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Flocking systems are known to be strongly out of equilibrium. Energy input occurs at the individual level to ensure self-propulsion, and the individual motility in turn contributes to ordering, enhancing information propagation and strengthening collective motion. However, even beyond ordering, a crucial feature of natural aggregations is response. How, then, do off-equilibrium features affect the response of the system? In this work, we consider a minimal model of flocking and investigate response behavior under directional perturbations. We show that equilibrium dynamical fluctuation-dissipation relations between response and correlations are violated, both at the local and at the global level. The amount of violation peaks at the ordering transition, exactly as for the entropy production rate. Entropy is always produced locally and connected to the local fluctuation-dissipation violation via Harada-Sasa relationships. However, cooperative mechanisms close to the transition spread off-equilibrium effects to the whole system, producing an out of equilibrium response on the global scale. Our findings elucidate the role of activity and interactions in the cost repartition of collective behavior and explain what observed in experiments on natural living groups., Comment: 15 pages, 6 figures

Experiments on bird flocks and midge swarms reveal that these natural systems are well described by an active theory in which conservation laws play a crucial role. By building a symplectic structure that couples the particles' velocities to the generator of their internal rotations (spin), the Inertial Spin Model (ISM) reinstates a second-order temporal dynamics that captures many phenomenological traits of flocks and swarms. The reversible structure of the ISM predicts that the total spin is a constant of motion, the central conservation law responsible for all the novel dynamical features of the model. However, fluctuations and dissipation introduced in the original model to make it relax, violate the spin conservation law, so that the ISM aligns with the biophysical phenomenology only within finite-size regimes, beyond which the overdamped dynamics characteristic of the Vicsek model takes over. Here, we introduce a novel version of the ISM, in which the irreversible terms needed to relax the dynamics strictly respect the conservation of the spin. We perform a numerical investigation of the fully conservative model, exploring both the fixed-network case, which belongs to the equilibrium class of Model G, and the active case, characterized by self-propulsion of the agents and an out-of-equilibrium reshuffling of the underlying interaction network. Our simulations not only capture the correct spin wave phenomenology of the ordered phase, but they also yield dynamical critical exponents in the near-ordering phase that agree very well with the theoretical predictions., Comment: 13 pages, 6 figures

Numerical analysis of conserved field dynamics has been generally performed with pseudo spectral methods. Finite differences integration, the common procedure for non-conserved field dynamics, indeed struggles to implement a conservative noise in the discrete spatial domain. In this work, we present a novel method to generate a conservative noise in the finite differences framework, which works for any discrete topology and boundary conditions. We apply it to numerically solve the conserved Kardar-Parisi-Zhang (cKPZ) equation, widely used to describe surface roughening when the number of particles is conserved. Our numerical simulations recover the correct scaling exponents $\alpha$, $\beta$, and $z$ in $d=1$ and in $d=2$. To illustrate the potentiality of the method, we further consider the cKPZ equation on different kinds of non-standard lattices and on the random Euclidean graph. This is the first numerical study of conserved field dynamics on an irregular topology, paving the way to a broad spectrum of possible applications., Comment: Multiplicative noise case added, 6 figures

Living groups move in complex environments and are constantly subject to external stimuli, predatory attacks and disturbances. An efficient response to such perturbations is vital to maintain the group's coherence and cohesion. Perturbations are often local, i.e. they are initially perceived only by few individuals in the group, but can elicit a global response. This is the case of starling flocks, that can turn very quickly to evade predators. In this paper, we investigate the conditions under which a global change of direction can occur upon local perturbations. Using minimal models of self-propelled particles, we show that a collective directional response occurs on timescales that grow with the system size and it is, therefore, a finite-size effect. The larger the group is, the longer it will take to turn. We also show that global coherent turns can only take place if i) the mechanism for information propagation is efficient enough to transmit the local reaction undamped through the whole group; and if ii) motility is not too strong, to avoid that the perturbed individual leaves the group before the turn is complete. No compliance with such conditions results in the group's fragmentation or in a non-efficient response., Comment: 23 pages, 7 figures

