Supplementary Figure 3: Empirical assessment of RT scale invariance.

When two distributions in time are related to each other through a rigid stretching of the time axis, their percentiles are proportional. To see this, assume that distribution ρ’(t) is obtained from ρ(t) by stretching of the time axis by a factor α, so that ρ(t) = α ρ’ (αt). It follows that. \(\mathop {\smallint }\limits_0^\tau dt\rho \left( t \right) = \alpha \mathop {\smallint }\limits_0^\tau dt\rho \prime \left( {\alpha t} \right) = \mathop {\smallint }\limits_0^{\alpha \tau } dt\prime \rho \prime \left( {t\prime } \right)\). for all τ⊡ If the value of the previous integrals is Q/100, then τ is the Q^th percentile PC_ρ(Q) of ρ(t) and ατ is the Q^th percentile PC_ρ’(Q) of ρ'(t), i.e., their percentiles are proportional. This fact allows us to use the slope of a linear fit of the percentiles of the two distributions to identify the (putative) temporal rescaling factor α corresponding to a given change in ABL, and the R² of the fit to quantify its precision. (a) Each plot shows, for each difficulty, PC_ABL(Q)-PC_60dB(Q) as a function of PC_{60 dB}(Q), where PC_ABL(Q) is the 100Q^th percentile of the RT distribution for the corresponding ABL of that difficulty. Each RTD contains all data for that condition from all rats (n = 5 rats). Although, for each pair of RTDs, the linear fit was made directly between their percentiles, we plot in the figure the difference between the two percentiles against one of them, because it graphically displays the linear relationship more clearly. Only RTs > RT_min are considered in this analysis. Dashed line is the outcome of the linear fit. Shaded regions represent three standard deviations of the sampling distribution of the percentiles (see Methods). Note that higher percentiles have more error. Because of this, we used weighted least squares to do the linear fit. (b) Since difficulty has been suggested to also lead to an approximate rescaling of the RTD (Wagenmakers, E. and Brown, S., Psychological Rev., 114(3):830, 2007; Ratcliff, R. and McKoon, G., Neural Computation, 20(4):873-922, 2008; Srivastava, V., et al., Journal of Mathematical Psychology, 75:96-109, 2016), we performed the same analysis as in (a) (n = 5 rats) for pairs of distributions corresponding to different difficulties (for each ABL separately). (c) R² of the weighted least squares fit vs slope of the fit for the effect of changes on ABL (gray) or ILD (black) on the RTDs from (a) and (b). Lines show linear fits. The accuracy of the scale invariance of the RTDs induced by changes in ABL is independent of the magnitude of the slope (slope of the linear fit not significantly different from zero; p = 0.6, Permutation test) and thus of the difference between the two RTDs being compared. In contrast, for changes in difficulty, the difference in shape (not just scale) of the RTDs becomes larger the larger the ILD-induced change in RT, as evident by statistically significant negative slope of a linear fit of R² versus slope (p = 0.003, Permutation test), showing that R²s in this case are only large because the speed accuracy trade-off in the task is weak. These results are in agreement with the claims made in the main text, whereby ABL-induced scale invariance is exact, whereas difficulty-induced scale invariance is approximate and degrades with the difference in difficulty between the two RTDs. For a thorough treatment of the nature and form of the changes in the RTD induced by difficulty, see Section 4 of the Supplementary Note.