Computational and robotic modeling reveal parsimonious combinations of interactions between individuals in schooling fish

Body orientation change of neighbors leads to scale-free correlation in collective motion

Collective motion, such as milling, flocking, and collective turning, is a common and captivating phenomenon in nature, which arises in a group of many self-propelled individuals using local interaction mechanisms. Recently, vision-based mechanisms, which establish the relationship between visual inputs and motion decisions, have been applied to model and better understand the emergence of collective motion. However, previous studies often characterize the visual input as a transient Boolean-like sensory stream, which makes it challenging to capture the salient movements of neighbors. This further hinders the onset of the collective response in vision-based mechanisms and increases demands on visual sensing devices in robotic swarms. An explicit and context-related visual cue serving as the sensory input for decision-making in vision-based mechanisms is still lacking. Here, we hypothesize that body orientation change (BOC) is a significant visual cue characterizing the motion salience of neighbors, facilitating the emergence of the collective response. To test our hypothesis, we reveal the significant role of BOC during collective U-turn behaviors in fish schools by reconstructing scenes from the view of individual fish. We find that an individual with the larger BOC often takes on the leading role during U-turns. To further explore this empirical finding, we build a pairwise interaction mechanism on the basis of the BOC. Then, we conduct experiments of collective spin and collective turn with a real-time physics simulator to investigate the dynamics of information transfer in BOC-based interaction and further validate its effectiveness on 50 real miniature swarm robots. The experimental results show that BOC-based interaction not only facilitates the directional information transfer within the group but also leads to scale-free correlation within the swarm. Our study highlights the practicability of interaction governed by the neighbor's body orientation change in swarm robotics and the effect of scale-free correlation in enhancing collective response.

Speckle statistics as a tool to distinguish collective behaviors of Zebrafish shoals

Zebrafish have become an important model animal for studying the emergence of collective behavior in nature. Here, we show how to properly analyze the polarization statistics to distinguish shoal regimes. In analogy with the statistical properties of optical speckles, we show that exponential and Rayleigh distributions emerge in shoals with many fish with uncorrelated velocity directions. In the opposite limit of just two fish, the polarization distribution peaks at high polarity, with the average value being a decreasing function of the shoal's size, even in the absence of correlations. We also perform a set of experiments unveiling two shoaling regimes. Large shoals behave as small domains with strong intra-domain and weak inter-domain correlations. A strongly correlated regime develops for small shoals. The reported polarization statistical features shall guide future automated neuroscience, pharmacological, toxicological, and embryogenesis-motivated experiments aiming to explore the collective behavior of fish shoals.

Perception of motion salience shapes the emergence of collective motions

Despite the profound implications of self-organization in animal groups for collective behaviors, understanding the fundamental principles and applying them to swarm robotics remains incomplete. Here we propose a heuristic measure of perception of motion salience (MS) to quantify relative motion changes of neighbors from first-person view. Leveraging three large bird-flocking datasets, we explore how this perception of MS relates to the structure of leader-follower (LF) relations, and further perform an individual-level correlation analysis between past perception of MS and future change rate of velocity consensus. We observe prevalence of the positive correlations in real flocks, which demonstrates that individuals will accelerate the convergence of velocity with neighbors who have higher MS. This empirical finding motivates us to introduce the concept of adaptive MS-based (AMS) interaction in swarm model. Finally, we implement AMS in a swarm of ~102 miniature robots. Swarm experiments show the significant advantage of AMS in enhancing self-organization of the swarm for smooth evacuations from confined environments.

