A neuronal circuit for vector computation builds an allocentric traveling-direction signal in the Drosophila fan-shaped body

Many behavioral tasks require the manipulation of mathematical vectors, but, outside of computational models1-8, it is not known how brains perform vector operations. Here we show how the Drosophila central complex, a region implicated in goal-directed navigation8-14, performs vector arithmetic. First, we describe neural signals in the fan-shaped body that explicitly track a flys allocentric traveling direction, that is, the traveling direction in reference to external cues. Past work has identified neurons in Drosophila12,15-17 and mammals18,19 that track allocentric heading (e.g., head-direction cells), but these new signals illuminate how the sense of space is properly updated when traveling and heading angles differ. We then characterize a neuronal circuit that rotates, scales, and adds four vectors related to the flys egocentric traveling direction-- the traveling angle referenced to the body axis--to compute the allocentric traveling direction. Each two-dimensional vector is explicitly represented by a sinusoidal activity pattern across a distinct neuronal population, with the sinusoids amplitude representing the vectors length and its phase representing the vectors angle. The principles of this circuit, which performs an egocentric-to-allocentric coordinate transformation, may generalize to other brains and to domains beyond navigation where vector operations or reference-frame transformations are required.