A well-established numerical technique to study the dynamics of spin systems in which symmetries and conservation laws play an important role is to microcanonically integrate their reversible equations of motion, obtaining thermalization through initial conditions drawn with the canonical distribution. In order to achieve a more realistic relaxation of the magnetic energy, numerically expensive methods that explicitly couple the spins to the underlying lattice are normally employed. Here we introduce a stochastic conservative thermostat that relaxes the magnetic energy while preserving the constant of motions, thus turning microcanonical spin dynamics into a conservative canonical dynamics, without actually simulating the lattice. We test the thermostat on the Heisenberg antiferromagnet in d=3 and show that the method reproduces the exact values of the static and dynamic critical exponents, while in the low-temperature phase it yields the correct spin wave phenomenology. Finally, we demonstrate that the relaxation coefficient of the new thermostat is quantitatively connected to the microscopic parameters of the spin-lattice coupling., Comment: Revised version. 30 pages, 7 figures

Flocking in $d=2$ is a genuine non-equilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady state distribution of the flock's microstates. They occur as consequences of the time reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data., Comment: 8 pages + Appendix; 6 figures

The dynamical critical exponent $z$ of natural swarms of insects is calculated using the renormalization group to order $\epsilon = 4-d$. A novel fixed point emerges, where both activity and inertia are relevant. In three dimensions the critical exponent at the new fixed point is $z = 1.35$, in agreement with both experiments ($1.37 \pm 0.11$) and numerical simulations ($1.35 \pm 0.04$)., Comment: Numerical simulations to validate RG predictions added. Experimental assesment of the dynamic critical exponent revised. 72 pages, 24 figures, 1 table

The recent inflow of empirical data about the collective behaviour of strongly correlated biological systems has brought field theory and the renormalization group into the biophysical arena. Experiments on bird flocks and insect swarms show that social forces act on the particles' velocity through the generator of its rotations, namely the spin, indicating that mode-coupling field theories are necessary to reproduce the correct dynamical behaviour. Unfortunately, a theory for three coupled fields - density, velocity and spin - has a prohibitive degree of intricacy. A simplifying path consists in getting rid of density fluctuations by studying incompressible systems. This requires imposing a solenoidal constraint on the primary field, an unsolved problem even for equilibrium mode-coupling theories. Here, we perform an equilibrium dynamic renormalization group analysis of a mode-coupling field theory subject to a solenoidal constraint; using the classification of Halperin and Hohenberg, we can dub this case as a solenoidal Model G. We demonstrate that the constraint produces a new vertex that mixes static and dynamical coupling constants, and that this vertex is essential to grant the closure of the renormalization group structure and the consistency of dynamics with statics. Interestingly, although the solenoidal constraint leads to a modification of the static universality class, we find that it does not change the dynamical universality class, a result that seems to represent an exception to the general rule that dynamical universality classes are narrower than static ones. Our results constitute a solid stepping stone in the admittedly large chasm towards developing an off-equilibrium mode-coupling theory of biological groups.

Speed fluctuations of individual birds in natural flocks are moderate, due to the aerodynamic and biomechanical constraints of flight. Yet the spatial correlations of such fluctuations are scale-free, namely they have a range as wide as the entire group, a property linked to the capacity of the system to collectively respond to external perturbations. Scale-free correlations and moderate fluctuations set conflicting constraints on the mechanism controlling the speed of each agent, as the factors boosting correlation amplify fluctuations, and vice versa. Here, using a statistical field theory approach, we suggest that a marginal speed confinement that ignores small deviations from the natural reference value while ferociously suppressing larger speed fluctuations, is able to reconcile scale-free correlations with biologically acceptable group's speed. We validate our theoretical predictions by comparing them with field experimental data on starling flocks with group sizes spanning an unprecedented interval of over two orders of magnitude., Comment: 18 pages, 5 figures. Latest version before publication uploaded, journal reference added

Discretization of continuous stochastic processes is needed to numerically simulate them or to infer models from experimental time series. However, depending on the nature of the process, the same discretization scheme, if not accurate enough, may perform very differently for the two tasks. Exact discretizations, which work equally well at any scale, are characterized by the property of invariance under coarse-graining. Motivated by this observation, we build an explicit Renormalization Group approach for Gaussian time series generated by auto-regressive models. We show that the RG fixed points correspond to discretizations of linear SDEs, and only come in the form of first order Markov processes or non-Markovian ones. This fact provides an alternative explanation of why standard delay-vector embedding procedures fail in reconstructing partially observed noise-driven systems. We also suggest a possible effective Markovian discretization for the inference of partially observed underdamped equilibrium processes based on the exploitation of the Einstein relation., Comment: 13 pages, 3 figures, 1 table

