Experiments with 1 to 30 fish were performed in shallow circular swimming pools that let the fish form quasi 2-dimensional schools (see Fig. 1A and Video S1, S2, S3, S4). At the collective level, we observe a transition from schooling to shoaling behavior when the density of fish increases in the tank: the group polarization , which measures the degree of alignment, is high in groups of two and five fish, even if sometimes we do observe some breaks in the synchronization, while in larger groups, when , it remains low (Fig. 1B). Within each group size, we notice some variability, the most striking effect being an increase of the synchronization level with the individuals velocity in groups of two fish.

For every group size, fish move continuously and quickly synchronize their speed to a well defined, but replicate-dependent value (Fig. S1). The fish trajectories are smooth, differentiable and the instantaneous speed has a well-defined mean and root mean square fluctuations of about 10–20% which are found to be uncorrelated to , the angular velocity of the fish orientation (Fig. S2). On this basis, fish can be modeled as self-propelled particles moving in 2D space at constant speed and the only dynamical variable retained is . Moreover, since the recorded trajectories, be they extracted from a single fish or from small groups in the tank, are always irregular/stochastic, our model takes the form of coupled stochastic differential equations for the angular velocities of each fish. Note that if noise acts on rather than the fish position or heading, trajectories are smooth and differentiable, as observed.

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 [29]. When the fish is away from the tank wall, the distribution of is nearly Gaussian with zero mean and variance , where is the characteristic time of the (exponentially decaying) autocorrelation function of . To avoid collisions with the tank walls, we found that a single fish adjusts its current turning speed towards a (time-dependent) target value where is a parameter, is the distance to the point of impact on the wall should the fish continue moving straight ahead, and is the angle between the current heading of the fish and the normal to the point of impact (see Fig. 2A). In short, obeys the stochastic differential equation:(1)where is a Wiener process of variance reflecting the stochasticity of the behavioral response. Non-linear regression analysis of the above model against our experimental data yielded excellent agreement and accurate estimations of and . Note that in the present work we adopted a slightly different form for the wall avoidance term with regards to the exponentially decreasing one of Ref. [29], since it actually prevents fish from crossing the tank boundary, while both ansatz are similar as fish moves away from tank walls (Fig. S8A).

The stimulus/response function of a single fish in the tank is directly expressed by how varies with the relative position of the fish and the wall. We now assume that this framework holds when two fish and are present in the tank by defining how, for fish , its turning speed is modulated by the combined stimuli due to the wall and to fish . Almost all existing fish behavior models, on the basis of common sense, intuition, and sometimes experimental evidence [30]–[37], offer a combination of three basic ingredients: short distance repulsion (to avoid collisions), alignment for intermediate distances, and attraction up to some maximal range. Here, we dispose of repulsion not only because we want to allow for the rare experimentally observed over- and under-passings events, but mostly because we do not need to incorporate it explicitly to avoid collisions (see below and Video S1, S2, S3, S4). In contrast with most existing “zonal” models, and because there is little cognitive/physiological evidence for a sudden switch between alignment and attraction, we want to allow for continuous, distance-dependent weighting between alignment and attraction in agreement with the recent findings of Katz et al. [28]. These two factors a priori depend on the geometrical quantities defining the location of fish from the viewpoint of fish : their distance is involved, but also , the angular position of fish with respect to , the current heading of fish , as well as their relative heading difference (Fig. 2A). The main angular variable for explicit alignment is, as usual, , whereas for attraction it is ; both may also depend on . The stimulus/response function of fish thus combines a priori wall avoidance, alignment and attraction in some unknown function with parameters and (reaction to the wall), , and : .

Next, in the spirit of an expansion around the no-interaction case, we write the expression for above as the sum of three terms:(2)where the “main” variables have been placed first for each term. The wall avoidance term depends explicitly on to reflect a possible screening of the wall by the other fish. We have tested the influence of this by introducing a dependence in the wall avoidance term determined for the single-fish behavior. Essentially, was made smaller for . But this brought no significant improvement, so we keep as found previously.

On general grounds, one expects that the relative importance of the positional interaction (attraction) to the velocity interaction (alignment) increases with . Given that the fish are constrained in a rather small tank, a limited range of inter-distances is effectively explored. In the spirit, again, of a small-distance expansion, a satisfactory choice is given by a linear dependence of on , while is independent of . Of course, such a functional choice cannot be correct at large distances since then would take large unrealistic values, meaning that the fish would spend enormous amounts of energy turning toward a distant “neighbor” (see the Discussion for more comments on this point).

