Fig. S4

Drift in positional homeostasis and PID control theory, related to Figure 1

(A) Average trajectories showing drift in the swim period that can be slowly forward or slowly backward or minimal. Fish were not imaged. When fish were imaged and exposed to blue light-sheet laser light, drift tended to be larger. One unparalyzed fish was tested; this fish drifted forward. Despite the drift, average trajectories converged for every fish; average trajectories are shown in the same color for each fish for each of the three possible pre-displacements, with the different pre-displacements shown in distinct colors in the inset and in Figure S1C.

(B) Diagram of PID control system (see STAR Methods). Swimming is modeled as a continuous variable, a weighted linear combination of a velocity term and a displacement term, and constrained to be non-negative. The displacement term is a leaky integral of fish velocity with memory time constant τ.

(C) Example trajectories during simulated behavior with τ = 16 s for four (α,β) combinations, shown as raw position (left) and centered to the no-displacement trajectory (right). For this short time constant, trajectories converge only about halfway.

(D) For a longer memory time constant τ = 256 s, trajectories converge much more. Importantly, trajectories can drift forward or backward despite the fact that positional memory is quite accurate.

(E) For the four (α,β) combinations, the final trajectory difference is mainly a function of the memory time constant. In the real data, trajectories converge within about 5%–20%, suggesting a memory time constant of about a minute or more (gray shaded area).

(F) The average accumulated absolute drift over (α,β) pairs.