We study the crossover between equilibrium and off-equilibrium dynamical universality classes in the Vicsek model near its ordering transition. Starting from the incompressible hydrodynamic theory of Chen et al \cite{chen2015critical}, we show that increasing the activity leads to a renormalization group (RG) crossover between the equilibrium ferromagnetic fixed point, with dynamical critical exponent $z = 2$, and the off-equilibrium active fixed point, with $z = 1.7$ (in $d=3$). We run simulations of the classic Vicsek model in the near-ordering regime and find that critical slowing down indeed changes with activity, displaying two exponents that are in remarkable agreement with the RG prediction. The equilibrium-to-off-equilibrium crossover is ruled by a characteristic length scale beyond which active dynamics takes over. Such length scale is smaller the larger the activity, suggesting the existence of a general trade-off between activity and system's size in determining the dynamical universality class of active matter., Comment: 10 pages, 4 figures

Many living and complex systems exhibit second order emergent dynamics. Limited experimental access to the configurational degrees of freedom results in data that appears to be generated by a non-Markovian process. This poses a challenge in the quantitative reconstruction of the model from experimental data, even in the simple case of equilibrium Langevin dynamics of Hamiltonian systems. We develop a novel Bayesian inference approach to learn the parameters of such stochastic effective models from discrete finite length trajectories. We first discuss the failure of naive inference approaches based on the estimation of derivatives through finite differences, regardless of the time resolution and the length of the sampled trajectories. We then derive, adopting higher order discretization schemes, maximum likelihood estimators for the model parameters that provide excellent results even with moderately long trajectories. We apply our method to second order models of collective motion and show that our results also hold in the presence of interactions., Comment: we correct previous inaccuracy about a reference; 29 pages, 9 figures

In arXiv:1911.06042v1 M. Casiulis et al. study a Hamiltonian model in which rigid rotations of moving clusters give rise to scale-free correlations of velocity and speed. M. Casiulis et al. compare correlations in their model to those observed in real flocks of birds and claim that rigid-body rotations provide an explanation that stands in contrast with, and it is simpler than, previously proposed explanations of correlations in bird flocks, namely Goldstone modes in the velocity orientations and marginal (or near-critical) modes in the speed. Here, we show that the rigid rotation scenario is completely inconsistent with a large body of well-established experimental evidence on real flocks of birds, and it therefore does not provide an appropriate explanation for the observed phenomenology.

We study the critical behaviour of a model with non-dissipative couplings aimed at describing the collective behaviour of natural swarms, using the dynamical renormalization group. At one loop, we find a crossover between a conservative yet unstable fixed point, characterized by a dynamical critical exponent $z=d/2$, and a dissipative stable fixed point with $z=2$, a result we confirm through numerical simulations. The crossover is regulated by a conservation length scale that is larger the smaller the effective friction, so that in finite-size biological systems with low dissipation, dynamics is ruled by the conservative fixed point. In three dimensions this mechanism gives $z=3/2$, a value significantly closer to the experimental result $z\approx 1$ than the value $z\approx 2$ found in fully dissipative models, either at or off equilibrium. This result indicates that non-dissipative dynamical couplings are necessary to develop a theory of natural swarms fully consistent with experiments, Comment: 5 pages, 2 figure

Motivated by the collective behaviour of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. By performing a dynamical renormalization group calculation at one loop, we show that the violation of momentum conservation generates a crossover between a conservative yet IR-unstable fixed point, characterized by a dynamic critical exponent $z=d/2$, and a dissipative IR-stable fixed point with $z=2$. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent $\kappa=4/d$. Such crossover is regulated by a conservation length scale, $\mathcal R_0$, which is larger the smaller the dissipation: beyond $\mathcal R_0$ the dissipative fixed point dominates, while at shorter distances dynamics is ruled by the conservative fixed point and critical exponent, a behaviour which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents $z=3/2$ and $z=2$ in the critical slowing down of the system, confirming the renormalization group results. From the biophysical point of view, our calculation indicates that in finite-size biological groups mode-coupling terms in the equation of motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step towards reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breaking terms violating number conservation, as quantum magnets or photon gases., Comment: 33 pages, 7 figures

