Relevance of metric-free interactions in flocking phenomena

Flocking transitions in confluent tissues

Collective cell migration in dense tissues underlies important biological processes, such as embryonic development, wound healing and cancer invasion. While many aspects of single cell movements are now well established, the mechanisms leading to displacements of cohesive cell groups are still poorly understood. To elucidate the emergence of collective migration in mechanosensitive cells, we examine a self-propelled Voronoi (SPV) model of confluent tissues with an orientational feedback that aligns a cell's polarization with its local migration velocity. While shape and motility are known to regulate a density-independent liquid-solid transition in tissues, we find that aligning interactions facilitate collective motion and promote solidification, with transitions that can be predicted by extending statistical physics tools such as effective temperature to this far-from-equilibrium system. In addition to accounting for recent experimental observations obtained with epithelial monolayers, our model predicts structural and dynamical signatures of flocking, which may serve as gateway to a more quantitative characterization of collective motility.

A flocking algorithm for multi-agent systems with connectivity preservation under hybrid metric-topological interactions

In this paper, we propose a connectivity-preserving flocking algorithm for multi-agent systems in which the neighbor set of each agent is determined by the hybrid metric-topological distance so that the interaction topology can be represented as the range-limited Delaunay graph, which combines the properties of the commonly used disk graph and Delaunay graph. As a result, the proposed flocking algorithm has the following advantages over the existing ones. First, range-limited Delaunay graph is sparser than the disk graph so that the information exchange among agents is reduced significantly. Second, some links irrelevant to the connectivity can be dynamically deleted during the evolution of the system. Thus, the proposed flocking algorithm is more flexible than existing algorithms, where links are not allowed to be disconnected once they are created. Finally, the multi-agent system spontaneously generates a regular quasi-lattice formation without imposing the constraint on the ratio of the sensing range of the agent to the desired distance between two adjacent agents. With the interaction topology induced by the hybrid distance, the proposed flocking algorithm can still be implemented in a distributed manner. We prove that the proposed flocking algorithm can steer the multi-agent system to a stable flocking motion, provided the initial interaction topology of multi-agent systems is connected and the hysteresis in link addition is smaller than a derived upper bound. The correctness and effectiveness of the proposed algorithm are verified by extensive numerical simulations, where the flocking algorithms based on the disk and Delaunay graph are compared.

Distributed Algorithm for Voronoi Partition of Wireless Sensor Networks with a Limited Sensing Range

For Wireless Sensor Networks (WSNs), the Voronoi partition of a region is a challenging problem owing to the limited sensing ability of each sensor and the distributed organization of the network. In this paper, an algorithm is proposed for each sensor having a limited sensing range to compute its limited Voronoi cell autonomously, so that the limited Voronoi partition of the entire WSN is generated in a distributed manner. Inspired by Graham’s Scan (GS) algorithm used to compute the convex hull of a point set, the limited Voronoi cell of each sensor is obtained by sequentially scanning two consecutive bisectors between the sensor and its neighbors. The proposed algorithm called the Boundary Scan (BS) algorithm has a lower computational complexity than the existing Range-Constrained Voronoi Cell (RCVC) algorithm and reaches the lower bound of the computational complexity of the algorithms used to solve the problem of this kind. Moreover, it also improves the time efficiency of a key step in the Adjust-Sensing-Radius (ASR) algorithm used to compute the exact Voronoi cell. Extensive numerical simulations are performed to demonstrate the correctness and effectiveness of the BS algorithm. The distributed realization of the BS combined with a localization algorithm in WSNs is used to justify the WSN nature of the proposed algorithm.

Soft yet Sharp Interfaces in a Vertex Model of Confluent Tissue

How can dense biological tissue maintain sharp boundaries between coexisting cell populations? We explore this question within a simple vertex model for cells, focusing on the role of topology and tissue surface tension. We show that the ability of cells to independently regulate adhesivity and tension, together with neighbor-based interaction rules, lets them support strikingly unusual interfaces. In particular, we show that mechanical- and fluctuation-based measurements of the effective surface tension of a cellular aggregate yield different results, leading to mechanically soft interfaces that are nevertheless extremely sharp.

Identifying influential neighbors in animal flocking

Schools of fish and flocks of birds can move together in synchrony and decide on new directions of movement in a seamless way. This is possible because group members constantly share directional information with their neighbors. Although detecting the directionality of other group members is known to be important to maintain cohesion, it is not clear how many neighbors each individual can simultaneously track and pay attention to, and what the spatial distribution of these influential neighbors is. Here, we address these questions on shoals of Hemigrammus rhodostomus, a species of fish exhibiting strong schooling behavior. We adopt a data-driven analysis technique based on the study of short-term directional correlations to identify which neighbors have the strongest influence over the participation of an individual in a collective U-turn event. We find that fish mainly react to one or two neighbors at a time. Moreover, we find no correlation between the distance rank of a neighbor and its likelihood to be influential. We interpret our results in terms of fish allocating sequential and selective attention to their neighbors.

Kinetic Models for Topological Nearest-Neighbor Interactions

We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys 163:41-60, 2016). The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.

