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

['Weijia Wang', 'School Of Systems Science', 'Beijing Normal University', 'Beijing', 'Centre De Recherches Sur La Cognition Animale', 'Centre De Biologie Intégrative', 'Cbi', 'Centre National De La Recherche Scientifique', 'Cnrs', 'Université De Toulouse Paul Sabatier']

Date: 2022-04

The data-driven model considered in the present work aims at studying the diversity and plasticity of collective behavior in a specific species of fish (Hemigrammus rhodostomus). Yet, the general structure of our burst-and-coast model can be also exploited to describe other species of fish performing intermittent swimming. This is for instance the case of zebra fish (Danio rerio) for which the interaction functions have been measured in [45], and where the range of interactions were found to be much shorter, contributing to explain the fact that zebra fish display a weaker collective social behavior than H. rhodostomus.

Moreover, within a model for a specific species, it is certainly worthwhile to study the behavior of the model when changing the value of its parameters (intensity and range of the interactions, intensity of the cognitive noise), and in particular, to address the possible occurrence of non-trivial collective states. Not only this is relevant on a theoretical point of view, but it is also expected that actual fish can experience different effective interaction parameters depending on their age, activity, or their environment. For instance, it is shown in [31] that the intensity of the alignment interaction for Kuhlia mugil is roughly proportional to the mean velocity of the school. Moreover, water turbidity [46–48] and/or light intensity [49] have been shown to impact the interactions between fish. For instance, the authors of [49] have shown that in darkness and under low illuminance, H. rhodostomus move with lower polarization and a large mean nearest-neighbor distance that can span many body lengths. In the same work, the authors showed that it is possible to change the intensity of the attraction between fish by compromising the lateral-line system with a specific antibiotic treatment.

Hence, in the following, we explore the parameter space of the H. rhodostomus model by varying its main parameters, which are the intensity of the attraction and alignment interactions γ Att and γ Ali , their respective range l Att and l Ali , the intensity of the random heading fluctuation γ R (cognitive noise), and the group size N. Results are presented in the form of two-dimensional phase diagrams obtained by varying the intensity of the attraction and alignment interactions, γ Att and γ Ali , while keeping other parameters fixed (interaction ranges, cognitive noise, group size).

We first present the most illustrative cases where the regions corresponding to the different phases of collective behaviors can be easily identified. We then describe the impact of the other parameters on these characteristic collective phases.

In each case, two social interaction strategies are considered, each one determined by the k selected neighbors (with k = 1 and k = 2) that have the largest influence on the focal fish when it performs a new kick.

3.1 Characterization and analysis of the collective states

Fig 3 shows the color maps of dispersion, polarization, and milling as a function of the parameters γ Att and γ Ali where significative variations have been found, for the two social strategies k = 1 and k = 2, in the particular case where the social interaction ranges are l Att = l Ali = 0.28 m, the intensity of random fluctuations (noise) is γ R = 0.2, and the group size is N = 100. For k = 1, variations take place for (γ Att , γ Ali ) in [0, 0.1] × [0, 0.6], and for (γ Att , γ Ali ) in [0, 0.1] × [0, 0.4] for k = 2.

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TIFF original image Download: Fig 3. Dispersion, polarization, and milling as functions of the intensity of attraction γ Att and alignment γ Ali when fish interact with their most influential neighbor (k = 1) and their two most influential neighbors k = 2. (AD) Dispersion: color intensity is proportional to dispersion. White region means that the school of fish is highly cohesive. In the green region, attraction is too weak and fish quickly disperse. (BE) Polarization. (CF) Milling. Regions of high color intensity mean that fish frequently display either the characteristic behavioral patterns of schooling or milling. Social interaction ranges are l Att = l Ali = 0.28 m, intensity of random fluctuation (noise) is γ R = 0.2. Each pixel is the average of 20 runs of 2000 kicks per fish (in which the first 100 kicks have been discarded). https://doi.org/10.1371/journal.pcbi.1009437.g003

The first remarkable result is that, despite the discrete character of the burst-and-coast swimming mode, the group of fish remains cohesive in a large region of the parameter space, even when fish interact only with their most influential neighbor, as testified by the wide regions in Fig 3A and 3D. For both interaction strategies k = 1 and k = 2, the dispersion map consists of two regions of high and low values of D separated by a thin transition layer. In the green region, γ Att is too small and the attraction is not strong enough to preserve the cohesion of the group and fish disperse. The high value shown in the green regions of Fig 3A and 3D (higher than 5) is determined by the duration of the numerical simulations; longer runs give rise to higher values of D(t). This is not the case in the white region, where the group remains cohesive whatever the length of the simulation time.

