R E M E R C I E M E N T S

On a coutume de dire que la thèse constitue une expérience solitaire. La densité de la salle BIOMIM (donnée en brillant.e.s étudiant.e.s et expériences novatrices par mètre carré) semble pourtant apporter une contradiction expérimentale évidente à cette a rmation. J'y ai pour ma part trouvé un groupe scientifique plein de créativité, où les idées fusent et les discussions se tiennent sans cérémonie; j'y ai modestement fait mes premiers pas de scientifique en travaillant auprès de collègues passionnés; j'y ai surtout rencontré des ami.e.s.

Roméo, tu as été le premier doctorant que j'ai croisé, et mon premier guide au PMMH; merci pour l'exemple d'attention aux autres et de sens du service que tu m'as o ert. À mes compagnons de route Tristan et Gatien, avec qui j'ai partagé ces ans d'un bout à l'autre, merci. À Gatien pour son sourire indéfectible, à Tristan parce qu'il est le seul à comprendre et à partager ma quête quasi-maniaque de la colormap parfaite (et pour beaucoup d'autres choses). Un grand merci aussi à tous les membres de l'équipe BIOMIM, anciens et nouveaux, qui contribuent à faire de ce groupe un espace où on peut s'amuser et faire de la bonne science -la plupart du temps simultanément-. Je suis particulièrement reconnaissant aux stagiaires qui m'ont fait confiance et ont accepté de m'accompagner quelques mois au cours de ma thèse, en travaillant à mes côtés. Matthieu, Alexis, Jeanne et Paul, ce travail serait moins bon si vous n'y aviez pas participé.

À mes directeurs de thèse, Ramiro et Benjamin, un immense merci. Merci d'abord de m'avoir (ré)accueilli quand, après une tentative infructueuse sur le plateau, je suis revenu vous demander un projet de thèse. Merci d'avoir vu le potentiel du sujet "poissons", de m'avoir proposé de travailler dessus (malgré, je dois le reconnaitre, une certaine méfiance initiale de ma part). Merci pour la liberté et la confiance que vous m'avez accordées tout au long de ces trois années. Benjamin, merci pour tes pep-talk proverbiaux, qui m'ont si souvent redonné du courage quand j'étais au creux de la vague. Ramiro, pour ta présence quotidienne attentive, et pour ta patience et ta sérénité face aux di cultés du travail scientifique, qui sont des exemples pour moi qui en manque tant. vii Il serait vain d'essayer de remercier nommément toutes celles et ceux dont j'ai croisé le chemin au PMMH et qui m'ont apporté leur soutien, ou avec qui j'ai eu une discussion enrichissante. Je veux simplement dire ici à quel point j'ai été heureux de vivre pendant ma thèse des moments épanouissants de partage scientifique et humain, des Houches à Fontainebleau en passant par le Moulin d'Andé. Je remercie toutes celles et ceux qui se sont engagés et qui continueront de le faire pour organiser les petits et les grands évènements qui rythment la vie du PMMH, en se mettant au service du groupe -Chloé, Gauthier, Aliénor, Samantha, Antoine servers, poets, and scientists alike. Whether it is the graceful ballet of starlings, the precise social organization of ants, or the untamed power of a human crowd, there emerges a fundamental question: how can a collection of similar agents, each of which exhibiting rather straightforward actions when considered in isolation, give rise to intricate structures that manifest on temporal and spatial scales far beyond that of any single individual?

The present report focuses on schools of fish, as a framework for tackling the question of collective motion. In particular, its objective is to understand how collective patterns are altered when the environment in which fish move is disrupted; how factors such as illumination (Chapter ), confinement (Chapter ) or flow (Chapter ) influence organisation within a group of fish. We have chosen an approach based on simplified and highly controlled laboratory experiments: they enable us to operate at a middle ground, without getting caught up either in the intricacies of the natural world, or in pure mathematical abstraction. This document opens with an introductory chapter describing its background and context (Chapter ), and closes with some conclusions and perspectives (Chapter ). The appendices provide details of the experimental setup and additional informations (Appendix A and Appendix B). The reader is invited to progress linearly through the dissertation, although Chapter can be understood independently of the two preceding. -Vingt Mille Lieues sous les mers -Chapitre VII ( ), Jules Verne T his introductory chapter aims to provide an overview of the scientific context in which this thesis is set. Firstly (Section . ), it addresses the topic of collective motion, providing a concise summary of the diverse approaches, models and challenges involved in this vast discipline. Section . outlines the questions related to fish assemblies (or schools) in particular. Section . is centred on the model organism used in this work, Hemigrammus rhodostomus. The final Section . details the contents of this manuscript and the objectives pursued in each chapter.

. :

Defining the notion of collective motion is in itself a challenge. As the founder of the discipline in its modern form, Tamas Vicsek nevertheless ventured to do so by providing the following proposition (V and Z , ): it is the study of displacement in a system consisting of many similar units (or individuals, or agents) interacting through simple "social" rules. These rules can exist in the physical space, or in a underlying network connecting the agents. One of the key characteristics of collective motions is that the behavior of an individual unit is predominantly controlled by the impact of the "others" -causing the unit to exhibit a significantly distinct behavior from what it would display if it was moving alone.

We can see immediately that this statement paves the way for many other questions : What are those rules? What does "many", or "similar" really mean? However, the first issue that we would like to address here is "is this really a problem for physicists?". The study of collective movements is indeed a highly interdisciplinary field, occupying experts in many di erent areas .

. These are macroscopic groups, but the biological dimension of collective motion can also focus on microscopic systems. The emergence of coordinated patterns can be seen in the movements of bacteria (G et al., ; Z et al., ) and cells (A -E et al., ; P et al., ), among many others: the interactions underlying these motions are quite di erent from those among animals, and these specific features are a growing source of interest for experimental biologists (M and V , ).

. Fields of study of collective motions

Among animals capable of collective motions, there is one that particularly fascinates scientists: the human being, whose behaviour can be studied through the prism of crowds. Crowds can display a very wide range of collective structures, which in some cases are reminiscent of the motion of animal groups. For example, it is common to see phenomena of social contagion in panic movements (L et al., ). Here, as in the case of ethology studies, the mathematical representation of interactions between pedestrians makes it possible to understand well-known everyday phenomena, such as the formation of two-way lanes in the street (M , H , and T , ).

Interestingly, in certain species of ants or termites, the two-way tra c that is established between the nest and a food source is similarly organised in queues (C and F , ). A better understanding of interactions within human groups and the study and modelling of crowd dynamics can provide quantitative guidelines for crowd management in order to better anticipate deadly disasters.

Although it is not strictly speaking a case of collective motion, the synergy between physics and collective behaviour can be traced back to the Ising model (I , ). This is a statistical physics model used to characterise the behaviour of a set of two-state particles interacting with each other. This model shows that a set of simple rules can give rise to complex behaviour when the number of interacting particles becomes large. Subsequently, inspired by the innovative work of V et al. (

) and his numerical model of self-propelled particles, physicists have become increasingly involved in the study of collective motion since the s. In the Vicsek model, simple alignment interactions between self-propelled particles lead to the emergence of organised collective motions, reminiscent of the movements of groups of animals. Whether or not these collective states appear depends on the parameters of the simulation, in particular the density of the agents and the intrinsic noise level of their orientation.

Based on these numerical observations, a great deal of research over the last three decades has attempted to use the well-known equations governing the behaviour of matter, whether granular, liquid or solid, to model collective motion. This analogy has its limitations, since there are many di erences between the interactions in a group on the move and those between the constituent elements of matter, which we will describe in more detail later. However, this approach allows to use the methodology and powerful mathematical tools o ered by physics to study the emergence of order. It has proved e ective in many cases, including for instance the study of transitions between collective states (B et al., ).

It is interesting to note that this abundant interdisciplinarity allows the emergence of shared ideas between these major fields, which then feed o each other.

One example is the faster-is-slower e ect: paradoxically, it is quicker for a group of people to evacuate a room if they do not hurry. This counter-intuitive observation can be explained by a physical analogy with granular flows, which create jams when they pass through restrictions too quickly. One solution, already commonly used for evacuating groups of animals (particularly sheep) is to place an obstacle near the exit, which prevents clogging during emergency exit (G et al., ; H , F , and V , ).

.

. Statistical physics models for collective motion

By implementing the statistical physics approach to study collective motions, the objective is to use scaling, modelling, and simulation tools to reach a general understanding of collective motion at all scales. To achieve this, various hypotheses must be formulated to set up the theoretical framework of what we call a system in collective motion. Drawing inspiration from V and Z ( ), we can suggest the subsequent guidelines for the component units of such a system:

• They are capable of self-propulsion through internal energy consumption or energy input from their environment

• They can change direction by interacting with neighbors

• There is some inherent noise associated with their motion

Starting from these basic hypothesis, numerous numerical models have been developed with the shared aim of providing a universal description of collective behaviour, using a minimal set of simple equations of motion. We present in the following paragraphs the three models that are most commonly used today.

This model (or class of models) was a precursor in the study of collective motion. It describes the motion of an individual i, with a constant . :

velocity norm v 0 . Each individual has an absolute orientation θ i (t) and absolute position r i (t) at time t. The motion is set by the following two equations describing the evolution of orientation and position between each time step δt in two dimensions:

θ i (t + δt) = θ j (t) |r i (t)-r j (t)|<R + η i (t) ( . )

r i (t + δt) = r i (t) + v 0 δte θ i ( . )
with e θ i = (cos θ i , sin θ i ). These equations can be interpreted as follows. Equation . expresses the fact that each individual aligns its orientation with the average orientation of the neighbors within a distance R. The orientation is also subject to a noise η i (drawn from an uniform distribution in the original model).

From Equation . , individuals simply move to their new position according to their updated orientation. This model exhibits a very rich phenomenology, with a transition between a disorganised state and a large-scale organised motion, depending on density (agents per unit area) and noise parameters. The model has been extensively studied (see for example C et al., ) and remains one of the standards in the field.

-

The Cucker-Smale model has been first proposed in (C and S , ). The main di erence between the Vicsek model and the Cucker-Smale model is that in the latter, the velocity vector of the agents can vary instead of only their orientation. The model postulates that every agent updates its velocity by averaging the opposite of the relative velocities of its neighbors within a given distance. For a group of N agents, the velocity of the i-th agent v i can be expressed by the following equation:

v i (t + δt) = v i (t) + δt N ∑ j K ij (v j (t) -v i (t)) ( . )
Figure . a illustrates this mechanism. Here K ij is an adjacency matrix that quantifies the way the agents influence each other. In the original paper, it is assumed that the intensity of the interaction decreases with the relative distance between agent, namely: • Individuals move away from the n r neighbors that are located in their zone of repulsion (zor):

K ij = 1 1 + |r i -r j | 2 β ( .
θ r,i (t + δt) = - n r ∑ j =i r i -r j ( . )
• Individuals align with the n o neighbors that are located in their zone of orientation (zoo):

θ o,i (t + δt) = n o ∑ j =i v j ( . )
• Individuals move toward the n a neighbors that are located in their zone of attraction (zoa): :

θ a,i (t + δt) = n a ∑ j =i r i -r j ( . )
The orientation is the sum of these three contributions: θ = θ r,i + θ o,i + θ a,i . A blind angle can be implemented, to prescribe a zone in which their is no social interaction (see Figure . b).

The most intuitive approach is to suppose that interactions depend on the metric distance between individuals, but experimental evidence tends to show that the topological distance is the one chosen by the animals during collective movements. Early research on starling flocks showed that each animal interacts with a fixed number of nearest neighbours ( or ), rather than all neighbours within a given range (B et al., ). This subtlety is crucial in numerical simulations of collective behaviour. Simulations have, for instance, demonstrated that interacting with one or two neighbours is su cient to generate complex collective behaviour (W et al., ). Another option, which is a variation of the topological distance, is to consider that the et al., ; F et al., ). Even if a large number of strong analogies can be established between collective phenomena as described by standard statistical physics and those existing in biological systems, there are essential di erences between these two approaches.