Patterned Disordered Cell Motion Ensures Vertebral Column Symmetry

The biomechanics of posterior embryonic growth must be dynamically regulated to ensure bilateral symmetry of the spinal column. Throughout vertebrate trunk elongation, motile mesodermal progenitors undergo an order-to-disorder transition via an epithelial-to-mesenchymal transition and sort symmetrically into the left and right paraxial mesoderm. We combine theoretical modeling of cell migration in a tail-bud-like geometry with experimental data analysis to assess the importance of ordered and disordered cell motion. We find that increasing order in cell motion causes a phase transition from symmetric to asymmetric body elongation. In silico and in vivo, overly ordered cell motion converts normal anisotropic fluxes into stable vortices near the posterior tail bud, contributing to asymmetric cell sorting. Thus, disorder is a physical mechanism that ensures the bilateral symmetry of the spinal column. These physical properties of the tissue connect across scales such that patterned disorder at the cellular level leads to the emergence of organism-level order.

Effect of Correlations in Swarms on Collective Response

Social interaction increases significantly the performance of a wide range of cooperative systems. However, evidence that natural swarms limit the number of interactions suggests potentially detrimental consequences of excessive interaction. Using a canonical model of collective motion, we find that the collective response to a dynamic localized perturbation-emulating a predator attack-is hindered when the number of interacting neighbors exceeds a certain threshold. Specifically, the effectiveness in avoiding the predator is enhanced by large integrated correlations, which are known to peak at a given level of interagent interaction. From the network-theoretic perspective, we uncover the same interplay between number of connections and effectiveness in group-level response for two distinct decision-making models of distributed consensus operating over a range of static networks. The effect of the number of connections on the collective response critically depends on the dynamics of the perturbation. While adding more connections improves the response to slow perturbations, the opposite is true for fast ones. These results have far-reaching implications for the design of artificial swarms or interaction networks.

Analysis of topological relationships and network properties in the interactions of human beings

In the animal world, various kinds of collective motions have been found and proven to be efficient ways of carrying out some activities such as searching for food and avoiding predators. Many scholars research the interactions of collective behaviors of human beings according to the rules of collective behaviors of animals. Based on the Lennard-Jones potential function and a self-organization process, our paper proposes a topological communication model to simulate the collective behaviors of human beings. In the results of simulations, we find various types of collective behavior and fission behavior and discover the threshold for the emergence of collective behavior, which is the range five to seven for the number of topology K. According to the analysis of network properties of the model, the in-degree of individuals is always equal to the number of topology. In the stable state, the out-degrees of individuals distribute around the value of the number of topology K, except that the out-degree of a single individual is approximately double the out-degrees of the other individuals. In addition, under different initial conditions, some features of different kinds of networks emerge from the model. We also find the leader and herd mentality effects in the characteristics of the behaviors of human beings in our model. Thus, this work could be used to discover how to promote the emergence of beneficial group behaviors and prevent the emergence of harmful behaviors.

Fore-aft asymmetric flocking

We show that fore-aft asymmetry, a generic feature of living organisms and some active matter systems, can have a strong influence on the collective properties of even the simplest flocking models. Specifically, an arbitrarily weak asymmetry favoring front neighbors changes qualitatively the phase diagram of the Vicsek model. A region where many sharp traveling band solutions coexist is present at low noise strength, below the Toner-Tu liquid, at odds with the phase-separation scenario well describing the usual isotropic model. Inside this region, a "banded-liquid" phase with algebraic density distribution coexists with band solutions. Linear stability analysis at the hydrodynamic level suggests that these results are generic and not specific to the Vicsek model.

The selfish herd: Noise effects in Local Crowded Horizon and Voronoi models

Randomness or noise in the motion of herding animals is more important than commonly assumed. We show through simulations that noise promotes the compactness of a herd by facilitating the breakup of small clusters in favor of a more compact whole, reducing the mean median distances to center (MMDC) for the herd. Adding noise to movement models can also decrease risk from predators emerging locally by reducing Domains of Danger (DODs); we specifically show that adding noise to the Voronoi (V) movement model gives favorable DOD results comparable to that of the Local Crowded Horizon (LCH) model. In addition, we show that adding noise to these models can significantly reduce risk from predators emerging outside the herding area through head-to-head competition in mixed herds, with effects that can be larger than the choice of model itself. Our results are consistent with recent observational studies in several different animal populations that suggest random motion plays a significant role in the movement of individuals within these groups. Because of its significance, noise must be considered whenever model effectiveness is discussed or comparisons between movement models are made.

Nonsymmetric Interactions Trigger Collective Swings in Globally Ordered Systems

Many systems in nature, from ferromagnets to flocks of birds, exhibit ordering phenomena on the large scale. In condensed matter systems, order is statistically robust for large enough dimensions, with relative fluctuations due to noise vanishing with system size. Several biological systems, however, are less stable and spontaneously change their global state on relatively short time scales. Here we show that there are two crucial ingredients in these systems that enhance the effect of noise, leading to collective changes of state on finite time scales and off-equilibrium behavior: the nonsymmetric nature of interactions between individuals, and the presence of local heterogeneities in the topology of the network. Our results might explain what is observed in several living systems and are consistent with recent experimental data on bird flocks and other animal groups.

