Determining individual-level interactions that govern highly coordinated motion in animal groups or cellular aggregates has been a long-standing challenge, central to understanding the mechanisms and evolution of collective behavior. Numerous models have been proposed, many of which display realistic-looking dynamics, but nonetheless rely on untested assumptions about how individuals integrate information to guide movement. Here we infer behavioral rules directly from experimental data. We begin by analyzing trajectories of golden shiners (Notemigonus crysoleucas) swimming in two-fish and three-fish shoals to map the mean effective forces as a function of fish positions and velocities. Speeding and turning responses are dynamically modulated and clearly delineated. Speed regulation is a dominant component of how fish interact, and changes in speed are transmitted to those both behind and ahead. Alignment emerges from attraction and repulsion, and fish tend to copy directional changes made by those ahead. We find no evidence for explicit matching of body orientation. By comparing data from two-fish and three-fish shoals, we challenge the standard assumption, ubiquitous in physics-inspired models of collective behavior, that individual motion results from averaging responses to each neighbor considered separately; three-body interactions make a substantial contribution to fish dynamics. However, pairwise interactions qualitatively capture the correct spatial interaction structure in small groups, and this structure persists in larger groups of 10 and 30 fish. The interactions revealed here may help account for the rapid changes in speed and direction that enable real animal groups to stay cohesive and amplify important social information.A fundamental problem in a wide range of biological disciplines is understanding how functional complexity at a macroscopic scale (such as the functioning of a biological tissue) results from the actions and interactions among the individual components (such as the cells forming the tissue). Animal groups such as bird flocks, fish schools, and insect swarms frequently exhibit complex and coordinated collective behaviors and present unrivaled opportunities to link the behavior of individuals with dynamic group-level properties. With the advent of tracking technologies such as computer vision and global positioning systems, group behavior can be reduced to a set of trajectories in space and time. Consequently, in principle, it is possible to deduce the individual interaction rules starting from the observed kinematics. However, calculating interindividual interactions from trajectories means solving a fundamental inverse problem that appears universally in many-body systems. In general, such problems are very hard to solve and, even if they can be solved, their solution is often not unique.To avoid solving these inverse problems (and because detailed kinematic data were not available until recently), many attempts have been made to replicate the patterns observed in animal groups by...

The spontaneous emergence of pattern formation is ubiquitous in nature, often arising as a collective phenomenon from interactions among a large number of individual constituents or sub-systems. Understanding, and controlling, collective behavior is dependent on determining the low-level dynamical principles from which spatial and temporal patterns emerge; a key question is whether different group-level patterns result from all components of a system responding to the same external factor, individual components changing behavior but in a distributed self-organized way, or whether multiple collective states co-exist for the same individual behaviors. Using schooling fish (golden shiners, in groups of 30 to 300 fish) as a model system, we demonstrate that collective motion can be effectively mapped onto a set of order parameters describing the macroscopic group structure, revealing the existence of at least three dynamically-stable collective states; swarm, milling and polarized groups. Swarms are characterized by slow individual motion and a relatively dense, disordered structure. Increasing swim speed is associated with a transition to one of two locally-ordered states, milling or highly-mobile polarized groups. The stability of the discrete collective behaviors exhibited by a group depends on the number of group members. Transitions between states are influenced by both external (boundary-driven) and internal (changing motion of group members) factors. Whereas transitions between locally-disordered and locally-ordered group states are speed dependent, analysis of local and global properties of groups suggests that, congruent with theory, milling and polarized states co-exist in a bistable regime with transitions largely driven by perturbations. Our study allows us to relate theoretical and empirical understanding of animal group behavior and emphasizes dynamic changes in the structure of such groups.

8In this paper we introduce a method for determining local interaction rules 9 in animal swarms. The method is based on the assumption that the behavior 10 of individuals in a swarm can be treated as a set of mechanistic rules. 11The principal idea behind the technique is to vary parameters that define a 12 set of hypothetical interactions to minimize the deviation between the forces 13 estimated from observed animal trajectories and the forces resulting from the 14 assumed rule set. We demonstrate the method by reconstructing the interac-15 tion rules from the trajectories produced by a computer simulation. 16 Key words: swarming, behavioral rules, reverse engineering, force match-17 ing. 18 1 The collective motion of living organisms, as manifested by flocking birds, 19 schooling fish, or swarming insects, presents a captivating phenomenon believed 20 to emerge mainly from local interactions between individual group members. In 21 part, the study of swarming and flocking aims to understand how animals use vi-22 sual, audial and other cues to orient themselves with respect to the swarm of which 23 they are part, and how the properties of the swarm as a whole depend on the be-24 haviors of the individual animals. Also when addressing evolutionary questions of 25 behaviour in swarms and flocks, such as the selective advantage of being bold or 26 shy in response to a predator, it is important to understand how the individuals be-27 have based on the relation to their neighbours in the swarm or flock. For example, 28if the question is "If the peripheral of the flock is more exposed to predators, do 29 some individuals cheat the others by staying at the center of the flock where they 30 are more protected?", knowing the effective rules would make it easier to address 31 questions regarding the evolutionary stability of the altruistic behavior. 32Because flocks cannot be understood by studying individuals in isolation, and 33 are difficult to conduct controlled experiments on, understanding the behavioural 34 patterns underlying flocking and swarming is especially challenging. Consequently, 35 collective behavior has been extensively modeled particularly using the agent-based 36 modeling framework, where simple mechanistic behavioral rules are used to gener-37 ate qualitatively realistic swarming behavior, (e.g Aoki, 1982; Reynolds, 1987; Vic-38 sek et al. Yates et al., 2009). The rules usually comprise three kinds of forces: A short-range 42 2 force to avoid collisions with obstacles or other animals; a force adjusting the veloc-43 ity to fit nearby individuals' velocities; and a force for avoiding being alone, e.g. by 44 moving towards the average position of the nearby individuals. However, see e.g. 45 Romanczuk et al. (2009) for an alternative formulation. In addition, drag forces 46 and noise are used to model the medium through which the individuals move, and 47 external forces can be used to model interactions with terrain or predators. 48 The main strength of the agent-based modeling framewor...

Understanding the organization of collective motion in biological systems is an ongoing challenge. In this paper we consider a minimal model of self-propelled particles with variable speed. Inspired by experimental data from schooling fish, we introduce a power-law dependency of the speed of each particle on the degree of polarization order in its neighborhood. We derive analytically a coarse-grained continuous approximation for this model and find that, while the specific variable speed rule used does not change the details of the ordering transition leading to collective motion, it induces an inverse power-law correlation between the speed or the local polarization order and the local density. Using numerical simulations, we verify the range of validity of this continuous description and explore regimes beyond it. We discover, in disordered states close to the transition, a phase-segregated regime where most particles cluster into almost static groups surrounded by isolated high-speed particles. We argue that the mechanism responsible for this regime could be present in a wide range of collective motion dynamics.