The transition layer is almost vertical and very thin (width ≈ 0.005). Its location is mostly determined by the value of γ Att , which is of about 0.03 when k = 1 and of about 0.015 when k = 2, showing that cohesion is mostly mediated by attraction, alignment having only a marginal contribution. When k = 2, an horizontal light green region where D ≈ 1 is visible in Panel D when γ Ali > 0.3. In this region, the value of γ Ali is so large that the contribution of the alignment to the heading angle change δϕ in Eq 4 makes the fish to rotate an angle often larger than π, as if the new heading angle was chosen randomly. For these values, the fish behave more or less like slow random walkers. This region of slow dispersion is not visible when k = 1 because in this case Eq 4 has only one contribution to the increase of δϕ. The same kind of dispersion is observed when γ Att is very large, for both k = 1 and k = 2. We have tried to determine the onset of this region in the interval of values of γ Att for k = 1. Increasing the value of γ Att produces a stronger attraction and should in principle give rise to a more cohesive group. However, we found that an optimal value of the attraction strength exists, , for which the distance to the nearest neighbor is minimal. This means that, above , increasing γ Att leads to an increase in the distance to the nearest neighbor, which is a clear indicator that cohesion begins to weaken. We have calculated this optimal value for γ Ali ∈ [0, 0.3], finding that it is more or less the same.

The second remarkable result is that significantly high values of polarization (P > 0.5 in Fig 3B and 3E and milling (M > 0.4 in Fig 3C and 3F) are observed in wide regions of the parameter space. This result shows that groups of fish are able to display the characteristic collective behaviors of schooling and milling, even if each fish only interacts with their most influential neighbor. Both the schooling and milling regions are mostly contained in the region of high cohesion, although the schooling region seems to slightly overlap with the dispersion region when γ Att is small. This corresponds to situations where the group is initially aligned (attraction is weak, so interaction is mostly alignment) and then splits in several subgroups. Although fish move away from each other, they remain more or less aligned inside each subgroup and also with respect to other subgroups, due to the persistence of the initial alignment, thus contributing to produce high values of P(t). As mentioned above for the region of dispersion, longer simulations would give rise to a lost of this persistence and thus to the shrinking of the region of high polarization, reducing it to the points contained in the white region of dispersion in Fig 3A.

We performed a series of very long simulations of our model for values of γ Att and γ Ali across the region of high polarization in Fig 3B, that is, for γ Att going from 0.026 to 0.06 and for a fixed value of γ Ali = 0.2 (20 times longer simulations, until t = 20.000 s, each point is the average of 100 runs). S4 Fig shows that the polarization is maximal and remains at its highest value P ≈ 0.85 for a small interval of the attraction strength γ Att ∈ [0.034, 0.04]. Outside this interval, high polarization is lost, and this occurs in two different ways, depending on if γ Att is smaller or higher than the values contained in this interval. For values of γ Att below 0.034, the polarization decreases because the attraction strength is so small that the group splits in subgroups (S5 Fig). Even if fish may be still aligned inside each subgroup, they are not necessarily aligned with fish from other subgroups, and consequently the polarization of the total group decreases. For values of γ Att higher than 0.04, the attraction strength is sufficiently high to maintain the group cohesive (there is only one group, as shown in S5 Fig). In turn, this strong attraction makes the fish to rotate excessively towards their neighbor, thus unbalancing the alignment between fish and consequently the polarization of the group. This effect is stronger the higher γ Att is, and is a direct consequence of the discrete nature of the model. For larger values of γ Att , above 0.06, the polarization has disappeared and the group is in a swarming state.

The transition from high to low polarization as γ Att decreases depends on the duration of the simulations. Each line in S4 Fig represents more than 5 hours of real swimming of the whole fish group, a time during which environmental conditions are expected to vary considerably, at least in light and temperature. In Fig 3 (and the rest of simulations in the paper), each point is the result of simulations of ∼16.6 minutes of swimming time, a reasonable time duration during which changes in environmental conditions are considered as not affecting fish behavior.

