Information transfer and behavioural inertia in starling flocks

Abstract

Collective decision-making in biological systems requires all individuals in the group to go through a behavioural change of state. During this transition fast and robust transfer of information is essential to prevent cohesion loss. The mechanism by which natural groups achieve such robustness, though, is not clear. Here we present an experimental study of starling flocks performing collective turns. We find that information about direction changes propagates across the flock with a linear dispersion law and negligible attenuation, hence minimizing group decoherence. These results contrast starkly with current models of collective motion, which predict diffusive transport of information. Building on spontaneous symmetry breaking and conservation laws arguments, we formulate a new theory that correctly reproduces linear and undamped propagation. Essential to the new framework is the inclusion of the birds' behavioural inertia. The new theory not only explains the data, but also predicts that information transfer must be faster the stronger the group's orientational order, a prediction accurately verified by the data. Our results suggest that swift decision-making may be the adaptive drive for the strong behavioural polarization observed in many living groups.

FIG. 1. Birds trajectories and turning delays

a, Reconstructed 3d trajectories of three birds belonging to a flock performing a collective turn. Inset: zig-zag due to wing flapping. b, c, Trajectories of all N = 176 birds in a flock. Each trajectory lies approximately on a plane. d, The radial acceleration of a turning bird displays a maximum as a function of time. This is in fact our very definition of a turn. Given two birds, i and j, we define the mutual turning delay τ_ij as the time we have to shift the full curve of the radial acceleration a_j(t) to maximally overlap it with a_i(t) (Methods). e, In the absence of experimental noise we must have, τ_ik + τ_kj = τ_ij: if i turns 20ms before k, and k turns 15ms before j, then i turns 35ms before j. Due to noise, time ordering will not hold strictly, but we still want it to be correct on average for τ_ij to make biological sense. We consider all triplets of birds and plot τ_ik + τ_kj vs. τ_ij. The data fall on the identity line with relatively small spread, confirming the temporal consistency of the turning delays (see also Methods and SI-Fig. S2).

FIG. 2. Propagation of the turn across the flock

a, The rank r of each bird, i.e. its order in the turning sequence, is plotted vs its absolute turning delay t, i.e. the delay with respect to the first bird to turn (ranking curves for all turning events are presented in Supplementary Information Fig. S3). b, The maximum mutual distance D between the top 5 birds in the rank does not increase with the linear size of the flock, L, hence indicating that the first birds to turn are actually close to each other in space. The result does not change if we use a different number of top birds, as long as this number is much smaller than the flock’s size. Inset: the actual position of the top 5 birds (red) within a real flock. c, The distance x traveled by the information in a time t is proportional to the radius of the sphere containing the first r(t) birds in the rank, namely x(t) = [r(t)/ρ]^1/3. The speed of propagation, c_s, is the slope of the linear regime of x(t) for early and intermediate times (black lines are linear fits - see Supplementary Information, Appendix A for later time saturation). d, The intensity of the peak of the radial acceleration, a^max, (solid symbols) decreases very weakly in passing from the first to the last turning birds. In the inset, we plot $a_i^{\max }$ vs the rank r_i for each bird. Hence, information propagates through the flock with negligible attenuation.

FIG. 3. Prediction of the new theory

a, The new theory predicts that the rescaled speed of propagation of the turn, c_s = a, must be a linear function of $1∕\sqrt{1-\Phi }$, where Φ is the polarization. The prediction is verified by the empirical data (P-value: P = 3.1 × 10⁻⁴; correlation coefficient: R² = 0.74). Each point is a different turning flock. Error bars on c_s are obtained from its variability under changing the linear fitting regime of x(t). c_s/a has the dimensions of sec⁻¹. The slope of this line is equal to $1∕\sqrt{\beta \chi }$ - equation (10). b, Polarization as a function of time in three different turning flocks. The value of Φ reported in panel a corresponds to the time average over the entire duration of the turn.