(v j -v i ) dt v i (t) v i (t+dt) v j (t)
The first of these is the non-symmetry of interactions between agents. In a classical Newtonian system, two particles interact with each other symmetrically: the force of A on B is the opposite of the force of B on A. Clearly, this assertion cannot be true in the case of social forces, since there is no a priori reason why an individual A should influence B in the same way as B influences A. These interactions may depend, for example, on the relative orientation of the two agents, their velocity vectors, or even internal rules. Another way of considering this di erence is to observe that the system's energy is not conserved during "collisions", i.e. interactions between agents leading to a change in their state (typically, their velocity).

Here again, this dissipation of energy contradicts the classical principles adopted by statistical physics.

Phase transitions in statistical physics are only described rigorously in the theoretical limit of a number of particles that tends towards infinity. This hypothesis is relatively well validated for many microscopic physical systems, where the number of interacting particles is typically 23 . In contrast, biological systems in general are made up of between and individuals, with rare extreme cases of 6 individuals (R and D , ). This finite size hinders the classical theoretical approach and challenges the classical predictions of statistical physics, for instance on the behaviour of the system close to criticality.

Another consequence of the 'small' size of biological systems is the role of the boundaries of the domain in which the system evolves. Necessarily, this role sometimes becomes significant in systems with a limited number of interacting agents, whereas statistical physics generally considers it to be negligible.

. . State transitions

We have seen that one of the characteristics of collective motion is the emergence of particular group structures during movement, in other words, a large-scale organisation of the displacement of individuals. These structures, also known as collective states, are characterised by synchronised motion at the scale of the group, which appear through self-organisation: they are not caused by the decisions of each individual, but rather by short-range interactions between a large number of agents.

-This emergence of collective patterns organised at the level of a group on the move is generally referred to as self-organisation. In nature, collective states present highly di erent geometries. They can exist in one dimension (like the lines of ants or humans), two dimensions (in the motions of flocks of sheep or swarms of marching locusts for example) or be three-dimensional (in most schools of fish or swarms of birds or insects). Research in collective motion usually classifies these states into three main categories:

• swarming states: these are essentially disorganised collective motions, where the speeds and directions of movement of each agent are uncorrelated or only weakly correlated. The members of the group move in all directions in space without favouring a common direction.

• polarised states: the whole group moves in a coordinated fashion in a preferred direction. The speeds are aligned and the centre of mass of the group thus moves at a speed close to the average speed of the individuals.

• These structures exist in systems in their natural state, in laboratory experiments or can be observed in numerical simulations. Figure . shows some examples from a numerical simulations. More complex collective movements exist in nature (rotation around a point other than the centre of mass, for example), but these categories provide a basis for describing the geometry of the organisation of collective movements, since all structures can be considered linear combinations of these three main states.

A phase transition is the process during which a large number of elements constituting a system -often atoms -change their organisation on a global scale, as a result of a variation of an external physical parameter. A well-known example is the phase transition from a disorganised liquid state to an organised solid state when temperature decreases. By observing the similarity between the collective states described above and the phases of matter as described by statistical physics (gas, solid, liquid), it has been suggested that a parallel can be drawn between transitions between collective states and phase transitions in thermodynamical systems (B et al., ; B et al., ; G and P , ).

In fact, it is possible to observe transitions between collective states when the external parameters of a group of moving individuals are modified. . ). Collective behaviour in fish provides many benefits and plays an important role in their behavioural ecology. Here we propose a summary of the main identified advantages of schooling. ). On the other hand, the transfer of information is improved by the high level of synchronisation in the school. This allows for an assessment of the danger and a rapid reaction speed without each fish needing to confirm independently the nature or distance of the predator (M , ). In the case of direct attacks on schools of gobies, individuals benefiting from social information respond even more quickly than individuals exposed directly to the threat (W ,

). Finally, schooling can be an evasion strategy in the event of an attack. The capture rate is lower for a group of fish operating close to a critical point, i.e. in a state where multiple behavioural patterns coexist (K and R , ).

Numerical studies show that a collective strategy is more e ective than a selfish escape (Z et al., ).

Energetic advantages of schooling have been first hypothesized by B ( ) and subsequently explored through a theoretical model developed by W ( ) for a two-dimensional school of fish that swim synchronously in a single plane (see Figure . a.). He argued that it is advantageous for a fish to position itself behind and midway between two adjacent fish, in a diamond-like configuration. In this location, the follower could take advantage of the vortices in the Von Karman street produced by its predecessors. In an infinite school, individual are also believed to profit from a channeling e ect induced by their lateral neighbors (F , ; L , ). However, this swimming configuration is rarely observed in schools of fish in their natural habitat (P and P , ). Experimentally, two metrics are employed to evaluate the energy consumption of fish. Firstly, assessing the quantity of dissolved oxygen in the water o ers a direct evaluation of the metabolic rate, i.e. the energy consummed during swimming. Additionally, the frequency of tail beats serves as an indicator of the 'transport cost'. By combining these two measurements in a swimming tunnel (flume), a study on grey mullets showed that the fish saved energy by swimming in a group rather than alone, regardless of their position in a shoal of eight individuals (M et al.,

). An experiment on Hemigrammus rhodostomus swimming in a tunnel, based on kinematic measurements, revealed that the fish chose a phalanx structure (aligned perpendicular to the flow, see Figure . b.) when the incident velocity was high, in contrast to the Weihs model (A et al., ). In this position, the tail beats are synchronised, suggesting that vortex phase matching can be an energy-saving mechanism (see also Figure . c. and L et al., ).

A recent numerical study of a minimal two-fish school suggests that this structure may results from a trade-o between the classically considered wake vortex har-vesting strategy and the avoidance of zones that are energetically detrimental (L et al., ).

a. b.

U = 0.8 BL/s U = 3.9 BL/s c. .

. Sensory mechanisms of fish

To interact with their neighbours or their environment in order to swim in schools, fish need to collect information about their surroundings. In particular, knowledge of the distance and relative orientation of neighbours is an essential component in the spontaneous emergence of collective movements (C et al., ). This kinematic input enables a given fish to synchronise with its neighbours and is transmitted within the group from close to close. The sum of this information, known as social information, enables each individual to make decisions without necessarily having access to all the environmental cues available. To acquire this information, fish use all the sensory mechanisms that enable them to interact with their environment. We report here a summary of these sensory systems, inspired by that proposed by P and K ( ).

In fish, vision functions in a similar way to terrestrial mammals or birds:

most fish species have a retina with cones and rods that enable them to see colours and, in some cases, to be sensitive to ultraviolet light (B and M ,

). the visual sensory modality is crucial in regulating the behavior schooling of fish, and plays a central role in maintaining school cohesion (P and K , ). Several experiments have consistently demonstrated the significance of the visual system in schooling behavior. Early studies, such as those by Parr in the s (P , , ), revealed that fish with impaired vision were unable to participate in schools, and their ability to school was restored when their vision was recovered. Similar findings were observed in various fish species, emphasizing the critical role of the visual system in schooling (B , ). Fish spiecies exhibiting stronger schooling tendencies were more inclined to respond to their mirror image (S , ). One specificity of the vision sensory mechanism is that it allows individuals to access information about their conspecifics in a quasi-instantaneous manner, which makes it preponderant in the realisation of rapid group movements with a high degree of synchronicity. a standard length of to cm. The red-nose tetra has a silvery body with a bright red spot that extends across the head to the gills (see Figure . ). Hemigrammus rhodostomus are commonly found in river sections where obstacles such as decaying leaf litter on the riverbed or present, giving it a dark appearance known as "blackwaters". These fish tend to swim in the middle or lower part of the water column (K , M , and F , ). They also show a preference for darker areas, likely due to their eggs and fry being sensitive to light. These fish are a very common species found in the aquarium trade due to their bright colors and because they are small, easy to maintain and breed in an artificial environment. The rummy-nose tetra, in its natural habitat, exhibits a social and cohesive behavior, often forming groups known as schools. These schools can consist of anywhere from about to more than fish. Rummy-nose tetras groups are known to be able to navigate in noisy environments. Interestingly, when introduced to the same environment, di erent species of Hemigrammus will shoal together. ). This visual interaction is essential for the emergence of synchronisation and coordination in groups. However, time is needed for an individual to examine its visual field and, in the presence of motion, the likelihood of detecting critical information in a complex and evolving scene decreases. The ability to slow down from time to time enhances attention to the visual field and thus the ability to process the positions and directions of neighbours (K and M L , ).

Canal Neuromasts Superfical Neuromasts

Another important feature of intermittent locomotion, which plays a key role in collective movement, is the reduction of perturbations of the sensory system. For example, for fish that use hydrodynamic sensing (measurement of flow around the The main objective of this work is to improve our understanding of the collective motion of fish swimming under perturbed conditions, through direct experimental measurements and modelling. Our aim is to characterise the e ects of environmental parameters on the interactions within a school of fish and their impact on the resulting collective states. We investigated two collective swimming scenarios: free swimming (without any flow) and forced swimming (with an incoming flow).

Chapter examines the role of ambient light on the collective behaviour of freeswimming fish, using experiments on large schools of individuals. It is shown that light controls the emergence of collective behaviour and the transition between polarisation and milling, depending on the intensity to which the group is exposed.

In Chapter the issue of confinement is addressed, again in free swimming situation. The school is seen as a bistable system, where polarisation and milling states coexist. We observe experimentally that confinement, measured in fish per square meter, controls the transition from polarization to milling, and more particularly the statistics of the transition times between these two states.

Chapter looks at the behaviour of rheotaxis (swimming against the current), comparing fish swimming alone and pairs of fish, considered as the minimum constitutive cell of a school. The experiments of forced swimming are carried out for di erent flow speeds and either in a lit or totally dark envrionnement. We show that sity. We show that, at low light levels, the individuals within the group are unable to form a cohesive group, while at higher illuminance the degree of alignment of the school correlates with the light intensity. When increasing the illuminance, the school structure is successively characterized by a polarized state followed by a highly regular and stable rotational configuration (milling). Our study shows that vision is necessary to achieve cohesive collective motion for free swimming fish schools, while the short-range lateral line sensing is insu cient in this situation. .

From a practical point of view, fish schooling involves, for each individual in a group, a knowledge of both position in space and kinematics of close neighbors (C et al., ; F et al., ). In order to get this information, fish rely on vision, sensing of hydrodynamic disturbances and chemo-olfactory cues (H , ; P and K ,

). The role of each of these senses is not clearly elucidated today (L et al.,

), but it is generally accepted that vision and hydrodynamic sensing are the most predominant (P and P , ; P and P , ).

To sense hydrodynamic disturbances, fish use their lateral line system (B ,

). This ability has been suggested to be a factor in the formation of fish schools have also addressed the problem of vision with larger schools ( -fish), and showed that fish wearing opaque eye covers were able to maintain collective motion, using their lateral line system only. However, in these experiments, only one fish was blinded and placed back in a normal school, which limits the conclusions in terms of collective motion.

It has been found that fish reduce or completely suppress schooling behavior below a certain light threshold, that can vary across species (R and O , ; W , ).

However, these experiments were conducted on to fish and therefore do not provide evidence for specific behaviors that may occur when increasing the number of individuals in the school. Furthermore, the question was tackled in terms of an abrupt limit between a cohesive and a non-cohesive state, without considering the e ect of an increase in light level over a wide range once these thresholds are exceeded.

In this chapter, we go further in addressing the role of vision in the formation of large groups of fish, by altering the vision of all individuals at once. For that purpose, we chose to work with a species of highly cohesive fish, Hemigrammus rhodostomus, freely swimming in a large and shallow water tank. The available visual information is altered by modifying the illumination, either fixed in time or with two cycles of increasing then decreasing ramps. In addition to quantifying the role of vision, our study allows us to evaluate the role of the lateral line in a non-invasive way (typically, the response of fish in an experiment without light informs on their hydrodynamic sensing capabilities). Moreover, the .

progressive nature of the light variation enables us to fully resolve the transition from non-cohesive to cohesive motion. In contrast with similar previous studies, we work on schools composed of a large number of individuals (around ), allowing for a robust statistical analysis of the collective behavior parameters.