The schooling region also overlaps with the milling region, meaning that both P and M have high average values for the same combination of parameters. This corresponds to situations where the group alternates between schooling and milling, and where wide turns (of radius equal to several times the radius of the group) can also be observed.

In order to analyze the relative size and location of the collective phases and to describe the impact of the interaction strategies and the other parameters on these behavioral phases, we synthesize the information in a phase diagram that puts together the distinct behavioral phases (see Fig 3).

We define the following four phases of collective behavior:

I). P ≥ 0.4 and M ≤ 0.4 correspond to the Schooling phase, in red. II). P ≤ 0.4 and M ≥ 0.4 correspond to the Milling phase, in blue. I-II). P > 0.4 and M > 0.4 correspond to the Intermittent region, in cyan. III). P < 0.4 and M < 0.4 correspond to the Swarming phase, in green.

The typical behavioral patterns displayed by the group in each of the four phases are illustrated in Fig 4A and in S3–S7 Videos.

Fig 4 shows the resulting phase diagram corresponding to each strategy in Panels B and C, with the borders of the regions overlaying the dispersion map in Panels D and E. Phase diagrams are centered on the region of interest, (γ Att , γ Ali ) ∈ [0, 0.08] × [0, 0.4], using the same scale for both strategies k = 1 and k = 2.

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TIFF original image Download: Fig 4. Impact of interaction strategies on collective states. (A) Spatial configuration of the characteristic collective states displayed by a group of N = 100 fish when each fish interacts with its most influential neighbor (k = 1). (BC) Phase diagrams showing four distinct behavioral regions: Schooling (I) in red, with highly polarized group (P > 0.4, M < 0.4), Milling (II) in blue, with low polarized group frequently displaying milling behavior (P < 0.4, M > 0.4), Swarming (III) in green, with small polarization and milling (P < 0.4, M < 0.4), intermittence between schooling and milling (I-II) in cyan (P > 0.4, M > 0.4) when fish interact with their most influential neighbor (B) and their two most influential neighbors (C). Gray regions correspond to excessively weak or excessively strong interactions, giving rise to fish dispersion (D > 5 m). Boundary between high dispersion region and swarming region (both in uniform color) is determined by the transition layer from high to low dispersion shown in Panels D and E. (DE): Contour lines of the collective behavior phases shown in Panels (BC), overlaying the dispersion map, from green (low dispersion) to gray (high dispersion). In panel (D), small circles denote the combination of parameters (γ Att , γ Ali ) corresponding to the time series shown in Fig 5, and yellow lines denote the vertical and horizontal cuts described in Fig 6. Social interaction ranges are l Att = l Ali = 0.28 m, intensity of random fluctuation (noise) is γ R = 0.2. https://doi.org/10.1371/journal.pcbi.1009437.g004

Fig 4B shows that the Schooling and Milling phases are precisely located in the zone between the region of high dispersion (gray region) and the region of high cohesion (green). When attraction is too small, fish are dispersed. When the attraction is larger than γ Att ≈ 0.03 for k = 1 (Fig 4D) and γ Att ≈ 0.015 for k = 2 (Fig 4E), fish reaction to its neighbor becomes sufficiently strong to coordinate collective movements, whose particular form depends on the intensity of the alignment.

Let us describe first the case k = 1 for growing values of γ Att . For γ Att ≈ 0.03 and 0.01 < γ Ali < 0.1, the fish turn around their barycenter in a typical milling formation shown in Fig 4A; for 0.12 < γ Ali < 0.3, the strength of the alignment induces the fish to swim almost all in the same direction, producing a fish schooling formation (Fig 4A(I)). For higher values of the alignment strength (i.e., when γ Ali > 0.3), the school is disorganized and forms a swarm (Fig 4A(III)). Indeed, because of the discrete nature of the model, when the intensity of the alignment becomes too strong, the heading change of the focal fish becomes too large and effectively random, which leads to a decrease in the group polarization and even to a dispersion of the group for γ Ali of order 1.

