Statistical multimoment bifurcations in random-delay coupled swarms

Spatial flocking: Control by speed, distance, noise and delay

Fish, birds, insects and robots frequently swim or fly in groups. During their three dimensional collective motion, these agents do not stop, they avoid collisions by strong short-range repulsion, and achieve group cohesion by weak long-range attraction. In a minimal model that is isotropic, and continuous in both space and time, we demonstrate that (i) adjusting speed to a preferred value, combined with (ii) radial repulsion and an (iii) effective long-range attraction are sufficient for the stable ordering of autonomously moving agents in space. Our results imply that beyond these three rules ordering in space requires no further rules, for example, explicit velocity alignment, anisotropy of the interactions or the frequent reversal of the direction of motion, friction, elastic interactions, sticky surfaces, a viscous medium, or vertical separation that prefers interactions within horizontal layers. Noise and delays are inherent to the communication and decisions of all moving agents. Thus, next we investigate their effects on ordering in the model. First, we find that the amount of noise necessary for preventing the ordering of agents is not sufficient for destroying order. In other words, for realistic noise amplitudes the transition between order and disorder is rapid. Second, we demonstrate that ordering is more sensitive to displacements caused by delayed interactions than to uncorrelated noise (random errors). Third, we find that with changing interaction delays the ordered state disappears at roughly the same rate, whereas it emerges with different rates. In summary, we find that the model discussed here is simple enough to allow a fair understanding of the modeled phenomena, yet sufficiently detailed for the description and management of large flocks with noisy and delayed interactions. Our code is available at http://github.com/fij/floc.

Effects of cell cycle noise on excitable gene circuits

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a three-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

Collective motion patterns of swarms with delay coupling: Theory and experiment

The formation of coherent patterns in swarms of interacting self-propelled autonomous agents is a subject of great interest in a wide range of application areas, ranging from engineering and physics to biology. In this paper, we model and experimentally realize a mixed-reality large-scale swarm of delay-coupled agents. The coupling term is modeled as a delayed communication relay of position. Our analyses, assuming agents communicating over an Erdös-Renyi network, demonstrate the existence of stable coherent patterns that can be achieved only with delay coupling and that are robust to decreasing network connectivity and heterogeneity in agent dynamics. We also show how the bifurcation structure for emergence of different patterns changes with heterogeneity in agent acceleration capabilities and limited connectivity in the network as a function of coupling strength and delay. Our results are verified through simulation as well as preliminary experimental results of delay-induced pattern formation in a mixed-reality swarm.

Stochastic switching in delay-coupled oscillators

A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and different oscillation patterns. For coupled phase oscillators we reduce the delay system to a nondelayed Langevin equation, which allows us to analytically compute the distribution of frequencies and their corresponding residence times. The number of stable periodic orbits scales with the roundtrip delay time and coupling strength, but the noisy system visits only a fraction of the orbits, which scales with the square root of the delay time and is independent of the coupling strength. In contrast, the residence time in the different orbits is mainly determined by the coupling strength and the number of oscillators, and only weakly dependent on the coupling delay. Finally we investigate the effect of a detuning between the oscillators. We demonstrate the generality of our results with delay-coupled FitzHugh-Nagumo oscillators.

Noise Induced Pattern Switching in Randomly Distributed Delayed Swarms

We study the effects of noise on the dynamics of a system of coupled self-propelling particles in the case where the coupling is time-delayed, and the delays are discrete and randomly generated. Previous work has demonstrated that the stability of a class of emerging patterns depends upon all moments of the time delay distribution, and predicts their bifurcation parameter ranges. Near the bifurcations of these patterns, noise may induce a transition from one type of pattern to another. We study the onset of these noise-induced swarm re-organizations by numerically simulating the system over a range of noise intensities and for various distributions of the delays. Interestingly, there is a critical noise threshold above which the system is forced to transition from a less organized state to a more organized one. We explore this phenomenon by quantifying this critical noise threshold, and note that transition time between states varies as a function of both the noise intensity and delay distribution.