Extended Data Fig. 4: Analysis of the statistical methodologies for measuring temporal integration.

(a) Sampling scheme for generating data sets. \(n\) samples were generated according to the given cumulative distribution function, \({\sigma }_{{p}_{0},\tau }\), and these were used to fit estimates for the generating \(\tau \) and \({p}_{0}\). The value of \(n\) was varied logarithmically from 10 to 1000 to evaluate what sample size would be necessary to accurately estimate the parameters of the distribution. (b) The cumulative distribution function can be qualitatively reconstructed with samples of size ~ 100 across a wide range of cumulative distribution function shapes. Smaller sample sizes (e.g. ~30) are highly variable, especially when the overall number of flies terminating the mating during the stimulation is low (top row). (c) The sensitivity of the inference of the value of \(\tau \) to sample size across a range of \({p}_{0}\) values. The closer a point is to the diagonal, the more likely the fitting procedure is to capture the correct \(\tau \). The fitting procedure overestimates \(\tau \) at low sample sizes, especially when the true value for \(\tau \) is small. This may, to some extent, be explained by the fact that termination times are rounded to the nearest second (we find it is impossible to judge the time of termination more precisely than this value, given the complex motor sequence of terminating the mating). For larger sample sizes, the estimate is much better, so long as a large number of flies terminate the mating during the stimulation. When \({p}_{0}\) and \(\tau \) are both small, however, the inference is considerably less reliable, because these conditions correspond to cases in which very few flies terminate the mating during the stimulus, providing very little information about \(\tau \). (d) As in ©, but instead examining the sensitivity of the estimate of \({p}_{0}\). The parameter \({p}_{0}\) is easier to estimate, because even flies that do not terminate the mating during the stimulation are still informative about its value to some extent (see Methods). However, we find that \({p}_{0}\) is systematically underestimated due to the bias towards overestimating \(\tau \) and the fact that the two estimates show substantial anticovariance (elaborated in panel (e)). (e) Covariance of \({p}_{0}\) and \(\tau \) for various sample sizes. Dashed lines indicate the true parameter values, while independent points show individual sample estimates. The two parameters always anticovary, as indicated by the diagonal slant of each distribution. This reflects the fact that \({p}_{0}\) only appears in the cumulative distribution with \(\tau \) in the form \({p}_{0}\tau \), and so this term is easier to fit than either value alone. If \(\tau \) is overestimated, \({p}_{0}\) will tend to be underestimated to compensate. The multiplicative relationship is clear from the approximately linear covariance of the logarithm of the two parameters. When the data is more informative about \(\tau \), i.e. many flies terminate the mating during the experiment, the cluster is much smaller (e.g. the orange data set). We therefore restricted our experiments to those conditions that would generate reliable estimates of the parameters, especially in cases where we expected \(\tau \) or \({p}_{0}\) to be very small.

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