Extended Data Fig. 1: Model predictions.

a, A resolver measures the current angle of some object (\(\theta \)_c, e.g., the angular position of a shaft) and resolves that angle into its Cartesian components, x_c and y_c. The goal angle \(\theta \)_g is similarly resolved into its Cartesian components, x_g and y_g. These components are cross-multiplied, and their difference is used to generate a rotational velocity command dθ/dt ∝ x_cy_g - x_gy_c. We treat positive velocity values as clockwise (CW) rotations. In this example, the current angle is rotated CW relative to the goal, meaning a positive directional error. This drives a CCW rotation. But because the error at this point (●) is almost 180°, rotational speed will be small. Mittelstaedt suggested that a similar process might be implemented in the brain’s navigation centres to control an organism’s heading, and thus its path through the environment; this is known as “Mittelstaedt ‘s bicomponent model” of steering control⁷. Arrowhead denotes the system’s stable fixed point. b, Any vector can be represented as a sinusoidal function whose amplitude represents the magnitude of the vector, and whose phase represents the angle of the vector. Although it is convenient to represent these sinusoids as continuous functions, they can also be discretized into spatial activity patterns over neural space^15,16. Adding these sinusoids is equivalent to performing vector addition. c, Model elements shown in Fig. 1d, here schematized as spatial activity patterns over neural space. The horizontal axis of this space represents the horizontal axis of the fan-shaped body. d, Model: goal input to PFL2&3 cells (left). When this spatial pattern is shifted leftward, this produces a clockwise shift in the model’s rotational velocity as a function of head direction (right). Arrowheads denote the system’s stable fixed point. e, Model: shifts in the spatial phase of goal input to PFL cells produce equal shifts in the head direction corresponding to the system’s stable fixed point. This is true for all values of S > 0. f, Model: The effect of silencing PFL2 cells on rotational velocity is similar to the effect of removing the indirect pathway (compare with Fig. 5c). In both cases, the rotational velocity function becomes equally steep around the goal and the anti-goal. g, Model: The effect of silencing PFL2 cells on steering dynamics is similar to the effect of removing the indirect pathway (compare with Fig. 5d). In both cases, the system oscillates when S is high.