Selective social interactions and speed-induced leadership in schooling fish

Abstract

Animals moving together in groups are believed to interact among each other with effective social forces, such as attraction, repulsion, and alignment. Such forces can be inferred using "force maps," i.e., by analyzing the dependency of the acceleration of a focal individual on relevant variables. Here, we introduce a force map technique suitable for the analysis of the alignment forces experienced by individuals. After validating it using an agent-based model, we apply the force map to experimental data of schooling fish. We observe signatures of an effective alignment force with faster neighbors and an unexpected antialignment with slower neighbors. Instead of an explicit antialignment behavior, we suggest that the observed pattern is the result of a selective attention mechanism, where fish pay less attention to slower neighbors. This mechanism implies the existence of temporal leadership interactions based on relative speeds between neighbors. We present support for this hypothesis both from agent-based modeling as well as from exploring leader-follower relationships in the experimental data.

Keywords: attention switch; collective behavior; leadership; schooling fish; social interactions.

Fig. 1.

Attraction–repulsion and alignment force maps for the standard model. The force maps are able to recover qualitatively the behavior of the corresponding forces. Average (A) attraction–repulsion force and (B) acceleration (attraction–repulsion force map) acting on an individual $i$ depending on its relative position with the nearest neighbor $\mathit{NN}$, ${\overrightarrow{x}}_i-{\overrightarrow{x}}_{\mathit{NN}}$. We observe for small distances repulsion, for intermediate distances no net force and for large distances attraction. Average (C) alignment force and (D) acceleration (alignment force map) acting on an individual $i$ depending on its relative velocity with the nearest neighbor $\mathit{NN}$, ${\overrightarrow{v}}_i-{\overrightarrow{v}}_{\mathit{NN}}$. In all plots, the $y$-axis is along the direction of motion of the focal individual, the colormap displays the modulus of the force and the arrows its components. Forces are expressed in units of mass $m=1$. We employ values for all focal individuals and time steps in the simulation.

Fig. 2.

Force maps for the experimental data: (A) attraction–repulsion and (B) alignment force map. While the attraction–repulsion force map is similar to the standard model, the alignment force map displays a striking different behavior: In the lower-half region, arrows point inward indicating alignment, but in the upper-half region, arrows point outward indicating antialignment.

Fig. 3.

Model variations with alignment and antialignment regions in the alignment force map. Alignment force maps for (A) the explicit antialignment model and (B) the selective interactions model.

Fig. 4.

The selective interactions model is more similar to the experimental data than the standard model for a set of observables. Probability density functions (PDF) of (A) polarization $\varphi$, (B) instantaneous average nearest neighbor distance $〈d_{\mathit{NN}}〉$, (C) area of the convex hull normalized by the number of individuals $s$ and (D) contact duration time between Voronoi neighbors $t_V$. The error bands are calculated from the SD of a Bernoulli distribution, with the probability given by the fraction of counts in each bin.

Fig. 5.

Surrogate data of the experimental school with known leader–follower interactions. (A) We compare the individual $i$ with itself at a delayed time $\tau$. (B) and (C) Alignment force maps comparing the individual $i$ with itself at a (B) positive and (C) negative delayed time $\tau =0.2$ s. When the delay is positive, the individual at present acts as a follower and we see pure alignment. When the delay is negative, the individual at present acts as a leader and we see pure antialignment.

Fig. 6.

Relative speeds in neighbors describe better leadership in schooling fish than frontal distances along the direction of motion of the individual. Correlations in (A) the orientation and (B) speed (Pearson correlation coefficient $r$) between an individual $i$ and its nearest neighbor $\mathit{NN}$ at a delayed time $\tau$ for the experimental data. We filter for different states of the individual compared to the neighbor (legend). The markers denote the maximum of each line. We find correlations are higher for faster and slower individuals than frontal and rear individuals. Frontal and rear individuals are more similar to the baseline correlations given by all individuals.