Speckle statistics as a tool to distinguish collective behaviors of Zebrafish shoals.

Zebrafish have become an important model animal for studying the emergence of collective behavior in nature. Here, we show how to properly analyze the polarization statistics to distinguish shoal regimes. In analogy with the statistical properties of optical speckles, we show that exponential and Rayleigh distributions emerge in shoals with many fish with uncorrelated velocity directions. In the opposite limit of just two fish, the polarization distribution peaks at high polarity, with the average value being a decreasing function of the shoal's size, even in the absence of correlations. We also perform a set of experiments unveiling two shoaling regimes. Large shoals behave as small domains with strong intra-domain and weak inter-domain correlations. A strongly correlated regime develops for small shoals. The reported polarization statistical features shall guide future automated neuroscience, pharmacological, toxicological, and embryogenesis-motivated experiments aiming to explore the collective behavior of fish shoals.

Multifractal fluctuations in zebrafish (Danio rerio) polarization time series.

In this work, we study the polarization time series obtained from experimental observation of a group of zebrafish (Danio rerio) confined in a circular tank. The complex dynamics of the individual trajectory evolution lead to the appearance of multiple characteristic scales. Employing the Multifractal Detrended Fluctuation Analysis (MF-DFA), we found distinct behaviors according to the parameters used. The polarization time series are multifractal at low fish densities and their average scales with ρ - 1 / 4 . On the other hand, they tend to be monofractal, and their average scales with ρ - 1 / 2 for high fish densities. These two regimes overlap at critical density ρ c , suggesting the existence of a phase transition separating them. We also observed that for low densities, the polarization velocity shows a non-Gaussian behavior with heavy tails associated with long-range correlation and becomes Gaussian for high densities, presenting an uncorrelated regime.

Lyapunov exponent in the Vicsek model.

The well-known Vicsek model describes the dynamics of a flock of self-propelled particles (SPPs). Surprisingly, there is no direct measure of the chaotic behavior of such systems. Here we discuss the dynamical phase transition present in Vicsek systems in light of the largest Lyapunov exponent (LLE), which is numerically computed by following the dynamical evolution in tangent space for up to two million SPPs. As discontinuities in the neighbor weighting factor hinder the computations, we propose a smooth form of the Vicsek model. We find a chaotic regime for the collective behavior of the SPPs based on the LLE. The dependence of LLE with the applied noise, used as a control parameter, changes sensibly in the vicinity of the well-known transition points of the Vicsek model.