Discrete turn strategies emerge in information-limited navigation

TL;DR

It is found that, without directional information on which way to turn, behavioural strategies which make sudden turns perform better than gradual steering.

Abstract

Navigation up a sensory gradient is one of the simplest behaviours, and the simplest strategy is run and tumble. But some organisms use other strategies, such as reversing direction or turning by some angle. Here we ask what drives the choice of strategy, which we frame as maximising up-gradient speed using a given amount of sensory information per unit time. We find that, without directional information on which way to turn, behavioural strategies which make sudden turns perform better than gradual steering. We see various transitions where a different strategy becomes optimal, such as a switch from reversing direction to fully re-orienting tumbles as more information becomes available. And, among more complex re-orientation strategies, we show that discrete turn angles are best, and see transitions in how many such angles the optimal strategy employs.

Figures (10)

• Figure 1: Performance of strategies for two-dimensional navigation, on the speed-information plane. The continuous steering strategy achieves the highest performance, provided the sign of heading $\theta$ is visible (red). Among strategies not using the sign, fully re-orienting instantaneous tumbles (blue) perform well at high information rates, but at lower rates reversing (green) needs half as much information for the same speed. All low-information limits are $v\sim\sqrt{i}$, and high-information limits $v/v_{0}\to1$ except reverse, which is limited to $v/v_{0}<2/\pi\approx0.64$. Plot points are numerical, and square points (blue & green) use ansatz $\lambda_{\mathrm{strong}}(\theta)$. Black stars are from figure \ref{['fig:Any-jump']}.
• Figure 2: Numerical optimal strategies for run & tumble, steering, and reversing. (A) Tumble rate $\lambda(\theta)$ in blue, and steady-state $p(\theta)$ in grey, for a low-information case, and a high-information case using $\lambda_{\mathrm{strong}}(\theta)$. (B) The reverse strategy achieves $\sqrt{2}$ times the speed at similar information rate $i/D_{r}\approx0.01$, by acting at half the rate of the tumble strategy. But its high-information case saturates at speed $2/\pi$, when $p(\theta)$ is uniform on $-\pi/2\leq\theta\leq\pi/2$. (C) Steering rate $\mu(\theta)$ in red, and again $p(\theta)$ in grey. Second panel translates $\mu(\theta)$ and controller noise $D_{c}=1$ to $\lambda(\Delta\theta,\theta)$ using two angles $\Delta\theta=\pm\alpha$. (D) Steering rate $\mu(\left|\theta\right|)$, assuming the sign of $\theta$ is not observable. At low information rate, again $i/D_{r}\approx0.01$, the agent always turns in one direction but modulates its turning speed. At high information rate, this strategy creates stable & unstable fixed points at some $\pm\theta_{\star}$. (A-D) Dashed lines are analytic results at low information, from the text. (E) Sample trajectories for three strategies, all with $i/D_{r}\approx0.1$ (times $0<t<50$ in units $D_{r}=v_{0}=1$, wrapped to $-5<x<5$). More trajectories are shown in the appendix.
• Figure 3: Discrete optimal strategies for three information rates. We impose that rate $\lambda(\Delta\theta,\theta)$ is even in $\Delta\theta$, which implies that the strategy ignores the sign of $\theta$. At low information rate (left) we recover the reverse strategy, $\Delta\theta=\pi$, but with increasing information it bifurcates to use three angles (centre), and then five (right). Alongside the rate, we plot the contact function $\Psi(\Delta\theta)$ and a mean distribution $\bar{q}(\Delta\theta) \propto \bar{\lambda}(\Delta\theta)$. The rate $\lambda(\Delta\theta,\theta)$ is nonzero precisely where $\Psi(\Delta\theta)=1$. Notice that $\Psi(0)=1$, but we do not allow turns of $\Delta\theta=0$.
• Figure 4: Navigation in three dimensions. (A) Performance of tumble and reverse are qualitatively similar to figure \ref{['fig:Performance-of-steering-vs-tumbles']}, with different prefactors in the $v\propto\sqrt{i}$ scaling at low information rates. Flick produces similar performance to tumble. The red steering points are a strategy which knows which way to turn. The orange points are now a steering strategy which does not know which way to turn: the agent suffers diffusion $D_{\psi}$ in its roll angle $\psi$ (in addition to diffusion of its heading, $D_{r}$), and this is much worse, with $v\propto i$ at low information rates. (B) Optimal $\lambda(\Delta,\theta)$ for a strategy allowed any turn angle $\Delta$. Like figure \ref{['fig:Any-jump']} we see a progression from reverse, to reverse & flick, to something more complicated.
• Figure S1: Performance of strategies for two-dimensional navigation, relative to steering. Shows most of the data in figure \ref{['fig:Performance-of-steering-vs-tumbles']}, but adds a lines for the exact left-right steering solution (dark red, section \ref{['sec:signonly']}), and numerical points for the flick solution (purple) and the optimal symmetric $\lambda(\Delta\theta|\theta)$ (black stars). If we ignore the black stars, then notice that three different unsigned solutions are optimal in turn -- reverse at low information rate, then flick, then tumble. As before, square plot points indicate the use of the ansatz $\lambda_\mathrm{strong}(\theta)$, while others solve for the whole function.

Theorems & Definitions (1)

• proof