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Abstract Fish schooling has attracted the interest of the scientific community for centuries. Energy savings have been long posited to be a key determinant for the emergence of schooling patterns. Yet, current methodologies do not allow the precise quantification of the metabolic rate of specific individuals within the school, typically leaving researchers with only a single, global measurement of metabolic rate for the collective. In this paper, we demonstrate the feasibility of inferring metabolic rate of swimming fish using the mouth‐opening frequency, a simple proxy that can be scored utilizing video recordings in the laboratory or in the field, even for small fish. The mouth‐opening frequency is independent of hydrodynamic interactions within the school, thereby mitigating potential confounding factors that arise when using locomotory measures associated with tail‐beat motion. We assessed the reliability of mouth‐opening frequency as a proxy for metabolic rate by conducting experiments on zebrafish (Danio rerio) using swimming respirometry. We varied the flow speed from 0.8 to 3.2 body lengths per second and extracted tail‐beat motion and mouth opening from video recordings. Our results revealed a strong correlation between oxygen uptake and mouth‐opening frequency for nonzero flow speeds but not in quiescent water. Contrary to our expectations, we did not find evidence in favor of the use of tail‐beat frequency as a proxy for metabolic rate. Overall, our results open the door to the study of individual metabolic rates in fish schools without confounding factors related to hydrodynamic interactions. 

Abstract Transfer entropy is emerging as the statistical approach of choice to support the inference of causal interactions in complex systems from time-series of their individual units. With reference to a simple dyadic system composed of two coupled units, the successful application of net transfer entropy-based inference relies on unidirectional coupling between the units and their homogeneous dynamics. What happens when the units are bidirectionally coupled and have different dynamics? Through analytical and numerical insights, we show that net transfer entropy may lead to erroneous inference of the dominant direction of influence that stems from its dependence on the units’ individual dynamics. To control for these confounding effects, one should incorporate further knowledge about the units’ time-histories through the recent framework offered by momentary information transfer. In this realm, we demonstrate the use of two measures: controlled and fully controlled transfer entropies, which consistently yield the correct direction of dominant coupling irrespective of the sources and targets individual dynamics. Through the study of two real-world examples, we identify critical limitations with respect to the use of net transfer entropy in the inference of causal mechanisms that warrant prudence by the community. 

Abstract Pairwise interactions are critical to collective dynamics of natural and technological systems. Information theory is the gold standard to study these interactions, but recent work has identified pitfalls in the way information flow is appraised through classical metrics—time-delayed mutual information and transfer entropy. These pitfalls have prompted the introduction of intrinsic mutual information to precisely measure information flow. However, little is known regarding the potential use of intrinsic mutual information in the inference of directional influences to diagnose interactions from time-series of individual units. We explore this possibility within a minimalistic, mathematically tractable leader–follower model, for which we document an excess of false inferences of intrinsic mutual information compared to transfer entropy. This unexpected finding is linked to a fundamental limitation of intrinsic mutual information, which suffers from the same sins of time-delayed mutual information: a thin tail of the null distribution that favors the rejection of the null-hypothesis of independence. 

Ongoing efforts seek to unravel theories that can make simple, quantitative and reasonably accurate predictions of the morphological adaptive changes that arise with the size variation. Yet, relatively scant attention has been directed towards lateral undulatory locomotion. In the current study, we explore: (i) the constraints imposed by the variation of length and mass in viscous and dry friction environments on the cost of transport (COT) of lateral undulatory locomotion and (ii) the role of the body, environment and input oscillations in such an intricate interplay. In a dry friction environment, minimum COT correlates with stiffer and longer bodies, higher frictional anisotropy and angular amplitudes greater than approximately 10^o. Conversely, a viscous environment favours flexible long bodies, higher frictional anisotropy and angular amplitudes lower than approximately 30^o. In both environments, optimizing mass and maintaining low angular frequencies minimizes COT. Our conclusions are applicable only in the low-Reynolds-number regime, and it is essential to consider the interdependence of parameters when applying the generalized results. Our findings highlight musculoskeletal and biomechanical adaptations that animals may use to mitigate the consequences of size variation and to meet the energetic demands of lateral undulatory locomotion. These insights enhance foundational biomechanics knowledge while offering practical applications in robotics and ecology. 

The phenomenon of fish schooling - coordinated swimming of fish in polarized groups of specific spatial formations- is commonly observed in several species of fish. Fish schooling may even provide hydrodynamic advantages reducing the overall swimming cost of the group. To date, the role of hydrodynamics in coordinated swimming is not completely understood as it is difficult to separately study the role of hydrodynamic interaction from other forms of interaction between the fish. Here, we propose a statistical methodology based on information theoretic tools and flow velocity measurements, that can potentially tease out the hydrodynamic interaction pathways from visual and tactile ones. To avoid experimental confounds from bidirectional interactions and objectively understand cause-and-effect relationships, we design a robotic platform that mimics the behavior of two fish swimming in-line in a controlled setup inside a water channel. We examine the response of a flag to the fish-like unsteady wake generated by an actively pitching airfoil located upstream. We systematically quantify the passive hydrodynamic effect by studying the flapping motion of the flag located downstream of the airfoil in response to both periodic pitching and less predictable, random startling motion of the upstream airfoil. The study integrates experimental biomimetics with information theory to establish a deeper understanding of hydrodynamics in fish schooling. 

Fish often swim in crystallized group formations (schooling) and orient themselves against the incoming flow (rheotaxis). At the intersection of these two phenomena, we investigate the emergence of unique schooling patterns through passive hydrodynamic mechanisms in a fish pair, the simplest subsystem of a school. First, we develop a fluid dynamics-based mathematical model for the positions and orientations of two fish swimming against a flow in an infinite channel, modelling the effect of the self-propelling motion of each fish as a point-dipole. The resulting system of equations is studied to gain an understanding of the properties of the dynamical system, its equilibria and their stability. The system is found to have five types of equilibria, out of which only upstream swimming in in-line and staggered formations can be stable. A stable near-wall configuration is observed only in limiting cases. It is shown that the stability of these equilibria depends on the flow curvature and streamwise interfish distance, below critical values of which, the system may not have a stable equilibrium. The study reveals that simply through passive fluid dynamics, in the absence of any other feedback/sensing, we can justify rheotaxis and the existence of stable in-line and staggered schooling configurations.