. . . Free-swimming tank with controlled illumination

The experimental setup consists of a large and shallow glass tank with a working area of by cm. This area is lit by visible light produced by a video projector (BenQ, × pixels) placed approximately cm above the water surface (see Figure C. ). This device makes it simple to control the illumination in the tank, either fixed as a constant or varying over time: simple homogeneous images of di erent grey levels are projected, ranging from white ( , maximum light intensity) to black ( , no illumination). We measured with a luxmeter that the corresponding light intensity in the tank ranges from to lux (± ). These levels of light intensity are comparable to those existing in the natural environment of origin of the tetra fish, as well as in their breeding conditions. For example, lux corresponds to the illumination on a clear sunny day. In order to visualize the fish regardless of the visible light conditions, the tank was also back-lit from the bottom by a powerful custom-made infrared LED panel ( W, λ nm, LEDpoint). We ensured that the chosen infrared wavelength was large enough to be invisible to the fish (C et al., ).

The entire setup is placed in an enclosure surrounded by opaque curtains, in order to avoid any light or visual disturbance due for example to the presence of the experimenter.

The shallow water depth (h cm) induces swimming motion essentially in D without causing stress to the animals. The four side walls of the tank are covered with opaque, non-reflective plastic sheeting, to prevent visual interaction with the outside of the tank as much as possible, but also to limit the natural tendency of fish (like many other animals)

to concentrate and move along the walls of the space allocated to them. Figure . shows a picture of the setup under a bright illumination.

. .

Data acquisition

The fish schools are recorded over long periods of time (typically between min and h), at a framerate ranging from to images per second, using a Mpx Basler greyscale overhead camera. The frame rate is chosen to allow for long acquisition period without generating excessive data, while still being su cient to accurately decompose the sometimes irregular movements of the fish, with their characteristic "burst and coast" motion .

. Experimental procedure

In each experimental run, a group of fish was allowed to swim freely in the tank while the illuminance was varied. Two types of experiments were carried out :

• Experiments during which the light intensity was varied continuously over time

• Experiments with a constant illumination, as a control

For the continuous experiments, the light variation was slow enough that the experiment could be considered 'quasi-static'In this way, the fish were not stressed by sudden changes in light, and the structure of the school changed on a time scale shorter than the light variation. For each experiment, a group of fish is removed from the rearing tanks and placed in the swimming tank, which has previously been filled with clear water and brought to a temperature of • C using thermostats and pumps. The fish are initially kept in medium ambient light (the same as in the aquarium) for minutes to allow them to acclimatize and to alleviate any stress caused by the change of environment. Both types of experiment lasted for hour and were recorded at frames per second.

Figure . : View of approximately

Hemigrammus rhodostomus in the free-swimming experimental setup. Here the swiming area is reduced to a : aspect ratio thanks to removable lateral walls.

Removable walls, covered with the same opaque material as the side walls, are placed on every side of the tank to create a uniform enclosure. Heaters are left on at the sides of the tank and an electric heater is started in the experimental enclosure. These precautions are taken to limit variations in water temperature to which the fish may be sensitive (C , C , and G , ).

. .

Data processing

The raw data obtained from the video recordings were processed according to the following pipeline, illustrated in Figure . :

. The background is first computed, as the maximum of randomly selected frames in the total stack of frames. The fish being darker than the background and moving constantly, this operation results in an image of the tank without any fish, that is then subtracted to every frame.

. The resulting movie is processed with the python-based open-source tracking software FastTrack (G and C , ), that first binarizes the frames and filters the resulting pixel clusters according to their area, then reconstructs trajectories thanks to the Hungarian algorithm (K , ), with a dedicated algorithm to handle crossing events.

. The data was finally exported and interpolated: missing positions were filled with a linear interpolation algorithm. The physical quantities we consider in our analysis are at the scale of the group, which means their are only weakly sensitive to potential switching of fish identities.

Original frame

Background frame 1. Background subtraction 2. Binary detection and filtering 3. Tracking .

. Physical quantities of interest

In order to quantify the relationship between the organisation of fish schools and the level of illumination, we compute two physical quantities inspired by statistical order parameters that characterise the degree of order and cohesion within the group, in terms of alignment and rotation. The alignment (polarisation P) and rotation (milling M) order parameters are defined as follows:

P = v i v i i∈1..N ( . ) M = r i × v i r i v i i∈1..N, r i <L ( . )
where v i (resp. r i ) is the instantaneous velocity vector (resp. the position with respect to the school instantaneous center of mass of the i-th fish) (see Figure . ). • denotes the averaging operator over all fish in the school. These parameters both range from to and quantify how much the individuals within the school are aligned along the same direction (P) or rotating around the center of mass of the group (M). We use a slightly . modified expression of M to discard cases where the fish are spread over all the tank area and swim along the borders, artificially producing high values of the milling parameter M, even when the group is not cohesive and no collective milling motion is observed in reality. M is thus defined in such a way that we only consider contributions from fish whose distance from the center of mass r i is less than a threshold value L, chosen to be half the short length of the tank (L cm).

O In addition, anticipating slightly onto Chapter , we note that the stable milling motion observed here with su cient lighting may in fact be induced by the interaction with walls (T et al.,

v i r i
). Figure C. A shows that the group polarization starts decreasing after exceeding the visual threshold. This decrease is coupled with the amplification of the milling parameter characterizing the group rotation around its center of mass. Thus, considering that fish tend to align with each other as their ability to see other individuals in the group is enhanced by a brighter environment, the milling behavior could be the consequence of being aligned in a confined space. Indeed, the alignment can be either quantified by the polarization or milling parameter, both having the same role in that particular geometry: while the polarization quantifies the alignment along lines, the milling parameters can be understood as a measure of an alignment along circles around the center of the group. The e ect of confinement on this transition from polarization to milling is the subject of the following Chapter.

.

. Preliminary results on heterogeneous illuminance

In the experiments detailed in this chapter, the illumination in the tank was uniformly adjusted. However, the question arises as to the influence of heterogeneous lighting on the schooling behaviour of fish. This question has been examined in various studies. It .

is generally observed that the majority of fish display negative phototaxis, which means they move towards the darkest regions along the negative light gradient. Experiments on groups of golden shiners and Hemigrammus r. indicate that the e ectiveness of phototaxis increases with an increase in the number of fish in the school (B et al., ).

Additionally, other findings attempted to establish thermodynamic equivalents for fish schools weakly confined by dark circles of various sizes by calculating an equivalent of kinetic pressure (G and P , ).

This chapter compares the sensory mechanisms of vision and lateral line. One way to investigate this question further is to expose groups of fish to equivalent stimuli in isolation from one another. This can be achieved by studying the navigation of a school of fish in a jet: the fish are exposed to a liquid jet without light, or to a luminous "jet" projected into the tank, like the one in Figure . . Such an experiment may reveal the relative e ectiveness of vision and lateral line in the context of navigation. We conducted preliminary experiments in our setup, the results of which can be seen in Figure . . Within a × cm square tank, groups of N = 50 fish swim under the same conditions as in the previous experiments. However, the illumination is now heterogeneous: a rectangular or toroidal shadow zone is included in the illumination (the illuminated zone is set at an intensity Ē , the dark zone Ē ). The dark zone is not as dim as in the case of homogeneous illumination with Ē = 0 since part of the light reflects, but the di erences in lighting are su cient to induce collective behaviour that are di erent from the homogeneous case. The key finding, evidenced in Figure . , is that fish density is significantly greater in dark areas, which aligns with Hemigrammus natural tendency to prefer areas shaded from light. The presence of dark areas leads to a weak confinement of the group, as noted by G and P ( ). Specifically, the higher density presence can be attributed to lower average speeds in the dark regions.

While the previous situations are binary in terms of light intensity, we wanted to study the behaviour of schools of fish when other intensities are available in their environment.

A supplementary test was conducted with two regions of varying intensities: the "dark" area, on the left in Figure . , is set at Ē = 0, whereas the "grey" region on the right is set at a marginally greater intensity ( Ē = 0.3). A slightly higher presence density is visually observed in the darker zone, but further experiments would be needed to statistically confirm this e ect.

Overall, these preliminary experiments confirm the existence of an interaction between heterogeneous lighting and collective behavior. They pave the way for more quantitative trials to analyze the respective roles of vision and lateral line in fish collective navigation.

cm

Dark area 10 cm ). These studies focus on the role of the number of fish in a confined space, but also mention that the proximity to the walls of the swimming arena can be decisive in the apparition of certain type of transition, such as milling to polarization transition (T et al., ).

We can unify these observations by examining the issue of state transition from the perspective of group confinement -in the sense of how crowded the available area is-which covers two distinct factors: the number of individuals in the group, and the surface of its arena. The impact of such confinement and the e ect of boundaries have been previously studied for systems of non-living (L et al., ) or microscopic active matter (from bacteria (B et al., ; W , L , and G , ) to cells (M and V , )), but seems to be a missing ingredient in our current understanding of state transitions for groups of live animals.

In particular, we still lack knowledge on how the reduction of the space given to the group influences its collective behavior. The study by (T et al., ) indicates that the swimming area is non-significant in the state transitions between swarming, milling, and polarized states in groups of golden shiners. However, this conclusion was based on a single experiment carried out for a di erent swimming area.

As of yet, no study has systematically investigated how confinement, both in terms of the number of individuals and the arena surface, impacts groups of animals on the move. Notably, the link between confinement and behavioral state transitions remains unsettled.

Here we report quantitative results on the collective dynamics of groups of rummy-nose tetras (Hemigrammus rhodostomus), a highly cohesive fresh water fish, under controlled experimental conditions. In first approximation, if one neglects the exact shape of the tank, the notion of confinement can be simply quantified by a "confinement density", which is the number of fish per unit area of the tank. Our analysis provides novel experimental evidence for a continuous state transition governed by confinement density. During this transition, the school is experiencing a bistable state, in which both polarization and milling states coexist; the group can fall randomly into one of these two states. Bouts of variable durations of either state intercede one another alternately over time. By measuring the distribution of transition times between states, we show that the statistics of this bistability is also directly influenced by the confinement density. Through these results, we show that the walls and the group size play a comparable role in the emergence of collective dynamics.

. . . Experimental setup

The experiment consists of recording the motion of free-swimming schools of Hemigrammus rhodostomus in a tank, using the same setup as described in the previous chapter, for di erent swimming areas S and school sizes N (number of individuals in the group). We systematically investigated the role of the confine-ment density, or mean density, ρ, defined as the ratio of these two values ρ = N/S in fish/m 2 . The water depth in the tank is su ciently shallow ( cm) to constrain the trajectories in two dimensions. The average body length (BL) of the fish used here is mm.

The swimming area S can be modified thanks to a system of movable partition walls. Three di erent surfaces are considered: , . and . m 2 . We focus mainly on arenas with a square shape (aspect ratio AR ), but we also discuss the implications of a di erent aspect ratio. We studied di erent group sizes ranging from to fish, resulting in densities between and fish/m 2 . During the experiments, the tank is brightly lit from above with visible light ( lux): as observed in Chapter , the milling state is predominant for a school of fish under these lighting conditions.

.

. Data processing

The acquisition frequency of the camera is frames per second (∆t = 0.02 s

method v i (t) = [x i (t + ∆t) -x i (t -∆t)]/2∆t
, where i = 1, 2, ..., N is the fish label.

For each confinement density ρ, at least di erent pairs of values of N and S corresponding to that density were tested (except for the two extreme density values and fish/m 2 ). We conducted at least di erent trials of min for every pair of values (swimming area, school size). In total, distinct experiments have been conducted.

