Previous studies (Bradley et al., 1996; Shenoy et al., 1999; Shenoy et al., 2002; Bremmer et al., 2010; Kaminiarz et al., 2014) evaluated heading tuning curves in a narrow range around straight ahead (white region around 90° in panel A). (A) Two hypothetical tuning curves with different bandwidths and amplitudes. (B) Correcting for the difference in response amplitudes reveals a clear difference in bandwidths, which may reflect a lack of compensation for rotation. The rank order of the responses (1–1, 2–2, 3–3) would be identical for the two curves, which previous methods would erroneously interpret as evidence for rotation-invariance (Bremmer et al., 2010; Kaminiarz et al., 2014). (C) Simulated tuning curves with peaks roughly near straight ahead. The underlying von Mises functions (to which Poisson noise was added) have peaks at headings of 80°, 100° and 120°, resulting in a simulated shift of 20°. (D) The cross-correlation function between the translation only (black) and rotation-added (red—leftward rotation, blue—rightward rotation) tuning curves. Reflecting the true shift of 20° that was introduced into the tuning curves (before noise was added), the cross-correlation functions peak near a lag of +20°. (E) Simulated tuning curves with peaks at 180° and different bandwidths. The difference in the width of the full tuning curves corresponded to a 20° shift (at half-height). (F) Cross-correlation functions for the simulated neurons with lateral heading preferences are quite flat and show no evidence of a peak near a lag of +20°. (G) Shifts from 10 sets of simulated tuning curves (with different noise samples) were measured using cross-correlation for heading preferences ranging from 0° to 180°. The gain and offsets of the tuning curves were randomized for each set. The mean shift (black markers) approaches the true shift (20°, dashed line) for tuning curves with heading preferences near 90°, but the mean shifts are grossly inaccurate for simulated neurons with lateral heading preferences. In contrast, Figure 4–figure supplement 1E shows that our analysis method correctly estimates tuning curve shifts regardless of heading preference.