Dynamics of collective U-turns

Hemigrammus rhodostomus performs burst-and-coast swimming behavior that consists of sudden heading changes combined with brief accelerations followed by quasi-passive, straight decelerations [15]. Moreover, fish spend most of their time swimming in a single group along the wall of the tank. Fish regularly change their position within the group [28], so that every individual fish can be found at the front of the group.

A typical collective U-turn event starts with the spontaneous turnaround of a single fish (hereafter called the initiator), mostly located at the front of the group [28]. This sudden change of behavior triggers a collective reaction in which all the other individuals in the group make a U-turn themselves, so that, after a short transient, all individuals adopt the same final direction of motion as the initiator. Overall, we analyzed 1586 U-turns of which 1111 were observed in groups of 2 fish and 475 in groups of 5 fish. Fig 1 shows two examples of collective U-turns in groups of N = 2 (left column, panels ABC) and N = 5 fish (right column, panels DEF; see also supplementary S8 Fig and supplementary S1 and S2 Videos in the Supplementary Information).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Collective U-turns in groups of two and five fish.

Fish trajectories (solid lines) with successive positions (circles) equispaced in time every 0.04s. (ABC): N = 2, (DEF): N = 5. The top row (AD) displays the collective U-turn one second before it starts, t ∈ [t_s − 1 s, t_s], where t_s denotes the time at which the collective U-turn starts. The middle row (BE) displays the collective U-turn, t ∈ [t_s, t_e], where t_e denotes the end time of the collective U-turn. The bottom row (CF) displays the movement data 0.5 s after collective U-turn’s end, t ∈ [t_e, t_e + 0.5 s]. For visual convenience solid lines indicate the actual fish trajectories before t_s − 1 s and after t_e + 0.5 s. Arrows indicate the direction of motion. The grey thick line represents the tank border of radius 35 cm.

https://doi.org/10.1371/journal.pcbi.1005822.g001

Fig 1A shows a first fish F₁ (red color) swimming close to the upper-left region of the tank, followed by a second fish F₂ (purple color) at a distance d₁₂ ≈ 8.5 cm, swimming in the same direction. Right before the U-turn starts (Fig 1A), fish F₁ reduces its speed (circles become closer to each other), the distance d₁₂ decreases (to ≈ 5.1 cm), and F₂ also reduces its speed. Then, both fish perform a change of direction which lasts about 1 second and during which fish F₂ clearly follows fish F₁ (see the corresponding circles at each instant of time in Fig 1B). Once the U-turn is completed (Fig 1C), F₁ accelerates again, and so does F₂, which also adopts the direction of motion of F₁. The distance d₁₂ increases again (≈ 9.5 cm), due to the larger velocities, and remains of the same order along the depicted trajectory.

The situation is less clear when we try to describe collective U-turns in larger groups. Fig 1D, 1E and 1F show a collective U-turn for the case where N = 5. Before the U-turn, fish F₂ (orange) seems to be the fish that the rest of the group follows, the first circle of its trajectory being the most advanced one in the direction of motion. In fact, a position order can be inferred from Fig 1D: F₂, F₃, F₅, F₁ and F₄. However, it is rather complicated to extract from Panel E a precise information about which fish is the initiator of the U-turn, in which order the other fish follow, and therefore, who is influencing whom, especially if time-delays and reaction times are taken into account. The same happens with the information about fish’s positions after the U-turn, provided by Panel F.

In order to describe rigorously the individual behavior of the N fish during a U-turn, we introduce the angle ϕ_i(t) as an instantaneous measure of the direction of motion of a fish F_i; see Fig 2. We assume that the instantaneous heading of a fish F_i can be defined in terms of the velocity vector , so that . The heading of a fish ϕ_i allows us to characterize the angle of incidence of the fish relative to the wall, θ_wi = ϕ_i − ψ_i, where ψ_i is the angle formed by the position vector of the fish with the horizontal line (see Fig 2). The angle of incidence θ_wi is an individual measure that doesn’t depend on the heading of another fish. When a fish F_i is swimming along the wall, the value of θ_wi is around ±90° (we choose, by convention, the positive sign for the anticlockwise angle). In our experiments, most of the time the absolute value of the angle of incidence is close to 90°; equivalently, |sin(θ_wi(t))| ≈ 1. When the motion is perpendicular to the wall, the incidence is zero if the fish points towards the wall (θ_wi = 0°), and maximal if the fish points towards the center of the tank (θ_wi = 180°); in both cases, sin(θ_wi(t)) = 0.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Angles and lengths characterizing the relative position of two fish.

Angle ψ_j denotes the angular position of fish F_j with respect to the horizontal (positive values fixed in the anticlockwise direction); angle ϕ_i is the heading of fish F_i; θ_wi is the angle of incidence of fish F_i with respect to the outer wall; d_ij is the distance between F_i and F_j; θ_ij is the viewing angle of F_i with respect to F_j (not necessarily equal to θ_ji), and ϕ_ij = ϕ_j − ϕ_i is the heading difference of F_i with respect to F_j.

https://doi.org/10.1371/journal.pcbi.1005822.g002

The change of sign of angle θ_wi can serve as an indicator that a U-turn has taken place. In fact, this allows us to delimit the individual U-turns with precision and, consequently, to determine the start and the end of a collective U-turn.

We define the start and end times t_s,i and t_e,i of the individual U-turn of fish F_i in terms of the absolute value of the angle of incidence, |θ_wi(t)|. Once a U-turn has been detected, we obtain the time t_s,i at which |θ_wi(t)| has decreased (from approximately 90°) below a given threshold , and the time t_e,i at which |θ_wi(t)| has increased again and is above another given threshold (see Materials and methods for more details).

Thus, the start of a collective U-turn is determined by the time t_s at which the first individual U-turn starts, while the end of a collective U-turn is given by the time t_e at which the last individual U-turn finishes. That is: (1) For each collective U-turn, we have made a convenient time shift so that t_s = 0. Then, t_e denotes not only the end time but also the duration of the collective U-turn.

We also introduce an instantaneous measure of how similar the direction of motion of individual fish are across the group. We define the instantaneous group polarization P(t) as the following function of normalized fish velocity vectors: (2) where . When all the fish have the same direction then the polarization is maximal and P(t) = 1. The minimum value P(t) = 0 is reached instead when the velocity vectors cancel.

