Extended Data Fig. 3: Relationship between FC2 activity and fly behaviour.

a, Correlation between EPG phase or FC2 phase and fly heading. Each dot represents one fly. Mean ± s.e.m. across flies is indicated. b, Cross-correlation between phase velocity and behavioural turning velocity. FC2 data are in purple and EPG data are in grey. A positive lag means that a change in heading precedes a change in the neuronal signal. Mean ± s.e.m. across flies is shown. c, Individual ±90° rotation trials for 113 trials from 9 flies in which we imaged EPG neurons. In contrast to Fig. 1, here we did not require for a trial to occur within a menotaxis bout (see Methods) or require that the fly return within 45° from its heading before the bar jump. Thick lines show the mean across flies. d, Same as panel c but for 140 trials from 15 flies in which we imaged FC2 neurons. Note that, on average, the FC2 phase slowly drifts away from its initial position. This small drift may be due to trials where the fly’s goal angle genuinely drifted to the fly’s new heading angle after the bar jump, which seems plausible given that on many trials analyzed here the fly did not turn so as to reorient themselves along their previous heading. e, Mean phase value during final 1 s of the open-loop period in panels c and d. Each dot is the mean for one fly. Horizontal lines show the mean ± s.e.m. across flies. V-test for EPG flies: μ = 90°, p = 2.49 × 10⁻⁵. V-test for FC2 flies: μ = 0°, p = 7.69 × 10⁻⁸. f, Example trace showing an abrupt change in the position of the FC2 bump in the fan-shaped body. g, Left: Each thin line shows an algorithmically-detected rapid change in the FC2 phase position, zeroed to the onset of the change in phase. Right: bar position, zeroed to the onset of the change in phase, during these moments. Thick lines show the mean across 141 transients from 15 flies. That the FC2 phase has the capacity to move by more than 90° within less than 2 s (the magnitude and duration of our bar jumps) suggests that the stability of the FC2 phase during virtual rotations was not due to the FC2 phase simply reflecting a low-pass filtered estimate of the fly’s heading. h, Left: example FC2 ΔF/F₀ signal and behavioural traces from a fly that occasionally deviated from its goal angle. The teal arrow marks a moment when the FC2 phase did not remain stable, but the fly nonetheless returned to its putative goal direction. One interpretation of the moment marked in teal is that inputs other than the longer-term menotaxis goal input to the FC2 system briefly dominated, which led the FC2 phase to drift. However, once the fly re-entered a menotaxis behavioural state and wished to progress forward, the FC2 phase locked back in to the menotaxis angle, communicating it to the PFL3 population to guide steering. In this view, the fan-shaped body may encode multiple potential goals, with the actual goal chosen from this set in a state-dependent manner and the FC2 calcium signal might be best viewed as a conduit between these long-term navigational goals and the central-complex’s pre-motor output. The red arrow marks an occasion when the FC2 phase remained stable throughout a brief deviation in heading direction. Right: expanded view of time period marked by teal box and red box. i, Example FC2 ΔF/F₀ signal and behavioural traces from a fly that was rotating in time and not stabilizing a consistent heading direction. This trace highlights that the FC2 phase can be well-estimated during moments where our algorithm would not detect that the fly is performing menotaxis. j, FC2 activity across the fan-shaped body from a single timeframe. k, Schematic of how we computed the population vector average (PVA) strength from FC2 activity. Each fan-shaped body column region-of-interest (ROI) is treated as a vector (thin arrows). The angle of each vector is determined by the position of the column in the fan-shaped body and the length of the vector is determined using the ΔF/F₀ value. The PVA strength is the length of the resulting mean vector (thick arrow). l, Difference between the mean ΔF/F₀ two seconds before and during the bar jump for EPG neurons in the bridge, and FC2 neurons in the fan-shaped body. Each dot is the mean across trials for an individual fly. Mean ± s.e.m. across flies shown (5 EPG and 7 FC2 flies). m, Same as panel l but for the difference in max-min ΔF/F₀. n, Same as panel l but for the difference in PVA strength. o, Trajectory of a fly color-coded by the vector strength of the fly’s mean heading direction, R (not to be confused with the FC2 PVA strength), calculated with a 60 s window (see Methods). p, FC2 activity as a function of R, computed using either a 30, 60 or 120 s time window. Mean ± s.e.m. across flies shown (n = 15). q, FC2 activity as a function of the fly’s forward walking velocity (left) and turning velocity (right). Mean ± s.e.m. across flies shown (n = 15).

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