Synchronization of multi-agent systems with metric-topological interactions

A hybrid multi-agent systems model integrating the advantages of both metric interaction and topological interaction rules, called the metric-topological model, is developed. This model describes planar motions of mobile agents, where each agent can interact with all the agents within a circle of a constant radius, and can furthermore interact with some distant agents to reach a pre-assigned number of neighbors, if needed. Some sufficient conditions imposed only on system parameters and agent initial states are presented, which ensure achieving synchronization of the whole group of agents. It reveals the intrinsic relationships among the interaction range, the speed, the initial heading, and the density of the group. Moreover, robustness against variations of interaction range, density, and speed are investigated by comparing the motion patterns and performances of the hybrid metric-topological interaction model with the conventional metric-only and topological-only interaction models. Practically in all cases, the hybrid metric-topological interaction model has the best performance in the sense of achieving highest frequency of synchronization, fastest convergent rate, and smallest heading difference.

Nontoxic colloidal particles impede antibiotic resistance of swarming bacteria by disrupting collective motion and speed

A monolayer of swarming B. subtilis on semisolid agar is shown to display enhanced resistance against antibacterial drugs due to their collective behavior and motility. The dynamics of swarming motion, visualized in real time using time-lapse microscopy, prevents the bacteria from prolonged exposure to lethal drug concentrations. The elevated drug resistance is significantly reduced when the collective motion of bacteria is judiciously disrupted using nontoxic polystyrene colloidal particles immobilized on the agar surface. The colloidal particles block and hinder the motion of the cells, and force large swarming rafts to break up into smaller packs in order to maneuver across narrow spaces between densely packed particles. In this manner, cohesive rafts rapidly lose their collectivity, speed, and group dynamics, and the cells become vulnerable to the drugs. The antibiotic resistance capability of swarming B. subtilis is experimentally observed to be negatively correlated with the number density of colloidal particles on the engineered surface. This relationship is further tested using an improved self-propelled particle model that takes into account interparticle alignment and hard-core repulsion. This work has pertinent implications on the design of optimal methods to treat drug resistant bacteria commonly found in swarming colonies.

Topological-distance-dependent transition in flocks with binary interactions

We have studied a flocking model with binary interactions (binary flock), where the velocity of an agent depends on the velocity of only another agent and its own velocity, topped by the angular noise. The other agent is selected as the nth topological neighbor; the specific value of n being a fixed parameter of the problem. On the basis of extensive numerical simulation results, we argue that for n = 1, the phase transition from the ordered to the disordered phase of the flock is a special kind of discontinuous transition. Here, the order parameter does not flip-flop between multiple metastable states. It continues its initial disordered state for a period t(c), then switches over to the ordered state and remains in this state ever after. For n = 2, it is the usual discontinuous transition between two metastable states. Beyond this range, the continuous transitions are observed for n≥3. Such a system of binary flocks has been further studied using the hydrodynamic equations of motion. Linear stability analysis of the homogeneous polarized state shows that such a state is unstable close to the critical point and above some critical speed, which increases as we increase n. The critical noise strengths, which depend on the average correlation between a pair of topological neighbors, are estimated for five different values of n, which match well with their simulated values.

Emergent behavioural phenotypes of swarming models revealed by mimicking a frustrated anti-ferromagnet

Self-propelled particle (SPP) models are often compared with animal swarms. However, the collective animal behaviour observed in experiments often leaves considerable unconstrained freedom in the structure of a proposed model. Essentially, multiple models can describe the observed behaviour of animal swarms in simple environments. To tackle this degeneracy, we study swarms of SPPs in non-trivial environments as a new approach to distinguish between candidate models. We restrict swarms of SPPs to circular (periodic) channels where they polarize in one of two directions (like spins) and permit information to pass through windows between neighbouring channels. Co-alignment between particles then couples the channels (anti-ferromagnetically) so that they tend to counter-rotate. We study channels arranged to mimic a geometrically frustrated anti-ferromagnet and show how the effects of this frustration allow us to better distinguish between SPP models. Similar experiments could therefore improve our understanding of collective motion in animals. Finally, we discuss how the spin analogy can be exploited to construct universal logic gates, and therefore swarming systems that can function as Turing machines.

Flocking with discrete symmetry: The two-dimensional active Ising model

We study in detail the active Ising model, a stochastic lattice gas where collective motion emerges from the spontaneous breaking of a discrete symmetry. On a two-dimensional lattice, active particles undergo a diffusion biased in one of two possible directions (left and right) and align ferromagnetically their direction of motion, hence yielding a minimal flocking model with discrete rotational symmetry. We show that the transition to collective motion amounts in this model to a bona fide liquid-gas phase transition in the canonical ensemble. The phase diagram in the density-velocity parameter plane has a critical point at zero velocity which belongs to the Ising universality class. In the density-temperature "canonical" ensemble, the usual critical point of the equilibrium liquid-gas transition is sent to infinite density because the different symmetries between liquid and gas phases preclude a supercritical region. We build a continuum theory which reproduces qualitatively the behavior of the microscopic model. In particular, we predict analytically the shapes of the phase diagrams in the vicinity of the critical points, the binodal and spinodal densities at coexistence, and the speeds and shapes of the phase-separated profiles.