Figs 3 and 4 depict the two U-turns introduced in Fig 1, in terms of the polarization P(t) and the sine of the angle of incidence of each fish with respect to the outer wall θ_wi(t). The duration of the two illustrated collective U-turns is t_e = 0.94 s for N = 2 and t_e = 1.5 s for N = 5.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Spatial and temporal dynamics of a collective U-turn for N = 2.

(A) Individual fish trajectories in the tank during the U-turn. Each individual is represented by a unique color. The temporal sequence is indicated by circles equally spaced over time with a time-step of 0.04 s (empty circles) and 0.1 s (filled circles). Arrows denote direction of motion. Grey wide line is the tank’s border. (B) Group polarization P(t), with a minimum value P_min ≈ 0.27 reached at t ≈ 0.46 s. (C) Sine of the angle of incidence of fish to the wall: when parallel to the wall, sin(θ_w) = 1 (anti-clockwise direction) or sin(θ_w) = −1 (clockwise). The three vertical lines of each color indicate for each fish the beginning, the middle and the end of its U-turn, with the middle representing the time when a fish has finally reversed its original direction. (D) Interaction with influential neighbors: arrows point from influential neighbors to the focal fish and with the same color as the focal fish. (E) Fish bursting activity and their influential neighbors. Dots at i = 1, 2 correspond to bursting activity, blank corresponds to coasting. Dots at i − 0.5 represent bursting activity of the neighbor influencing fish i.

https://doi.org/10.1371/journal.pcbi.1005822.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Spatial and temporal dynamics of a collective U-turn for N = 5.

The displayed temporal sequence is drawn from the fish trajectories one second before the U-turn begins till one second after its end. Symbols in all panels are the same as in Fig 3. (A) Individuals trajectories in the tank during the U-turn. (B) Group polarization with a minimum value P_min ≈ 0.59 reached at t ≈ 0.66 s. (C) Sine of the angle of incidence of fish to the wall θ_w. The three vertical lines of each color indicate for each fish the beginning, the middle and the end of its U-turn. Here the middle time means the instant where sin(θ_w) = 0. (D) Interactions with influential neighbors: arrows point from influential neighbors to the focal fish and with the same color as the focal fish. (E) Fish bursting activity and their influential neighbors. If there is more than one influential neighbor, F_j with largest index value j is shown. Grey lines in Panels BCDE denote the start and end of the collective U-turn.

https://doi.org/10.1371/journal.pcbi.1005822.g004

For both group sizes, the group polarization (Figs 3B and 4B) before and after the U-turn is quite close to 1, showing that before and after the collective U-turn, all individual fish maintain essentially the same common direction. During the U-turn, the polarization decreases, describing a sharp V-form with a minimum at P(t) ≈ 0.27 for N = 2 and P(t) ≈ 0.60 for N = 5. The minimum is reached at approximately half the duration of the collective U-turn, t_m = (t_s + t_e)/2: t_m = 0.47 s for N = 2 and t_m = 0.75 s for N = 5.

Figs 3C and 4C show the change of direction individually for each fish in both U-turns: from anticlockwise to clockwise direction for N = 2, and vice versa for N = 5. Fig 3C clearly indicates that at t ≈ 0.3 s, the fish F₁ has almost completed its individual U-turn, while F₂ has just started to change direction: sin(θ_w2(0.3)) ≈ 0.98, while sin(θ_w1(0.3)) ≈ −0.5.

In Fig 4C, a similar ordering can be inferred from the times of departure from the bottom line at ordinate sin(θ_wi) = −1 + δ, where δ > 0 is a small parameter with respect to the range of ordinate values; we used δ = 0.1. Thus, the order is 2-3-1-5-4. However, the order in which individual fish change the sign of their angle of incidence θ_wi is different, 2-1-3-5-4, and also different is the arrival order to the top line at ordinate sin(θ_wi) = 1 − δ: 2-5-1-4-3. Moreover, some of these departure and arrival times are almost identical (see, e.g., F₁ and F₄), and the behavior of the fish during the U-turn is completely different. These difficulties in establishing a consistent order show that another criterion is necessary to identify the relation of influence between fish.

We have based our criterion to decide if a fish is an influential neighbor of another fish on the average value of the time-dependent directional correlation between the two fish along a time window.

For each pair of fish F_i and F_j, we define the directional correlation H_ij as a function of the heading of F_i evaluated at time t and the heading of F_j evaluated at a delayed time t − τ, where τ is the time-delay [26]: (3) The function H_ij(t, τ) is in fact the cosine of the angle formed by the headings and , and is a measure of how aligned is fish F_i at time t with fish F_j at time t − τ. The values of H_ij(t, τ) are between −1 (when fish swim in opposite directions) and 1 (when fish have the same direction), and equals zero when fish have perpendicular directions.

By averaging H_ij(t, τ) along a time-window of length (2w + 1)Δt, we are able to quantify how much the focal fish F_i is copying the moving direction of its neighbor with a time-delay τ by means of the following function [26] (4) where t_k = kΔt (the time-step in our experiments is Δt = 0.02s). The time-window parameter length w has been determined by means of a sensitivity analysis (pairwise similarity matrix), finding that w = 2 yields the more satisfactory results; see Section “Parameter selection” in Materials and methods and S5 Fig.

The average directional correlation C_ij(t, τ, w) allows us to characterize a fish F_j as an influential neighbor of a focal fish F_i at time t with time-delay τ, if the value of C_ij(t, τ, w) is larger than a given threshold C_min. Details on how w and C_min are obtained are given in Sections “Optimal setting parameters for influential neighbors identification” and “Parameter selection” in Material and Methods.

Fig 5 shows the directional correlation H₁₂ and its time-average C₁₂ between fish F₁ and F₂ along the collective U-turn depicted in Fig 3. Left (resp. right) panels aim to indicate the alignment of fish F₁ (resp. F₂) at each time t with respect to the alignment of fish F₂ (resp. F₁) at an earlier time t − τ. Panels A and C show respectively that for all τ, there is always an interval of time during which H₁₂(t, τ) ≈ −1 and C₁₂(t, τ) ≈ −1 (dark region), meaning that for all time-delays there is always an interval of time in which fish have opposite directions. Moreover, the larger the time-delay, the wider the black region where the direction of F₁ is opposite to the direction of F₂ at the previous time.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Directional correlations between fish F₁ and F₂.

