The authors have declared that no competing interests exist.
Conceived and designed the experiments: MS GT. Performed the experiments: MS GT. Analyzed the data: JG FG RF SB HC. Wrote the paper: JG FG RF SB HC GT. Designed the software used in analysis: JG RF SB.
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.
Collective motion occurs across a variety of scales in nature, offering a wealth of fascinating phenomena which have attracted a lot of attention
On the other hand, recent studies within the physics community of simple, minimal models for collective motion have revealed an emerging picture of universality classes
Significant features nevertheless may be altered when a qualitatively important feature is changed, such as the symmetry of the aligning interaction, or added, as when local attraction/repulsion between individuals is also considered
So, it remains important to know how individuals make behavioral choices when interacting with others, not only from a social ethology and cognitive viewpoint, but also because i) different behavioral rules may make a difference in small enough groups and ii) the analysis of local-scale data that this requires may lead to discover features eventually found to give rise to different qualitative collective properties. A recent instance can be found in the results on the structure of starling flocks gathered by Ballerini et al.
Here, we assess the power of a bottom-up methodology to build models for animal group motion from data gathered at the individual scale in groups of increasing sizes. We use data obtained by recording the motion of barred flagtails (
Our analysis is incremental: in a previous work we characterized the spontaneous behavior of a single fish, including wall-avoidance behavior
Experiments with 1 to 30 fish were performed in shallow circular swimming pools that let the fish form quasi 2-dimensional schools (see
(A) Illustrations of typical fish trajectories in the tank, in groups of 2, 5 and 10 fish, over 9, 5 and 3 seconds respectively. The similarity of trajectories reflect schooling behavior. (B) Time series of the group polarization
For every group size, fish move continuously and quickly synchronize their speed to a well defined, but replicate-dependent value (
We have shown elsewhere that single fish trajectories in barred flagtails are very well described by an Ornstein-Uhlenbeck process acting on the instantaneous curvature, or, equivalently, on
(A) The distance
The stimulus/response function of a single fish in the tank is directly expressed by how
Next, in the spirit of an expansion around the no-interaction case, we write the expression for
On general grounds, one expects that the relative importance of the positional interaction
The attraction interaction
The alignment interaction is mostly characterized by its functional dependence on
To summarize the case of two fish
Using nonlinear regression analysis, the faithfulness to our data of the model consisting of
(A) Determination of the parameters of the model defined by
Note that these results mean also that the wall avoidance is actually governed by
To validate this experimental finding, these parameter values were used in simulations of the model which were compared directly to the data. Good agreement is found not only for statistical quantifiers of the emergent synchronization between the two fish (see
Can multiple-fish interactions be factorized into pairs? This is often taken for granted, following a typical physics approach where this assumption is routinely made. However, recent work has suggested that this is not valid when describing pedestrian interactions in a crowd
For the larger group sizes, all-to-all equal-weight coupling quickly becomes unrealistic, and one must determine the set of neighbors a fish interacts with. In principle, abundant data recorded in larger tanks would allow to discriminate between alternative choices, but our experimental recordings are too short for this. Nevertheless, many choices can be eliminated: the usual one, which consists in cutting off interactions at fixed distances (zonal models), is inconsistent with our continuous weighting of alignment and attraction with fish inter-distance. Based on an analysis of starling flocks, Ballerini et al. have argued that these birds actually pay attention to their 6–8 closest neighbors, irrespective of the density of the flock
This is however not true anymore for larger groups which display too high a polarization when using the
(A) The five parameters
Characterizing and modeling the interactions between individuals and their behavioral consequences is a crucial step to understand the emergence of complex collective animal behaviors. With the recent progress in tracking technologies, high precision datasets on moving animal groups are now available, thus opening the way to a fine-scale analysis of individual behavior
We explored this point further, still considering the statistical behavior of each fish separately, but only using the data corresponding to the large-group experiments. We concluded that our model could still grasp the observed individual and collective features but with smaller positional and alignment coefficients. We believe that this decrease in reactivity to neighbors is a consequence of the high density already imposed by confinement effects. Indeed, our model predicts that large groups adopting the high neighbor reactivity found in smaller groups would remain polarized also in open space, keeping group cohesion with an average distance to neighbors of about two body lengths (
Our approach yielded a novel type of fish school model whose main features are its built-in balancing mechanism between positional and orientational information, a topological interaction neighborhood, and explicit dependencies on fish speed. Note that similar features were recently uncovered for another species thanks to a novel data analysis procedure
The speed dependence of the parameters, directly derived from our data, is in contrast with most previous fish school models. It leads to an increase of group polarization with swimming speed, a direct consequence of the predominance of alignment at high speed (see Video S7). In natural conditions, this mechanism could be involved in the transitions from shoaling at low speed often associated with feeding behavior to polarized schooling at high speed associated with searching for food. Such speed change could also be elicited by the detection of a threat and abrupt transitions can occur when fish suddenly increase their speed, for instance generating a flash expansion (see Video S8). The question of whether the propagation of such an excitation wave within large schools can generate an efficient collective evasion call for further experimental tests
