Model-free information-theoretic approach to infer leadership in pairs of zebrafish

Collective behavior affords several advantages to fish in avoiding predators, foraging, mating, and swimming. Although fish schools have been traditionally considered egalitarian superorganisms, a number of empirical observations suggest the emergence of leadership in gregarious groups. Detecting and classifying leader-follower relationships is central to elucidate the behavioral and physiological causes of leadership and understand its consequences. Here, we demonstrate an information-theoretic approach to infer leadership from positional data of fish swimming. In this framework, we measure social interactions between fish pairs through the mathematical construct of transfer entropy, which quantifies the predictive power of a time series to anticipate another, possibly coupled, time series. We focus on the zebrafish model organism, which is rapidly emerging as a species of choice in preclinical research for its genetic similarity to humans and reduced neurobiological complexity with respect to mammals. To overcome experimental confounds and generate test data sets on which we can thoroughly assess our approach, we adapt and calibrate a data-driven stochastic model of zebrafish motion for the simulation of a coupled dynamical system of zebrafish pairs. In this synthetic data set, the extent and direction of the coupling between the fish are systematically varied across a wide parameter range to demonstrate the accuracy and reliability of transfer entropy in inferring leadership. Our approach is expected to aid in the analysis of collective behavior, providing a data-driven perspective to understand social interactions.

Figure 1

Transfer entropy quantifies the relationship between two processes $X$ and $Y$, where knowledge of one process reduces the uncertainty in the other. In the example above, arrows denote the direction of information flow, where process $Y$ is said to cause process $X$ in the sense of Granger [47], where past values of $Y$ help predict future values of $X$.

Figure 2

Extreme-event synchronization quantifies the synchronicity between extreme events in two processes with varying delays. Two extreme events within $\xi$ time of each other are considered synchronous.

Figure 3

Schematic of a zebrafish pair with superimposed notation.

Figure 4

Sample time trace trajectories of simulated fish, 10 s long, for (a) leader-follower and (b) leaderless pairs, with corresponding turn rates in the inset. Arrows are marked at the same instants in time for each fish.

Figure 5

Net transfer entropy $\left(T_{\mathrm{leader}\to \mathrm{follower}}-T_{\mathrm{follower}\to \mathrm{leader}}\right)$ at different bin widths for simulations with leader-follower (red) and leaderless (green) pairs. Error envelopes denote standard deviation over 100 simulations each.

Figure 6

ROC curves for event delay from extreme-event synchronization (squares), time lag from cross-correlation (circles), and net transfer entropy (crosses). Points closer to the diagonal (dashed) are less accurate than points towards the top and left borders. Net transfer entropy is the most accurate at 87.5% for a threshold of 0.17.

Figure 7

Net transfer entropy, event delay $q_{\xi }$, and time lag as a function of attraction and alignment gains $k_p$ and $k_v$.

Figure 8

Sample fish trajectories, 10 s long, within the 90 cm diameter circular experimental area, with corresponding turn rates in the inset. Locations where turn rate is more than three standard deviations of the average turn rate are marked with red circles. The data set was obtained from [57].

Figure 9

Verification of the interaction model through the computation of the turning acceleration with respect to fish position in the body frame [18, 57] across all (a) experimental and (b) simulated data. In the 2D histogram, (0,0) denotes the focal fish position, and the ordinate points in the direction of the focal fish direction of motion. Each point $(x,y)$ on the histogram represents an average of the turning acceleration of the focal fish across all times and all data sets, when the neighboring fish is in the $(x,y)$ location. A simulation with the same model parameters obtained after the calibration is run per experimental data set. (c) Mean turning acceleration versus left-right distance is shown for experimental (black) and simulated (blue) data, where each point on the graph represents a mean of all values across front-back distance for a given left-right distance from the focal fish. The color bar has the same scale as the plot in (c), from $-0.04$ to $0.06 \mathrm{m}{\mathrm{s}}^{-2}$.

Figure 10

Conditional entropy $H\left(X_k\right|X_{k-1})$ as a function of sampling time between steps. The conditional entropy increases significantly with the increase in sampling time until 0.2 s. Error bars denote mean and standard deviation over eight simulations.

Figure 11

Accuracy for each of the three classifiers, net transfer entropy, event delay, and time lag, as a function of threshold value.

Figure 12

Correlation of net transfer entropy, event delay, and time lag with alignment and attraction coupling gains. Correlation with alignment gain is performed over a range of attraction gains for each value. Similarly, correlation with attraction gain is computed over a range of alignment gains for each value. For alignment gain, all classifiers except time lag are correlated. For attraction gain, none of the classifiers are correlated.