(A) Directional correlations H₁₂(t, τ) and (B) H₂₁(t, τ) for t ∈ [−0.5, 2] and τ ∈ [0, 2] and their corresponding average over a time-window of width 2w = 0.4 s: (C) C₁₂(t, τ, w), (D) C₂₁(t, τ, w). Yellow regions: H_ij ≈ 1 and C_ij ≈ 1, i.e., fish have the same direction with a time-delay τ. Dark regions: H_ij ≈ −1 and C_ij ≈ −1, i.e., fish have opposite directions. The white upper-left corners indicate the τ is larger than the minimal time considered in this data set.

https://doi.org/10.1371/journal.pcbi.1005822.g005

On the other hand, the figures of the directional correlation of F₂ with F₁, especially Panel D, show a connected region in which the correlation C₂₁(t, τ) remains positive and above the threshold (yellow in the figure) around τ ≈ 0.42 s where H₂₁ ≈ 1 during all the time interval [−0.5, 2 s]. This strongly suggests that, during this time interval, F₂ is copying the behavior of F₁ with a 0.42 s time-delay, denoted τ_2,1 for this specific U-turn. Thus, one can consider that F₁ is influencing F₂ with time-delay τ_2,1, while F₂ is not influencing F₁ in this specific case. This influence dynamics is illustrated in Fig 3D by drawing an arrow at time t from F_j to F_i when F_j satisfies the condition C_ij(t, τ, w) > C_min for being an influential neighbor of F_i at time t, which in turn receives this influence and responds by copying the exhibited heading with a time-delay τ.

Using the same procedure for the N = 5 case depicted in Fig 4, we draw Fig 6 that shows F₁ copying F₂ with a time-delay τ_1,2 ≈ 0.5 s (Panels A and E). F₁ also copies F₃ and F₅ with, respectively, τ_1,3 ≈ 0.2 s (Panels B and F) and τ_1,5 ≈ 0.1 s (Panels D and H), but it doesn’t copy F₄ (Panels C and G). The influential neighbors of F₁ are thus F₂, F₃ and F₅, at different times and with different time-delays. We have calculated the rest of the correlations for all pairs of fish (see S1 Fig for an overview of all the heading correlations). As for the N = 2 case, these relations are illustrated by arrows going from the influential neighbors to the reacting fish in Fig 4D.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Directional correlation of fish F₁ with the other fish F_j, j = 2, …, 5.

Directional correlations H_1j(t, τ) (panels ABCD) for t ∈ [−0.5, 2] and τ ∈ [0, 2], and their corresponding average C_1j(t, τ) (panels EFGH) over a time-window of width 2w = 0.4 s. Yellow regions: H_ij ≈ 1 and C_ij ≈ 1 (fish have the same direction with a time-delay τ). Dark regions: H_ij ≈ −1 and C_ij ≈ −1 (fish have opposite directions). In the upper-left corners the white color indicates that τ is larger than the minimal time considered in this data set.

https://doi.org/10.1371/journal.pcbi.1005822.g006

Effect of bursting on the transmission of information

The specific behavior of H. rhodostomus, namely, the successive alternation of bursts and coasts [15], leads us to ask whether these abrupt changes of acceleration and speed can provide information that other fish could use to adjust their own movement. To address this aspect we study whether there is any correlation between the bursting activity of one fish at time t and the fact that this fish is an influential neighbor of another fish shortly after time t.

A burst corresponds to a brief phase of acceleration during which most changes in fish heading occur [15]. Panels E in Figs 3 and 4 show the bursting activity of each fish F_i, i = 1, …, N, and that of its influential neighbors. For each fish F_i, we draw a dot at time t and ordinate i if fish F_i is displaying a burst precisely at time t. Dot color at ordinate i corresponds to fish F_i’s color. The absence of a dot at a given time denotes that the fish is in a coasting phase at that time.

A second row of colored dots is drawn at ordinate i − 0.5 for some values of t when two conditions are met: (1) Fish F_i is being influenced at those times by one or more fish F_j, j ∈ {1, …, N}, j ≠ i, whose identity is given by the color of the dots, and (2) the influential fish F_j was bursting when it was influencing F_i at time t − τ earlier. If F_i has more than one influential neighbor at time t, the dot drawn at time t in row i − 0.5 has the color of the F_j fish with the highest index j.

In Fig 3E, red dots at i = 1 mean that fish F₁ is bursting at those time-steps and coasting at the other time-steps, and red dots at i − 0.5 = 1.5 indicate that, first, F₁ is the influential fish of F₂ at those time-steps, and second, F₁ was bursting when it was earlier influencing F₂. In turn, there are two possible reasons to explain the absence of red dots at i − 0.5 = 1.5 for certain time values: either F₂ has no influential neighbor, or F₁ was coasting. To assess which of the two explanations is valid, one needs to look at Fig 3D. For example, the absence of dots at i − 0.5 = 1.5 during 0.57 s and 0.62 s is due to F₂ having no influential neighbors, while the absence of dots in the same row between 0.75 s and 0.81 s results from the fact that F₁, which is the influential neighbor of F₂, is in a coasting phase at time t − τ (in this example the delay was found to be τ = 0.42 s).

Fig 3E shows that the bursting activities of both the focal fish and its influential neighbor are not directly correlated, suggesting that the primary source of information for fish to adjust their movements is the distance, orientation and angular position of their neighbors [15]. The same conclusion is obtained for N = 5. By focusing on fish F₂ for example, Fig 4E shows that there is no systematic overlap between the yellow dots at i = 2 and those at i − 0.5 for i ≠ 2, suggesting that the correlation between the bursting activity of a fish and that of their influential neighbors is marginal.

Number of influential neighbors

For all U-turns, we have counted the number of frames in which a fish is an influential neighbor, that is, the number of frames where the above described condition for identifying influential neighbors is met. When there are only two fish, a fish is found to be the influential neighbor 30% of the time spent in a U-turn. In groups of five fish, this proportion grows up to 62%.