The reason why our approach was fruitful in spite of the limited amount of data available lies largely in the suitable properties of the behavior of the fish studied: the smooth fluctuations of tangential speed and their de-correlation from angular velocity variations were essential in limiting the number of variables at play but also allowed for a faithful account of single fish behavior by a simple Ornstein-Uhlenbeck process. Clearly it is likely that more complicated solutions will be needed for other species where tangential and angular accelerations are intimately coupled and/or the underlying stochastic process is not as transparent
Our experiments were all carried out in full accordance with the ethical guidelines of our research institutions and comply with the European legislation for animal welfare. The welfare of fishes in the tanks was optimized with a continuous seawater flow, a suitable temperature, and oxygen content. The maximum density in the holding tank was lower than
The experiments were performed from April to June 2001 at the Sea Turtle Survey and Discovery Centre of Reunion Island. Barred flagtail Kuhlia mugil (Forster) were caught in March 2001 in the coastal area around Reunion Island. 80–100 fishes were conveyed to the marine station and housed in a holding tank of 4 m diameter and 1.2 m depth. Fishes were fed daily ad libitum with a mixture of aquaria flake-food and pieces of fish flesh. Fishes were considered acclimatized when all of them feed on the aquaria flake-food. This weaning period lasted 15 days. Experiments were performed in a circular tank similar to the holding tank. Opaque curtains were placed around and above the tank to obtain diffuse lighting and to reduce external disturbances from the environment. The tank was supplied with a continuous flow of seawater
Model parameters were estimated from each fish time series separately (typical series are shown on
The model was simulated within a virtual tank, using the estimates of behavioral parameters extracted by statistical analysis from
For each
This yielded an estimation of the expected measures distribution under model hypothesis and over the typical observation time of experiments. We then computed the mean and
By construction, our method does not “learn the parameters to make the model fit”, contrasting with a more usual procedure which consists in stating an a priori model and searching a best set of free parameters that optimizes its collective patterns towards the observed collective properties (namely, make the model fit at the collective scale). In such cases, it is known that several models can adjust the data at the collective scale (because the search for best match is unconstrained and can be performed for each model, so that the collective level underdetermines the individual level).
In the present study, once the model has been formulated, that is, once we identified in the experiments with pairs of fish the nature of stimuli (the orientation and relative position of neighboring fish, and how they combine to determine the response of a focal fish), we estimated the values of 5 parameters at the individual scale. So for each fish, we measured its behavioral response (i.e. the change of its turning speed) for each configuration of stimuli encountered in its path.
Only then, we tested whether these parameters measured at the individual level can explain the observations at the collective scale
Distance travelled by fish as a function of time in 3 different experiments with N = 2 fish (left panel), one N = 5 and one N = 10 experiment (middle and right panel). In any given experiment, fish synchronize their speed, but this value is replicate-dependent.
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Swimming speed and angular velocity of one fish. Left: Time series of instantaneous speed
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Comparisons between experimental data at all group sizes and predictions of the model, using the
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Example of experimental time-series used to estimate the N = 2 parameters. From top to bottom: turning speed response, wall effect stimulus, positional stimulus and directional stimulus. This shows that the tracking yielded a very good signal to noise ratio.
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Distribution of residuals for the N = 2 parameters estimation. For each fish in the groups with
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Model predictions in open space, using the N = 2 parameters for every group size and the first shell of Voronoi neighbors. (horizontal bars: predicted medians, vertical bars:
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Tests of the alternative neighborhood definition, based on K-Nearest neighbors with
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Validation of the ansatzes. (A) Strength of the wall avoidance term
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Video recording of an experiment with N = 2 fish swimming at low speed (
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Video recording of an experiment with N = 2 fish swimming at a higher speed (
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Video recording of an experiment with N = 5 fish swimming at high speed (
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Video recording of an experiment with N = 15 fish swimming at high speed (
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Simulation of interactions in a group of N = 2 fish swimming at low speed (
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Simulation of interactions in a group of N = 2 fish swimming at higher speed (
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Transition in group polarization induced by velocity change. The simulation was performed in unbounded conditions with a group of 100 fish with parameters
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Transition in group polarization induced by a sudden velocity increase. The simulation was performed in the same conditions and with the same parameters as those used in Video S7, but with a different time profile of the change of the swimming speed (from
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We thank Guillaume Viallon for field assistance, P. Degond, S. Motsch, C. Huepe and A. Cavagna for inspiring discussions, A. Campo and S. Martin for help at early stages of this project, and C. Jost for comments on this paper. We also thank three anonymous reviewers for their critical and constructive comments.