Behavioral transition of a fish school in a crowded environment

In open water, social fish gather to form schools, in which fish generally align with each other. In this work, we study how this social behavior evolves when perturbed by artificial obstacles. We measure the behavior of a group of zebrafish in the presence of a periodic array of pillars. When the pillar density is low, the fish regroup with a typical interdistance and a well-polarized state with parallel orientations, similarly to their behavior in open-water conditions. Above a critical density of pillars, their social interactions, which are mostly based on vision, are screened and the fish spread randomly through the aquarium, orienting themselves along the free axes of the pillar lattice. The abrupt transition from natural to artificial orientation happens when the pillar interdistance is comparable to the social distance of the fish, i.e., their most probable interdistance. We develop a stochastic model of the relative orientation between fish pairs, taking into account alignment, antialignment, and tumbling, from a distribution biased by the environment. This model provides a good description of the experimental probability distribution of the relative orientation between the fish and captures the behavioral transition. Using the model to fit the experimental data provides qualitative information on the evolution of cognitive parameters, such as the alignment or the tumbling rates, as the pillar density increases. At high pillar density, we find that the artificial environment imposes its geometrical constraints to the fish school, drastically increasing the tumbling rate.

Multifractal fluctuations in zebrafish (Danio rerio) polarization time series

In this work, we study the polarization time series obtained from experimental observation of a group of zebrafish (Danio rerio) confined in a circular tank. The complex dynamics of the individual trajectory evolution lead to the appearance of multiple characteristic scales. Employing the Multifractal Detrended Fluctuation Analysis (MF-DFA), we found distinct behaviors according to the parameters used. The polarization time series are multifractal at low fish densities and their average scales with ρ - 1 / 4 . On the other hand, they tend to be monofractal, and their average scales with ρ - 1 / 2 for high fish densities. These two regimes overlap at critical density ρ c , suggesting the existence of a phase transition separating them. We also observed that for low densities, the polarization velocity shows a non-Gaussian behavior with heavy tails associated with long-range correlation and becomes Gaussian for high densities, presenting an uncorrelated regime.

Selective social interactions and speed-induced leadership in schooling fish

Animals moving together in groups are believed to interact among each other with effective social forces, such as attraction, repulsion, and alignment. Such forces can be inferred using "force maps," i.e., by analyzing the dependency of the acceleration of a focal individual on relevant variables. Here, we introduce a force map technique suitable for the analysis of the alignment forces experienced by individuals. After validating it using an agent-based model, we apply the force map to experimental data of schooling fish. We observe signatures of an effective alignment force with faster neighbors and an unexpected antialignment with slower neighbors. Instead of an explicit antialignment behavior, we suggest that the observed pattern is the result of a selective attention mechanism, where fish pay less attention to slower neighbors. This mechanism implies the existence of temporal leadership interactions based on relative speeds between neighbors. We present support for this hypothesis both from agent-based modeling as well as from exploring leader-follower relationships in the experimental data.

Predicting the long-term collective behaviour of fish pairs with deep learning

Modern computing has enhanced our understanding of how social interactions shape collective behaviour in animal societies. Although analytical models dominate in studying collective behaviour, this study introduces a deep learning model to assess social interactions in the fish species Hemigrammus rhodostomus. We compare the results of our deep learning approach with experiments and with the results of a state-of-the-art analytical model. To that end, we propose a systematic methodology to assess the faithfulness of a collective motion model, exploiting a set of stringent individual and collective spatio-temporal observables. We demonstrate that machine learning (ML) models of social interactions can directly compete with their analytical counterparts in reproducing subtle experimental observables. Moreover, this work emphasizes the need for consistent validation across different timescales, and identifies key design aspects that enable our deep learning approach to capture both short- and long-term dynamics. We also show that our approach can be extended to larger groups without any retraining, and to other fish species, while retaining the same architecture of the deep learning network. Finally, we discuss the added value of ML in the context of the study of collective motion in animal groups and its potential as a complementary approach to analytical models.