We have counted the number of influential neighbors N_if a fish F_i has during a U-turn in groups of five fish, finding that in most cases, a fish has only one or two influential neighbors (for 58% of the time spent in a U-turn N_if = 1 or 2); see Fig 7A. The most frequent case is N_if = 1 (43%). Having more than one influential neighbor is frequent (19%), but less than having no influential neighbors (38%). The cases where there are more than two influential neighbors are negligible (less than 4% of the total time spent in U-turns).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Number, location and temporal occurrence of influential neighbors.

Cumulative analysis of collective U-turns of over 475 experimental (blue) and 1000 artificial (red) observations in groups of N = 5 fish. (A) Number of influential neighbors.(B) Distance rank of influential neighbors with respect to the focal fish. (C) Position rank of influential neighbors in the group. (D) Turning rank of influential neighbors. Histograms represent the proportion of time during which influential neighbors have been observed in a given class. The procedure to construct the artificial observations is presented later and in the section “Null model” in Materials and Methods.

https://doi.org/10.1371/journal.pcbi.1005822.g007

For each fish F_i, we have calculated the respective distance d_ij(t) at which the other N − 1 fish F_j are from F_i during the U-turns, thus establishing a rank order among the neighbors influenced by F_i. We have then compared the influence of close neighbors with those of distant neighbors, finding no correlation between the distance rank of a neighbor and the influence it exerts on the focal fish. This is shown in Fig 7B, where we have depicted the distribution of the distance rank of influential neighbors with respect to a focal fish. The figure shows that fish spent the same proportion of time (≈ 25%) being an influential neighbor of a focal fish independently of their distance rank. In other words, influential neighbors are not necessarily the closest ones.

When trying to identify events of causal influence by means of correlations, it is crucial to keep in mind that correlation does not imply causation. We thus have controlled the effects of potential chains of influence, where e.g. fish F₁ is highly correlated with F₃ not because F₁ is directly influencing F₃, but because F₁ is influencing fish F₂, which in turn is influencing F₃. To check the impact of these chains of influence on our results, we have removed from our data all the pairwise influence data that correspond to the following situation: if F₁ is influenced by both F₂ and F₃ and F₂ is simultaneously influenced by F₃ (or F₃ is influenced by F₂), then we removed the pairwise correlation (focal fish, influential neighbor) corresponding to (F₁, F₂) (or (F₁, F₃)). After removing 7172 out of 69703 data points and recomputing the results with the remaining data, we found that our results remain practically unchanged.

We have also calculated the position rank that each fish occupies in the group during a collective U-turn, finding that influential neighbors are mostly located in the front region of the group: 32% in the leading most advanced position, and 20% in the second place; see Fig 7C. Noticeably, influential neighbors can be found in the back of the group (in 29% of the cases in the fourth or fifth position), and even in the last position (a non-negligible 13% of cases).

We also paid attention to the order in which each fish starts its individual U-turn during a collective U-turn, finding that influential neighbors are those that most frequently turn earlier (32% of the cases), and that this relation decreases linearly; see Fig 7D. It is again noticeable that influential fish can be found to be the last turning fish (in 8% of the cases).

The apparently surprising fact that influential fish can be found in the back of the group and that the last fish turning can be an influential fish is due to the anisotropic perception of fish and their relative orientations during U-turns. But these findings have to be understood in the light of our specific time-dependent characterization of influential neighbor. If, for instance, F₁ turns first and influences F₂, F₂ will turn with some time-delay after F₁. Then, when F₂ is at half of its individual turning process, F₂ can be rotating in the same direction as F₁ in such a way that F₁, influenced by F₂, slightly adjusts its direction. We would then say that F₂, which is the last turning fish, has influenced F₁, the first turning fish.

In order to compare different collective U-turns, we define a normalized time in terms of the actual time t and the starting and ending time of each U-turn, so that the duration of a U-turn is now . Thus, corresponds to a time as long as the U-turn duration previous to the start of the U-turn, and corresponds to a time as long as the U-turn duration after the end of the U-turn. We have calculated the instantaneous value of the average speed , the average group polarization and the average number of influential neighbors . Here, angle brackets refer to the average across all fish in the U-turn along a time-window containing the collective U-turn.

Fig 8A and 8B show respectively the time evolution of and during the collective U-turns in groups of 5 fish. The description of the specific U-turn presented in Fig 4 is also valid for the general case: the speed decreases before the U-turn (from mm/s to mm/s), it reaches a minimum at half the U-turn duration ( mm/s), and it then grows to a higher value after the U-turn ( mm/s). A very similar behavior was found in groups of 2, 4, 8 and 10 fish of the same species in [28]. At the same time, the polarization is very high and almost constant outside the U-turn (), and exhibits a perfect V-shape during the U-turn, with the high values () reached at exactly the instants where the start and end of the U-turn takes place and , and the minimum value () at the middle of the U-turn. As expected, the average group polarization significantly decreases during the U-turn to almost half the value it has outside the U-turn. Right after reaching this minimum, there is a sharp increase of speed and polarization as more fish adopt the new direction of motion.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. U-turn dynamics in groups of N = 5 fish.

We depict here the temporal dynamics for the average velocity, average polarization, number of influential neighbors and its variation in over 475 experimental (blue) and 1000 artificial (red) recordings of collective U-turns. (A) Average speed . (B) Average group polarization . (C) Average number of influential neighbor per focal fish. (D) Average of the absolute variation in the number of influential neighbors |ΔN_if| divided by the number of influential neighbors N_if, defined in Eq (5): 〈η(t)〉. Horizontal axis denotes normalized time , where t_s and t_e denote the start and end of the collective U-turn respectively. The procedure to construct the artificial observations is presented later and in the section “Null model” in Materials and Methods.

https://doi.org/10.1371/journal.pcbi.1005822.g008

Fig 8C shows that before the U-turn the average number of influential neighbors increases until a maximum value is reached right before the start of the U-turn (). During more than one half of the U-turn, decreases until a minimum (), and grows again beyond the end of the U-turn until a second maximum (, twice the height of the minimum). After that, all fish have completed their U-turns and decreases again.

