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

Abstract

AbstractMany 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 fly’s 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 fly’s 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 sinusoid’s amplitude representing the vector’s length and its phase representing the vector’s 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.