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Feedback between motion and sensation provides nonlinear boost in run-and-tumble navigation

Fig 2

Non-normal dynamics enables large asymmetric transients in internal state.

(A) Phase space diagram of Eq (8) when the positive feedback dominates, τ_E = 0.1. White: streamlines without noise; magenta: the r-nullcline where dr/dτ = 0; black: the two v-nullclines where dv/dτ = 0. Heat map: noise magnitude of dv/dτ ( in Eq (8)). (B) Two example trajectories starting in positive (cyan) or negative (magenta) direction. Each trajectory starts from black and lasts over the same time period of τ = 10. See also S1 Movie. (C,D) Same as A,B except in the negative-feedback-dominated regime, τ_E = 3. When the positive feedback dominates (τ_E = 0.1, A), the streamlines (white) are highly asymmetric around the fixed point. They tend to bring the system transiently towards r = 1 and v = 1—a result of both non-normal dynamics (non-orthogonal eigenvectors near the fixed point) and nonlinear positive feedback (growth towards v = 1 away from the fixed point)—before eventually falling back to the fixed point. High noise near the fixed point causes the system to quickly move away from it (magenta in B). Low noise in the upper right corner (r = 1 and v = 1) facilitates longer runs in the correct direction (cyan in B). Taken together, these effects result in a fast “ratchet-like” gradient climbing behavior. In contrast, when the negative feedback dominates (τ_E = 0.1, C) the streamlines all point back directly to the fixed point and small deviations do not grow (cyan and magenta in D). Details in Methods.

doi: https://doi.org/10.1371/journal.pcbi.1005429.g002