Larger values of γ Ali , not appearing in the phase diagram, correspond to the transition to the slow dispersion region described above. For given values of γ Ali in the narrow interval [0.12, 0.13], the group alternates milling and schooling along time. Intermediate states appear, in which fish are more or less aligned, forming a curved group that describes wide turns of a radius larger than several times the width of the group when it is in a milling formation. In both phases, these intervals of γ Ali become smaller as larger values of γ Att are considered. Thus, the Schooling phase lasts until γ Att ≈ 0.05 for γ Ali ≈ 0.2, and the Milling phase goes up to γ Att ≈ 0.065 for γ Ali ≈ 0.05. The transition from schooling and milling phases to the swarming phase results from the unbalance of attraction and alignment interactions. For instance, when attraction is too strong, or, equivalently, alignment is too weak, fish turn excessively towards their neighbors, in detriment of the alignment required to preserve group configuration. For larger values of γ Att , not appearing in the phase diagram, we find a region of slow dispersion where fish move as if their heading angle varies in a random way.

Extending the number of interacting neighbors from k = 1 to 2 strongly reduces the size of the Milling phase, which shrinks in width from about 0.045 to about 0.022 and in height from 0.1 to 0.026 (Fig 4C). This is due to the fact that fish now pay attention to two neighbors, thus destabilizing the milling formation, in benefit of the schooling formation, which absorbs most of the milling region. The Schooling phase is indeed much wider than when k = 1, growing in width from 0.02 to 0.065, for almost the same height of about 0.15, but for a smaller range of the alignment: when k = 1, schooling takes place for γ Ali ∈ [0.125, 0.3], while when k = 2, schooling appears when γ Ali ∈ [0.025, 0.2].

In order to understand how the transition between collective states take place, we explored in detail the time series of the three observables D(t), P(t), and M(t), in a number of representative cases. Fig 5 shows the time series of polarization (red) and milling (blue) for the four sets of parameters (γ Att , γ Ali ) represented in Fig 4D, one for each distinct behavioral phase. Panels A, B, and D of Fig 5 clearly correspond to what is expected from the phase diagram. Note that these time series are 5 times longer than those used to calculate the average values plotted in the phase diagram. Less obvious is what happens in the transition region between schooling and milling. Panel C indicates that different collective states can emerge with the same set of parameters, and that the averaging process used to draw the phase diagram can hide complex combinations of patterns.

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TIFF original image Download: Fig 5. Time series of polarization (red line) and milling (blue line), representative of each characteristic collective state by a group of N = 100 fish when each fish only interacts with its most influential neighbor (k = 1). (A) Schooling state: (γ Att , γ Ali ) = (0.04, 0.2), high polarization P(t) ≈ 0.9, low milling M(t) ≈ 0.22. (B) Milling state: (γ Att , γ Ali ) = (0.05, 0.05), low polarization P(t) ≈ 0.16, high milling M(t) ≈ 0.6. (C) Intermittent state: (γ Att , γ Ali ) = (0.033, 0.11), region where high polarization and high milling alternate: P(t) ≈ 0.98 with M(t) ≈ 0.22, and P(t) ≈ 0.18 with M(t) ≈ 0.6. (D) Swarming state: (γ Att , γ Ali ) = (0.06, 0.125), low values of both polarization and milling: P(t) ≈ 0.2, M(t) ≈ 0.2. Social interaction ranges are l Att = l Ali = 0.28 m, intensity of random fluctuation (noise) is γ R = 0.2. Time interval is [0, 10000] s, i.e., more than 2.5 hours. https://doi.org/10.1371/journal.pcbi.1009437.g005

In order to explore how frequent these patterns of bistability are, we computed the probability density functions (PDF) of polarization and milling along three cuts crossing the identified phases regions in Fig 4D (yellow lines): one vertical cut at γ Att = 0.0325, and two horizontal cuts at γ Ali = 0.2 and γ Ali = 0.05. The PDFs are calculated over the 20 runs of 2000 kicks per fish, initial transient excluded.