. . . Experimental results

When first qualitatively examining the dynamics of the fish at any di erent combinations of number of fish N and tank surface S, we see that the group of tetras spontaneously and repeatedly alternate between milling and polarized states as can clearly be seen in Figure . . At the smallest density values ( -fish/m 2 ), we observe that the school remains in a polarized state for almost the entire recording duration, with only short-lived (a few seconds) incursions to the milling state that quickly revert to the polarized state. This behavior is illustrated in Figure . , for S = 1 m 2 and N = 10. As we further increase the confinement density by adding more fish to the school or reducing the swimming area (typically for densities between and fish/m 2 ), the bistability can be more clearly observed, as milling bouts last for longer. The group state shows an increase in fluctuations, with frequent shifts observed between the polarized and milling states, although the proportion of time in the polarized state remains more significant than the time spent in the milling state for densities lower than fish/m 2 ; as the density increases again, the milling state becomes more predominant (see Figure . , for S = 0.5 m 2 for exemple). Finally, when approaching densities of fish/m 2 and above, the groups mostly display a milling behaviour that lasts for long durations of time, still punctuated with occasional very short periods of polarization (typically less than a few seconds). This is shown by the panel corresponding to S = 0.25 m 2 and N = 30 ), and is reminiscent of a simple dynamical system oscillating between two stable states. This leads us to define these two states objectively: we say that the school is in the milling state (resp. in the polarised state) when M > 0.5 (resp. when P > 0.5). D can be understood as the major axis length of the smallest ellipse surrounding the group. Figure . shows the values of D with respect to the number of fish in the school N, for the di erent swimming areas S. The first observation is that variations of S have no influence on the groupe size, and that the group size in-creases slowly with N. We fit the experimental data with a square root function, tanks to a non-linear least squares method:

The colored bar above the time signal in

D = S 0 N ( . )
with S 0 a constant fitting parameter homogeneous to a surface. The evolution of D is well captured by this fit, which suggests that whatever the experimental conditions, we can consider that the fish school maintains a constant local density.

The best fitting parameter value is found to be S 0 = 4.98 BL 2 : this surface can be interpreted as a minimal 'comfort' area kept by each individual around itself.

In order to quantify the observed intermittency in states we first investigate it on the time scale of each experiment ( minutes). .

. Two-state model

We introduce a simple two-state Markov process at the scale of the school to gain insight on the transition between the polarized and milling states. In this scenario, the school is considered as a bistable system; we denote α (resp. β) the rate of transition from the polarized to milling states (resp. from milling to polarized). The transition probability from polarized to milling (resp. milling to polarized) during a time δt is therefore δtα (resp. δtβ). This probability is considered to depend only on the current state and is independant of the history of the system. In this case, the time between switching events (i.e the duration of the bout spent in either state)

follows an exponential distribution with a rate parameter specific to that state. We have the following probability density functions for the duration of milling and polarized bouts:

µ p (τ p = τ) = α exp(-ατ) ( . ) µ m (τ m = τ) = β exp(-βτ) ( . )
We With this approach we extract α and β for every set of experiments, that is for all pairs of (N, S). The variations of the two rates are presented on .

. Influence of the aspect ratio of the tank

The swimming area variations presented until now have been conducted using a constant unitary aspect ratio (AR), i.e with the tank retaining a square shape. However, because the interaction with the tank walls has been identified as a critical . parameter in the observed transition, it is possible that the aspect ratio also plays a significant role. To assess the influence of the aspect ratio, we conducted additional qualitative experiments with di erent AR values, specifically AR { , , , }. In this set of four additional experiments, the height of the tank remains fixed at m, while the width is varied to , , and cm. All other experimental parameters, such as frame rate, water depth, and ambient light intensity, were kept consistent with the conditions used in the initial experiments. This additional data set consists of one video recording of minutes of schooling for each value of AR.

Because of the proximity imposed on the fish in this setup, trajectory crossings are common and periods where fish are superimposed are regularly observed; therefore, tracking accuracy drops drastically and renders quantitative analysis with the tracking pipeline used for this study unreliable. However, a qualitative analysis of collective behaviour is still possible from the raw images. (no longer at all for AR = 10), and the fish move back and forth between the two ends of the tank, remaining aligned along its long side. We note that in this case the transition is exclusively due to geometric constraints, since at the confinement density considered here (respectively , , fish/m 2 for AR = , , ), we are well above the threshold after which we observe only milling in the case of a square tank. It is also interesting to point out that, starting from a situation where the milling is suppressed (for example the tank with an aspect ratio of ), it is still possible to make the milling 'reappear' by reducing the aspect ratio, as demonstrated by an additional experiment carried out in a × cm square tank (see Figure . e). We hypothesize that these observations explain the experimental results obtained by (T et al., ): they report that, when reducing the swimming area, higher confinement density does not lead to increased time spent milling. However, this reduction of area was conducted at an aspect ratio of approximately ( × cm), which means that they might have observed the same phenomenon of geometrical suppression of milling that we describe here.

.

Although in this work, the confinement of the group of fish is carried out using simple walls, analogies can be drawn with systems in the wild. Confinement in cases of predation can take di erent forms: in some cases real boundaries are introduced, as with humpback whales caging o schools of fish, trapping them (S and D , ), in others the boundary may be e ective like with dusky dolphins herding fish into 'prey balls' (V et al., ). Although these analogies are mostly qualitative, the parallel one can draw between geometric confinement and predation pressure opens an interesting avenue for research. Better understanding these similarities can help us break down complex behaviors observed in the wild.

The experiments reported here constitute the first quantitative laboratory study of the influence of confinement on the social behavior of live animals. Along with the works previously reported by (L et al., ), the results described above demonstrate that the collective states depend both on interactions between individuals and the environment of the fish group. Fundamentally, the coupling between the collective state and the surroundings is unsurprising, as the evolutionary pressure that has given rise to these states is likely highly environmentdependant.

.

In the case of confinement, we showed here that the nature of the bistability between polarization and milling of the group of Hemigrammus rhodostomus was controlled by confinement density. Whether it be the overall proportion of time spent in either state, or the statistical nature of bout durations, the intensity of the confinement seems to play a pivotal role. Furthermore, the qualitative investigation of di erent aspect ratios has shown that the boundary conditions can remove the bistability and select either state. These experimental results can be useful for future numerical models or the development of theory as an empirical benchmark to which these could be compared.

.

Since the shape of the tank appears significant in the behavioural state shift described in this section, it is reasonable to consider the impact of a circular tank on milling. Further experimentation was carried out in a circular tank with a diameter of cm (S . m 2 ) using a di erent fish species, zebrafish (Danio rerio)

. While zebrafish are less cohesive than rummy-nose tetras, they are similar in to approximately . for fish/m 2 . However, the milling parameter outcome di ers for this geometry, as its value remains constant at around . , irrespective of the confinement density. This di erence of collective behavior can be explained by the discrete nature of the zebrafish trajectories, which move using a burst and coast gait. At the end of a each burst period, an individual selects an orientation angle. Experimentally, it can be shown that the distribution of these angles is a wrapped Gaussian centered around (C et al., ). However, when an individual is close to a wall with a curvature, like in this experiment, it is somehow 'trapped' by the edge of the tank because part of the accessible angle distribution is truncated. Thus, in fish swimming in burst and coast, the presence of a curved boundary favours the behaviour of alignment with the walls and therefore the emergence of milling. -Saint-John Perse, Eloges VII ( ) T his chapter examines the determinants of rheotaxic behaviour, i.e the ability to orient in the direction of a flow, using a swimming tunnel. In particular, we study the station-holding behaviour of fish subjected to di erent flow speeds.

In order to identify the sensory mechanisms that enable this type of response to emerge, experiments were repeated in an illuminated tank and in a tank placed in total darkness. Little is known about the role of inter-individual interactions in this type of situation: to provide some insight on this issue, we compared fish swimming alone with 'minimal schools' of fish.

. Rheotaxis refers to the ability of aquatic organisms to orient themselves in response to a flow. One refers to rheotaxis as negative if the animal moves preferentially in the opposite direction to the flow (if it is swimming against the current) and positive in the opposite case. In particular, negative rheotaxis is a behaviour common to numerous fish species and employed in diverse contexts, such as locating upstream sources of odours carried by currents, intercepting invertebrate drift during foraging, and conserving energy while avoiding downstream displacement (C , B -C , and M , ).

Typically, negative rheotaxis is manifested by station-holding behaviour, in which the fish (or group of fish) maintains a constant mean position in the laboratory reference frame (an external observer sees the animal as stationary). This be- two linear sections and two turns. In the first section, the water is set in motion by a propeller rotating around a horizontal axis. The rotation is generated by a mechanical transmission system that transmits the rotation of a motor positioned vertically above the tank. This motor is a DC geared motor (RS Pro) capable of delivering up to W of mechanical power, with a nominal speed of revolutions per second at V. D printed corner modules are placed between the two sections to minimise the curvature of the water path, allowing the flow to turn while reducing flow separation and turbulence in the test section. Thin guide walls are D printed vertically in these corner modules to laminarize the flow (wall thickness: . mm, distance between walls mm). The swim channel, or test section, is located in the other linear section. It is enclosed upstream and downstream by two honeycombs blocks used to reduce the velocity gradients and the turbulence rate of the flow in the swimming area. Two fine meshes are placed in front of the honeycombs (HDPE,

haviour
. mm mesh size) to prevent fish from escaping through.

Overall, the swim section has the following dimensions: × cm in the longitudinal plane, with a depth of cm. A floor can be placed at any height to control the depth available to the fish. The entire swim tank is isolated by an upper wall with an access hatch to the swim section, allowing the operator to place the fish in this space. The flow in the channel has been extensively characterised using PIV measurements. Qualitatively, we found that the flow was unidirectional, without separation, and that the velocity was essentially in the longitudinal plane. The turbulence rate (ratio of velocity fluctuations to the average velocity) varies from . to .

depending on the prescribed flow velocity (see the dedicated section in Annexes for details on the calibration of the swim tunnel). By controlling the speed of the motor, the flow velocity can be varied from cm/s to cm/s (i.e. from . to . BL/s, BL . cm). A MPx camera (Basler ACa ) was used to record the position and kinematics of the fish placed in the test section, placed approximately m above the tank.

It records frames per second and is equipped with a fixed mm lens (Kowa, F . ) su cient to observe the entire longitudinal surface of the tunnel. The lighting for the camera is provided by a custom-made infrared LED panel ( W, × cm, LEDs, LEDpoint) placed under the tank, allowing to obtain images with very good contrast without disturbing the fish with visible light, similar to the free-swimming experiment described in previous chapters. The entire experimental setup is surrounded by opaque walls. The sides are covered with sheets of opaque black plastic and the top with a rigid sheet of PMMA (plexiglass) that is only transparent to infrared light. In this way, no visible external light can enter the chamber, but the camera can record images thanks to infrared light. Inside the chamber, a strip of white LEDs controls the visible light to which the fish are exposed.

. . Experimental procedure

A typical experiments consists in recording forced swimming periods lasting from to seconds at fps (dt . ×10 -3 s). We evaluate the role of di erent parameters on the kinematics and trajectories of the fish :

• The number of fish N is either or

• The incoming flow velocity U is varied between . and . BL/s

• The visible light is either turned o or on When the light is o , the swimming enclosure is completely dark. The light intensity measured in this case is below the sensitivity of the luxmeter ( . lux). If the light is switched on ( V voltage applied to the LED strip used for lighting), an ambient luminosity of lux is measured.

Initially, before starting an experiment, N or fish are selected at random from the rearing tanks. These fish are placed at rest in the swimming section without flow for min, with medium light intensity ( lux). During the experiment, the individuals are subjected to incident flows at di erent velocities U randomly chosen between . and . BL/s. Only swimming events during which the fish swim continuously for at least seconds are recorded. Acquisitions in which the fish explore the tank (particularly for incident velocities of less than BL/s) or do not swim are discarded. After the experiment, the individuals studied are returned to a separate tank; they are only used for an experiment once a week at most. For all the experiments presented here, the floor of the swimming section was raised and the available depth was cm. This depth is chosen to limit the crossing between fish or out-of-plane schooling during the two-fish experiments, making tracking and kinematic measurement impractical. However, this depth is still su cient not to induce stress in the animals and is consistent with the usual values reported in the literature. At least replicates of each experiment with a given triplet of parameters (N, U, light) is carried out, except for experiments with the lowest and highest values of U.