We introduce a new ferromagnetic model capable of reproducing one of the most intriguing properties of collective behaviour in starling flocks, namely the fact that strong collective order of the system coexists with scale-free correlations of the modulus of the microscopic degrees of freedom, that is the birds' speeds. The key idea of the new theory is that the single-particle potential needed to bound the modulus of the microscopic degrees of freedom around a finite value, is marginal, that is has zero curvature. We study the model by using mean-field approximation and Monte Carlo simulations in three dimensions, complemented by finite-size scaling analysis. While at the standard critical temperature, $T_c$, the properties of the marginal model are exactly the same as a normal ferromagnet with continuous symmetry-breaking, our results show that a novel zero-temperature critical point emerges, so that in its deeply ordered phase the marginal model develops divergent susceptibility and correlation length of the modulus of the microscopic degrees of freedom, in complete analogy with experimental data on natural flocks of starlings., Comment: 14 pages, 8 pdf figures

Efficient collective response to external perturbations is one of the most striking abilities of a biological system. Signal propagation through the group is an important condition for the imple- mentation of such a response. Information transfer has been experimentally observed in the turning mechanism of birds flocks. In this context it is well-known also the existence of density waves: birds under predation, attempting to escape, give rise to self-organized density waves that propagates linearly on the flock. Most aspects of this phenomenon are still not fully captured by theoretical models. In this work we present a new model for the propagation of the speed (the modulus of the velocity) fluctuations inside a flock, which is the simplest way to reproduce the observed density waves. We have studied the full solution of the model in d = 1 and we found that there is a line in the parameter space along which the system relaxes in the fastest way with no oscillation after a signal has passed. This is the critical damping condition. By analyzing the parameters plane we show that critical damping represents an attractor for a steepest descent dynamics of the return time of the system. Finally we propose a method to test the validity of the model through through future experiments.

Information transfer is an essential factor in determining the robustness of collective behaviour in biological systems with distributed control. The most direct way to study the information transfer mechanisms is to experimentally detect the propagation across the system of a signal triggered by some perturbation. However, for field experiments this method is inefficient, as the possibilities of the observer to perturb the group are limited and empirical observations must rely on rare natural perturbations. An alternative way is to use spatio-temporal correlations to assess the information transfer mechanism directly from the spontaneous fluctuations of the system, without the need to have an actual propagating signal on record. We test the approach on ground truth data provided by numerical simulations in three dimensions of two models of collective behaviour characterized by very different dynamical equations and information transfer mechanisms: the classic Vicsek model, describing an overdamped noninertial dynamics and the inertial spin model, characterized by an un- derdamped inertial dynamics. By using dynamical finite size scaling, we show that spatio-temporal correlations are able to distinguish unambiguously the diffusive information transfer mechanism of the Vicsek model from the linear mechanism of the inertial spin model., Comment: key references about dynamical scaling added; a few typos corrected

Many systems in nature, from ferromagnets to flocks of birds, exhibit ordering phenomena on the large scale. In physical systems order is statistically robust for large enough dimensions, with relative fluctuations due to noise vanishing with system size. Several biological systems, however, are less stable than their physical analogues and spontaneously change their global state on relatively short timescales. In this paper we show that there are two crucial ingredients in these systems that enhance the effect of noise, leading to collective changes of state: the non-symmetric nature of interactions between individuals, and the presence of local heterogeneities in the topology of the network. The consequences of these features can be larger the larger the system size leading to a localization of the fluctuation modes and a relaxation time that remains finite in the thermodynamic limit. The system keeps changing its global state in time, being constantly driven out of equilibrium by spontaneous fluctuations. Our results explain what is observed in several living and social systems and are consistent with recent experimental data on bird flocks and other animal groups.