Fig 6 shows the resulting PDFs, with the mean value depicted in the phase diagram superimposed to each PDF. Panels A and B show respectively the PDF of polarization and milling along the vertical cut for γ Ali ∈ [0, 0.4]. Each phase is clearly recognizable: Milling phase for 0.02 < γ Ali < 0.08 (low values of P below 0.15, high values of M above 0.5 and up to 0.8), Schooling phase for 0.14 < γ Ali < 0.28 (high values of P above 0.5 and up to 0.9, low values of M below 0.18), and Swarming phase for γ Ali < 0.02 and γ Ali > 0.3 (values of P below 0.5 and values of M below 0.2). One can notice a first sharp transition to milling state at γ Ali ≈ 0.02, then milling decreases until γ Ali ≈ 0.1, and then there is another sharp transition to schooling state at γ Ali ≈ 0.12.

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TIFF original image Download: Fig 6. Polarization (red) and milling (blue) probability distribution functions and mean values (solid lines) along one vertical and two horizontal cuts of the phase diagram for the case k = 1. (AB) Vertical cut at γ Att = 0.0325, (CD) Horizontal cut at γ Ali = 0.2, and (EF) Horizontal cut at γ Ali = 0.05. Each vertical line is the probability distribution function (PDF) of the corresponding value of the parameter shown in the x-axis, for the parameter of each figure. Color intensity is proportional to the frequence of a value in the vertical axis. Solid lines are the corresponding mean value of each PDF. Interaction ranges are l Att = l Ali = 0.28 m and noise intensity is γ R = 0.2. https://doi.org/10.1371/journal.pcbi.1009437.g006

In the Schooling phase, the level of polarization is smaller for higher values of the alignment strength. This is a consequence of the higher amplitude that angular changes have in the swimming direction of fish when γ Ali is high, thus decreasing the group polarization. This effect, which at first glance may appear paradoxical, reveals the impact on collective behaviors of a burst-and-coast swimming mode in which there is a discontinuous influence of social interactions on the direction of swimming of fish.

For larger values of the alignment strength (i.e., when γ Ali > 0.3), the heading angle change δϕ is so large that fish turn excessively and can hardly synchronize their movements with their neighbors, making the group to adopt a swarming phase.

Of particular interest is the intermediate region where 0.08 < γ Ali < 0.14 (see Fig 6A and 6B). There, the PDFs of both P and M exhibit two peaks, thus revealing the alternation between the high polarization and high milling intervals, as shown in the time series in Fig 5C. Moreover, the polarization is higher in this intermittent region (P ≈ 0.95) than in the pure schooling phase (P ≈ 0.85).

Phases are equally easily identifiable in the horizontal cuts (Fig 6C–6F). No ordered phases appear when the strength of attraction is too small (i.e., when γ Att < 0.02), for both values of γ Ali . The Schooling phase shows a high value of P ≈ 0.85 from γ Att ≈ 0.028 to 0.048, where the PDFs start to exhibit a second wide peak at low values of P ≈ 0.2, corresponding to the transition from the schooling phase to the swarming phase. The Milling phase, crossed by the lower horizontal cut, starts at a higher value of γ Att ≈ 0.032, and spans a wider interval up to γ Att ≈ 0.064, where, as for the schooling phase, the transition to the swarming phase starts.

It is remarkable that cohesive and even ordered (schooling; milling) collective phases are obtained when fish only interact with their most influential neighbor (k = 1), whereas only interacting with the nearest neighbor would not even permit the cohesiveness of the school, as shown in [60]. In S6 and S7 Figs, we respectively show the PDF of the number of different most influential neighbors (DMIN) and of the number of different first nearest neighbors (DFNN), at any given time, for a school of N = 100 fish, and for interaction parameters chosen deep in the 3 different collective phases (swarming; schooling; milling). Note that for a physical attractive interaction only depending on the distance between interacting particles (like gravity), DFNN would exactly coincide with the DMIN. We find that the mean number of DMIN (39 for schooling and swarming states; 46 for the milling state) is very significantly smaller than the number of DFNN (mean 71 in the 3 collective phases), showing that a typical DMIN is linked (and interacts) with more fish than a DFNN (2.2–2.6 fish linked to each DMIN; 1.4 fish linked to each DFNN). Therefore, the dynamical notion of most influential neighbors is more effective at propagating information within the school, and hence ensuring cohesiveness and a possible long-range order (schooling and milling). Interestingly, the milling phase is characterized by more DMIN than the 2 other phases (which are indiscernible under this criterion), possibly because more information exchange within the school is needed to ensure the cohesiveness, alignment, and collective rotation of the school.

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