. . Data processing

A typical raw greyscale image acquired during experiments is represented on Figure . . The fish shape is obtained by binarizing greyscale images, providing a fish silhouette. For each fish, we extract the contour which consists of M points in the D plane, that we denote x n , y n for n = 1 . . . M, describing the codes developed for the data processing described in this section are open-source and made available on github the boundaries of the black shape. Then, we compute the Fourier transform S(k)

of the signal x 0 + iy 0 , . . . , x M + iy M , which reads :

S(k) = M-1 ∑ n=0 A n e -i2πk n M ( . )
and keep only the first M Fourier coe cients A n , with M = 16. This method reduces the amount of data stored, smoothes the contour of the fish and obtains data whose size remain constant (unlike the initially extracted contour, that contains a variable number of points depending on the deformation of the fish's body).

An exemple illustrating the extraction of the contour is shown on Figure . .

To accurately capture the kinematics of fish swimming, it is useful to know the position of the fish's centerline over time. The centerline is a curve running from the nose to the tail tip of an individual, passing through the centre of its body; it is used to describe the deformation of the fish's body over time.

In order to extract the position of the central line, we compute the Voronoi diagram (V , ) whose seeds are the contour points obtained previously. In this way, we define a set of regions in the plane such that each region contains all the points that are closer to one of the contour points than to any other point. In particular, the intersections of these Voronoi regions contained within the contour of the fish constitute a good approximation of the centerline (see Figure . for a visual explanation, and Figure . for an illustration of the method on experimental data). Finally, the midline is smoothed using the Savitzky-Golay convolution

filter. An example of the extracted midline deformation over time is provided on If the number of fish swimming in the experiment is N , a tracking is also carried out, i.e. at each time step we check whether the identity of the fish has been preserved compared with the previous time step. Given the small number of individuals, this poses no particular technical issue: we simply assign to fish i the position

x(t + dt), y(t + dt) that is closest to its position at time t.

. . Measured quantities

The points describing the contour and midline of each fish are extracted using the methods described above. From these physical quantities, we then compute derived kinematic measures that quantify the level of interaction between fish, the swim- Each midline plotted is obtained from snapshots separated by . s in time.

On this graph, the midlines are shifted so that the first point (nose of the fish) is always located in ( , ).

ming e ciency and the rheotactic accuracy of the individuals (ability to maintain a stable position when swimming against a flow).

To measure the characteristics of the tailbeat of each fish, we start by rectifying the positions of the points obtained for the midline with respect to the global orientation of the fish. If we consider a midline constituted of M points in D, the global orientation of the fish is considered to be the main direction of the first M/3 points of the midline (starting from the nose of the fish). By fitting a straight line to these first M/3 points, we find an orientation angle with the horizontal, that corresponds to the slope measured in this way. In this way, the displacement of each point along the midline can be broken down into two components: a transverse component (perpendicular to the overall orientation of the fish) and a longitudinal component (in the direction of the overall orientation of the fish). In the following, we will refer to the transverse component of the displacement of the tail tip as the "tail displacement". The tailbeat frequency and amplitude obtained from the tail displacement signal are proxies for the energy expenditure (cost of transport) of a fish (B , ; S , S , and A , ).

The tail displacement signal is a complex numerical signal (see We consider that the position of each fish is given by the position of the nose over time, (x(t), y(t)) its displacement with respect to its initial position r(t) is thus:

r(t) = [x(t) -x(0)] 2 + [y(t) -y(0)] 2 ( . )
Figure . provides an exemple of a typical experimental timeseries of x(t), y(t)

and r(t). To assess the ability of the fish to maintain its average position when swimming, the amplitude of the variation in position of individuals (or positional variability) is evaluated using the standard deviation of r, denoted a. The expression for a is :

a = 1 N T N T ∑ i=0 (r(i dt) -r) 2 ( . )
where N T is the number of time steps in the experiment and r is the average value of r(t) over the duration of the experiment. Finally, for experiments with N fish, we also compute the distance d between individuals as a measurement of the level of interaction.

A metric commonly used to assess the intensity of rheotaxis behaviour is the rheotactic index RI (B -C et al., ; E and C , ) which is a measure derived from the angular deviation from the upstream direction. However, in the case of our experiments, this angular deviation remains mainly negligible, typically limited to angles of less than , which makes this measure irrelevant here.

. . . General observations

During the swimming experiments, we observe that the fish maintain or attempted to maintain a constant average position in the laboratory reference frame, oscillating around this position mostly in the streamwise direction x. The variations in position remain small in relation to the size of the tank (of the order of / to BL,

i.e approximately to mm). As shown in Figure . , the variations in position are .

negligible in comparison with the dimensions of the swimming tunnel, ruling out a priori the hypothesis that they would be influenced by the confinement of the walls. These streamwise position variations were strongly periodic, with a typical frequency of Hz, due to intermittent swimming.

No favoured positioning was observed in the reference frame of the swimming section: the fish chose an average position close to the inlet or outlet of the tunnel with the same probability; generally, individuals did not touch the side walls and the upstream and downstream grids while swimming. Regarding the tailbeat frequency f tail , as shown in Figure . , a positive correlation with flow velocity U is also observed: the higher U, the more important is the frequency. f tail varies from Hz for the weakest flow velocities (U ≈ -BL/s) to -Hz for the strongest ones ( BL/s). This is a classic observation that is verified in most aquatic species (G , A , and M , ; H and S , ). However, it is noteworthy that, in our experiments, this re-lationship does not seem to depend on either the number of individuals N or the illumination of the swimming channel.

St = f tail A tail U ( . )
is relatively constant here, as can be seen from Figure . . This observation is commonly made for animal locomotion, particularly aquatic oscillatory locomotion (E , ), and confirms the quality of our measurements. The St values reported here are in the lower range of those reported in the literature (usually .

St . ).

Tailbeat amplitude and frequency are indirect metrics of the fish cost of transport, i.e. their energy consumption necessary to maintain their position in the flow.

These experimental results on swimming kinematics show that energy expenditure increases as flow velocity increases, and that the number of fish N or illuminance do not modify this trend.

.

. Rheotactic performance

The rheotactic accuracy is measured as a function of the experimental parameters.

In particular, the positional variability a, which corresponds to the standard deviation of the distance from the initial position r, is an indicator of this accuracy:

the lower a is, the more precise the rheotaxis. Variations of a with respect to to the experimental parameters are shown in Figure C. . Here, the data is divided into three categories according to the incident flow velocity U: low velocity (U BL/s), cruising velocity (U between and BL/s) and high velocity (U BL/s). We also assessed the impact of collective swimming by comparing the cases with one or two fish. In all cases, we observed that the precision was restricted to values between . and . BL on average. The distribution of data is wider for experiments involving fish.

Strikingly, for N , the positional variability a decreases in the non-illuminated cases. This is true for cruising speeds (p .

) and for high speeds (p . ), while the di erence is statistically non-significant in the low speed cases. For N , there is no statistically significant di erences if the light is on or o in terms of positional variability. ), the distance between fish is higher when the light is turned o . When the light is o , the fish swim at a typical distance d . BL from each other, with distance values ranging from to BL. Conversely, in cases where the light is switched on, the fish pairs are found at much smaller distances and much less dispersed, of the order of BL on average, and BL at the maximum. they are swimming, and adapt their gait accordingly to maintain a stable position.

By limiting the amplitude of positional variation, the individuals are better able to use their lateral line to assess the average speed and structures in the flow. This may explain the low values of the positional variability a found in the unlit cases for solitary fish. This hypothesis is consistent with the observation in Figure .   that the tail beat amplitude is reduced in the unlit case and for N . Indeed, this behaviour potentially makes it possible to limit the disturbances generated by the individual itself on its environment, and therefore to improve hydrodynamic sensing. E and C ( ) noted a di erent trend for similar experiments: their study reported greater position variability for fish with impaired vision (swimming in a dark environment) than for fish with visual cues. However, there is notable differences between this study and the present results, since the fish in Elder's study oscillate essentially in the span wise direction, and much more slowly (in order of magnitude: a few oscillations per minute).

The data concerning the distance d between fish suggests a priori a decrease in interaction between individuals for cases with no light, since d increases in these situations. However, the fact that the follower fish places itself behind the fish in front, without modifying the lateral distance, indicate that interaction may still occur via hydrodynamic sensing in the absence of visual information.

Overall, the data show no clear di erence in terms of energy consumption (assessed indirectly via the kinematic quantities A tail and f tail ) for solitary fish or pairs

G E N E R A L C O N C L U S I O N A N D P E R S P E C T I V E S
This thesis focused on collective motions in schools of fish. In particular, we attempted to shed light on how variations in environmental parameters can influence interactions between individuals within a group, and consequently the organisation of that group. Using simplified laboratory experiments, we varied these environmental parameters in a controlled manner and observed the motions of groups of fish in response to these perturbations. The model species chosen for these experiments is Hemigrammus rhodostomus, a small gregarious tropical fish.

We were able to show that light plays a major role in the emergence of collective motion: in the absence of lighting, schools of around fish were unable to swim in a coordinated fashion. Conversely, as the light intensity gradually increases, the school goes through di erent behavioural phases, with an initially highly polarised organisation followed by a state of rotation (milling) that is stable over time above a light level threshold. The limits of the swimming range also play a part in determining the collective state observed for a group of interacting agents. In the case of schools of to fish, it has been shown that reducing the available swimming area and increasing the number of fish in the school have equivalent e ects:

the greater the confinement (measured by the density of fish per unit area), the greater the probability of moving from a polarised state to a rotating state. Finally, the behaviour of rheotaxis was studied, i.e. the aptitude of a fish or a group of fish to swim against the current. In particular, the ability to maintain a stable position for single fish and pairs of fish, considered as the minimum constitutive cell of a school, was measured during forced swimming experiments in a swim tunnel.

The tests were repeated in illuminated and dark environments. The results suggest that flow sensing is used by fish in unlit conditions, but is not essential in lit conditions.

One of the most promising avenues for extending the experimental results presented here is to carry out numerical simulations of self-propelled particles interacting with each other through a certain number of social rules, with the aim of reproducing the behaviour observed in fish. In particular, we know that light influences collective behaviour indirectly, by modifying the visual information available. However, we do not fully understand its role on viewing angle, viewing distance or the number of neighbours considered by each neighbour in its in- This produces a D velocity field, with a vector attached to each window. In our case, we obtain a field of × vectors, since the recorded images are × pixel in size. This procedure is repeated for each pair of frames, thus producing a time-dependant velocity vector field in D. We used the OpenPIV module in Python to process images and extract velocity fields.

The laser is positioned so as to illuminate a part of the horizontal plane of the test section (a × cm rectangle), at two di erent heights from the bottom floor of the section h (h cm, approximately in the middle of the height of the section, and h cm, almost at the top of the section). A camera positioned above the system records greyscale images at a rate of frames per second.

The aim here is to calibrate the flow, i.e:

• establish the relationship between the average flow velocity in the section and the input voltage to the motor rotating the propeller responsible for setting the water in motion

• observe the flow structure to ensure that there is no excessive velocity gradient, that the turbulent intensity remains acceptable (typically less than )

and that there are no recirculation zones.

V

12 V 20 V In order to perform this calibration, PIV flow recordings are repeated for input voltages between and V, at the two chosen heights h, each recording lasting s. In all the experiments, the floor used to reduce the depth available to the fish is left in place, to faithfully reproduce the experimental conditions of the forced swimming tests as described in Chapter . From the velocity fields u thereby obtained, we can also compute the vorticity field |∇ × u|, which provides local in-formation about the intensity of the vortical structures in the flow. This graphs confirms the presence of a boundary layer, with zero velocity at the walls. It can also be seen that the profil is characterised by noise as the motor input voltage (and hence the average velocity of the flow) increases, with greater variations in central part of the velocity profil.