The correlated motion of flocks is an instance of global order emerging from local interactions. An essential difference with analogous ferromagnetic systems is that flocks are active: animals move relative to each other, dynamically rearranging their interaction network. The effect of this off-equilibrium element is well studied theoretically, but its impact on actual biological groups deserves more experimental attention. Here, we introduce a novel dynamical inference technique, based on the principle of maximum entropy, which accodomates network rearrangements and overcomes the problem of slow experimental sampling rates. We use this method to infer the strength and range of alignment forces from data of starling flocks. We find that local bird alignment happens on a much faster timescale than neighbour rearrangement. Accordingly, equilibrium inference, which assumes a fixed interaction network, gives results consistent with dynamical inference. We conclude that bird orientations are in a state of local quasi-equilibrium over the interaction length scale, providing firm ground for the applicability of statistical physics in certain active systems.

Three-dimensional tracking of animal systems is the key to the comprehension of collective behavior. Experimental data collected via a stereo camera system allow the reconstruction of the 3d trajectories of each individual in the group. Trajectories can then be used to compute some quantities of interest to better understand collective motion, such as velocities, distances between individuals and correlation functions. The reliability of the retrieved trajectories is strictly related to the accuracy of the 3d reconstruction. In this paper, we perform a careful analysis of the most significant errors affecting 3d reconstruction, showing how the accuracy depends on the camera system set-up and on the precision of the calibration parameters., Comment: 14 pages, 9 figures

When birds come together to form a flock, the distribution of their individual velocities narrows around the mean velocity of the flock. We argue that, in a broad class of models for the joint distribution of positions and velocities, this narrowing generates an entropic force that opposes the cohesion of the flock. The strength of this force depends strongly on the nature of the interactions among birds: if birds are coupled to a fixed number of neighbors, the entropic forces are weak, while if they couple to all other birds within a fixed distance, the entropic forces are sufficient to tear a flock apart. Similar entropic forces should occur in other non-equilibrium systems. For the joint distribution of protein structures and amino-acid sequences, these forces favor the occurrence of "highly designable" structures.

One of the most impressive features of moving animal groups is their ability to perform sudden coherent changes in travel direction. While this collective decision can be a response to an external perturbation, such as the presence of a predator, recent studies show that such directional switching can also emerge from the intrinsic fluctuations in the individual behaviour. However, the cause and the mechanism by which such collective changes of direction occur are not fully understood yet. Here, we present an experimental study of spontaneous collective turns in natural flocks of starlings. We employ a recently developed tracking algorithm to reconstruct three-dimensional trajectories of each individual bird in the flock for the whole duration of a turning event. Our approach enables us to analyze changes in the individual behavior of every group member and reveal the emergent dynamics of turning. We show that spontaneous turns start from individuals located at the elongated edges of the flocks, and then propagate through the group. We find that birds on the edges deviate from the mean direction of motion much more frequently than other individuals, indicating that persistent localized fluctuations are the crucial ingredient for triggering a collective directional change. Finally, we quantitatively show that birds follow equal radius paths during turning allowing the flock to change orientation and redistribute risky locations among group members. The whole process of turning is a remarkable example of how a self-organized system can sustain collective changes and reorganize, while retaining coherence., Comment: 18 pages, 2 Videos added

Experiments find coherent information transfer through biological groups on length and time scales distinctly below those on which asymptotically correct hydrodynamic theories apply. We present here a new continuum theory of collective motion coupling the velocity and density fields of Toner and Tu to the inertial spin field recently introduced to describe information propagation in natural flocks of birds. The long-wavelength limit of the new equations reproduces Toner-Tu theory, while at shorter wavelengths (or, equivalently, smaller damping), spin fluctuations dominate over density fluctuations and second sound propagation of the kind observed in real flocks emerges. We study the dispersion relation of the new theory and find that when the speed of second sound is large, a gap sharply separates first from second sound modes. This gap implies the existence of `silent' flocks, namely medium-sized systems across which neither first nor second sound can propagate.