When the polarization is very high, the time-delay with which influential neighbors are detected is often too small in comparison with biologically realistic reaction times τ_R, so that these influential neighbors are not taken into account (we used τ_R = 0.04 s; see Section “Optimal setting parameters for influential neighbors identification” in Materials and methods). This is the reason why the average number of influential neighbors appears to be smaller in regions outside the U-turn, than when the U-turn is just about to start ( or slightly after its end (). Meanwhile, the decrease of in the middle of the U-turn has a different origin: once a fish has started to turn around, there is no real need of updating its alignment according to all its neighbors. That fish can safely reverse its motion by keeping the alignment with only one of those neighbors and even not paying attention to them for some period of time.

Another indicator of how fish make decisions while turning is how frequently a focal fish pays attention to other individuals. We define the relative variation of the number of influential neighbor per fish N_if(t) between two successive time-steps as follows: (5) denoting by Δt the time-step between frames (Δt = 0.02 s).

We have depicted the time-evolution of the average 〈η(t)〉 in Fig 8D, finding that 〈η(t)〉 remains essentially constant before, during and after the U-turn event, the amplitude of its variation being smaller than 10% of the signal (0.007 and 0.08, respectively).

Since the average number of influential neighbors is smaller when fish are engaged in the U-turn than right before or right after the U-turn, a constant average 〈η(t)〉 suggests that fish adjust their heading more frequently during the U-turn than outside the U-turn. Indeed, in the middle of a U-turn, no real common direction of motion exists (), that is, there is a high diversity of headings, so that fish have to frequently update their direction by paying attention to different neighbors.

Spatial organization of influential neighbors

We are now interested in determining the dynamical spatial organization of the influential neighbors of a focal fish. The relative state of a fish F_j with respect to a focal fish F_i is characterized by several parameters: the relative position of the neighbor , where is the vector position of F_i in cartesian coordinates, the distance between them , the viewing angle of F_j relative to the direction of F_i [26], which is the angle θ_ij with which F_i perceives F_j (note that θ_ij is not necessarily equal to θ_ji), the relative velocity , and the relative heading ϕ_ij = ϕ_j − ϕ_i. All these quantities are time-dependent. We have calculated their average value for all the U-turns in a uniform spatial grid of square cells to facilitate the interpretation of the vector field of these continuous variables. Each square cell, of side 20 mm, shows the average of the arbitrarily different number of values contained in the cell.

Fig 9A shows the density map of the relative position of the influential neighbor with respect to the focal fish when N = 2. The intensity of color is proportional to the frequency of occupation of the grid cell, showing that the influential neighbor is mostly located in front of the focal fish and at a distance of one to three body lengths from the focal fish. The same information is quantified in Panel B with a heat map in polar coordinates, highlighting the most frequent location of the influential neighbor.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Spatial and velocity distributions of influential neighbors around a focal fish in groups of 2 individuals.

(A) Density map of influential neighbors’ location (blue) and their average relative velocity field (arrows) with respect to the focal fish (red arrow). (B) Average spatial distribution of influential neighbors in polar coordinates (red: highest frequency; dark blue: low frequency; white: frequency equals zero). (C) and (D): Distributions of the distance d_ij and the angle of exposure θ_ij respectively. Blue histograms: F_j is an influential neighbor of F_i; orange: F_j is a neighbor of F_i, not necessarily influencing F_i; dark pink: overlap between the two.

https://doi.org/10.1371/journal.pcbi.1005822.g009

The average relative velocity is shown in Fig 9A (arrows), superimposed to the density map. The vector field shows that when the influential neighbor is in front of or behind the focal fish (sin〈θ_ij〉 ≈ 0), both fish move at similar speed although the focal fish is a little bit faster (the small black arrows are pointing in the opposite direction to the red one) and the difference in heading is also small. However, when the influential neighbor is on the sides of the focal fish, relative speed and heading difference tend to vary more as the distance between them increases.

The distributions of distances d_ij and exposure angles θ_ij between a focal fish and its neighbors are depicted in Panels C and D of Fig 9 respectively. We find, on the one hand, that their most frequent separation is 62.6 mm ± 29.7 mm (mean and standard deviation of histogram in Fig 9C), a value that is consistent with previous results where it was shown that the behavioral reactions of a fish depend on the angular position of its neighbors, as a consequence of the anisotropic perception of the environment [15].

On the other hand, the distribution of the exposure angle of fish F_j to the focal fish F_i is narrower when F_j is influencing F_i than when F_j is a neighbor of F_i, not necessarily influencing F_i. As both distributions are centered on θ_ij = 0, this shows that F_j is more frequently located in front of F_i when F_j is an influential neighbor of F_i than in the case when F_j is just a neighbor of F_i.

Fig 10 shows similar results for groups of N = 5 fish. Influential neighbors are more frequently located in front of the focal fish (although with a slight shift to the right; see Panels A and B) and at a mean distance of 67.5 mm ± 40.6 mm (Panel C).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10. Spatial and velocity distribution of influential neighbors around a focal fish in groups of 5 individuals.

(A) Density map of influential neighbors’ location (blue) and their average relative velocity field (arrows) with respect to the focal fish (red arrow). (B) Average spatial distribution of influential neighbors in polar coordinates (red: highest frequency; dark blue: low frequency; white: frequency equals zero). (C) and (D): Distributions of the distance d_ij and the angle of exposure θ_ij respectively. Blue histograms: F_j is an influential neighbor of F_i; orange: F_j is a neighbor of F_i, not necessarily influencing F_i; dark pink: overlap between the two.

https://doi.org/10.1371/journal.pcbi.1005822.g010

In turn, the velocity field has a smaller intensity and is much more homogeneous than in the case where N = 2. A slight asymmetry can also be observed (not noticed when N = 2) with fish located in front and slightly to the right of the focal fish having a higher velocity than those located elsewhere. Moreover, the distribution of exposure angles is more dispersed than in the case of two fish, meaning that influential neighbors are exposed to the focal fish with a larger diversity of angles, something that is simply due to the higher number of fish.

The difference in the homogeneity of the velocity field between groups of 5 and 2 individuals is not necessarily the result of averaging over a larger number of individuals. Although averaging over fish data pairs may reduce the uncertainty in the extracted parameter values, it is well-known that the level of homogeneity in the direction of motion of the school increases with group size [29]. But one also ought to consider that specific values of delay and curvature the individuals adopt during the U-turns could help to limit variability in coordinating the group. Some theoretical studies support this idea: simplified models of velocity alignment with additive noise have shown semi-analytically the existence of delay and rate of turn values that minimise the fluctuations in the variance of the individual speed [30], and flocking models of self-propelled particles have also shown that delay can be tuned to increase stability and alignment of the group [31].