To assess the level of velocity fluctuations in the flow, we evaluate the turbulent intensity τ, defined as follows:

τ = 1 3 ( ũx + ũy + ũz ) U (A. )
where Ū is the mean velocity of the flow, and ũx = u x -U (resp. ũy = u y -U) is the velocity fluctuation in the streamwise (resp. spanwise) direction. Here we we do not have access to the last velocity component u z with the PIV, but we make the hypothesis that u z = u y because of the symmetries of the channel geometry.

The values of τ with respect to the input voltage are reported on Figure A. . The overall trend observed is that of an increase in τ as the motor voltage increases.

We note that τ remains limited to values below , which is within the low range of the standards for swimming channels and water flumes.

B D I R E C T M E A S U R E M E N T O F T H E C O S T O F T R A N S P O R T F O R D I F F E R E N T F I S H S C H O O L S I Z E S B Y R E S P I R O M E T R Y
As mentionned in Section . , schooling of fish is considered to be a possible way .

In order to assess whether the number N of fish in a school subjected to a flow influences the average metabolic rate of the individuals, we use a swimming tank setup (Loligo systems), similar to that described in Section . . The swimming tank itself is placed in a large bu er tank where the temperature, quality and salinity of the water is controlled. The dimensions of test section of the swimming tank in the vertical plane are × cm. The upper lid of the tank is watertight and gastight, which allows for measurements of the dioxygen (O 2 ) concentration in the water contained in the tank, over time. For this measurement, we use a sensor consisting of an optical fiber probe, immersed in the flow where the group swims. The small size of the sensor ( cm diameter, cm height cylinder) limits the disturbance created in the flow. A sealed hole in the lid of the tank ensures that the watertightness of the swimming tank is not compromised when the sensor is in place.

m O 2 (t) = V (c O 2 (t) -c O 2 (0)) (B. )
where t is the starting time of the experiment. In the litterature, the metabolic rate is usually given in milligrams of O 2 consummed per kilograms of fish body mass, per hour (mg O 2 /kg fish /h). We thus weight the fish used before each experiment to evaluate this quantity precisely (mass m ranging from m . g to . g, average m . g, standard deviation σ . g, for individuals). This propeller sets the water in motion, generating flow speeds of to BL/s in the test section ( BL . cm ± . cm, measured on average over individuals).

The experimental parameters controlled in this experiment are the following:

• The flow speed U, from to BL/s, by steps of BL/s

• The number of fish in the school N, ranging from to individuals. We selected values for N ( , ,,,,,, ).

An experiment consists of letting N fish swim in the test section, for increasing flow speeds U, starting with U BL/s. For each speed, the school swims for min at the set speed, then the flow is cut o for a recovery period of at least min ( to min in practice). This procedure is repeated until the speed U BL/s is reached, or a fatigue behaviour is observed in at least one of the fish in the group (typically the fish stops swimming and drifts into the flow). During this experiment, the oxygen concentration in the water is continuously monitored. An example of an experimental signal of the dioxygen consumption for N fish is shown on The values measured are typically ranging from to mg O 2 /kg fish /h, which is consistent with previous similar studies (C et al., ).

Measurements of the metabolic rate are a ected by the background rate. This is a decrease in the concentration of dioxygen in the tank water, even in the absence of fish. This phenomenon is due to two factors: ( ) the presence of micro-organisms in the water, which can consume some of the available dioxygen, and ( ) any leaks in the equipment, in particular the hole through which the dioxygen probe passes. The background rate also depends on the flow velocity U. To evaluate this background rate, we carry out an experiment similar to the swimming experiment, but in the absence of fish: we measure the time evolution of the dioxygen concentration in the water, for flow speeds U from to BL/s.

The measurement period is min for each U. A dioxygen depletion rate as a function of U is extracted from these measurements, with the method described in the Indeed, a MO 2 curve as a function of speed is expected to have a quadratic shape, with a minimum reached at a positive speed, known as the optimum speed (speed at which the COT is minimal). Here we obtain an optimal swimming speed of BL/s. .

In this appendix, a respirometry technique was used to measure the energetic cost of transport of groups of fish subjected to a flow, for di erent numbers of fish in the school. It was observed that this COT does not seem to be influenced by N, with the exception of the specific case of a fish swimming alone N , which appears to have a higher COT than fish swimming in a group, whatever the number of fish in the group. These results are consistent with those reported for Hemigrammus rhodostomus (A et al., ). Two hypotheses can be made to explain them: ( ) there is a mechanism of interaction between individuals in a school, for example via the harvesting of vortices, in accordance with the Weihs hypothesis (W , ). However, this hypothesis should lead to a decrease in COT, not just for N compared with N ≤ , but a decrease that becomes more significant as N increases, since more fish are interacting. ( ) It is possible that the solitary swimming situation creates a stress response in the isolated individual, which would explain the increase in its metabolic rate.

In view of the dispersion of the measurement points for N , the complexity of the measurements, the inherent noise associated with them and the limited number of replicates carried out, it is however complicated to draw definitive conclusions. More experiments would be useful, as well as kinematic measurements of individuals in the school, which would make it possible to quantify both indirect markers of energy expenditure (tailbeat frequency and amplitude) and distances and organisation within the school. In this way, we would be more able to assess whether interactions between individuals do indeed lead to energy savings, and in which cases.

G E N E R A L S U M M A R Y

Collective motion, fish schooling, confinement, sensory mechanisms, rheotaxis This thesis is focused on collective motions in schools of fish. In particular, we attempted to shed light on how variations in environmental parameters can influence interactions between individuals within a group, and consequently the organisation of that group. Using simplified laboratory experiments, we varied these environmental parameters in a controlled manner and observed the motions of groups of fish in response to these perturbations. The model species chosen for these experiments is Hemigrammus rhodostomus, a small gregarious tropical fish.

We were able to show that light plays a major role in the emergence of collective motion: in the absence of lighting, schools of around fish were unable to swim in a coordinated fashion. Conversely, as the light intensity gradually increases, the school goes through di erent behavioural phases, with an initially highly polarised organisation followed by a state of rotation (milling) that is stable over time above a light level threshold.

The limits of the swimming range also play a part in determining the collective state observed for a group of interacting agents. In the case of schools of to fish, it has been shown that reducing the available swimming area and increasing the number of fish in the school have equivalent e ects: the greater the confinement (measured by the density of fish per unit area), the greater the probability of moving from a polarised state to a rotating state.

Finally, the behaviour of rheotaxis was studied, i.e. the aptitude of a fish or a group of fish to swim against the current. In particular, the ability to maintain a stable position for single fish and pairs of fish, considered as the minimum constitutive cell of a school, was measured during forced swimming experiments in a swim tunnel. The tests were repeated in illuminated and dark environments. The results suggest that flow sensing is used by fish in unlit conditions, but is not essential in lit conditions.

In summary, this research underscores the role of light and spatial constraints in shaping collective behaviors but also sheds light on the interplay between environmental factors and group dynamics in schools of fish. These insights contribute Final Version as of February ,

et

  Jeanne et tant d'autres. . . -. Merci à la direction du PMMH, Damien Vandembroucq et à Philippe Petitjeans, de rendre possible cette dynamique de laboratoire par leur vision et leur action en faveur d'une recherche ouverte sur le monde et libre, et d'un gouvernance horizontale. Cette liberté serait un vain mot sans Frederique Auger et Claudette Barez, qui font, par leur travail de l'ombre, tourner la boutique. Merci à elles de m'avoir aidé à remplir sur ma route les quêtes annexes du travail administratif. Ma gratitude va aussi aux membres de mon jury, pour l'intérêt et le temps consacrés à mon travail; merci Aurélie, Fernando, Agnese, Olivier et Guy d'avoir mis votre expertise et votre expérience au service d'une relecture attentive de mon manuscrit; merci pour vos questions, qui m'ont ouvert des horizons aussi nombreux qu'inattendus. Finalement, je veux ici remercier mes parents et mes grands parents, parce qu'il m'ont transmis le goût, le respect et l'admiration pour le savoir qui m'ont mené jusqu'à cette thèse. À ma chère épouse, qui n'a eu de cesse de m'écouter me plaindre du Reviewer , ou des poissons qui ne font pas ce qu'on leur demande: elle a partagé toutes mes nombreuses joies et mes quelques tristesses de thésard et de mari -depuis le labo, jusqu'au bateau-. Merci ma Sev ! viii F O R E W O R D C oordinated motion of animals have always captivated the imagination of ob-

  an overview . . . . . . . . . . . . . . . . . . . . . . . . . Fields of study of collective motions . . . . . . . . . . . . . . . 1 S T A T E O F T H E A R T A N D C O N T E X T Et si je ne pus observer ni miralets, ni balistes, ni tétrodons, ni hippocampes, ni jouans, ni centrisques, ni blennies, ni surmulets, ni labres, ni éperlans, ni exocets, ni anchois, ni pagels, ni bogues, ni orphes, ni tous ces principaux représentants de l'ordre des pleuronectes, les limandes, les flez, les plies, les soles, les carrelets, communs à l'Atlantique et à la Méditerranée, il faut en accuser la vertigineuse vitesse qui emportait le Nautilus à travers ces eaux opulentes.

  Biology and zoology form the basis of research on collective motion. Scientists have traditionally approached this topic by studying the intriguing motions of animal populations. The flocks of migratory birds (the most studied systems by ethologists, who are interested in the behavior of animals in their natural environment. From a biological point of view, there is particular interest in the physiological nature of the interactions between individuals that lead to the emergence of synchronized motion at the scale of the group. Among numerous examples, the role of pheromones in the emergence of threads in ants, or the comparative role of vision, olfaction and the sensation of flow in fish (an interesting review of what the study of collective movements borrows from di erent branches of research on his Youtube Channel F ( ). His video has inspired the following summary.

  Figure . b). Like in the Vicsek model, the velocity of the agents is a constant. At each time step, the orientation is updated according to the following rules, where • denotes the normalization operator (x = x/|x|) :

  influential neighbours are those located in the adjacent cells of the Voronoi diagram (the set of points in the plane closer to the focal individual than to any other individual, see Figure . c) (C

Figure

  Figure . : a. Visual illustration of the governing equation of the Cucker-Smale model b. Zones of repulsion (zor), orientation (zoo) and alignment (zoa) around a fish, and representation of its blind angle α in C et al. ( ). c. A school of numerical fish from C et al. ( ). The areas depicted here are the cells of the Voronoi diagram whose seeds are the fish positions, i.e the are the set sets of points that are closest to a given fish than to any other. In red is the cell of the focal fish and in orange those of its Voronoi neighbours (individuals located in cells sharing a border with the focal fish's cell).

  milling states: in such structures, individuals move by rotating around the group's centre of mass, in two or three dimensions of space. The speed of each individual is perpendicular to the vector linking its position and the centre of mass.

Figure . :

 . Figure . : The three typical collective states exhibited by the Couzin numerical model in D, for self-propelled particles (from C et al. ( )). a. swarming, b. milling, c. polarized state.

Figure

  Figure . : Fossil of a school of juvenile fish (Erismatopterus levatus) found in the Green River Formation (Arizona, USA). The fossil is dated to around million years ago.

Figure

  Figure . : a. Original schematic from W ( ) showing the diamond-shaped school. Black dots represent the vortices in the wake of predecessor fish, with their rotation direction. b. Experimental snapshots from A et al. ( ) showing diamond-shaped schooling at low speed and the phalanx structure at high speed. The fish swim in a water flume with a flow from left to right. Flow speeds are given in fish Body Length (BL) per second. c. Spatial configuration of a pair of fish illustrating the voretx phase matching strategy: energy saving is possible if the two fish synchronize their tailbeat in-phase or out-of-phase, depending on the streamwise distance between the individual (L et al., ).