Bird flocks are a paradigmatic example of collective motion. One of the prominent experimental traits discovered about flocks is the presence of long range velocity correlations between individuals, which allow them to influence each other over the large scales, keeping a high level of group coordination. A crucial question is to understand what is the mutual interaction between birds generating such nontrivial correlations. Here we use the Maximum Entropy (ME) approach to infer from experimental data of natural flocks the effective interactions between birds. Compared to previous studies, we make a significant step forward as we retrieve the full functional dependence of the interaction on distance and find that it decays exponentially over a range of a few individuals. The fact that ME gives a short-range interaction even though its experimental input is the long-range correlation function, shows that the method is able to discriminate the relevant information encoded in such correlations and single out a minimal number of effective parameters. Finally, we show how the method can be used to capture the degree of anisotropy of mutual interactions., Comment: 21 pages, 7 figures, 1 table

Birds in a flock move in a correlated way, resulting in large polarization of velocities. A good understanding of this collective behavior exists for linear motion of the flock. Yet observing actual birds, the center of mass of the group often turns giving rise to more complicated dynamics, still keeping strong polarization of the flock. Here we propose novel dynamical equations for the collective motion of polarized animal groups that account for correlated turning including solely social forces. We exploit rotational symmetries and conservation laws of the problem to formulate a theory in terms of generalized coordinates of motion for the velocity directions akin to a Hamiltonian formulation for rotations. We explicitly derive the correspondence between this formulation and the dynamics of the individual velocities, thus obtaining a new model of collective motion. In the appropriate overdamped limit we recover the well-known Vicsek model, which dissipates rotational information and does not allow for polarized turns. Although the new model has its most vivid success in describing turning groups, its dynamics is intrinsically different from previous ones in a wide dynamical regime, while reducing to the hydrodynamic description of Toner and Tu at very large length-scales. The derived framework is therefore general and it may describe the collective motion of any strongly polarized active matter system., Comment: Accepted for the Special Issue of the Journal of Statistical Physics: Collective Behavior in Biological Systems, 17 pages, 4 figures, 3 videos

We derive a new method to infer from data the out-of-equilibrium alignment dynamics of collectively moving animal groups, by considering the maximum entropy distribution consistent with temporal and spatial correlations of flight direction. When bird neighborhoods evolve rapidly, this dynamical inference correctly learns the parameters of the model, while a static one relying only on the spatial correlations fails. When neighbors change slowly and detailed balance is satisfied, we recover the static procedure. We demonstrate the validity of the method on simulated data. The approach is applicable to other systems of active matter.

Flocks of birds exhibit a remarkable degree of coordination and collective response. It is not just that thousands of individuals fly, on average, in the same direction and at the same speed, but that even the fluctuations around the mean velocity are correlated over long distances. Quantitative measurements on flocks of starlings, in particular, show that these fluctuations are scale-free, with effective correlation lengths proportional to the linear size of the flock. Here we construct models for the joint distribution of velocities in the flock that reproduce the observed local correlations between individuals and their neighbors, as well as the variance of flight speeds across individuals, but otherwise have as little structure as possible. These minimally structured, or maximum entropy models provide quantitative, parameter-free predictions for the spread of correlations throughout the flock, and these are in excellent agreement with the data. These models are mathematically equivalent to statistical physics models for ordering in magnets, and the correct prediction of scale-free correlations arises because the parameters - completely determined by the data - are in the critical regime. In biological terms, criticality allows the flock to achieve maximal correlation across long distances with limited speed fluctuations.

Tracking multiple moving targets allows quantitative measure of the dynamic behavior in systems as diverse as animal groups in biology, turbulence in fluid dynamics and crowd and traffic control. In three dimensions, tracking several targets becomes increasingly hard since optical occlusions are very likely, i.e. two featureless targets frequently overlap for several frames. Occlusions are particularly frequent in biological groups such as bird flocks, fish schools, and insect swarms, a fact that has severely limited collective animal behavior field studies in the past. This paper presents a 3D tracking method that is robust in the case of severe occlusions. To ensure robustness, we adopt a global optimization approach that works on all objects and frames at once. To achieve practicality and scalability, we employ a divide and conquer formulation, thanks to which the computational complexity of the problem is reduced by orders of magnitude. We tested our algorithm with synthetic data, with experimental data of bird flocks and insect swarms and with public benchmark datasets, and show that our system yields high quality trajectories for hundreds of moving targets with severe overlap. The results obtained on very heterogeneous data show the potential applicability of our method to the most diverse experimental situations., Comment: 13 pages, 6 figures, 3 tables. Version 3 was slightly shortened, and new comprative results on the public datasets (thermal infrared videos of flying bats) by Z. Wu and coworkers (2014) were included. in A. Attanasi et al., "GReTA - A Novel Global and Recursive Tracking Algorithm in Three Dimensions", IEEE Trans. Pattern Anal. Mach. Intell., vol.37 (2015)

