Extended Data Fig. 8: Impact of stable and unstable fixed points on drift and velocity integration.

a-b, Repeated from Fig. 3c–d, with regimes colored according to drift speed (grayscale). c, Input velocity shifts the orientations of the fixed points. Top: as the input velocity increases from 0, the orientations of the stable and unstable fixed points shift toward the boundary between regimes. At a threshold velocity (Eq. (5)), the two fixed points will meet at the boundary; this threshold velocity is the minimum input velocity needed to move the bump continuously. Bottom: For velocities below this threshold, the bump will be driven to the stable fixed point, regardless of its initial orientation. d, The orientations of stable and unstable fixed points (turquoise and orange lines, respectively) shift with increasing velocity (darker shades). The rate of these shifts is set by the drift speeds in the stable and unstable regimes (see panel b): lower drift speeds lead to faster shifts (marked by the large spacing between turquoise lines at the left of the panel, and between orange lines at the right of the panel). The precise values of these drift speeds ensure that the pair of stable and unstable fixed points will meet at the boundary between regimes at the same threshold velocity, given a fixed value of J_E. e, Top: as the input velocity increases above the threshold velocity, the stable and unstable fixed points move beyond their respective regimes. Bottom: When in the stable regime, the bump is pulled from ahead toward the stable fixed point. However, before reaching the stable fixed point, the bump transitions into the unstable regime, and is pushed from behind by the unstable fixed point. This push and pull causes the bump to slow down and speed up as it moves through the stable and unstable regimes, respectively; the closer the fixed points are to the boundary, the stronger this effect. f, Above the threshold velocity, the stable and unstable fixed points move beyond their respective regimes, and they continue to shift with velocity at the same rate as shown in panel d. g, Example bump trajectories in the absence (top row) and presence (bottom row) of velocity input.