Many behavioral tasks require the manipulation of mathematical vectors, but, outside of computational models 1-8 , it is not known how brains perform vector operations. Here we show how the Drosophila central complex, a region implicated in goal-directed navigation 8-14 , 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 Drosophila 12,15-17 and mammals 18,19 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 twodimensional 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.

Vision influences behavior, but ongoing behavior also modulates vision in animals ranging from insects to primates. The function and biophysical mechanisms of most such modulations remain unresolved. Here, we combine behavioral genetics, electrophysiology, and high-speed videography to advance a function for behavioral modulations of visual processing in Drosophila. We argue that a set of motion-sensitive visual neurons regulate gaze-stabilizing head movements. We describe how, during flight turns, Drosophila perform a set of head movements that require silencing their gaze-stability reflexes along the primary rotation axis of the turn. Consistent with this behavioral requirement, we find pervasive motor-related inputs to the visual neurons, which quantitatively silence their predicted visual responses to rotations around the relevant axis while preserving sensitivity around other axes. This work proposes a function for a behavioral modulation of visual processing and illustrates how the brain can remove one sensory signal from a circuit carrying multiple related signals.

Rational engineering of biological systems is often complicated by the complex but unwanted interactions between cellular components at multiple levels. Here we address this issue at the level of prokaryotic transcription by insulating minimal promoters and operators to prevent their interaction and enable the biophysical modeling of synthetic transcription without free parameters. This approach allows genetic circuit design with extraordinary precision and diversity, and consequently simplifies the design-build-test-learn cycle of circuit engineering to a mix-and-match workflow. As a demonstration, combinatorial promoters encoding NOT-gate functions were designed from scratch with mean errors of <1.5-fold and a success rate of >96% using our insulated transcription elements. Furthermore, four-node transcriptional networks with incoherent feed-forward loops that execute stripe-forming functions were obtained without any trial-and-error work. This insulation-based engineering strategy improves the resolution of genetic circuit technology and provides a simple approach for designing genetic circuits for systems and synthetic biology.

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