Intermittent collective dynamics emerge from conflicting imperatives in sheep herds

Among the many fascinating examples of collective behavior exhibited by animal groups, some species are known to alternate slow group dispersion in space with rapid aggregation phenomena induced by a sudden behavioral shift at the individual level. We study this phenomenon quantitatively in large groups of grazing Merino sheep under controlled experimental conditions. Our analysis reveals strongly intermittent collective dynamics consisting of fast, avalanche-like regrouping events distributed on all experimentally accessible scales. As a proof of principle, we introduce an agent-based model with individual behavioral shifts, which we show to account faithfully for all collective properties observed. This offers, in turn, an insight on the individual stimulus/response functions that can generate such intermittent behavior. In particular, the intensity of sheep allelomimetic behavior plays a key role in the group's ability to increase the per capita grazing surface while minimizing the time needed to regroup into a tightly packed configuration. We conclude that the emergent behavior reported probably arises from the necessity to balance two conflicting imperatives: (i) the exploration of foraging space by individuals and (ii) the protection from predators offered by being part of large, cohesive groups. We discuss our results in the context of the current debate about criticality in biology.

Emergence of Collective Motion in a Model of Interacting Brownian Particles

By studying a system of Brownian particles that interact among themselves only through a local velocity-alignment force that does not affect their speed, we show that self-propulsion is not a necessary feature for the flocking transition to take place as long as underdamped particle dynamics can be guaranteed. Moreover, the system transits from stationary phases close to thermal equilibrium, with no net flux of particles, to far-from-equilibrium ones exhibiting collective motion, phase coexistence, long-range order, and giant number fluctuations, features typically associated with ordered phases of models where self-propelled particles with overdamped dynamics are considered.

Short-range interactions versus long-range correlations in bird flocks

Bird flocks are a paradigmatic example of collective motion. One of the prominent traits of flocking is the presence of long range velocity correlations between individuals, which allow them to influence each other over the large scales, keeping a high level of group coordination. A crucial question is to understand what is the mutual interaction between birds generating such nontrivial correlations. Here we use the maximum entropy (ME) approach to infer from experimental data of natural flocks the effective interactions between individuals. Compared to previous studies, we make a significant step forward as we retrieve the full functional dependence of the interaction on distance, and find that it decays exponentially over a range of a few individuals. The fact that ME gives a short-range interaction even though its experimental input is the long-range correlation function, shows that the method is able to discriminate the relevant information encoded in such correlations and single out a minimal number of effective parameters. Finally, we show how the method can be used to capture the degree of anisotropy of mutual interactions.

Active matter beyond mean-field: ring-kinetic theory for self-propelled particles

Recently, Hanke et al. [Phys. Rev. E 88, 052309 (2013)] showed that mean-field kinetic theory fails to describe collective motion in soft active colloids and that correlations must not be neglected. Correlation effects are also expected to be essential in systems of biofilaments driven by molecular motors and in swarms of midges. To obtain correlations in an active matter system from first principles, we derive a ring-kinetic theory for Vicsek-style models of self-propelled agents from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle precollisional correlations and cluster formation, which are both important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small-density expansion of the collision operator and derive the first two equations of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. An algorithm is presented that numerically solves the evolution equation for the two-particle correlations on a lattice. Agent-based simulations are performed and informative quantities such as orientational and density correlation functions are compared with those obtained by ring-kinetic theory. Excellent quantitative agreement between simulations and theory is found at not-too-small noises and mean free paths. This shows that there are parameter ranges in Vicsek-like models where the correlated closure of the BBGKY hierarchy gives correct and nontrivial results. We calculate the dependence of the orientational correlations on distance in the disordered phase and find that it seems to be consistent with a power law with an exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed.

Information transfer and behavioural inertia in starling flocks

Collective decision-making in biological systems requires all individuals in the group to go through a behavioural change of state. During this transition fast and robust transfer of information is essential to prevent cohesion loss. The mechanism by which natural groups achieve such robustness, though, is not clear. Here we present an experimental study of starling flocks performing collective turns. We find that information about direction changes propagates across the flock with a linear dispersion law and negligible attenuation, hence minimizing group decoherence. These results contrast starkly with current models of collective motion, which predict diffusive transport of information. Building on spontaneous symmetry breaking and conservation laws arguments, we formulate a new theory that correctly reproduces linear and undamped propagation. Essential to the new framework is the inclusion of the birds' behavioural inertia. The new theory not only explains the data, but also predicts that information transfer must be faster the stronger the group's orientational order, a prediction accurately verified by the data. Our results suggest that swift decision-making may be the adaptive drive for the strong behavioural polarization observed in many living groups.

