Extended Data Fig. 13: Relationship between the frequency of wing-beat and of internal representations.

a, Average power spectral density (PSD, normalized to unit area) for the decoding error during flight (n = 659 flights from 16 sessions and 6 bats). Dashed line shows fit with exponential decay. The inset shows the residual after subtracting the exponential decay. Note the peak around 8 Hz. b, Scatter plot and histograms for the peak of the residual PSD of the decoding error (after subtracting the exponential decay) versus average wing-beat frequency for single flights (dots, n = 659 flights from 16 sessions and 6 bats; Methods). Inset shows the same data at higher magnification. c, Same as b, but for the estimated sweep frequency for consecutive sweeps (example in the topmost inset, calculated as the inverse of their time interval, n = 360 from 15 sessions, 5 bats) versus average wing-beat frequency calculated from the time average of the instantaneous wing-beat frequency on the same time interval between sweeps (Methods). Inset shows the same data at higher magnification. d, Average P value (calculated over 30 repetitions, shaded area indicates s.d.) for the Spearman correlation between simulated sweep frequency versus empirical wing-beat frequency, plotted against imposed correlation values. Simulated sweep frequency (f) was generated from the empirical wing-beat frequency values (x) as f = r·zscore(x) + √(1 − r2) ·z + mean(x), where z ≈ N(0,1). Inset shows data for one simulated sample at r = 0.5. Simulations suggest that even under idealized conditions, detecting weak correlations (e.g., r ≈ 0.2, similar to the value observed between stepping and theta in rats²⁷) could be challenging, given the little wing-beat frequency variability.