Disentangling and modeling interactions in fish with burst-and-coast swimming reveal distinct alignment and attraction behaviors

Abstract

The development of tracking methods for automatically quantifying individual behavior and social interactions in animal groups has open up new perspectives for building quantitative and predictive models of collective behavior. In this work, we combine extensive data analyses with a modeling approach to measure, disentangle, and reconstruct the actual functional form of interactions involved in the coordination of swimming in Rummy-nose tetra (Hemigrammus rhodostomus). This species of fish performs burst-and-coast swimming behavior that consists of sudden heading changes combined with brief accelerations followed by quasi-passive, straight decelerations. We quantify the spontaneous stochastic behavior of a fish and the interactions that govern wall avoidance and the reaction to a neighboring fish, the latter by exploiting general symmetry constraints for the interactions. In contrast with previous experimental works, we find that both attraction and alignment behaviors control the reaction of fish to a neighbor. We then exploit these results to build a model of spontaneous burst-and-coast swimming and interactions of fish, with all parameters being estimated or directly measured from experiments. This model quantitatively reproduces the key features of the motion and spatial distributions observed in experiments with a single fish and with two fish. This demonstrates the power of our method that exploits large amounts of data for disentangling and fully characterizing the interactions that govern collective behaviors in animals groups.

Fig 1. Burst-and-coast motion.

Trajectories along with the bursts (circles) of a fish swimming alone (A) and a group of 2 fish (B). The color of trajectories indicates instantaneous speed. The corresponding speed time series are shown in C and D, along with the acceleration/burst phase delimited by red and blue vertical lines. E defines the variables r_w and θ_w (distance and relative orientation to the wall) in order to describe the fish interaction with the wall. F defines the relevant variables d, ψ, and Δϕ (distance, viewing angle, relative orientation of the focal fish with respect to the other fish) in order to describe the influence of the blue fish on the red one. G and H show respectively the probability distribution function (PDF) of the duration and distance traveled between two kicks as measured in the one (black) and two (red) fish experiments (tank of radius R = 250 mm). Insets show the corresponding graphs in semi-log scale.

Fig 2. Quantification of the spatial distribution and motion of a fish swimming alone.

Experimental (A; full lines) and theoretical (B; dashed lines) PDF of the distance to the wall r_w after a kick in the three arenas of radius R = 176, 250, 353 mm. C: experimental (full line) and theoretical (dashed line) PDF of the relative angle of the fish with the wall θ_w (R = 353 mm). D: PDF of the signed angle variation δϕ₊ = δϕ×Sign(θ_w) after each kick (R = 353 mm). The inset shows the distribution of δϕ₊ when the fish is near the center of the tank (r_w > 60 mm), for R = 176, 250, 353 mm (colored dots), which becomes centered at δϕ₊ = 0° and Gaussian of width ≈ 20° (full line).

Fig 3. Average decay of the fish speed right after a kick.

This decay can be reasonably described by an exponential decay with a relaxation time τ₀ ≈ 0.80s (violet dashed line).

Fig 4. Interaction of a fish with the tank wall.

Its intensity is shown as a function of the fish distance r_w (A) and relative orientation to the wall θ_w (B) as measured experimentally in the three tanks of radius R = 176 mm (black), R = 250 mm (blue), R = 353 mm (red). The full lines correspond to the analytic forms of f_w(r_w) and O_w(θ_w) given in the text. In particular, f_w(r_w) is well approximated by a Gaussian of width l_w ≈ 2 BL ∼ 60 mm.

Fig 5. Quantification of the spatial distribution and motion in groups of two fish.

In all graphs, full lines correspond to experimental results and dashed lines to numerical simulations of the model. A: PDF of the distance to the wall, for the geometrical leader (red) and follower (blue) fish; the inset displays the PDF of the distance d between the two fish. B: PDF of the relative orientation Δϕ = ϕ₂−ϕ₁ between the two fish (black) and PDF of the viewing angle ψ of the follower (blue). C: PDF of the relative angle to the wall θ_w for the leader (red) and follower fish (blue). D: PDF (averaged over both fish) of the signed angle variation δϕ₊ = δϕ×Sign(θ_w) after each kick.

Fig 6. Quantification and modeling of interactions between pairs of fish.

A: we plot the average signed angle change after a kick δϕ₊ = δϕ×Sign(ψ) vs Δϕ×Sign(ψ) (red) and δϕ₊ = δϕ×Sign(Δϕ) vs ψ×Sign(Δϕ) (blue) (see text). B: dependence of the attraction (F_Att(d) in red) and alignment (F_Ali(d) in blue) interactions with the distance d between fish. The full lines correspond to the physically motivated form of Eq (14) (red), and the fit proposed in the text for F_Ali(d) (blue). C: O_Att(ψ) (odd function in red) and E_Att(Δϕ) (even function in orange) characterize the angular dependence of the attraction interaction, and are defined in Eq (12). D: O_Ali(Δϕ) (odd function in blue) and E_Ali(ψ) (even function in violet), defined in Eq (13), characterize the angular dependence of the alignment interaction. Dots in B, C, and D correspond to the results of applying the procedure explained in Materials and Methods to extract the interaction functions from experimental data.

Fig 7. A group of two fish from the study species Hemigrammus rhodostomus.

Credits to David Villa ScienceImage/CBI/CNRS, Toulouse, 2015.

Fig 8. Tracking.

A: Background image in grayscale extracted from a video file of an experiment with the biggest tank (radius R = 353 mm). B: Arena estimated from user-defined mask. The outer bold circle of radius R is derived from the mask drawn by the user of the tracking software and defining the area where tracking occurs. Inner dashed circle has arbitrary radius 0.85×R. C: Estimated walls of the tank. D: Estimation of the radius along the circle. Red line stands for cubic spline smoothing and extrapolation over 2000 angular points. The signal is repeated 3 times to improve estimation on limits (at 0 and 2π). The second period is kept to compute wall distances. Estimation of local polynomials is done on 30 equally spaced ranges over one period. Dashed line shows the average radius. E: Distribution of estimated radius in pixels. Red line stands for estimation of the average, used as radius approximation to compute the ratio of pixel to millimeters (PixelsToMm ratio is equal to 0.71 for this video). F: Trajectory of a fish during 40 seconds (2000 points) reported inside the estimated walls. Filled and empty circle respectively stand for start and end points.