Role of projection in the control of bird flocks

Swarming is a conspicuous behavioral trait observed in bird flocks, fish shoals, insect swarms, and mammal herds. It is thought to improve collective awareness and offer protection from predators. Many current models involve the hypothesis that information coordinating motion is exchanged among neighbors. We argue that such local interactions alone are insufficient to explain the organization of large flocks of birds and that the mechanism for the exchange of long-range information necessary to control their density remains unknown. We show that large flocks self-organize to the maximum density at which a typical individual still can see out of the flock in many directions. Such flocks are marginally opaque--an external observer also still can see a substantial fraction of sky through the flock. Although this seems intuitive, we show it need not be the case; flocks might easily be highly diffuse or entirely opaque. The emergence of marginal opacity strongly constrains how individuals interact with one another within large swarms. It also provides a mechanism for global interactions: an individual can respond to the projection of the flock that it sees. This provides for faster information transfer and hence rapid flock dynamics, another advantage over local models. From a behavioral perspective, it optimizes the information available to each bird while maintaining the protection of a dense, coherent flock.

Visual sensory networks and effective information transfer in animal groups

Social transmission of information is vital for many group-living animals, allowing coordination of motion and effective response to complex environments. Revealing the interaction networks underlying information flow within these groups is a central challenge. Previous work has modeled interactions between individuals based directly on their relative spatial positions: each individual is considered to interact with all neighbors within a fixed distance (metric range), a fixed number of nearest neighbors (topological range), a 'shell' of near neighbors (Voronoi range), or some combination (Figure 1A). However, conclusive evidence to support these assumptions is lacking. Here, we employ a novel approach that considers individual movement decisions to be based explicitly on the sensory information available to the organism. In other words, we consider that while spatial relations do inform interactions between individuals, they do so indirectly, through individuals' detection of sensory cues. We reconstruct computationally the visual field of each individual throughout experiments designed to investigate information propagation within fish schools (golden shiners, Notemigonus crysoleucas). Explicitly considering visual sensing allows us to more accurately predict the propagation of behavioral change in these groups during leadership events. Furthermore, we find that structural properties of visual interaction networks differ markedly from those of metric and topological counterparts, suggesting that previous assumptions may not appropriately reflect information flow in animal groups.

Dynamical maximum entropy approach to flocking

We derive a new method to infer from data the out-of-equilibrium alignment dynamics of collectively moving animal groups, by considering the maximum entropy model distribution consistent with temporal and spatial correlations of flight direction. When bird neighborhoods evolve rapidly, this dynamical inference correctly learns the parameters of the model, while a static one relying only on the spatial correlations fails. When neighbors change slowly and the detailed balance is satisfied, we recover the static procedure. We demonstrate the validity of the method on simulated data. The approach is applicable to other systems of active matter.

Promoting collective motion of self-propelled agents by distance-based influence

We propose a dynamic model for a system consisting of self-propelled agents in which the influence of an agent on another agent is weighted by geographical distance. A parameter α is introduced to adjust the influence: The smaller value of α means that the closer neighbors have a stronger influence on the moving direction. We find that there exists an optimal value of α leading to the highest degree of direction consensus. The value of optimal α increases as the system size increases, while it decreases as the absolute velocity, the sensing radius, and the noise amplitude increase.

Self-organized vortices of circling self-propelled particles and curved active flagella

Self-propelled pointlike particles move along circular trajectories when their translocation velocity is constant and the angular velocity related to their orientation vector is also constant. We investigate the collective behavior of ensembles of such circle swimmers by Brownian dynamics simulations. If the particles interact via a "velocity-trajectory coordination" rule within neighboring particles, a self-organized vortex pattern emerges. This vortex pattern is characterized by its particle-density correlation function Gρ, the density correlation function Gc of trajectory centers, and an order parameter S representing the degree of the aggregation of the particles. Here we systematically vary the system parameters, such as the particle density and the interaction range, in order to reveal the transition of the system from a light-vortex-dominated to heavy-vortex-dominated state, where vortices contain mainly a single and many self-propelled particles, respectively. We also study a semidilute solution of curved, sinusoidal-beating flagella, as an example of circling self-propelled particles with explicit propulsion mechanism and excluded-volume interactions. Our simulation results are compared with previous experimental results for the vortices in sea-urchin sperm solutions near a wall. The properties of the vortices in simulations and experiments are found to agree quantitatively.

Evidence of a robust universality class in the critical behavior of self-propelled agents: metric versus topological interactions

The nature of the interactions among self-propelled agents (SPA), i.e., topological versus metric or a combination of both types, is a relevant open question in the field of self-organization phenomena. We studied the critical behavior of a Vicsek-like system of SPA given by a group of agents moving at constant speed and interacting among themselves under the action of a topological rule: each agent aligns itself with the average direction of its seven nearest neighbors, independent of the distance, under the influence of some noise. Based on both stationary and dynamic measurements, we provide strong evidence that both types of interactions are manifestations of the same phenomenon, which defines a robust universality class. Also, the cluster size distribution evaluated at the critical point shows a power-law behavior, and the exponent corresponding to the topological model is in excellent agreement with that of the metric one, further reinforcing our claim. Furthermore, we found that with topological interactions the average distance of influence between agents undergoes large fluctuations that diverge at the critical noise, thus providing clues about a mechanism that could be implemented by the agents to change their moving strategy.