Figure

  Figure . : Lateral line system of the bitterling Rhodeus sericeus amarus. (adapted from B and Z ( ))

Figure . : a .

 a Figure . : a. Hemigrammus rhodostomus schooling in the lab b. Region of origin of Hemigrammus rhodostomus in the lower Amazon basin (from B , ). c. Hemigrammus rhodostomus schooling in their natural habitat (credit: Below Water YouTube channel)

Figure . :

 . Figure . : top. Typical discrete trajectory of a Hemigrammus rhodostomus swimming alone in a large tank for a few seconds. The color represents the instantaneous speed, given here in Body Length -BL-per second ( BL ∼ . cm). The acquisition rate is images per second. Note the sharp changes in direction followed

2

  I L L U M I N A N C E -T U N E D C O L L E C T I V E M O T I O N [...] and the melody of the song was the gray-green water and the little scuttling animals and the clouds of fish that flitted by and were gone. -The Pearl, John Steinbeck ( ) I n this chapter, we report experimental investigations on the role of illumination on the collective dynamics of a large school (ca. individuals) of Hemigrammus rhodostomus. The structure of the group is quantified while progressively altering the visual range of the fish through controlled cycles of ambient light inten-

(

  succession of propulsion and gliding phases): at fps, a time step represents on average a displacement of . ± . BL. A filter letting only infrared light pass through (IR-

Figure . :

 . Figure . : Groups of fish swim freely in a large, shallow tank where the light level can be adjusted using a video projector. The whole system is backlit by a custommade infrared LED panel, and the fish's trajectories are filmed by a overhanging camera that records only infrared wavelengths at frames per second (visible light is filtered so that its variations do not alter the lighting conditions of the videos).

Figure . :

 . Figure . : Data processing pipeline for a typical experiment on a school of fish

Figure . :

 : Figure . : Snapshot of fish swimming under an illuminance E of lx. The group motion is captured at fps with a high-resolution camera. From the videos, we reconstruct the individual trajectories of each animal ( D position and velocity at each time step). Zoom-in: previous positions ( s-period, grey dots) and current velocity vectors from tracking data. For the i-th fish, we denote v i the instantaneous velocity and r i the position with respect to the school current center of mass O. Additionally, we also quantify two intrinsic characteristic lengths of the fish school: the Nearest-Neighbour Distance (NN-D) and the Inter-Individual Distance (II-D). For a given individual, the NN-D is the distance to the closest fish and the II-D is the average distance to all the other fish. If we denote d ij the distance between the i-th and the j-th fish: II-D = d ij j∈1.. N i∈1.. N(. )

Figure C .

 C Figure C. displays the averaged results obtained over di erent cycles of the illuminance varying experiments (see Materials and Methods). The top graphic (Figure C. .A) shows the evolutions of both M and P with respect to normalized illuminance level Ē. The behaviour described for a single experiment in Figure . is found again in the average curves of Figure C. .A: while the milling parameter increases monotonically with illuminance, the polarization parameter peaks rapidly and decreases afterwards to a plateau.

Figure C .

 C Figure C. .B shows the Nearest-Neighbor Distance (NN-D) and the Inter-Individual Distance(II-D) in Body Lengths ( BL ≈ . cm). It is worth noting that in its averaged form, the II-D gives a good approximation of the characteristic size of the school. As can be observed, both characteristic lengths are large when there is no illumination: the average distance between the fish is about BL and the distance to the nearest neighbor is . BL at the most. This case corresponds to a swarming behavior, without cohesion in the group, as confirmed by . .A. The fish are distributed throughout the tank space and swim independently with respect to each other. These quantities rapidly decrease, showing that the individuals within the group get closer to each other as light intensity increases. II-D eventually saturates to a constant value above an illuminance threshold around E . . As can be observed, the average distance stabilizes around BL, while the NN-D increases progressively before stabilizing around . BL after E . , which shows that, within a group of a given size (quantified by the II-D), a better quality of the visual information lets these highly cohesive fish reorganize, finding more regular patterns leaving more space between themselves and their nearest neighbor. It is known that fish can exhibit changes in their behavior in experiments over time (S , ). We thus conducted an additional set of experiments at fixed illumination levels for one hour allowed to confirm the dynamical

Figure . :

 . Figure . : Fish school order parameters and distances with respect to light intensity. Solid lines show values averaged over every trials (N = 6) and every light ramps (either increasing or decreasing, for each trial), which represents replicates. (The shaded region is the confidence interval for the mean). A Polarization and milling parameter. B Nearest-Neighbour Distance (NN-D) and Inter-Individual Distance (II-D) in Body Lengths. For a given individual, the NN-D is the distance to the closest fish and the II-D is the average distance to all the other fish. Values displayed here are averaged over all individuals in the school. (The scales for NN-D and II-D are di erent)

Figure . :

 . Figure . : Comparison of experiments with continuous illuminance variation and experiments with constant illuminance for the milling and polarization parameters (M and P, top), and NN-D and II-D (bottom). Continuous data are the same as in Figure C. . The boxplots show the distribution over the replicates for each light level (dots represent the median of the distribution, and the error bars show the confidence interval).

Figure . :

 . Figure . : Example of turbulent jet image that could be used to compare the navigation ability of a school with mechanosensory cues from to lateral line and with vision (credit: E. Villermaux)

Figure . :

 . Figure . : Spatial density of presence of fish in heterogeneously lit tanks. The graphs on the right (resp. on the top) represent the distribution averaged in the leftright direction (resp. in the bottom-top direction).

Figure . :

 . Figure . : Examples of animal groups in milling configuration in the wild. a. A "death circle" of fire ants, b. A barracuda mill, c. A reindeer "vortex". The transition to milling state is widespread in nature and has been extensively described for a number of other species: marine worms (F et al., ), planktonic crustaceans (O , B , and M , ), sheeps...

  Figure . : Example of an experimental result for N = 50 fish schooling in a square tank of swimming area S = 1 m 2 . Snapshot of the school in (a.) milling configuration (M > 0.5) and in (b.) polarized configuration (P > 0.5). The photographs show the entire surface of the tank. (c.) The time signal of M and P shows clearly distinct periods of milling state followed by periods of polarized state of respective duration τ m and τ p . The colored bar on the top represents the current state of the school.

in

  Figure . : State transitions over min experiments for various school sizes N and swimming area S.

  Figure . c shows that the two states follow each other almost without interruption. Inspired by thermodynamics (Gand P ,), the bistability of the fish group can be described by a potential landscape: transitions take place when the fluctuations of the system are greater than the corresponding potential barrier.

Figure . :

 . Figure . : Phase diagram of the collective state transition in the parameter space (swimming area S, school size N). On the left (resp. on the left): fraction of experimental time spent in the milling state (resp. in the polarized state) by the school.Each dot represents at least repeated experiments of min each.

Figure . :

 . Figure . : Group size (average of the largest interindividual distances, in Body Lengths, BL) with respect to the number of fish in the school N. The colors represent the size of the swimming area S.

  Figure C. shows the time spent in either state plotted as a function of confinement density ρ. First, it corroborates the observations made on individual experiments, where time spent milling goes up with the density, and oppositely for time spent in the polarized state. Most strikingly Figure C. shows a clear collapse of the data of a total of experiments with di erent (N, S) combinations when plotted against the confinement density. For each confinement density (with the exception of and fish/m -2 ), experiments with at least two (N, S) combinations were carried out, to ensure that they yielded equivalent results if the N/S ratio was the same. This collapse clearly highlights that the proportion of time spent by the group of Rummy-nose tetras in either state is solely a function of the confinement density, rather than of the number of fish alone, as suggested by previous studies on golden shiners (T et al., ).

  density ρ [fish.m -2 ]

Figure . :

 . Figure . : Fraction of time spend by fish schools in the milling state (a.) and in the polarized state (b.) over total experiment duration, with respect to confinement density, i.e the number of fish per unit area of the tank in fish.m -2 . The size of the markers indicates the area of the tank, while the colors show the number of fish in the school.

  experimentally measured the distributions of the time spent in the milling and polarized states. The results, reported in Figure C. a-b show an exemple of the empirical distribution of the durations between state transitions, for a confinement density at which fish spend approximately the same time in the milling and polarized states (ρ = 30 fish/m 2 , for N = 30 and S = 1 m 2 ). At low densities, where transitions are frequent and the experimental statistics is therefore good, fitting an exponential distribution to the data shows good agreement. It should be noted that as density rises, and milling bouts become very long, the number of transitions falls sharply, and the number of bouts observed is much lower. For these high densities, the fit quality deteriorates, likely due to the insu cient statistics behind the experimental distribution. From the fit parameters we can directly obtain the mean time to transition for each state, with τ m = β -1 and τ p = α -1 . The good agreement demonstrated by the distributions in Figure C. a-b, which closely fit toan exponential distribution is the hallmark of a memoryless process and suggests that the school operates as a system near a pseudo-critical point (

  Figure C. c, with respect to the confinement density. We see that the stability of the milling state increases with density, while that of the polarised state decreases.

Figure . :

 . Figure . : Loglog plot of the probability density function of the duration of milling state τ m (a.) and polarized state τ p (b.) for a confinement density ρ = 30 fish/m 2 (N = 30, S = 1 m 2 , replicates, transition events). Small dots are generated by di erentiating the complementary cumulative density function obtained from sorting the transition times, they do not depend on the binning. Large points are the histogram obtained by binning data into linearly spaced time intervals. Dashed lines show the best exponential law fit with parameter α = τ m -1 and β = τ p -1 (c) Transition rates with respect to confinement density. Error bars represent the standard deviation over all replicates of experiments at a given density.

Figure

  Figure . : E ect of a variation of the aspect ratio AR of the tank, from AR = 1 to AR = 10. The height of the swimming area remains fixed at m while its width changes: (a.) m (b.) . m (c.) . m (d.) . m. Panel (e.) shows an experiment in a . m × . m (AR = 1). Note that the milling motion is totally suppressed for AR > 2.

  size and share a gregarious nature and a burst and coast swimming gait. In these experiments, the images were captured at frames per second for a duration of one minute. The results presented here are exploratory and open the door to the question of the role of tank shape in confinement transitions.

Figure . :

 . Figure . : Images of experiments in a circular arena with fish (a.) and fish (b.)

Figure . :

 . Figure . : Average values of milling parameter M and polarization parameter P for experiments in a circular tank, with respect to confinement density ρ. The error bars show the standard deviation of the distributions of M and P for each experiment.

  is achieved by the interaction of di erent sensory mechanisms: the lateral line, especially for the slowest flow speeds (B some cases it has also been suggested that the vestibular system (responsible for balance) and the sense of touch are also used by fish during rheotaxis (behaviour is robust and is maintained in the absence of one or more of the sensory mechanisms mentioned above. For example, it has been shown experimentally that rheotaxis functions without visual cues, in complete darkness(B -C et al., ). Similarly, fish whose lateral line has been chemically deactivated can maintain this type of behaviour (investigated the role of environmental disturbances on the rheotactic abilities of di erent fish species. For example, the turbulent or nonturbulent nature of the flow to which fish are exposed can, depending on the case, However, no study to date has focused specifically on understanding the influence of collective behaviour on rheotaxis and station keeping. In this chapter, we compare the rheotactic performance of fish swimming alone and of fish pairs, considered as the minimum constitutive cell of a school. which one or more Hemigrammus rhodostomus can be subjected to a chosen flow velocity. The swim tunnel and its constituents are shown on Figure C. . It consists of a large x cm glass tank filled with water at a controlled temperature ( • C ± • C). The tunnel itself consists in a secondary tank placed in this large tank. It is a closed-loop in transparent PMMA (plexiglass), with

  Figure . : Schematic of the experimental setup (swim tunnel only, the large main tank is not represented). Tu blue arrows show the direction of the water current.

Figure . :

 . Figure . : Examples of raw images obtained during swimming experiments. left N

Figure . .

 . Figure . . This process is repeated for each time step of the experiment.

Figure . :

 . Figure . : Example of the Voronoi diagram of a set of D points. The cyan dots are the seeds of the tilling, the black lines show the limits of the Voronoi cells (points that are closer from a given seed than to any other seed). Dashed lines represent borders that have infinite length.

Figure . :

 . Figure . : Midline deformation over time for a single fish swimming at . BL/s (no light).