Tuning social interactions' strength drives collective response to light intensity in schooling fish

Schooling fish heavily rely on visual cues to interact with neighbors and avoid obstacles. The availability of sensory information is influenced by environmental conditions and changes in the physical environment that can alter the sensory environment of the fish, which in turn affects individual and group movements. In this study, we combine experiments and data-driven modeling to investigate the impact of varying levels of light intensity on social interactions and collective behavior in rummy-nose tetra fish. The trajectories of single fish and groups of fish swimming in a tank under different lighting conditions were analyzed to quantify their movements and spatial distribution. Interaction functions between two individuals and the fish interaction with the tank wall were reconstructed and modeled for each light condition. Our results demonstrate that light intensity strongly modulates social interactions between fish and their reactions to obstacles, which then impact collective motion patterns that emerge at the group level.

Nearest Neighbour Node Deployment Algorithm for Mobile Sensor Networks

Many animal aggregations display remarkable collective coordinated movements on a large scale, which emerge as a result of distributed local decision-making by individuals. The recent advances in modelling the collective motion of animals through the utilisation of Nearest Neighbour rules, without the need for centralised coordination, resulted in the development of self-deployment algorithms in Mobile Sensor Networks (MSNs) to achieve various types of coverage essential for different applications. However, the energy consumption associated with sensor movement to achieve the desired coverage remains a significant concern for the majority of algorithms reported in the literature. In this paper, the Nearest Neighbour Node Deployment (NNND) algorithm is proposed to efficiently provide blanket coverage across a given area while minimising energy consumption and enhancing fault tolerance. In contrast to other algorithms that sequentially move sensors, NNND leverages the power of parallelism by employing multiple streams of sensor motions, each directed towards a distinct section of the area. The cohesion of each stream is maintained by adaptively choosing a leader for each stream while collision avoidance is also ensured. These properties contribute to minimising the travel distance within each stream, resulting in decreased energy consumption. Additionally, the utilisation of multiple leaders in NNND eliminates the presence of a single point of failure, hence enhancing the fault tolerance of the area coverage. The results of our extensive simulation study demonstrate that NNND not only achieves lower energy consumption but also a higher percentage of k-coverage.

An examination of force maps targeted at orientation interactions in moving groups

Force mapping is an established method for inferring the underlying interaction rules thought to govern collective motion from trajectory data. Here we examine the ability of force maps to reconstruct interactions that govern individual's tendency to orient, or align, their heading within a moving group, one of the primary factors thought to drive collective motion, using data from three established general collective motion models. Specifically, our force maps extract how individuals adjust their direction of motion on average as a function of the distance to neighbours and relative alignment in heading with these neighbours, or in more detail as a function of the relative coordinates and relative headings of neighbours. We also examine the association between plots of local alignment and underlying alignment rules. We find that the simpler force maps that examined changes in heading as a function of neighbour distances and differences in heading can qualitatively reconstruct the form of orientation interactions, but also overestimate the spatial range over which these interactions apply. More complex force maps that examine heading changes as a function of the relative coordinates of neighbours (in two spatial dimensions), can also reveal underlying orientation interactions in some cases, but are relatively harder to interpret. Responses to neighbours in both the simpler and more complex force maps are affected by group-level patterns of motion. We also find a correlation between the sizes of regions of high alignment in local alignment plots and the size of the region over which alignment rules apply when only an alignment interaction rule is in action. However, when data derived from more complex models is analysed, the shapes of regions of high alignment are clearly influenced by emergent patterns of motion, and these regions of high alignment can appear even when there is no explicit direct mechanism that governs alignment.

Emergence of intelligent collective motion in a group of agents with memory

Intelligent agents collect and process information from their dynamically evolving neighborhood to efficiently navigate through it. However, agent-level intelligence does not guarantee that at the level of a collective; a common example is the jamming we observe in traffic flows. In this study, we ask: how and when do the interactions between intelligent agents translate to desirable or intelligent collective outcomes? To explore this question, we choose a collective consisting of two kinds of agents with opposing desired directions of movement. Agents in this collective are minimally intelligent: they possess only a single facet of intelligence, viz., memory, where the agents remember how well they were able to travel in their desired directions and make up for their non-optimal past. We find that dynamics due to the agent's memory influences the collective, giving rise to diverse outcomes at the level of the group: from those that are undesirable to those that can be called "intelligent." When memory is short term, local rearrangement of agents leads to the formation of symmetrically jammed arrangements that take longer to unjam. However, when agents remember across longer time-scales, their dynamics become sensitive to small differences in their movement history. This gives rise to heterogeneity in the movement that causes agents to unjam more readily and form lanes.

