided by the linear encoded information depicted in a.). f-j. The same as a-e., but for the case of unbounded tuning curve centers. Note that in the unbounded case, boundary-dependent variability (h.) does not contribute to overall variability, as predicted by the theory. In both cases, as $N\to \mathrm{\infty }$, overall variability is driven towards 0 (e., j.), as predicted by the theory. This quenching in variability in larger populations drives the measure used in [8] to become indistinguishable from ours in larger populations.