The Dance Performance of Fish schools
Thanks to the development of new tracking techniques and behavioural data analysis, scientists have managed to reconstruct and model the social interactions between fish in a shoal. These interactions govern their collective movements.
How can hundreds of fish or birds move together in a coordinated way and behave as if they were the same single superorganism? Until recently, this was one of the most captivating mysteries observable in nature. Collective motion can be found at all levels of organization in living systems, be it in colonies of bacteria, swarms of insects, shoals of fish, flocks of birds or even herds of ungulates. All these systems comprise a large number of individuals whose interactions generate complex collective dynamics. For a dozen years or so, a growing number of scientists from a variety of fields such as ethology and statistical physics have been combining their methods to decode the mechanisms governing these phenomena. Thus, by analysing and modelling behaviours at the individual and collective scales, ethologists have been able to identify the social interactions at work in coordinating the movements of individuals and in sharing information within a group. For their part, physicists are increasingly looking at “active matter”. This term is used to denote physical objects or living organisms which expend energy to move and adopt various forms of collective motion. The goal of “active matter” physicists is to identify the universal properties of these phenomena and from them, deduce general laws.
For a long time, studies of the collective motion of groups of animals relied mainly on building and analysing mathematical models. But these studies had often no direct link with experimental observations. Why? Because it was difficult to obtain accurate data and model these phenomena. The first notable model, the Aoki-Couzin model, the initial version of which was introduced in the 1980s by the Japanese biologist Ichiro Aoki from Japan’s Tokyo University, included individual behavioural rules of avoidance, alignment and attraction, and reproduced some of the characteristics of collective movements in fish schools (see (1) and here). This model postulates the existence of three concentric and increasingly large zones around one individual. The presence of other fish in each of these zones determines the behaviour of this one individual. The first, in the immediate environment of the fish, is the repulsion zone; when other fish penetrate this zone, the individual moves away by changing direction. The second is known as the alignment zone. This is because the fish aligns itself with the average direction followed by all the fish in this zone. Finally, the last zone is the zone of attraction. The individual moves towards the average position occupied by the fish in this zone. In this model, the behavioural rule that has the highest priority is the rule of repulsion, as it enables collisions between fish to be avoided.
RADIUS VALUE
Numerical simulations of this model show that by maintaining the radius values of the attraction and repulsion zones constant, and by simply changing the value of the radius of the alignment zone, very significant qualitative changes in the types of collective movements of a school of fish are obtained. So, if this value is zero, the alignment zone disappears; the fish attract over a long distance and repulse each other when they are too close, and a disordered collective movement known as “swarming” can be observed. When the radius of the alignment zone increases, circular collective movements appear, and the fish swim in a circle around an empty central area (“milling” phase). Finally, when the radius of the alignment zone exceeds a certain threshold, the school is highly polarised; all the fish align and move in a common direction. Numerous other phenomenological models developed over the last decades (especially in the 1980s-1990s) and based on specific rules, can qualitatively reproduce the collective movements of schools of fish that resemble those observed in nature. However, the main problem remains the lack of experimental validation of the hypotheses these models are based on, and particularly those concerning the behavioural rules and interactions at the individual scale. In fact, very different rules can produce similar collective dynamics.
In rummy-nose tetras, the reactions of an individual to his neighbor result from a combination of alignment and attraction (and repulsion at short distance). These interactions have been measured and depend on the distance between the two fish and their relative position (angle of view and orientation). © G. Theraulaz - C. Sire - V. Lecheval / CNRS
However, for a few years now, new methods based on learning algorithms have enabled automated tracking of the trajectories and behaviour of animals moving in groups. These techniques have been used to analyse the collective movements of swarms of midges, schools of fish, flocks of starlings, herds of sheep or crowds of humans. They have improved the accuracy of the available data on social interactions and their effects on the behaviour of individuals, thus opening the way to build models of collective movements that are both quantitative and predictive. The main difficulty in building these models lies in the close entanglement of the interactions between the animal and its physical environment (i.e. physical obstacles), and also its social environment (its congeners).