Data-driven discovery of stochastic dynamical equations of collective motion

Coarse-grained descriptions of collective motion of flocking systems are often derived for the macroscopic or the thermodynamic limit. However, the size of many real flocks falls within 'mesoscopic' scales (10 to 100 individuals), where stochasticity arising from the finite flock sizes is important. Previous studies on mesoscopic models have typically focused on non-spatial models. Developing mesoscopic scale equations, typically in the form of stochastic differential equations, can be challenging even for the simplest of the collective motion models that explicitly account for space. To address this gap, here, we take a novel data-driven equation learning approach to construct the stochastic mesoscopic descriptions of a simple, spatial, self-propelled particle (SPP) model of collective motion. In the spatial model, a focal individual can interact withkrandomly chosen neighbours within an interaction radius. We considerk = 1 (called stochastic pairwise interactions),k = 2 (stochastic ternary interactions), andkequalling all available neighbours within the interaction radius (equivalent to Vicsek-like local averaging). For the stochastic pairwise interaction model, the data-driven mesoscopic equations reveal that the collective order is driven by a multiplicative noise term (hence termed, noise-induced flocking). In contrast, for higher order interactions (k > 1), including Vicsek-like averaging interactions, models yield collective order driven by a combination of deterministic and stochastic forces. We find that the relation between the parameters of the mesoscopic equations describing the dynamics and the population size are sensitive to the density and to the interaction radius, exhibiting deviations from mean-field theoretical expectations. We provide semi-analytic arguments potentially explaining these observed deviations. In summary, our study emphasises the importance of mesoscopic descriptions of flocking systems and demonstrates the potential of the data-driven equation discovery methods for complex systems studies.

Is the neighborhood of interaction in human crowds metric, topological, or visual?

Global patterns of collective motion in bird flocks, fish schools, and human crowds are thought to emerge from local interactions within a neighborhood of interaction, the zone in which an individual is influenced by their neighbors. Both metric and topological neighborhoods have been reported in animal groups, but this question has not been addressed for human crowds. The answer has important implications for modeling crowd behavior and predicting crowd disasters such as jams, crushes, and stampedes. In a metric neighborhood, an individual is influenced by all neighbors within a fixed radius, whereas in a topological neighborhood, an individual is influenced by a fixed number of nearest neighbors, regardless of their physical distance. A recently proposed alternative is a visual neighborhood, in which an individual is influenced by the optical motions of all visible neighbors. We test these hypotheses experimentally by asking participants to walk in real and virtual crowds and manipulating the crowd's density. Our results rule out a topological neighborhood, are approximated by a metric neighborhood, but are best explained by a visual neighborhood that has elements of both. We conclude that the neighborhood of interaction in human crowds follows naturally from the laws of optics and suggest that previously observed "topological" and "metric" interactions might be a consequence of the visual neighborhood.

Collective motions of fish originate from balanced local perceptual interactions and individual stochastics

The movement of a group of biological individuals, such as fish schools, can evolve from disordered motions to synergistic movements or even ordered patterns. However, the physical origins behind such emergent phenomena of complex systems remain elusive. Here, we established a high-precision protocol for studying the collective behavior of biological groups in quasi-two-dimensional systems. Based on our video recording of ∼600h of fish movements, we extracted a force map of the interactions between fish from their trajectories using the convolution neural network. Presumably, this force implies the fish's perception of the surrounding individuals, the environment, and their response to social information. Interestingly, the fish in our experiments were predominantly in a seemingly disordered swarm state, but their local interactions were clearly specific. Combining such local interactions with the inherent stochasticity of the fish movements, we reproduced the collective motions of the fish through simulations. We demonstrated that a delicate balance between the specific local force and the intrinsic stochasticity is essential for ordered movements. This study presents implications for self-organized systems that use basic physical characterization to produce higher-level sophistication.