Collective decision-making in biological systems requires all individuals in the group to go through a behavioural change of state. During this transition, the efficiency of information transport is a key factor to prevent cohesion loss and preserve robustness. The precise mechanism by which natural groups achieve such efficiency, though, is currently not fully understood. Here, we present an experimental study of starling flocks performing collective turns in the field. We find that the information to change direction propagates across the flock linearly in time with negligible attenuation, hence keeping group decoherence to a minimum. This result contrasts with current theories of collective motion, which predict a slower and dissipative transport of directional information. We propose a novel theory whose cornerstone is the existence of a conserved spin current generated by the gauge symmetry of the system. The theory turns out to be mathematically identical to that of superfluid transport in liquid helium and it explains the dissipationless propagating mode observed in turning flocks. Superfluidity also provides a quantitative expression for the speed of propagation of the information, according to which transport must be swifter the stronger the group's orientational order. This prediction is verified by the data. We argue that the link between strong order and efficient decision-making required by superfluidity may be the adaptive drive for the high degree of behavioural polarization observed in many living groups. The mathematical equivalence between superfluid liquids and turning flocks is a compelling demonstration of the far-reaching consequences of symmetry and conservation laws across different natural systems.

Flocks of starlings exhibit a remarkable ability to maintain cohesion as a group in highly uncertain environments and with limited, noisy information. Recent work demonstrated that individual starlings within large flocks respond to a fixed number of nearest neighbors, but until now it was not understood why this number is seven. We analyze robustness to uncertainty of consensus in empirical data from multiple starling flocks and show that the flock interaction networks with six or seven neighbors optimize the trade-off between group cohesion and individual effort. We can distinguish these numbers of neighbors from fewer or greater numbers using our systems-theoretic approach to measuring robustness of interaction networks as a function of the network structure, i.e., who is sensing whom. The metric quantifies the disagreement within the network due to disturbances and noise during consensus behavior and can be evaluated over a parameterized family of hypothesized sensing strategies (here the parameter is number of neighbors). We use this approach to further show that for the range of flocks studied the optimal number of neighbors does not depend on the number of birds within a flock; rather, it depends on the shape, notably the thickness, of the flock. The results suggest that robustness to uncertainty may have been a factor in the evolution of flocking for starlings. More generally, our results elucidate the role of the interaction network on uncertainty management in collective behavior, and motivate the application of our approach to other biological networks., Comment: 19 pages, 3 figures, 9 supporting figures

The most conspicuous trait of collective animal behaviour is the emergence of highly ordered structures. Less obvious to the eye, but perhaps more profound a signature of self-organization, is the presence of long-range spatial correlations. Experimental data on starling flocks in 3d show that the exponent ruling the decay of the velocity correlation function, C(r) ~ 1/r^\gamma, is extremely small, \gamma << 1. This result can neither be explained by equilibrium field theory, nor by off-equilibrium theories and simulations of active systems. Here, by means of numerical simulations and theoretical calculations, we show that a dynamical field applied to the boundary of a set of Heisemberg spins on a 3d lattice, gives rise to a vanishing exponent \gamma, as in starling flocks. The effect of the dynamical field is to create an information inflow from border to bulk that triggers long range spin wave modes, thus giving rise to an anomalously long-ranged correlation. The biological origin of this phenomenon can be either exogenous - information produced by environmental perturbations is transferred from boundary to bulk of the flock - or endogenous - the flock keeps itself in a constant state of dynamical excitation that is beneficial to correlation and collective response.