This problem can be resolved by adopting a method that consists firstly in using experimental data to model the movement of an isolated individual, that meets no physical obstacles on its path. Then, this model is used to measure the effects on the individual’s movements of its interactions with obstacles or with another individual present in its neighbourhood. To do so, the trajectories predicted by the model are compared with those observed experimentally. This incremental method can be used to reconstruct and faithfully model the form of interactions between the animal and its physical and social environment. The model of individual movement including these interactions can then be validated using comparison (and in particular the quantitative agreement obtained) of the model’s predictions with the data collected from various experimental conditions, including experiments that were not used to build the model.
INTERMITTENT SWIMMING MODE
We successfully applied this method to analyse and model the movement and the social interactions of a small tropical fish measuring around 3 cm, the rummy-nose tetra (Hemigrammus rhodostomus), living in streams in the lower Amazon basin. This species swims in a highly synchronised and polarised manner, and it alternates phases of bursts in which the fish accelerates and changes direction, with gliding phases. This intermittent mode of swimming makes it possible to analyse a trajectory as a series of discrete behavioural decisions in time and space. By accumulating several tens of hours of data on fish swimming alone or in pairs in aquariums, we characterised the spontaneous movement of a fish, identified the potential stimuli it reacts to and measured their effects on its behavioural response.
In this species, an individual’s interactions with an obstacle (the wall of a tank) and with another fish trigger four main behaviours. The first is obstacle avoidance behaviour, the effect of which appears when the distance is less than two body lengths (about 6 centimetres). Then repulsion can be seen between close fish, when the distance separating them is less than one body-length. Moreover, attraction between far-off fish is maximal when the distance between fish is of the order of six to seven body lengths. Finally, an alignment behaviour with the direction of swimming of another fish appears; this is maximum when the distance between the fish is of the order of three body lengths. Unlike the arbitrary hypotheses of the above-mentioned phenomenological models, these behaviours are not confined to strict zones around the fish whose distances are fixed without experimental justification. Analysis of the experimental data shows that a fish’s reactions to its neighbour are a result of a continuous combination of attraction and alignment, the effects of which depend on the distance between the fish. More simply put, alignment takes priority over attraction when the fish are close (less than 2½ body lengths), whereas attraction dominates alignment when the distance between the fish is greater than this value.
Furthermore, the behavioural responses are considerably modulated by the perception the fish have of their environment, a phenomenon known as perception anisotropy. Thus, the effect of repulsion from the wall is maximum when the orientation of the fish is 45 degrees to the edge of the tank and minimum when it is parallel to the edge, which leads this species to swim near the edge of the aquarium. If the fish doesn’t react when parallel to the edge of the aquarium, it has a greater chance of maintaining this direction. In humans, this minimum repulsion reaction is found when the obstacle is behind the individual. Similarly, alignment is maximum when the other individual is located to the front left or front right of the fish being monitored and gradually vanishes when it moves behind it. Fish therefore interact strongly with fish located in front of them and only weakly with those behind.
The social interactions of repulsion (over short distances), attraction (longer distances) and alignment between fish may seem similar to the forces at play in the world of physics. For example, the van der Waals forces describing electrostatic interactions between atoms or molecules are repulsive over short distances and attractive at longer distances. Similarly, interactions between some magnetic atoms tend to align their spin (*), like compasses that align themselves with each other. It seems normal therefore to wonder whether the similarity between a school of aligned fish and a magnet in which the atomic spins line up collectively in a common direction is just qualitative or based on a deeper analogy.
Fundamental differences exist between social interactions and physical forces. As we have seen, the influence of a fish on another fish depends on the direction it is moving in, and especially the “angle of vision” with which it perceives the other fish or an obstacle. Also, the intensity of social interactions between two fish depends on the direction of their movement with respect to each other. This perception anisotropy leads to an asymmetry of the “social force” exerted on fish A by fish B and that exerted on fish B by fish A. Inversely, physical forces between two particles, such as gravity, magnetic or electrostatic forces (resulting in the van der Waals force), do not depend on the speed of the particles but simply on the distance separating them. A physical force exerted by particle A on particle B is exactly opposite to that exerted on B by A, a property known as Newton’s law of action and reaction. In physics, the notion of energy and its conservation in an isolated system derives directly from this law of action and reaction. The asymmetry in social interactions, due to the perception anisotropy fish (and most animals) have of their environment, breaks Newton’s law of action and reaction and prevents the definition of a “social energy”, thus complicating the quantitative measurement of social interactions.