The attraction interaction must depend on , the relative angle with the other fish position: it is reasonable to assume that a fish is not attracted much towards a neighbor located behind, and of course this term must be zero when the other fish is right ahead, yielding . A simple, compatible, trigonometric function representing the leading term of a Fourier expansion is the sine function. We thus write where is a parameter controlling the weight of the positional information. Finally, we neglect the possible dependence on : the way a fish would turn toward the position of a neighbor does not depend on the orientation of that fish. This is especially natural when this interaction dominates, i.e. when the neighbor is far away. Moreover knowing the other fish orientation is a cognitively expensive and/or time consuming process at larger distances.

The alignment interaction is mostly characterized by its functional dependence on . The main constraint here is that (the two fish are then already aligned). Here again, the simplest choice is as in most models [8]–[10]. Including higher harmonics (e.g. ) would allow to account for the few observed nematic alignment events where a fish remains anti-aligned with its neighbors. However, incorporating this term did not improve the faithfulness of the model to our dataset, so we keep only the leading sine function. In principle, the strength of alignment can also depend on : less attention may be paid to “back neighbors”. We have tested simple and reasonable choices for the dependence of on , e.g. , but this did not lead to significant improvement so we kept no angular position dependence in the alignment interaction. We thus write, finally: where is a parameter controlling the weight of the orientational information.

To summarize the case of two fish and , the stimulus/response function in the general evolution equation (1) is thus finally written:(3)

Using nonlinear regression analysis, the faithfulness to our data of the model consisting of Eqs. (1) and (3) was found very good for each of our two-fish recordings and the 5 parameters , , , and were estimated for each fish. We find clear dependences of the estimated parameters on , the average speed of each fish (see Fig. 3A). In particular, , , and are found proportional to , whereas and no significant -dependence appears for . Results regarding this last parameter are the least convincing, with a large dispersion of individual values. This is mostly due to the confinement of fish in the tank: the positional interaction never dominates alignment, preventing its accurate estimation. Nevertheless it is crucial to note here that without these positional interactions the model fails to match the data. Furthermore, we have tested a posteriori our ansatz by testing each contribution (either wall avoidance, neighbor position or neighbor orientation) after the other twos have been subtracted from the fish response according to Eq. (3). Results show an excellent agreement between our ansatz and the mean fish response (for more details see Fig. S8 B–D).

Note that these results mean also that the wall avoidance is actually governed by , the time it would take the fish to hit the wall, rather than the distance . Conversely, , the relaxation time of the angular velocity, is better expressed as the ratio between a characteristic length and the speed . These -dependences were then incorporated explicitly in the model:(4)with(5)where , and are now constants over all fish. Running again our nonlinear regressions using this form, and using data for all replicate, allows for a more accurate estimation of the parameters , , , and now the same for all fish. We find , , , and .

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 Fig. 3C), but in fact also for the dynamics: see for instance Video S1, S2, S5, S6 and the time series of polarization which show the same intermittent behavior (Fig. 3B). We emphasize that the model captures the experimental observation that the orientational order is lower when the swimming speed is lower, and is better in faster groups (Fig. 3B, C).

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 [38]. Even more recently, Katz et al. argued that this is also the case for groups of three golden shiners [28] (but see [39] for the case of birds). Here, our data set is too small to allow for an in-depth analysis of group behavior at the level of detail that was accomplished above for two fish, mostly because many more variables are involved, but the quality of the pair approximation can be evaluated a posteriori. Assuming that multiple fish interactions are indeed essentially made of the sum of the pair interactions involved, Eq. (5) is extended to(6)where is the (current) neighborhood of fish which contains individuals. In our observations with fish, individuals mostly stayed together, suggesting that individuals remains aware of all others. Using all-to-all, equal-weight coupling, we found good agreement between data and simulations of Eqs. (4) and (6) (see Fig. S3). This justifies a posteriori the factorization in pairs and the use of two-fish parameters for groups, but also the overall normalization factor in Eq. (6), which indicates that, in the stimulus response of a fish, wall avoidance and the averaged influence of neighbors keep, on average, the same relative importance irrespective of the group size. The raw, “force-like” un-normalized superposition would yield too strong a coupling.

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 [25]. Coming back to our observations, this non-metric choice of neighbors can, however, lead to unrealistic situations when, for instance, a fish is leading a small group, since then this fish will only pay attention to those behind, even if individuals are located at intermediate distances ahead (but see Fig. S7). A simple, reasonable, non-metric solution is that of neighbors determined by the Voronoi tessellation around each individual: this allows for continuous weighting between alignment and attraction and avoids the caveat mentioned above in the case of a fixed number of closest neighbors. Moreover, given the rather small inter-distances observed, individuals beyond the first shell of Voronoi neighbors are largely screened out, so that our final choice was that of the first shell of Voronoi neighbors (see Fig. 2B). Using this, the validation of the model simulated with fish using the parameters is again quite satisfactory (see Fig. S3).