Finally, we have analyzed the variation of the time-delay τ as a function of both the distance between the focal fish and its influential neighbors d_ij and the difference of heading ϕ_ij, finding that in both cases N = 2 and N = 5, the time-delay increases with respect to both the distance d_ij and the heading difference ϕ_ij (see Fig 11). This result can be understood because during a U-turn the fish speed is decreasing and two fish can display larger reaction times the more separated they are and the less aligned they are.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 11. Time-delay dependence on heading difference and separation distance.

Time delay τ extracted from the empirical observations as a function of heading difference ϕ_ij and separation distance d_ij. (A) N = 2, (B) N = 5. In both cases, the larger the heading difference and the distance, the longer the time-delay.

https://doi.org/10.1371/journal.pcbi.1005822.g011

A null model to detect spurious correlations

As already mentioned in the introduction, establishing causal influence on the basis of correlation measures requires controlling for spurious effects. Although our experimental data correspond to a specific collective behavior in which individuals influence each other, the relatively short time-windows over which cross-correlation are averaged and the use of several parameters through sensitivity analysis can weaken the accuracy of our results. To demonstrate that the particular detections of influential neighbors are not purely due to chance, we generated random artificial U-turns events by bootstrapping the data and applying the same procedure used to analyze collective U-turns in our experiments.

The null model is built for groups of 5 fish, for which our experimental data provide M = 2375 individual trajectories (5 × 475 collective U-turns). For every fish F_i, i = 1, …, M, the trajectory is rotated so that the individual turning point of the fish (where sin(θ_wi) = 0) is located in the upper part of the tank, by randomly sampling the new angular position ψ_i in the interval [π/2 − ξ, π/2 + ξ], where ξ is a small angle (we used ξ = π/12). Similarly, the time scale of each fish is shifted by sampling the instant of turning in the time interval [−ζ, ζ], where ζ is a short time (we have used ζ = 1 s). Then, five trajectories are randomly sampled, each one from a different randomly sampled collective U-turn, and mirrored if necessary so that the five individual U-turns are done in the same direction, clockwise or anti-clockwise. This way, the five fish of the artificial U-turn make their individual U-turn approximately at the same place and approximately the same time. For more details, see the section “Null model” in Materials and methods.

We have produced 1000 artificial collective U-turns; S9 Fig shows a collection of 10 of them. The results of our analysis are shown in red in Figs 7 and 8. As expected, they reveal clear differences between artificial and experimental U-turns.

Fig 7A shows that in artificial U-turns the proportion of time during which a focal fish has no influential neighbor is more than 63% of the time, while in the experiments it was less than 39%. The analysis also reveals that in artificial U-turns a focal fish has one influential neighbor for less than 28% of the time, while in the experiments, the proportion raises to 43%. Similarly, Fig 8C shows that the average number of influential neighbors is much smaller in artificial U-turns (≈ 0.4) than in real U-turns, where is almost always greater than 1. Note that the increase of during U-turns in artificial data is the consequence of the channeled motion of fish by the corridor. Moreover, the variation of along time, including the transients preceding and following the U-turn, decreases in artificial U-turns while it remains constant and with a higher value in experiments.

Fig 7B shows that distance rank has no significant effect on which fish is the influential one, both in experiments and in artificial U-turns. The decreasing number of influential neighbors comes from the fact that the tank is circular and the method we use. If the tunnel had been a straight corridor, we should have detected no decrease in our null model. However, in a circular tank, because of the geometrical constraints imposed by the curvature, even when two fish are both swimming in the same direction (i.e., clockwise or anti-clockwise), as the distance between fish increases, our method will detect a decrease of correlation. While Fig 7C confirms that influential neighbors are slightly more often ranked in the first position of the group, this effect is much more pronounced in the experiments. In fact, Figs 7B, 7C and 7D and 8A and 8B show that the selected null model satisfactorily reproduces the typical spatiotemporal behavioral patterns of real U-turns: the position and turning ranks are almost identical, as well as the variation of the average speed and the average group polarization, although the V-shape of the average polarization in real U-turns is significantly sharper than in artificial U-turns.

An additional, albeit expected, result of our null model is the homogeneous (isotropic) spatial distribution of “influential neighbors”, while in real collective U-turns influential neighbors are mostly located in front of the focal fish; see S10A and S10B Fig, compared with Fig 10A and 10B.

Ethics statement

Our experiments have been approved by the Ethics Committee for Animal Experimentation of the Toulouse Research Federation in Biology N°1 and comply with the European legislation for animal welfare.

Experimental procedures and data collection

Hemigrammus rhodostomus (rummy-nose tetras, Fig 12A) were purchased from Amazonie Labège (http://www.amazonie.com) in Toulouse, France. Fish were kept in 150 L aquariums on a 12:12 hour, dark:light photoperiod, at 27.7°C (±0.5°C) and were fed ad libitum with fish flakes. The average body length of the fish used in these experiments was 31 mm (± 2.5 mm). The experimental tank (120 × 120 cm) was made of glass and was set on top of a box to isolate fish from vibrations. The setup was placed in a chamber made by four opaque white curtains surrounded by four LED light panels to provide an isotropic lighting. A ring-shaped corridor was set inside the experimental tank filled with 7 cm of water of controlled quality (50% of water purified by reverse osmosis and 50% of water treated by activated carbon) heated at 28.1°C (±0.7°C) (Fig 12B). The corridor was made of a vertical circular outer wall of radius 35 cm and a circular inner wall with a conic shape of radius 25 cm at the bottom, so that the effective width of the corridor available to fish for swimming ranges from 10 cm at the bottom to 12 cm at the surface. The conic shape was chosen to avoid the occlusion on videos of fish swimming too close to the inner wall. Fish were randomly sampled from their breeding tank for a trial and were used at most in only one experiment per day. Groups of 2 or 5 fish were introduced in the experimental tank and acclimatized to their new environment for a period of 10 minutes. Their behavior was then recorded for one hour by a Sony HandyCam HD camera filming from above the setup at 50 images per second in HDTV resolution (1920x1080p). We performed 10 trials for each group size of 2 and 5 fish.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 12. Fish and experimental setup.