PHASE TRANSITIONS
This asymmetry also has a decisive impact on the kind of collective movements that emerge on the scale of a school of fish (for example, it helps stabilise the “milling” phase) and more generally, a group of animals.
There is however a formal analogy between the existence of disordered and ordered states in schools of fish and that of similar states or phases in the inert world of physics: disordered liquid, ordered crystal solid; magnets corresponding to the alignment of atomic spins on a macroscopic scale, and which become disordered and lose their magnetic properties at high temperature.
These systems display phase transitions, which are changes between the various states or phases. In particular, ordered states are characterised by a notion of order at the scale of the whole system which can be quantified by an order parameter, for example measuring the quality of the alignment/polarisation of the school of fish or the magnetisation/polarisation of the atomic spins in a magnet. For a magnet, this order parameter is maximum at low temperature and decreases when the temperature rises, until it vanishes at a certain critical temperature which depends on the material (respectively 1115°C and 354°C for cobalt and nickel), above which it loses its magnetisation. For a school of fish, polarisation in an ordered school falls with the intensity of the alignment interaction and vanishes below a critical value for this intensity. Then the school becomes disordered, being in a state known as the “swarming” phase. In practice, we have shown that the intensity of the alignment interaction increases with the speed of the fish, and a school can hence switch from the swarming to the polarised phase (“magnetised”) just by increasing its speed, and it should be noted, without the need for an “instruction” given by a hypothetical leader fish. Using a model built from experiments on Kuhlia mugil, a fish that lives along the coasts of the island of La Réunion, we showed that the three phases observed in real fish schools (i.e. swarming, schooling and milling) were indeed present when the intensities of the alignment and attraction interactions were varied.
These three phases are presented in a “phase diagram” according to these two intensities, in perfect analogy with the phase diagram for a material displaying the gaseous, liquid and solid phases, according to temperature and pressure (Fig. 1). Moreover, the typical intensity parameters for Kuhlia mugil were measured close to the point where the three phases meet, i.e. close to the critical lines (see inset). This phenomenon, which seems to occur in numerous animal species that move in groups (starlings, midges, etc.) apparently enable the group to adapt their collective behaviour more effectively to changes occurring in their environment: presence of food, prey, or predators!
By combining their methods and tools to understand the complex collective phenomena in animal societies, ethologists and physicists have not only enriched their own field of research, but they have also raised new common questions about the living world.
(*) Spin is a quantum property of elementary particles and atoms, an intrinsic form of angular momentum associated with a magnetic moment.
FIG. 1 - THREE PHASE OF COLLECTIVE MOVEMENTS
The interactions of attraction and alignment between fish within a bench produce different forms of collective movements. Depending on the respective intensity of these interactions, the movements may be disordered, so-called "swarming" (1), completely aligned (3), or form circular structures (2). When the interaction parameters are close to a critical line, a small change in alignment or attraction intensity suffices to move from one state to another. This explains how schools of fish can change shape so quickly.
CRITICALITY IN PHYSICAL AND BIOLOGICAL SYSTEMS
At the critical point (or on the critical line) separating two phases where an order parameter (for instance, quantifying polarisation in a magnet or a fish school) cancels out, the system (physical or biological) is said to be “critical” and the dynamic of its components is correlated at the scale of the whole system. A critical system is then characterised by very strong fluctuations in its order parameter; the direction of magnetisation for a magnet or the direction of swimming for a school of fish, fluctuate rapidly over time. Due to these strong correlations at the system scale, one can show that a critical system is sensitive to and responds extremely to the slightest disturbance; an infinitesimal magnetic field can strongly repolarise magnetic material; the behaviour of a small number of fish (who may have detected the presence of a predator) can very rapidly change the collective movement of a school from one phase to another. In the biological context, the fact that the natural parameters of interaction are close to a critical point or line is of crucial importance for a school of fish, or for any group of animals, as it enables it to efficiently change its behaviour according to its environment.
> AUTHORS
Guy Theraulaz
Ethologist
Guy Theraulaz is a research director at the CNRS.
Valentin Lecheval
Ethologist
Valentin Lecheval is a post-doctoral student.
Clément Sire
Physicist
Clément Sire is a CNRS research director at the at the laboratory of theoretical physics in Toulouse.