This is however not true anymore for larger groups which display too high a polarization when using the parameters (whereas distance predictions remains satisfactory, see Fig. S3). Our approach actually allows to further investigate this discrepancy. We estimate the parameters at the individual scale for each fish with our nonlinear least-square procedure using the Ito-integrated version of the Ornstein-Uhlenbeck process of Eqs (4) and (6) for each fish time series (see Statistical Analysis). Thanks to this parametric inversion strategy, we have been able to extract the parameter values for each replicate separately (Fig. 4A). The model predictions with these replicate-based parameters yield a near-perfect match with the data (Fig. 4B). The results confirm that, within the limits of statistical accuracy, the parameters and their v-dependence remain about the same up to N = 10, in agreement with the above findings ; but in larger groups there is a decreased tendency of fish to react to their neighbors, which both concerns the alignment and positional interactions (Fig. 4A).

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 . During the experiments, low mortality occurred (five individuals). At the end of the experiment, the fish were released at their capture site.

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 [49]. Since currents may influence fish behavior, the seawater inlet pipe was placed vertically and the water flow was stopped throughout the observation periods. A digital video camera (Sony model CDR-TRV 900E) was fixed at 5 meters above the tank and tilted at to observe the whole tank. The remotely operated video camera was fitted with a polarizing filter and a wide-angle lens. Groups of N = 1 to 30 fish were introduced in the experimental tank and acclimatized to their new environment for a period of 20 min. Their behavior was then recorded at 24 fps for 2 mins. Prior to each trial, the fish were deprived of food for 12 hours to standardize the hunger level and were transferred to the experimental tank. The relative shallowness of the water ensured quasi two-dimensional motion. Five replicates per group size using different individuals were performed. Eighty per cent of the trials were performed in the morning to avoid possible conditions of strong wind that may disturb the fish, and sunshine that may render light inside the tank unsuitable for video recording. A first data processing consisted in sampling 12 images per second out of the 24 images recorded by the video camera. A custom-made tracking software was then used to extract high-quality, smooth trajectories from the video recordings, with crossing ambiguities resolved by eyes (see Video S3, S4). In order to get even higher precision data, the head position and the orientation of each fish in groups of N = 2 were acquired with a manual tracking software (Video S1, S2).

Model parameters were estimated from each fish time series separately (typical series are shown on Fig. S4). In order to perform the estimation of the parameters , , , and in the stochastic differential equation (1), (3) and (5), we considered its discrete-time version using Ito integration over , assuming is small enough so that is constant [50]:(7)where i = 1,2 and is given by Eq. (3) or (5). Estimates for the parameters were obtained using a standard non-linear least squares procedure (we employed the nls package of the statistical environment R [51]) either separately for each fish using Eq. (3) or for all fish together using Eq. (5). Residuals given by were checked to be Gaussian-distributed (see Fig. S5) and their variance yielded .

The model was simulated within a virtual tank, using the estimates of behavioral parameters extracted by statistical analysis from time-series in groups of fish. The fish heading (direction of motion) and position were updated by Euler integration, following:(8)where .

For each value, numerical simulations were performed over 120 seconds (a time corresponding to the duration of individual experiments with real fish) with a time step . A transient time of was discarded before measuring statistical averages. We computed the mean value and the variance over time of the global polarization(9)and of the neighbor inter-distance(10)

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 confidence interval of such distributions, to obtain the expected mean and variance (with their confidence intervals) of alignment and of neighbor inter-distance. This provided the check of the model against experimental data. The above procedure was repeated varying the mean speed over the range covered by the experimental data, with the results plotted in Fig. 3C. The same procedure was adopted to make predictions for higher group sizes, using the stimulus/response function as determined by equation (5) with interacting neighbors defined by first neighbors in a Voronoi tessellation (For a set of points, Voronoi tessellation divides the space in different cells, each the locus of space closer to its center than to any other points in : at each time step space is divided in Voronoi cells centered around the fish position, with Voronoi neighbors being the fish lying in neighboring cells (Fig. 2B). For each experimental replicate, the same measures were repeated with the parameters extracted from the replicate, and the corresponding initial conditions (Fig. 4B).

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 with no free parameters. For each group independently, we thus checked that the model allows a quantitative matching concurrently at individual and collective scales. This confirmed that our model calibrated with the parameters estimated from the third derivative of the fish position (i.e. the change in the turning speed) was able to reproduce quantitatively the statistics resulting from the time integration of the coupling between fish (polarization, inter-distance). Moreover the same procedure applied separately on each group size revealed, on the one hand, the dependences of the estimated parameters on the swimming speed (using groups of N = 2 fish), and on the other hand, the modulation of interactions' strength with group size (in the largest groups).