Figure . :

 . Figure . : Illustration of the numerical method used to extract the tailbeat frequency and amplitude (experimental parameters here: U . BL/s, no light). top. Raw signal of the tail tip displacement perpendicular to the fish global direction. bottom. Continuous Wavlet Transform (CWT) of the tail displacement signal. The black line in this graph follows the maximum amplitude of the CWT, providing both the tailbeat frequency and amplitude with respect to time.

Figure . :

 . Figure . : Timeseries of the displacement of a fish in the direction parallel to the incoming flow (x) and perpendicular to the flow (y). r represents the distance to the initial position (r = x 2 = y 2 )

Figure . :

 . Figure . : Temporal evolution of the position of a fish (the position of the nose is represented here, the fish body is shown by the gray silhouette) in the swim tunnel (U . BL/s, light on). The black rectangle represents the swimming section limits, and the inset is a zoom on the red area. The flow in the swimming section is from left to right.

Figure

  Figure . : ( : light o -: light on) Tailbeat amplitude A tail in fraction of the Body Length (BL) with respect to the incoming flow velocity U, for N (left) and N (right).

Figure

  Figure . : ( : light o -: light on) Tailbeat frequency in Hz with respect to the incoming flow velocity U, for N (left) and N (right).

Figure

  Figure . : ( : light o -: light on) Strouhal number of swimming fish with respect to flow speed U.

  Figure . : Positional variability a (standard deviation of the distance to initial position r) with respect to the lighting of the swimming tank, for three di erent flow velocity regimes. The left column (resp. the right column) show the data for N fish (resp. for N fish). p-values of the Welch T-Test for the means: (ns) non-significant, ( ) p . , ( ) p . . The numbers on the right of the boxplots are the sample size.

Figure . shows the

  Figure.shows the decomposition of d in streamwise d x and spanwise d y contributions. This last graphs reveals that the variations of d are exclusively due to variations of the streamwise distance between fish, while the spanwise distance remains constant between lit or unlit conditions. This means that in non-illuminated cases, the fish are placed one behind the other and form a line in the streamwise direction.

Figure . :

 . Figure . : Distance between fish for experiments with N , with respect to lighting conditions. p-values of the Welch T-Test for the means: ( ) p . .

  teraction rules. By modulating these parameters in a numerical simulation, we can relate light intensity and physiological characteristics indirectly. Such simulations can also confirm the results obtained concerning collective motion in confinement, to o er a generalisation beyond fish schools. A P P E N D I X This appendix is divided into two sections: Appendix A provides details on the design of the swim tunnel and the characterisation of the flow in the test section of this tunnel by PIV. Appendix B reports additional direct measurements of the cost of transport of schooling fish using respirometry. the characterisation of the flow through the test section of the swim tunnel presented in Chapter . The Particle Image Velocitmetry (PIV) method is used to measure this flow. The principle of PIV is the following: a flow is seeded with plastic particles (polyamide in our case) of small size (typically to microns in water). The fluid is illuminated by a thin laser sheet, whose light is reflected by the particles. The size of the particles ensures that they follow the flow without disturbing it, while preventing them from settling. By recording with a video camera the light reflections created by the particles over time, an image correlation algorithm can be used to reconstruct a time-dependent velocity field in the plane of the laser sheet. In practice, the algorithm divides all the images into interrogation windows, in the present case of pixels by pixels. It compares two successive images by estimating, for each window, what is its most probable position in the next image.

Figure

  Figure A. : Time-averaged flow in the test section, for di erent voltages input to the motor ( , , and V), obtain with PIV measurements. The flow is going from right to left (arrows). left. Norm of the flow velocity |u|. right. Norm of the flow vorticity |∇ × u|.

Figure A .Figure

 A Figure A. shows the velocity norm |u| and the vorticity |∇ × u|, time-averaged over the whole duration of the recording ( s), for h cm. The flow is going from right to left (arrows in Figure A. ). Firstly, we note that the velocity is highly homogeneous in the test section, particularly in the streamwise x direction. In the spanwise y direction, the walls cause a boundary layer to develop, resulting in a lower velocity near them. This boundary layer induces shear in this area of the flow, which results in higher vorticity values, especially as the mean velocity U increases. Figure A. shows the profil of the norm of the velocity in the y-direction, averaged over time and over the streamwise direction x.

  Figure A. : Turbulent intensity in the test section of the swim tunnel, with respect to the input voltage of the motor.

Figure

  Figure A. : Flow speed in the test section (in cm/s or in Body Lengths BL/s) with respect to the motor input voltage, for the di erent depth h (height at wich the laser sheet is placed).

.

  This last photograph(Figure A. ) shows the swimming channel as a whole, to help the reader visualise the experimental setup as a whole.

Figure

  Figure A. : Global view of the swim tunnel experimental setup

  for individuals to reduce their cost of transport (COT), i.e. the energetic cost associated with their displacements. Recent studies (A et al., ) have indirectly measured (via tailbeat frequency) a reduction of the COT for groups of Hemigrammus rhodostomus forced to swim in a swimming canal, as a function of the number of individuals in the school. However, there is no systematic study directly measuring the energy consumption of individuals as a function of the size N of the group. The only experimental means of directly measuring COT is to assess the metabolic rate of the school, i.e. the oxygen consumed per unit of time. This method is called respirometry. In this section, we report experimental results on the measurement of metabolic rate of schools of sand smelts (Atherina presbyter, see Figure B. ) comprising N to individuals in forced swimming, for di erent flow speeds. These experiments were carried out during a collaborative exchange in the summer of in Valentina di Santo's team (Stockholm University).

Figure

  Figure B. : A specimen of sand smelt (Atherina presbyter)

Figure

  Figure B. . For each pair of values (N, U), replicates of the same experiment are carried out.In all cases, we ensure that the percentage of dissolved dioxygen in the water (the ratio between the measured concentration and the maximum concentration at • C, i.e . mg/L) is always greater than , in order to prevent any risk of hypoxia for the fish.

Figure

  Figure B. (top panel) shows a timeseries of the dioxygen consummed by this fish (here in mg/fish) for the different flow speeds (the recovery periods are removed from the signal). To measure the temporal rate of change of this quantity (metabolic rate MO 2 ), we assume that, for each swimming period of min at a given flow speed U, the dioxygen consumption is an a ne function of the time t. We thus extract the slope of this function with a standard least square method. The value of this slope (converted to the usual unit system) is the average metabolic rate (shown in Figure B. , bottom panel).

  previous paragraph; it is plotted in Figure B. . It should be noted that these values are not negligible, as the typical consumption of a single fish is of the order of mg O 2 /h, the background metabolic rate can thus correspond to the dioxygen consumption of approximately to fish depending on the value of U. These values of background rate are thus systematically subtracted from the measured values in each swimming experiment, for the corresponding value of U.

Figure]]

  Figure B. : top. Dioxygen consumption per individual in the tank for N fish as a function of time, for di erent flow speeds U represented by the colors. The recovery periods are removed from this time signal. bottom. Average metabolic rate with respect to U.

Figure

  Figure B. shows the respirometry data as a function of flow velocity U. The graph on the left confirms that at velocities U greater than BL/s, the metabolic rate is much higher for N than for any other number of fish (the increase is on average .for this range of flow velocities). In the graph on the right, where the data are averaged for all values of N, we find the classic U-shaped metabolism-speed curve phenomenon, previously reported in the literature (D S , K , and

  Figure B. : Metabolism-speed curves: metabolic rate MO 2 with respect to flow speed U. left. MO 2 for the di erent values of the number of fish N, color-coded. right. MO 2 averaged over all the di erent values of N. The errorbars show the standard deviation over the di erent N. All the data is averaged over the replicates.

Figure C .

 C Figure C. : Des groupes de poissons nagent librement dans un grand bassin peu profond dont le niveau de lumière peut être réglé à l'aide d'un vidéoprojecteur. L'ensemble est rétro-éclairé par un panneau LED infrarouge fabriqué sur mesure, et les trajectoires des poissons sont filmées par une caméra en surplomb.

Figure C .

 C Figure C. : Paramètres d'ordre des bancs de poissons et distances en fonction de l'intensité lumineuse. B Distance entre les plus proches voisins (NN-D) et distance interindividuelle (II-D) en longeur de poisson (Body Length BL). Pour un individu donné, la NN-D est la distance avec le poisson le plus proche et la II-D est la distance moyenne avec tous les autres poissons.

Figure C .

 C Figure C. : Fraction du temps passé par les bancs de poissons dans l'état de milling (a.) et dans l'état polarisé (b.) sur la durée totale de l'expérience ( min), en fonction de la densité de confinement, c'est-à-dire le nombre de poissons par unité de surface de l'aquarium en poissons/m -2 . La taille des marqueurs indique la surface de nage disponible, tandis que les couleurs indiquent le nombre de poissons dans le banc.

Figure C .

 C Figure C. : Fonction de densité de probabilité de la durée de l'état de milling τ m (a.) et de l'état polarisé τ p (b.) pour une densité de confinement ρ = 30 poissons/m 2 (N = 30, S = 1 m 2 , réplications, événements de transition). (c.) Taux de transition en fonction de la densité de confinement. Les barres d'erreur représentent l'écart-type sur toutes les répétitions d'expériences à une densité donnée.

  These changes in the environment can alter the nature or intensity of interactions between individuals: for example, variations in luminosity change the ability of individuals to see their neighbours. Another type of parameter that can trigger changes the in collective state are environmental constraints, such as the presence of obsta-

	cles or predators in the case of groups of animals (K		and R	,	;
	R	and O ,	). In some cases, intrinsic noise in the orientation of the in-
	dividual can also lead to the emergence of collective motion or to transitions be-
	tween states (B	, D	, and M K	,	; C	et al.,	; J
	et al.,
	Fish provide a typical example of self-organization, with a natural tendency to form
	ordered groups, known as swarms or schools (S	,	). More than	of fish
	species exhibit schooling motion during their lifetime (P	, G	, and
	H	,	), and this type of behaviour is a strategy that has been adopted for
	millions of years, as shown by fossils found in Arizona (see M	, M	, and
	P	,	and Figure

). Finally, the number of individuals in the group is knows to trigger collective state transitions in both live (T et al., ) and numerically simulated systems (C and R , ). Even if, as mentioned in Section . . , there are considerable di erences between thermodynamic systems and groups in collective motion, this parallel opens interesting avenues for studying complex biological systems by providing powerful pre-existing theoretical tools. . . . Advantages of schooling

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  located in one of the corners of the observation tank. For this reason, we have chosen not to quantitatively study the swarming state in this case, and will consider only the milling and polarized states. This absence of swarming can be explained by the nature of the illumination to which the fish are subjected: we

		The swarming state, where the school is disorgan-
	ised and orientations are isotropic, (M and P both small) reported by previous
	studies in similar setups (B	et al.,	; T	et al.,	) is nearly not ob-

served in our experimental conditions, and tends to only appear when the group . of fish is have observed in Chapter that above a certain luminosity threshold, schools of Hemigrammus rhodostomus are almost exclusively in polarized or milling states, with swarming only occurring below this limit. The threshold, identified at lux for a group of fish swimming in a . m 2 tank, is largely exceeded here.

  The oxygen sensor measures the O 2 concentration c O 2 in mg of O 2 per liter of water, every second (dt s). Its measurement range is to mg/L, with a precison of . mg/L. Since the exact volume of water in the tank is V . L, we get the quantity of O 2 consummed at time t in mg, m O 2 (t), with:

of fish, whatever the lighting conditions. Additional direct measurements of cost of transport for schools of di erent sizes are reported in Appendix B.

.

To complete the results presented here, it might be useful to repeat the experiments with a larger number of fish in order to assess whether the trends observed for positional variability a are maintained for larger groups.

One of the pitfalls of this experiment is that the fish may be aware of the structure and size of the channel, as well as the presence of neighbours, even in unlit conditions. Indeed, the individuals always have the possibility of obtaining visual information during their transfer into the tank. To solve this problem, it is possible to use blind fish, such as Astyanax mexicanus. This tetra, native from eastern Mexico, is a fresh water fish that regroup multiple populations, some of which evolved to completely lose vision depending on their habitat (surface or caves).