Modeling three-dimensional bait ball collective motion

Collective motion of animal groups such as fish schools and bird flocks in three-dimensional (3D) space are modeled by considering a topological (Voronoi) neighborhood. The tridimensionality of the group is quantified. Apart from the patterns of swarming, schooling, and milling, we identify a 3D bait ball around the phase transition boundary. More significantly, we find that by considering a blind angle in this topology based model, an individual interacts statistically with six to seven neighbors, consistent precisely with the previous field observations of the starling flocks. This model could be expected to enable more insightful investigation on realistic collective motion of shoals or flocks.

Automated optimization of multilevel models of collective behaviour: application to mixed society of animals and robots

Animal societies exhibit complex dynamics that require multi-level descriptions. They are difficult to model, as they encompass information at different levels of description, such as individual physiology, individual behaviour, group behaviour and features of the environment. The collective behaviour of a group of animals can be modelled as a dynamical system. Typically, models of behaviour are either macroscopic (differential equations of population dynamics) or microscopic (such as Markov chains, explicitly specifying the spatio-temporal state of each individual). These two kind of models offer distinct and complementary descriptions of the observed behaviour. Macroscopic models offer mean field description of the collective dynamics, where collective choices are considered as the stable steady states of a nonlinear system governed by control parameters leading to bifurcation diagrams. Microscopic models can be used to perform computer simulations or as building blocks for robot controllers, at the individual level, of the observed spatial behaviour of animals. Here, we present a methodology to translate a macroscopic model into different microscopic models. We automatically calibrate the microscopic models so that the resulting simulated collective dynamics fit the solutions of the reference macroscopic model for a set of parameter values corresponding to a bifurcation diagram leading to multiple steady states. We apply evolutionary algorithms to simultaneously optimize the parameters of the models at different levels of description. This methodology is applied, in simulation, to an experimentally validated shelter-selection problem solved by gregarious insects and robots. Our framework can be used for multi-level modelling of collective behaviour in animals and robots.

Beyond Bio-Inspired Robotics: How Multi-Robot Systems Can Support Research on Collective Animal Behavior

In the study of collective animal behavior, researchers usually rely on gathering empirical data from animals in the wild. While the data gathered can be highly accurate, researchers have limited control over both the test environment and the agents under study. Further aggravating the data gathering problem is the fact that empirical studies of animal groups typically involve a large number of conspecifics. In these groups, collective dynamics may occur over long periods of time interspersed with excessively rapid events such as collective evasive maneuvers following a predator’s attack. All these factors stress the steep challenges faced by biologists seeking to uncover the fundamental mechanisms and functions of social organization in a given taxon. Here, we argue that beyond commonly used simulations, experiments with multi-robot systems offer a powerful toolkit to deepen our understanding of various forms of swarming and other social animal organizations. Indeed, the advances in multi-robot systems and swarm robotics over the past decade pave the way for the development of a new hybrid form of scientific investigation of social organization in biology. We believe that by fostering such interdisciplinary research, a feedback loop can be created where agent behaviors designed and tested in robotico can assist in identifying hypotheses worth being validated through the observation of animal collectives in nature. In turn, these observations can be used as a novel source of inspiration for even more innovative behaviors in engineered systems, thereby perpetuating the feedback loop.