(A) A spontaneous U-turn initiated by a single fish in a group of five Hemigrammus rhodostomus fish. (B) Experimental ring-shaped tank, ©David Villa ScienceImage/CBI/CNRS, Toulouse.

https://doi.org/10.1371/journal.pcbi.1005822.g012

Data extraction and pre-processing

The positions of each fish on each frame were tracked with idTracker 2.1 [10]. Fish were sometimes misidentified by the tracking software, for instance when two fish were swimming too close to each other for a long period of time. In those cases, the missing positions were corrected manually. All sequences with 50 consecutive missing positions or less were interpolated. Larger sequences of missing values were checked by eye to determine whether interpolating was reasonable or not; if not, namely the trajectory doesn’t look like a straight line, then merging positions with closest neighbors were considered. Time series of positions were converted from pixels into meters. The origin of the coordinate system was set to the center of the ring-shaped tank. Body orientation of fish were measured using the first axis of a principal component analysis of the fish shapes detected by idTracker 2.1.

Detection and quantification of collective U-turns

Since the experiments were performed in an annular setup, the direction of rotation can be converted into a binary value: clockwise or anti-clockwise. We choose the anti-clockwise direction as the positive values for angular position. Before a U-turn event, all fish move in the same direction, say clockwise. Then, one fish, not necessarily the one located at the front of the group, changes its direction of motion to anti-clockwise direction. After a short transient, the other fish of the group display the same direction change, from clockwise to anti-clockwise. We defined the whole process of changing direction as a collective U-turn (see examples in Fig 1 and in S8 Fig). After data extraction and pre-processing, we found 1111 and 475 collective U-turns in groups of 2 and 5 fish, respectively. The duration distribution of collective U-turns in groups of 2 fish is shown in S3 Fig while the results for groups of 5 fish are shown in S4 Fig. Most of the collective U-turns last between 1 and 3 seconds, while the individual turning time usually lasts between 0.4 and 1 second.

The procedure used to define an individual U-turn for a fish F_i is as follows: we first determine the time t_m,i at which the sign of the angle of incidence of fish F_i changes sign (from negative to positive or vice versa). Then, starting from t_m,i, we reverse time step by step until the first time at which the absolute value of the angle of incidence is higher than a threshold is reached. We denote this time by t_s,i. Similarly, we start again from t_m,i and go forward step by step until the first time at which the absolute value of the angle of incidence is higher than a second threshold is reached. We denote this time by t_e,i. To determine the values of the thresholds and , we first compute the moving average of the angle of incidence over a period of 50 time steps (1s in real time), before and after the middle point t_m,i, with a window of 5 time steps (0.1s in real time), respectively. Then we set the threshold values as the maximum values of the absolute moving average. Doubling the length of the period of time over which the average is computed, or doubling the width of the window, do not affect the results. Finally, the time at which the collective U-turn starts (resp. ends) is defined by (resp. ).

Position rank in a group

The relative position of a fish F_i in a group of N fish is calculated by projecting the vector position of the fish on the average group velocity vector . This allows us to define a group centroid in the direction of , with respect to which the fish are ranked: the first fish in the group is the fish whose projection on is the most advanced one in the direction of motion of the group (given by ), the second fish in the group is the second most advanced, and so on. Relative distance between fish are not taken into account when establishing the rank.

Optimal setting parameters for influential neighbors identification

Four parameters are used to identify influential neighbors: the time-delay τ, the window size w, the correlation threshold C_min above which individuals are supposed to be interacting, and the threshold ε for selecting more than one influential fish.

The time delay must be specified along the whole trajectory of the focal fish: it is thus a series of values , where M is the number of time-steps or frames in the individual U-turn. The parameters C_min, ε and w are in turn given for all time and for all fish by means of a sensitivity analysis described in the next section.

Assume by now that the three values C_min, ε and w are known, and denote by F_i the focal fish and by F_j one of its neighbors. Then, the series of time-delays is built recursively as follows (actually only w is required to extract the time delays).

Denote by Γ_i(t_k) the highest value of the pairwise directional correlation C_ij of the velocity of fish F_i at time t_k with the velocity of F_j at each time-step in the range of the previous time-steps : (6) Then, the time-delays , k = 1, …, M_i, are determined by the smallest value of the time-delay τ_r ∈ R_k where Γ_i(t_k, w) reaches its maximum. For t₁, the maximum correlation is reached at , for some time-delay . We set for the initial value of the recurrence. For the rest of time-delays , k = 2, …, M_i, the size of R_k is based on the assumption that if, at some time t, F_i copies the behavior that F_j displayed at a previous time t − τ, then, after time t, F_i will not copy the behavior that F_j displayed at any time earlier than t − τ.

Time-delays obtained with more complicated and time consuming procedures such as the time-ordered technique developed in [26] or through the similarity analysis based on Fréchet distances [25] would in principle produce similar values.

Fig 13B shows the distribution of time-delays obtained with this procedure in groups of two fish. The distribution is clearly bimodal with a first peak when τ = 0 and a second one around τ = 0.4 s. Considering a reaction time threshold of 50-100 ms for a fish to integrate information and reach a decision [42], we cannot attribute small values of time-delays to situations where the behavioral decision of the focal fish has been influenced by its neighbors. This is confirmed by the analysis of the spatial distribution of the extracted time-delays (Fig 13A), where we show that the lowest average values of τ are found mostly when the neighbor was behind the focal fish, in a zone with the lowest perception [15], while the highest values of τ > 0.4 s are found when the neighbor is located in front of the focal fish. This has lead us to consider in our analyzes only situations where τ > τ_R = 0.04 s.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 13. Distribution of time-delay τ.

