Starting from a Langevin description of active particles that move with constant speed in infinite two-dimensional space and its corresponding Fokker-Planck equation, we develop a systematic method that allows us to obtain the coarse-grained probability density of finding a particle at a given location and at a given time in arbitrary short-time regimes. By going beyond the diffusive limit, we derive a generalization of the telegrapher equation. Such generalization preserves the hyperbolic structure of the equation and incorporates memory effects in the diffusive term. While no difference is observed for the mean-square displacement computed from the two-dimensional telegrapher equation and from our generalization, the kurtosis results in a sensible parameter that discriminates between both approximations. We carry out a comparative analysis in Fourier space that sheds light on why the standard telegrapher equation is not an appropriate model to describe the propagation of particles with constant speed in dispersive media.

We explore minimal navigation strategies for active particles in complex, dynamical, external fields, introducing a class of autonomous, self-propelled particles which we call Markovian robots (MR). These machines are equipped with a navigation control system (NCS) that triggers random changes in the direction of self-propulsion of the robots. The internal state of the NCS is described by a Boolean variable that adopts two values. The temporal dynamics of this Boolean variable is dictated by a closed Markov chain-ensuring the absence of fixed points in the dynamics-with transition rates that may depend exclusively on the instantaneous, local value of the external field. Importantly, the NCS does not store past measurements of this value in continuous, internal variables. We show that despite the strong constraints, it is possible to conceive closed Markov chain motifs that lead to nontrivial motility behaviors of the MR in one, two, and three dimensions. By analytically reducing the complexity of the NCS dynamics, we obtain an effective description of the long-time motility behavior of the MR that allows us to identify the minimum requirements in the design of NCS motifs and transition rates to perform complex navigation tasks such as adaptive gradient following, detection of minima or maxima, or selection of a desired value in a dynamical, external field. We put these ideas in practice by assembling a robot that operates by the proposed minimalistic NCS to evaluate the robustness of MR, providing a proof of concept that is possible to navigate through complex information landscapes with such a simple NCS whose internal state can be stored in one bit. These ideas may prove useful for the engineering of miniaturized robots.

The collective dynamics and structure of animal groups has attracted the attention of scientists across a broad range of fields. A variety of agent-based models have been developed to help understand the emergence of coordinated collective behavior from simple interaction rules. A common, simplifying assumption of such collective movement models, is that individual agents move with a constant speed. In this work we critically re-asses this assumption. First, we discuss experimental data showcasing the omnipresent speed variability observed in different species of live fish and artificial agents (RoboFish). Based on theoretical considerations accounting for inertia and rotational friction, we derive a functional dependence of the turning response of individuals on their instantaneous speed, which is confirmed by experimental data. We then investigate the interplay of variable speed and speed-dependent turning on self-organized collective behavior by implementing an agent-based model which accounts for both these effects. We show that, besides the average speed of individuals, the variability in individual speed can have a dramatic impact on the emergent collective dynamics: a group which differs to another only in a lower speed variability of its individuals (groups being identical in all other behavioral parameters), can be in the polarized state while the other group is disordered. We find that the local coupling between group polarization and individual speed is strongest at the order-disorder transition, and that, in contrast to fixed speed models, the group’s spatial extent does not have a maximum at the transition. Furthermore, we demonstrate a decrease in polarization with group size for groups of individuals with variable speed, and a sudden decrease in mean individual speed at a critical group size (N = 4 for Voronoi interactions) linked to a topological transition from an all-to-all to a distributed spatial interaction network. Overall, our work highlights the importance to account for fundamental kinematic constraints in general, and variable speed in particular, when modeling self-organized collective dynamics.

