Finding 3 of 4 · the sign problem

The stored vector points away from food. One unassigned neuron type, hΔM, is the only thing that turns it around

The problem. The store of finding 2 holds the fly's displacement from the food: "I am 2 north of the food". To walk home the steering neurons need the opposite vector: "walk 2 south". Somewhere between the store and the steering neurons the vector has to be rotated by 180°. The question is which neuron does that.

Why the store cells looked like the answer. Every hΔ neuron has its dendrites in one column of the fan-shaped body and its axon in the column 180° away. A signal that arrives on the dendrites at the "north" column is emitted at the "south" column: a built-in 180° rotation. The store candidates are hΔ cells, so the store would receive "north" and automatically emit "south", which is "go home". The rotation would come free with the anatomy.

Why that is wrong. The connectome shows hΔB's synapses onto the store cells land on the store cells' axons, not their dendrites. A signal that enters at the axon leaves from that same axon, in that same column. So the store receives "north" and emits "north". It passes the displacement along unchanged, pointing away from the food.

Case A · hΔB axon lands on the store's dendrite → inverted column T (travel dir.)column T + 180° hΔB out store in store out store outputs at T + 180°: −T reaches the goal layer Case B · hΔB axon lands on the store's axonal arbor → not inverted column T − 180°column T (travel dir.) store in hΔB out store out contact and output share column T: +T reaches the goal layer Measured (MaleCNS, replicated in hemibrain) hΔA: 92 % of hΔB input on its axonal arbor → case B hΔH: 95 % → case B hΔI: 82 % → case B hΔG: 73 % → case B hΔJ: 58 % / 42 % → both, net weak hΔK: 99 % on dendrite → case A (inverts) All four store candidates are case B. They relay the displacement; they do not flip it. Hulse 2021 (Fig. 37Biii) noted this two-band motif anatomically; its sign consequence was not drawn.
Same cell, opposite outcome, depending on which arbor hΔB reaches. Right: the measured share of hΔB synapses on each candidate's axon.

Where the rotation actually happens. From the store the signal goes to the goal neurons FC2, and from FC2 to the steering neurons PFL3. Both connections stay in the same column, so by that route PFL3 receives "north" and steers the fly further north, away from the food. There is one other route from FC2 to PFL3: through hΔM. FC2 contacts hΔM's dendrites, and hΔM's axon, 180° away, contacts PFL3. Through hΔM, PFL3 receives "south". hΔM is the only cell type between the goal and the steering neurons that does this, and nobody has assigned it a function.

store (hΔH/A/I/G)holds +D FC2 goalreceives +D +15° PFL3 ← +Dsteer away: keep going direct · 7,690 · 2° hΔM336° in, 202° out 1,073 PFL3 ← −Dsteer toward food: return 3,983 Both PFL3 boxes are the same neurons; which input dominates decides whether the fly continues or returns.FB5A (GABA) inhibits FC2, PFL3 and hΔM together and could set that balance. Total rotation via hΔM: 336° + 202° = 178°.
The goal reaches the steering neurons twice: unchanged through FC2 (about 7,700 synapses) and rotated by half a turn through hΔM (about 4,000). Only the rotated copy points home.
FC2 → PFL3 routesynapsesrotation
direct7,6902°
through hΔM (1,073 in, 3,983 out)3,983178°
through hΔM to PFL21,555178°

The simulation check. A model fly with the store wired to steering with no rotation walks away from the food on every run. With 180° rotation, or with hΔM's measured 178°, it returns.

rotation between store and PFL3returned to foodclosest approach
180° / 178° (hΔM)35 %1.05 / 1.36
90°20 %2.64
0° (direct route only)0 %10.3, walks away
no memory–6.76

Distances in units of the outbound walk length; median over 60 runs.

Consequence. PFL3 receives the same memory twice: "north" directly, meaning keep going, and "south" through hΔM, meaning go back. Whichever input wins decides whether the fly keeps exploring or returns. Flies that are dispersing do keep going for hours (Green 2019), and a GABA neuron, FB5A, inhibits FC2, PFL3 and hΔM together, so it is placed to set the balance. hΔM is the return path. PFL2, which oddly fires most when the fly faces away from its goal (Westeinde 2024), also receives the rotated goal from hΔM, which explains that tuning with no new anatomy.

What would settle it. Silence hΔM during a return to food and during straight heading maintenance: return should fail, heading should survive. Image hΔM and FC2 together: bumps half a turn apart. Image hΔH or hΔA during a return: bump along the outbound displacement, not toward the food.

Prior work. Hulse 2021 drew the two-arbor contact motif on hΔ neurons without following its sign. Liao 2025 describe an anti-goal circuit further downstream in the lateral accessory lobe. Nanni & Lee 2026 propose FB5A as the FC2 normaliser. The hΔM rotation and its necessity for return are not in print.

Wiring: two brainsNecessity of the rotation: simulatedFunction: predicted

Sources: docs/exhaustive-search.md §4; data/derived/fb_offset_stats.csv; simulations/closed_loop_kernel_controls.json. Hulse 2021; Lyu 2022; Mussells Pires 2024; Westeinde 2024; Liao 2025.