(A) Spatial distribution of time-delays obtained by selecting the maximum of the pairwise correlation between the focal fish and its neighbor. The color of each bin represent the mean value of all the cases in that bin. An angle of 0° degree corresponds to when the influential neighbor is in front of the focal individual. (B) Spatially integrated distribution of time-delays. The data here are the same as those used in panel A. The dotted line corresponds to the reaction time threshold τ_R = 0.04s.

https://doi.org/10.1371/journal.pcbi.1005822.g013

Parameter selection

Although the time-delays are determined once w is known, they also strongly depend on C_min and ε, as the value of these three parameters must be fixed at the same time. This is done by means of a sensitivity analysis in which we have tested the following 40 combinations of parameter values: w ∈ {0, 1, 2, 3, 4}, ε = {3, 5}, and C_min ∈ {0.995, 0.99, 0.95, 0.5}.

Each combination (C_min, ε, w) gives rise to four histograms like those depicted in Fig 7. These histograms constitute the solution of our method of analysis, and can be characterized by a vector in 19 dimensions: (i) the 5 proportions of the number of influential neighbors in groups of 5 fish, (ii) the 4 proportions of their distance rank, (iii) the 5 proportions of their position rank, and (iv) the 5 proportions of their turning rank. This allows us to determine how similar are the results arising from two combinations (C_min, ε, w) and , by computing the cosine similarity of the two vectors and .

The cosine similarity of two vectors and , denoted , is the cosine of the angle between these two vectors. Thus, two colinear vectors are such that independently of their magnitude, while two perpendicular vectors are such that . In our case, the components of the vectors are positive, so for all (C_min, ε, w) and . Moreover, as the components are proportions, colinearity implies identity, both in direction and magnitude. Thus, means that both results are identical, while means that they differ as much as possible.

S5 Fig shows the cosine similarity matrix for the 40 combinations we have tested. Note that the matrix is symmetric with respect to the diagonal, where . Except for C_min = 0.5, all similarity values are in the thin range [0.96, 1], showing that all combinations yield practically the same results. The higher dissimilarity is found in the white-yellow lines, where one of the combinations is (C_min, ε, w) = (0.5, 3, 2).

The selection of parameter values is thus done as follows.

We choose w = 2, which corresponds to the higher dissimilarity regions. The selected time window size is sufficiently large so that the jagged nature of the movement data is smoothed out but not too large so that the actual turns gets washed out from the data.

Using ε = 3 or ε = 5 yields very similar results and we have arbitrarily chosen ε = 3.

The selection of C_min is done by a specific procedure, which consists in calculating the number of data points that remain available for our analysis for each value of C_min. S6 and S7 Figs exhaustively demonstrate that the larger C_min is, the less data points remain available, and vice versa. We might be prone to choose a sufficiently small C_min in order to get the maximum number of data points. However, according to our definition of influential neighbor, C_min should be sufficiently large to select only the real influential neighbors. We have thus chosen the highest value which provides a sufficiently large number of data points, that is, the largest value before the fall of the number of data points in S11 Fig, C_min = 0.95. This value preserves 61% (23830) and 76% (69703) of data points for N = 2 and N = 5 respectively.

Null model of collective U-turns

We want to design artificial collective U-turns in groups of 5 fish where all fish perform an individual U-turn at more or less the same place and more or less the same time, and in the same direction (clockwise or anti-clockwise). Fish must coincide in time and space to constitute a “group”, but individual U-turns must happen in an absolutely independent way. Correlations at hand in this paper are thus reduced to a minimum, while preserving the general aspect of a group of fish changing direction.

Our experimental data provide us with 5 × 475 = 2375 trajectories of individual fish, which we have conveniently normalized and combined to build 1000 groups of 5 fish changing direction in the same spatiotemporal interval. This is done as follows.

The whole trajectory of a fish F_i during a U-turn takes place in an interval of time [t_s,i, t_e,i], where t_s,i is the instant at which the individual U-turn of fish F_i starts, and t_e,i is the time at which the individual U-turn ends. See the paragraph above Eq (1). The trajectory of fish F_i in radial coordinates is given by (7) where ρ_i(t_k) is the radius (distance of the fish from the center of the tank), ψ_i(t_k) the already defined angle position (computed anticlockwise as positive), and N_i is the number of time-steps t_k in the trajectory.

Denote by T_i the instant at which fish F_i effectively turns, i.e., F_i is perpendicular to the wall: sin(θ_wi(T_i)) = 0. In well defined individual U-turns as the ones we are using in our data, this happens only once per U-turn. Accordingly, (ρ_i(T_i), ψ_i(T_i)) denotes the fish position at time T_i.

Although we would like to have absolutely uncorrelated fish, it would not make sense to use groups of trajectories that do not reproduce a consistent U-turn, e.g., if one fish makes its U-turn much later than another, or on the other side of the tank. We thus try to decorrelate fish trajectories as much as possible, while preserving at the same time the typical spatiotemporal shape of real collective U-turns.

The decorrelation of all individual U-turns is done with the following two steps:

• Spatial rotation: For all individual fish F_i in all U-turns, we rotate its trajectory an angle −ψ_i(T_i) + π/2 + ξ_i, where ξ_i is a random number in [−π/12, π/12] sampled uniformly, so that the new location of fish F_i at the time T_i when it performs its individual U-turn is in the upper part of the tank around π/2, in [5π/12, 7π/12].
• Time shift: For all individual fish F_i in all U-turns, we shift the time scale a value −T_i + ζ_i, where ζ_i is a random number sampled uniformly in [−1, 1] s, so that F_i makes its individual U-turn at around time 0, in [−1, 1] seconds.

The artificial collective U-turn is thus built as follows:

1. Select randomly 5 real collective U-turns, and, from each collective U-turn, select randomly one trajectory. Rotate and time-shift trajectories according to the process described above.
2. Select randomly one of the 5 fish as the fish of reference F_ref for building the artificial U-turn. If necessary, mirror the trajectories of other fish so that all fish move in the same direction as F_ref with respect to the center of the tank, i.e., clockwise or anti-clockwise.

Then, the fish of reference of the artificial U-turn will make its individual U-turn at time ζ_ref ∈ [−1, 1] s and position (ρ_ref(T_ref), π/2 + ξ_ref). The other four fish F_j will make their individual U-turn at time ζ_j ∈ [−1, 1] s and position (ρ_j(T_j), π/2 + ξ_j) respectively, for j = 1, …, 5, j ≠ ref.

We have depicted in S9 Fig a set of artificial U-turns for comparison with the real U-turns shown in S8 Fig. Note that in these figures the time-scale has been shifted again so that collective U-turns start at t = 0 s.