The impact of individual perceptual and cognitive factors on collective states in a data-driven fish school model

In moving animal groups, social interactions play a key role in the ability of individuals to achieve coordinated motion. However, a large number of environmental and cognitive factors are able to modulate the expression of these interactions and the characteristics of the collective movements that result from these interactions. Here, we use a data-driven fish school model to quantitatively investigate the impact of perceptual and cognitive factors on coordination and collective swimming patterns. The model describes the interactions involved in the coordination of burst-and-coast swimming in groups of Hemigrammus rhodostomus. We perform a comprehensive investigation of the respective impacts of two interactions strategies between fish based on the selection of the most or the two most influential neighbors, of the range and intensity of social interactions, of the intensity of individual random behavioral fluctuations, and of the group size, on the ability of groups of fish to coordinate their movements. We find that fish are able to coordinate their movements when they interact with their most or two most influential neighbors, provided that a minimal level of attraction between fish exist to maintain group cohesion. A minimal level of alignment is also required to allow the formation of schooling and milling. However, increasing the strength of social interactions does not necessarily enhance group cohesion and coordination. When attraction and alignment strengths are too high, or when the heading random fluctuations are too large, schooling and milling can no longer be maintained and the school switches to a swarming phase. Increasing the interaction range between fish has a similar impact on collective dynamics as increasing the strengths of attraction and alignment. Finally, we find that coordination and schooling occurs for a wider range of attraction and alignment strength in small group sizes.

Randomness in the choice of neighbours promotes cohesion in mobile animal groups

Classic computational models of collective motion suggest that simple local averaging rules can promote many observed group-level patterns. Recent studies, however, suggest that rules simpler than local averaging may be at play in real organisms; for example, fish stochastically align towards only one randomly chosen neighbour and yet the schools are highly polarized. Here, we ask-how do organisms maintain group cohesion? Using a spatially explicit model, inspired from empirical investigations, we show that group cohesion can be achieved in finite groups even when organisms randomly choose only one neighbour to interact with. Cohesion is maintained even in the absence of local averaging that requires interactions with many neighbours. Furthermore, we show that choosing a neighbour randomly is a better way to achieve cohesion than interacting with just its closest neighbour. To understand how cohesion emerges from these random pairwise interactions, we turn to a graph-theoretic analysis of the underlying dynamic interaction networks. We find that randomness in choosing a neighbour gives rise to well-connected networks that essentially cause the groups to stay cohesive. We compare our findings with the canonical averaging models (analogous to the Vicsek model). In summary, we argue that randomness in the choice of interacting neighbours plays a crucial role in achieving cohesion.

Systematic Analysis of Emergent Collective Motion Produced by a 3D Hybrid Zonal Model

Emergent patterns of collective motion are thought to arise from local rules of interaction that govern how individuals adjust their velocity in response to the relative locations and velocities of near neighbours. Many models of collective motion apply rules of interaction over a metric scale, based on the distances to neighbouring group members. However, empirical work suggests that some species apply interactions over a topological scale, based on distance determined neighbour rank. Here, we modify an important metric model of collective motion (Couzin et al. in J Theor Biol 218(1):1-11, 2002), so that interactions relating to orienting movements with neighbours and attraction towards more distant neighbours operate over topological scales. We examine the emergent group movement patterns generated by the model as the numbers of neighbours that contribute to orientation- and attraction-based velocity adjustments vary. Like the metric form of the model, simulated groups can fragment (when interactions are influenced by less than 10-15% of the group), swarm and move in parallel, but milling does not occur. The model also generates other cohesive group movements including cases where groups exhibit directed motion without strong overall alignment of individuals. Multiple emergent states are possible for the same set of underlying model parameters in some cases, suggesting sensitivity to initial conditions, and there is evidence that emergent states of the system depend on the history of the system. Groups that do not fragment tend to stay relatively compact in terms of neighbour distances. Even if a group does fragment, individuals remain relatively close to near neighbours, avoiding complete isolation.

