Student Theses and Dissertations


Cheng Lyu

Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)

RU Laboratory

Maimon Laboratory


Many behavioral tasks require the manipulation of mathematical vectors, but, outside of computational models, it is not known how brains perform vector operations. Here we show how the Drosophila central complex, a region implicated in goal-directed navigation, performs vector arithmetic. First, we describe a neural signal in the fan-shaped body that explicitly tracks a fly's allocentric traveling angle, that is, the traveling angle in reference to external cues. Past work has identified neurons in Drosophila and mammals that track an animal's heading angle referenced to external cues (e.g., head-direction cells), but this new signal illuminates how the sense of space is properly updated when traveling and heading angles differ (e.g., when walking sideways). 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––to compute the allocentric traveling direction. This circuit operates by mapping spatial vectors onto sinusoidal patterns of activity across distinct neuronal populations, 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 and vector addition, may generalize to other brains and to domains beyond navigation where vector operations or reference-frame transformations are required.


A Thesis Presented to the Faculty of The Rockefeller University in Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy

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