Groups of animals can perform highly coordinated collective behaviours that confer benefits to the participating individuals by facilitating social information exchange and protection from predators1. Some of these characteristics could arise when groups operate at critical points between two structurally and functionally different states, leading to maximal responsiveness to external stimuli and effective propagation of information2,3. It has been proposed that animal groups constitute examples of self-organized systems at criticality2,3; however, direct empirical evidence of this hypothesis—in particular in the wild—is mostly absent. Here we show that highly conspicuous, repetitive and rhythmic collective dive cascades produced by many thousands of freshwater fish under high predation risk resemble a stochastic excitable system driven by environmental perturbations. Together with the results of an agent-based model of the system, this suggests that these fish shoals might operate at a critical point between a state of high individual diving activity and low overall diving activity. We show that the best fitting model, which is located at a critical point, allows information about external perturbations—such as predator attacks—to propagate most effectively through the shoal. Our results suggest that criticality might be a plausible principle of distributed information processing in large animal collectives.

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The ability of an individual to predict the outcome of actions of others and to change own behavior adaptively is called anticipation. There are many examples from mammalian species - including humans - that show anticipatory abilities in a social context, however, it is not clear to what extent fishes can anticipate the actions of their interaction partners and what are the underlying mechanisms for that anticipation. To answer these questions, we let live guppies (Poecilia reticulata) interact repeatedly with an open-loop (non-interactive) biomimetic robot that has been previously shown to be an accepted conspecific. The robot performed always the same zigzag trajectory in the experimental tank that ended in one of the corners, which gave the live fish the possibility to learn both the location of the final destination as well as the specific turning movementof the robot over three consecutive trials. The live fish’s reactions were categorized into a global anticipation which we defined as relative time to reach the robot’s final corner and a local anticipation which was the relative time and location of the live fish’s turns relative to robofish turns. As a proxy for global anticipation, we found that live fish in the last trial reached the robot’s destination corner significantly earlier than the robot. Overall, more than 50% of all fish arrived at the destination before the robot. This is more than a random walk model would predict and significantly more as compared to all other equidistant, yet unvisited corners. As a proxy for local anticipation, we found fish to change their turning behavior in response to the robot over the course of the trials. Initially the fish would turn after the robot, which was reversed in the end, as they began to turn slightly before the robot in the final trial. Our results indicate that live fish are able to anticipate predictably behaving social partners both in regard to final movement locations as well as movement dynamics. Given that fish have been found to exhibit consistent behavioral differences, anticipation in fish could have evolved as a mechanism to adapt to different social interaction partners.

Motivated by the observation of non-exponential run-time distributions of bacterial swimmers, we propose a minimal phenomenological model for taxis of active particles whose motion is controlled by an internal clock. The ticking of the clock depends on an external concentration field, e.g., a chemical substance. We demonstrate that these particles can detect concentration gradients and respond to them by moving up-or down-gradient depending on the clock design, albeit measurements of these fields are purely local in space and instantaneous in time. Altogether, our results open a new route in the study of directional navigation: we show that the use of a clock to control motility actions represents a generic and versatile toolbox to engineer behavioral responses to external cues, such as light, chemical, or temperature gradients.

A large number of biological systems — from bacteria to sheep — can be described as ensembles of self-propelled agents (active particles) with a complex internal dynamic that controls the agent’s behavior: resting, moving slow, moving fast, feeding, etc. In this study, we assume that such a complex internal dynamic can be described by a Markov chain, which controls the moving direction, speed, and internal state of the agent. We refer to this Markov chain as the Navigation Control System (NCS). Furthermore, we model that agents sense the environment by considering that the transition rates of the NCS depend on local (scalar) measurements of the environment such as e.g. chemical concentrations, light intensity, or temperature. Here, we investigate under which conditions the (asymptotic) behavior of the agents can be reduced to an effective convection–diffusion equation for the density of the agents, providing effective expressions for the drift and diffusion terms. We apply the developed generic framework to a series of specific examples to show that in order to obtain a drift term three necessary conditions should be fulfilled: (i) the NCS should possess two or more internal states, (ii) the NCS transition rates should depend on the agent’s position, and (iii) transition rates should be asymmetric. In addition, we indicate that the sign of the drift term — i.e. whether agents develop a positive or negative chemotactic response — can be changed by modifying the asymmetry of the NCS or by swapping the speed associated to the internal states. The developed theoretical framework paves the way to model a large variety of biological systems and provides a solid proof that chemotactic responses can be developed, counterintuitively, by agents that cannot measure gradients and lack memory as to store past measurements of the environment.