Collective predator evasion: Putting the criticality hypothesis to the test

According to the criticality hypothesis, collective biological systems should operate in a special parameter region, close to so-called critical points, where the collective behavior undergoes a qualitative change between different dynamical regimes. Critical systems exhibit unique properties, which may benefit collective information processing such as maximal responsiveness to external stimuli. Besides neuronal and gene-regulatory networks, recent empirical data suggests that also animal collectives may be examples of self-organized critical systems. However, open questions about self-organization mechanisms in animal groups remain: Evolutionary adaptation towards a group-level optimum (group-level selection), implicitly assumed in the "criticality hypothesis", appears in general not reasonable for fission-fusion groups composed of non-related individuals. Furthermore, previous theoretical work relies on non-spatial models, which ignore potentially important self-organization and spatial sorting effects. Using a generic, spatially-explicit model of schooling prey being attacked by a predator, we show first that schools operating at criticality perform best. However, this is not due to optimal response of the prey to the predator, as suggested by the "criticality hypothesis", but rather due to the spatial structure of the prey school at criticality. Secondly, by investigating individual-level evolution, we show that strong spatial self-sorting effects at the critical point lead to strong selection gradients, and make it an evolutionary unstable state. Our results demonstrate the decisive role of spatio-temporal phenomena in collective behavior, and that individual-level selection is in general not a viable mechanism for self-tuning of unrelated animal groups towards criticality.

Multi-scale analysis and modelling of collective migration in biological systems

Collective migration has become a paradigm for emergent behaviour in systems of moving and interacting individual units resulting in coherent motion. In biology, these units are cells or organisms. Collective cell migration is important in embryonic development, where it underlies tissue and organ formation, as well as pathological processes, such as cancer invasion and metastasis. In animal groups, collective movements may enhance individuals' decisions and facilitate navigation through complex environments and access to food resources. Mathematical models can extract unifying principles behind the diverse manifestations of collective migration. In biology, with a few exceptions, collective migration typically occurs at a 'mesoscopic scale' where the number of units ranges from only a few dozen to a few thousands, in contrast to the large systems treated by statistical mechanics. Recent developments in multi-scale analysis have allowed linkage of mesoscopic to micro- and macroscopic scales, and for different biological systems. The articles in this theme issue on 'Multi-scale analysis and modelling of collective migration' compile a range of mathematical modelling ideas and multi-scale methods for the analysis of collective migration. These approaches (i) uncover new unifying organization principles of collective behaviour, (ii) shed light on the transition from single to collective migration, and (iii) allow us to define similarities and differences of collective behaviour in groups of cells and organisms. As a common theme, self-organized collective migration is the result of ecological and evolutionary constraints both at the cell and organismic levels. Thereby, the rules governing physiological collective behaviours also underlie pathological processes, albeit with different upstream inputs and consequences for the group. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

A data-driven method for reconstructing and modelling social interactions in moving animal groups

Group-living organisms that collectively migrate range from cells and bacteria to human crowds, and include swarms of insects, schools of fish, and flocks of birds or ungulates. Unveiling the behavioural and cognitive mechanisms by which these groups coordinate their movements is a challenging task. These mechanisms take place at the individual scale and can be described as a combination of interactions between individuals and interactions between these individuals and the physical obstacles in the environment. Thanks to the development of novel tracking techniques that provide large and accurate datasets, the main characteristics of individual and collective behavioural patterns can be quantified with an unprecedented level of precision. However, in a large number of studies, social interactions are usually described by force map methods that only have a limited capacity of explanation and prediction, being rarely suitable for a direct implementation in a concise and explicit mathematical model. Here, we present a general method to extract the interactions between individuals that are involved in the coordination of collective movements in groups of organisms. We then apply this method to characterize social interactions in two species of shoaling fish, the rummy-nose tetra (Hemigrammus rhodostomus) and the zebrafish (Danio rerio), which both present a burst-and-coast motion. From the detailed quantitative description of individual-level interactions, it is thus possible to develop a quantitative model of the emergent dynamics observed at the group level, whose predictions can be checked against experimental results. This method can be applied to a wide range of biological and social systems. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.