Discrete turn strategies emerge in information-limited navigation

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.

• The paper introduces an optimization framework that maximizes mean up-gradient speed subject to strict sensory information limits.
• It reveals that discrete turn strategies, including run-and-reverse and arbitrary-angle jumps, outperform continuous steering in low-information regimes.
• Analytical and numerical analyses demonstrate a stepwise emergence of discrete behaviors that explains empirical navigation patterns in microorganisms and artificial agents.

Discrete Turn Strategies in Information-Constrained Navigation

Introduction

Navigational behaviors in microorganisms and small motile agents are often structured as distinct, repetitive actions—such as runs, tumbles, reversals, or discrete-angle turns—despite the stimuli being typically continuous in nature. The work titled "Discrete turn strategies emerge in information-limited navigation" (2602.23324) formalizes the optimization problem underlying the emergence of such discrete behaviors, proposing that these strategies maximize gradient-climbing performance subject to strict sensory information constraints. The framework abstracts away the detailed sensory mechanism, focusing instead on the information rate available for converting heading into behavioral actions, and systematically characterizes optimal navigation strategies across a spectrum of information rates and accessible sensory modalities.

Optimization Framework for Navigation Strategies

The central construct is an optimization: maximize the mean speed up a one- or two-dimensional gradient, subject to a bound on the mutual information rate between the instantaneous heading $\theta$ and the controlled change in heading $d\theta_c$. This formulation naturally applies to a wide range of agent systems, including microorganisms such as E. coli, which rely on run-and-tumble dynamics, and animals or agents capable of more sophisticated directional sensing and actuation.

Optimality is characterized on the "speed–information" plane, with each behavioral class—continuous steering, run-and-tumble, run-and-reverse, or arbitrary-angle jumps—tested for achievable $v$ vs. $i$, where $v$ is the population-mean up-gradient speed and $i$ is the end-to-end information rate (in bits per unit time). Notably, the analysis distinguishes between strategies using directional (sign of heading) vs. nondirectional information, mapping the fundamental limits of navigation as a function of accessible information.

Numerical and Analytical Results for Key Strategies

The study provides extensive analytical and numerical results for several minimalist behavioral classes:

• Continuous Steering (with and without sign of $\theta$): Access to sign information enables simple and efficient gradient climbing, with mean velocity scaling as $v/v_0 \approx \sqrt{i/2D_r}$ in the low-information regime ($D_r$ is rotational diffusion). However, if only the cosine of the heading is accessible (no sign), performance is degraded and further constrained by more symmetric steering rates.
• Run-and-Tumble and Run-and-Reverse: Without directional information, the run-and-tumble class achieves $v/v_0 \approx \sqrt{i/4D_r}$ at low $i$, while run-and-reverse achieves the same mean speed with half the required information rate, i.e., $v/v_0 \approx \sqrt{i/2D_r}$. This distinction arises from the reversibility constraint, with reverse strategies saturating at a lower maximal speed as information increases.
• Arbitrary Discrete-Angle Jumps: Remarkably, when the agent is permitted to choose jump angles from any distribution, the optimal rate function is found to be discrete—with the number of supported turning angles increasing stepwise with the information rate.

These results are illustrated with both analytical expressions (in perturbative limits) and detailed numerical optimization.

Figure 1: Numerical optimal strategies for run-and-tumble, steering, and reversing; panels illustrate transition rates and steady-state heading distributions for representative information-limited scenarios.

Figure 2: Performance of navigation strategies for two-dimensional movement, plotting relative mean speed as a function of information rate for steering, tumble, and reversal classes.

Emergence of Discrete Behavioral Strategies

A central result of the analysis is that discrete (rather than continuous) reorientation strategies dominate in the low- and intermediate-information regimes. The optimal solution for the turn-angle distribution is strictly discrete, with the number of support points increasing as information flow from the heading to the controller increases. This discreteness emerges generically from the analyticity and convexity properties of the optimization problem: within the class of admissible $\lambda(\Delta\theta,\theta)$ (turn rate for angle $\Delta\theta$ at heading $\theta$), the contact function $\Psi(\Delta\theta)$ enforces that optimal rates vanish except at isolated angles.

In low-information regimes, the optimal strategy is reversal ($\Delta\theta = \pi$). As the information budget increases, the strategy bifurcates to additional discrete turning angles, e.g., three- or five-angle strategies.

Biological and Theoretical Implications

The findings offer explanations for the empirical ubiquity of discrete turn-based navigation strategies in microbial and animal systems, wherever physical or informational limitations preclude precise continuous steering. For instance, the transition between reversal- and tumble-dominated optima provides a mechanistic rationale for the diversity of observed turning behaviors in bacteria residing in different physical or environmental regimes.

Discrete reorientation strategies are likely to be more robust and efficient when sensors cannot resolve full directional cues, mirroring principles from rational inattention theory and neural coding literature, where optimal encoding strategies under information constraints are similarly found to be quantized or discretized. The study’s inability to capture even higher-level strategies (e.g., time-memory, loitering, or group navigation) is acknowledged as a consequence of focusing exclusively on memoryless mappings from heading to action; incorporating such additional elements is identified as an important direction for future research.

From a theoretical standpoint, the work connects the classical channel capacity problem in information theory to the behavioral control of agents, showing that input constraints generically induce discrete optimal output alphabets for the controller, in analogy to the well-known phenomenon of discrete optimum solutions for channel input distributions under amplitude constraints.

Generalization and Future Directions

Three-dimensional analogs of the navigation problem are also considered, with similar qualitative results: directionally unbiased steering strategies are outperformed by discrete-angle strategies at low information rates. Higher-dimensional environments and more elaborate behavioral rules are expected to preserve the basic structure of these results, though the geometry of heading spaces and relationships among behavioral symmetries add new layers of complexity.

Potential future work includes formal integration of memory-based strategies, explicit energetic constraints and trade-offs, and the extension of these optimization principles to multi-agent and collective navigation problems. Understanding how discrete behavior emerges when memory, energy, and communication constraints interact remains a fundamental target for both theoretical biology and artificial agent modeling.

Conclusion

This work rigorously demonstrates that when navigation is limited by sensory information constraints, discrete turn strategies naturally emerge and often exceed continuous steering in efficacy, especially in the absence of rich directional cues. The optimal solutions—characterized by a discrete set of support points for behavioral actions—arise robustly from the geometry of the information–performance trade-off. These findings have broad applicability in understanding behavioral diversity across biological systems and hold implications for the design and theoretically-informed control of artificial, information-limited agents.