Resilience and controllability of dynamic collective behaviors

The network paradigm is used to gain insight into the structural root causes of the resilience of consensus in dynamic collective behaviors, and to analyze the controllability of the swarm dynamics. Here we devise the dynamic signaling network which is the information transfer channel underpinning the swarm dynamics of the directed interagent connectivity based on a topological neighborhood of interactions. The study of the connectedness of the swarm signaling network reveals the profound relationship between group size and number of interacting neighbors, which is found to be in good agreement with field observations on flock of starlings [Ballerini et al. (2008) Proc. Natl. Acad. Sci. USA, 105: 1232]. Using a dynamical model, we generate dynamic collective behaviors enabling us to uncover that the swarm signaling network is a homogeneous clustered small-world network, thus facilitating emergent outcomes if connectedness is maintained. Resilience of the emergent consensus is tested by introducing exogenous environmental noise, which ultimately stresses how deeply intertwined are the swarm dynamics in the physical and network spaces. The availability of the signaling network allows us to analytically establish for the first time the number of driver agents necessary to fully control the swarm dynamics.

Invasion-wave-induced first-order phase transition in systems of active particles

An instability near the transition to collective motion of self-propelled particles is studied numerically by Enskog-like kinetic theory. While hydrodynamics breaks down, the kinetic approach leads to steep solitonlike waves. These supersonic waves show hysteresis and lead to an abrupt jump of the global order parameter if the noise level is changed. Thus they provide a mean-field mechanism to change the second-order character of the phase transition to first order. The shape of the wave is shown to follow a scaling law and to quantitatively agree with agent-based simulations.

Density-independent model of self-propelled particles

We examine a density-independent modification of the Vicsek model in which a particle interacts with neighbors defined by Delaunay triangulation. To feasibly simulate the model, an algorithm for repairing the triangulation over time was developed. This algorithm may also be applied to any time varying two-dimensional Delaunay triangulation. This model exhibits a continuous phase transition with noise, and a distinct set of critical exponents were measured which satisfy a hyperscaling relationship. The critical exponents are found to vary between a low and high velocity regime, but they are robust under the inclusion of a repulsive interaction. We present evidence that the correlation length approximately scales with the size of the system even in the ordered phase.

Revisiting the flocking transition using active spins

We consider an active Ising model in which spins both diffuse and align on lattice in one and two dimensions. The diffusion is biased so that plus or minus spins hop preferably to the left or to the right, which generates a flocking transition at low temperature and high density. We construct a coarse-grained description of the model that predicts this transition to be a first-order liquid-gas transition in the temperature-density ensemble, with a critical density sent to infinity. In this first-order phase transition, the magnetization is proportional to the liquid fraction and thus varies continuously throughout the phase diagram. Using microscopic simulations, we show that this theoretical prediction holds in 2D whereas the fluctuations alter the transition in 1D, preventing, for instance, any spontaneous symmetry breaking.

Optimal noise maximizes collective motion in heterogeneous media

We study the effect of spatial heterogeneity on the collective motion of self-propelled particles (SPPs). The heterogeneity is modeled as a random distribution of either static or diffusive obstacles, which the SPPs avoid while trying to align their movements. We find that such obstacles have a dramatic effect on the collective dynamics of usual SPP models. In particular, we report about the existence of an optimal (angular) noise amplitude that maximizes collective motion. We also show that while at low obstacle densities the system exhibits long-range order, in strongly heterogeneous media collective motion is quasi-long-range and exists only for noise values in between two critical values, with the system being disordered at both large and low noise amplitudes. Since most real systems have spatial heterogeneities, the finding of an optimal noise intensity has immediate practical and fundamental implications for the design and evolution of collective motion strategies.

Collective dynamics in systems of active Brownian particles with dissipative interactions

We use computer simulations to study the onset of collective motion in systems of interacting active particles. Our model is a swarm of active Brownian particles with an internal energy depot and interactions inspired by the dissipative particle dynamics method, imposing pairwise friction force on the nearest neighbors. We study orientational ordering in a 2D system as a function of energy influx rate and particle density. The model demonstrates a transition into the ordered state on increasing the particle density and increasing the input power. Although both the alignment mechanism and the character of individual motion in our model differ from those in the well-studied Vicsek model, it demonstrates identical statistical properties and phase behavior.

Boundary information inflow enhances correlation in flocking

The most conspicuous trait of collective animal behavior is the emergence of highly ordered structures. Less obvious to the eye, but perhaps more profound a signature of self-organization, is the presence of long-range spatial correlations. Experimental data on starling flocks in 3D show that the exponent ruling the decay of the velocity correlation function, C(r)~1/r(γ), is extremely small, γ<<1. This result can neither be explained by equilibrium field theory nor by off-equilibrium theories and simulations of active systems. Here, by means of numerical simulations and theoretical calculations, we show that a dynamical field applied to the boundary of a set of Heisenberg spins on a 3D lattice gives rise to a vanishing exponent γ, as in starling flocks. The effect of the dynamical field is to create an information inflow from border to bulk that triggers long-range spin-wave modes, thus giving rise to an anomalously long-ranged correlation. The biological origin of this phenomenon can be either exogenous-information produced by environmental perturbations is transferred from boundary to bulk of the flock-or endogenous-the flock keeps itself in a constant state of dynamical excitation that is beneficial to correlation and collective response.

