Several analyses assessed changes in the median of repeatedly sampled data, e.g., average pupil size measured across multiple behavior blocks (Figure 1) or average state-dependent modulation measured across multiple units (Figures 3 and and558). In this case, significant differences were assessed using a hierarchical bootstrap test (Saravanan et al., 2020). The hierarchical bootstrap is nonparametric, like the more traditional Wilcoxon signed-rank test, but it accounts for potential bias resulting from the relatively small number of array recordings in the data set. A statistical test that treats each unit as an independent measure could potentially be biased if a single array recording produced spuriously large effects, and the hierarchical bootstrap controls for this possibility. The bootstrap analysis was run for 10,000 iterations, so that the minimum measurable p-value was 10−5. Thus, results that returned = 0 after this many iterations are reported as p < 10−5.

State-dependent changes in individual neurons were assessed using a combination of nested cross-validation and a jackknife t-test (see above, Efron and Tibshirani, 1986). To determine if any population-level effects depended on task performance or between animal differences, we used multivariate ANOVA (Figure 4).

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