How a home vector can live in synapse strengths

A worked example with four direction columns. The fly walks 6 north, then 4 south, ends up 2 north of the food, and comes back. Step through it. Left: the world. Middle: the strength of the synapse from the travel-direction bump onto each direction column. Right: what the steering neurons receive.

World north ↑ food (start) Synapse strengths (hΔB → store, per direction column) grow while walking (dopamine-gated) What steering receives NESW each column pushes toward its direction;pushes add as vectors

The three rules, and what is measured versus assumed

  1. Write. While the fly walks, the active hΔB column (its current travel direction) strengthens its synapse onto the matching store column, by an amount proportional to distance. Measured: hΔB synapses column-matched onto hΔH/hΔA/hΔI/hΔG; dopamine cells FB4M/FB1H, driven by the velocity system, contact every store cell. Assumed: that the synapses actually potentiate under that dopamine.
  2. Read. Whenever hΔB is active, each store cell fires in proportion to its synapse strength, and the steering neurons sum the columns as vectors, with opposite columns pulling opposite ways. 6 north and 4 south are both present; the sum is 2 north. Measured: this vector summation is the operation the fan-shaped body already uses to combine PFN inputs (Lyu 2022; Maimon & Abbott 2026). The sum points away from food; hΔM inverts it (finding 3).
  3. Cancel and reset. Walking back fills the opposite column until the pushes cancel and the fly stops; that is arrival. But the strengths are then 6/6, not 0/0, and would saturate over trips. Something must scale them all down at food. Measured: the reward octopamine neuron OA-VPM3 contacts the store cells. Assumed: that it depresses or renormalises the synapses. This is the least supported step and the first thing to test.

Path independence: any path with the same endpoint produces the same net vector, because only totals per direction are kept and opposite directions cancel at readout. The path itself is never stored.

Companion to finding 2 and finding 3. Numbers in the example are arbitrary; the circuit pages give the measured synapse counts.