Diffusion of individual birds in starling flocks

Flocking is a paradigmatic example of collective animal behaviour, where global order emerges out of self-organization. Each individual has a tendency to align its flight direction with those of neighbours, and such a simple form of interaction produces a state of collective motion of the group. When compared with other cases of collective ordering, a crucial feature of animal groups is that the interaction network is not fixed in time, as each individual moves and continuously changes its neighbours. The possibility to exchange neighbours strongly enhances the stability of global ordering and the way information is propagated through the group. Here, we assess the relevance of this mechanism in large flocks of starlings (Sturnus vulgaris). We find that birds move faster than Brownian walkers both with respect to the centre of mass of the flock, and with respect to each other. Moreover, this behaviour is strongly anisotropic with respect to the direction of motion of the flock. We also measure the amount of neighbours reshuffling and find that neighbours change in time exclusively as a consequence of the random fluctuations in the individual motion, so that no specific mechanism to keep one's neighbours seems to be enforced. On the contrary, our findings suggest that a more complex dynamical process occurs at the border of the flock.

Spatially balanced topological interaction grants optimal cohesion in flocking models

Models of self-propelled particles (SPPs) are an indispensable tool to investigate collective animal behaviour. Originally, SPP models were proposed with metric interactions, where each individual coordinates with neighbours within a fixed metric radius. However, recent experiments on bird flocks indicate that interactions are topological: each individual interacts with a fixed number of neighbours, irrespective of their distance. It has been argued that topological interactions are more robust than metric ones against external perturbations, a significant evolutionary advantage for systems under constant predatory pressure. Here, we test this hypothesis by comparing the stability of metric versus topological SPP models in three dimensions. We show that topological models are more stable than metric ones. We also show that a significantly better stability is achieved when neighbours are selected according to a spatially balanced topological rule, namely when interacting neighbours are evenly distributed in angle around the focal individual. Finally, we find that the minimal number of interacting neighbours needed to achieve fully stable cohesion in a spatially balanced model is compatible with the value observed in field experiments on starling flocks.

Competing ferromagnetic and nematic alignment in self-propelled polar particles

We study a Vicsek-style model of self-propelled particles where ferromagnetic and nematic alignment compete in both the usual "metric" version and in the "metric-free" case where a particle interacts with its Voronoi neighbors. We show that the phase diagram of this out-of-equilibrium XY model is similar to that of its equilibrium counterpart: The properties of the fully nematic model, studied before by Ginelli et al. [Phys. Rev. Lett. 104, 184502 (2010)], are thus robust to the introduction of a modest bias of interactions toward ferromagnetic alignment. The direct transitions between polar and nematic ordered phases are shown to be discontinuous in the metric case, and continuous, belonging to the Ising universality class, in the metric-free version.

From behavioural analyses to models of collective motion in fish schools

Fish schooling is a phenomenon of long-lasting interest in ethology and ecology, widely spread across taxa and ecological contexts, and has attracted much interest from statistical physics and theoretical biology as a case of self-organized behaviour. One topic of intense interest is the search of specific behavioural mechanisms at stake at the individual level and from which the school properties emerges. This is fundamental for understanding how selective pressure acting at the individual level promotes adaptive properties of schools and in trying to disambiguate functional properties from non-adaptive epiphenomena. Decades of studies on collective motion by means of individual-based modelling have allowed a qualitative understanding of the self-organization processes leading to collective properties at school level, and provided an insight into the behavioural mechanisms that result in coordinated motion. Here, we emphasize a set of paradigmatic modelling assumptions whose validity remains unclear, both from a behavioural point of view and in terms of quantitative agreement between model outcome and empirical data. We advocate for a specific and biologically oriented re-examination of these assumptions through experimental-based behavioural analysis and modelling.

Deciphering interactions in moving animal groups

Collective motion phenomena in large groups of social organisms have long fascinated the observer, especially in cases, such as bird flocks or fish schools, where large-scale highly coordinated actions emerge in the absence of obvious leaders. However, the mechanisms involved in this self-organized behavior are still poorly understood, because the individual-level interactions underlying them remain elusive. Here, we demonstrate the power of a bottom-up methodology to build models for animal group motion from data gathered at the individual scale. Using video tracks of fish shoal in a tank, we show how a careful, incremental analysis at the local scale allows for the determination of the stimulus/response function governing an individual's moving decisions. We find in particular that both positional and orientational effects are present, act upon the fish turning speed, and depend on the swimming speed, yielding a novel schooling model whose parameters are all estimated from data. Our approach also leads to identify a density-dependent effect that results in a behavioral change for the largest groups considered. This suggests that, in confined environment, the behavioral state of fish and their reaction patterns change with group size. We debate the applicability, beyond the particular case studied here, of this novel framework for deciphering interactions in moving animal groups.

Swarms of insects, schools of fish and flocks of birds display an impressive variety of collective patterns that emerge from local interactions among group members. These puzzling phenomena raise a variety of questions about the behavioral rules that govern the coordination of individuals' motions and the emergence of large-scale patterns. While numerous models have been proposed, there is still a strong need for detailed experimental studies to foster the biological understanding of such collective motion. Here, we use data recorded on fish barred flagtails moving in groups of increasing sizes in a water tank to demonstrate the power of an incremental methodology for building a fish behavior model completely based on interactions with the physical environment and neighboring fish. In contrast to previous works, our model revealed an implicit balancing of neighbors position and orientation on the turning speed of fish, an unexpected transition between shoaling and schooling induced by a change in the swimming speed, and a group-size effect which results in a decrease of social interactions among fish as density increases. An important feature of this model lies in its ability to allow a large palette of adaptive patterns with a great economy of means.

Continuous theory of active matter systems with metric-free interactions

We derive a hydrodynamic description of metric-free active matter: starting from self-propelled particles aligning with neighbors defined by "topological" rules, not metric zones-a situation advocated recently to be relevant for bird flocks, fish schools, and crowds-we use a kinetic approach to obtain well-controlled nonlinear field equations. We show that the density-independent collision rate per particle characteristic of topological interactions suppresses the linear instability of the homogeneous ordered phase and the nonlinear density segregation generically present near threshold in metric models, in agreement with microscopic simulations.

The modelling cycle for collective animal behaviour

Collective animal behaviour is the study of how interactions between individuals produce group level patterns, and why these interactions have evolved. This study has proved itself uniquely interdisciplinary, involving physicists, mathematicians, engineers as well as biologists. Almost all experimental work in this area is related directly or indirectly to mathematical models, with regular movement back and forth between models, experimental data and statistical fitting. In this paper, we describe how the modelling cycle works in the study of collective animal behaviour. We classify studies as addressing questions at different levels or linking different levels, i.e. as local, local to global, global to local or global. We also describe three distinct approaches—theory-driven, data-driven and model selection—to these questions. We show, with reference to our own research on species across different taxa, how we move between these different levels of description and how these various approaches can be applied to link levels together.

Kinetic theory for systems of self-propelled particles with metric-free interactions

A model of self-driven particles similar to the Vicsek model [Phys. Rev. Lett. 75, 1226 (1995)] but with metric-free interactions is studied by means of a novel Enskog-type kinetic theory. In this model, N particles of constant speed v(0) try to align their travel directions with the average direction of a fixed number of closest neighbors. At strong alignment a global flocking state forms. The alignment is defined by a stochastic rule, not by a Hamiltonian. The corresponding interactions are of genuine multibody nature. The theory is based on a Master equation in 3N-dimensional phase space, which is made tractable by means of the molecular chaos approximation. The phase diagram for the transition to collective motion is calculated and compared to direct numerical simulations. A linear stability analysis of a homogeneous ordered state is performed using the kinetic but not the hydrodynamic equations in order to achieve high accuracy. In contrast to the regular metric Vicsek-model no instabilities occur. This confirms previous direct simulations that, for Vicsek-like models with metric-free interactions, there is no formation of density bands and that the flocking transition is continuous.

Spatial structures and giant number fluctuations in models of active matter

The large scale fluctuations of the ordered state in active matter systems are usually characterized by studying the "giant number fluctuations" of particles in any finite volume, as compared to the expectations from the central limit theorem. However, in ordering systems, the fluctuations in density ordering are often captured through their structure functions deviating from Porod's law. In this Letter we study the relationship between giant number fluctuations and structure functions for different models of active matter as well as other nonequilibrium systems. A unified picture emerges, with different models falling in four distinct classes depending on the nature of their structure functions. For one class, we show that experimentalists may find Porod's law violation, by measuring subleading corrections to the number fluctuations.

Fluctuation-driven flocking movement in three dimensions and scale-free correlation

Recent advances in the study of flocking behavior have permitted more sophisticated analyses than previously possible. The concepts of "topological distances" and "scale-free correlations" are important developments that have contributed to this improvement. These concepts require us to reconsider the notion of a neighborhood when applied to theoretical models. Previous work has assumed that individuals interact with neighbors within a certain radius (called the "metric distance"). However, other work has shown that, assuming topological interactions, starlings interact on average with the six or seven nearest neighbors within a flock. Accounting for this observation, we previously proposed a metric-topological interaction model in two dimensions. The goal of our model was to unite these two interaction components, the metric distance and the topological distance, into one rule. In our previous study, we demonstrated that the metric-topological interaction model could explain a real bird flocking phenomenon called scale-free correlation, which was first reported by Cavagna et al. In this study, we extended our model to three dimensions while also accounting for variations in speed. This three-dimensional metric-topological interaction model displayed scale-free correlation for velocity and orientation. Finally, we introduced an additional new feature of the model, namely, that a flock can store and release its fluctuations.

Scale-invariant correlations in dynamic bacterial clusters

In Bacillus subtilis colonies, motile bacteria move collectively, spontaneously forming dynamic clusters. These bacterial clusters share similarities with other systems exhibiting polarized collective motion, such as bird flocks or fish schools. Here we study experimentally how velocity and orientation fluctuations within clusters are spatially correlated. For a range of cell density and cluster size, the correlation length is shown to be 30% of the spatial size of clusters, and the correlation functions collapse onto a master curve after rescaling the separation with correlation length. Our results demonstrate that correlations of velocity and orientation fluctuations are scale invariant in dynamic bacterial clusters.