Flocking is a paradigmatic example of collective animal behaviour, where decentralized interaction rules give rise to a globally ordered state. In the emergence of order out of self-organization we find similarities between biological systems, as bird flocks, and some physical systems, as ferromagnets. In both cases, the tendency of individuals to align to their neighbours gives rise to a polarized state. There is, however, one crucial difference: the interaction network within an animal group is not necessarily fixed in time, as each individual moves and may change its neighbours. Therefore, the dynamical interaction mechanism in biological and physical system can be quite different, not only due to the gross disparity in the complexity of the individual entities, but also because of the potential role of inter-individual motion. To assess the relevance of this mechanism it is necessary to gain quantitative experimental information about how much individuals move with respect to each other within the group. Here, by using data from field observations on starlings, we study the diffusion properties of individual birds within a flock and investigate the effect of diffusion on the dynamics of the interaction network. We find that birds diffuse faster than Brownian particles (superdiffusion) and in a strongly anisotropic way. We also find that neighbours change in time exclusively as a consequence of diffusion, so that no specific mechanism to keep one's neighbours seems to be enforced. Finally, we study the diffusion properties of birds at the border of the flock. We find that these individuals remain on the border significantly longer than what would be expected on the basis of a purely diffusional model, suggesting that there is a sort barrier a bird must cross to make the transition from border to interior of the flock., Comment: 22 pages, 10 figures

Interactions among neighboring birds in a flock cause an alignment of their flight directions. We show that the minimally structured (maximum entropy) model consistent with these local correlations correctly predicts the propagation of order throughout entire flocks of starlings, with no free parameters. These models are mathematically equivalent to the Heisenberg model of magnetism, and define an "energy" for each configuration of flight directions in the flock. Comparing flocks of different densities, the range of interactions that contribute to the energy involves a fixed number of (topological) neighbors, rather than a fixed (metric) spatial range. Comparing flocks of different sizes, the model correctly accounts for the observed scale invariance of long ranged correlations among the fluctuations in flight direction.

From bird flocks to fish schools, animal groups often seem to react to environmental perturbations as if of one mind. Most studies in collective animal behaviour have aimed to understand how a globally ordered state may emerge from simple behavioural rules. Less effort has been devoted to understanding the origin of collective response, namely the way the group as a whole reacts to its environment. Yet collective response is the adaptive key to survivor, especially when strong predatory pressure is present. Here we argue that collective response in animal groups is achieved through scale-free behavioural correlations. By reconstructing the three-dimensional position and velocity of individual birds in large flocks of starlings, we measured to what extent the velocity fluctuations of different birds are correlated to each other. We found that the range of such spatial correlation does not have a constant value, but it scales with the linear size of the flock. This result indicates that behavioural correlations are scale-free: the change in the behavioural state of one animal affects and is affected by that of all other animals in the group, no matter how large the group is. Scale-free correlations extend maximally the effective perception range of the individuals, thus compensating for the short-range nature of the direct inter-individual interaction and enhancing global response to perturbations. Our results suggest that flocks behave as critical systems, poised to respond maximally to environmental perturbations., Comment: Submitted to PNAS

Bird flocking is a striking example of collective animal behaviour. A vivid illustration of this phenomenon is provided by the aerial display of vast flocks of starlings gathering at dusk over the roost and swirling with extraordinary spatial coherence. Both the evolutionary justification and the mechanistic laws of flocking are poorly understood, arguably because of a lack of data on large flocks. Here, we report a quantitative study of aerial display. We measured the individual three-dimensional positions in compact flocks of up to 2700 birds. We investigated the main features of the flock as a whole - shape, movement, density and structure - and discuss these as emergent attributes of the grouping phenomenon. We find that flocks are relatively thin, with variable sizes, but constant proportions. They tend to slide parallel to the ground and, during turns, their orientation changes with respect to the direction of motion. Individual birds keep a minimum distance from each other that is comparable to their wingspan. The density within the aggregations is non-homogeneous, as birds are packed more tightly at the border compared to the centre of the flock. These results constitute the first set of large-scale data on three-dimensional animal aggregations. Current models and theories of collective animal behaviour can now be tested against these results., Comment: To be published in Animal Behaviour

The study of collective animal behaviour must progress through a comparison between the theoretical predictions of numerical models and data coming from empirical observations. To this aim it is important to develop methods of three-dimensional (3D) analysis that are at the same time informative about the structure of the group and suitable to empirical data. In fact, empirical data are considerably noisier than numerical data, and they are subject to several constraints. We review here the tools of analysis used by the STARFLAG project to characterise the 3D structure of large flocks of starlings in the field. We show how to avoid the most common pitfalls i the quantitative analysis of 3D animal groups, with particular attention to the problem of the bias introduced by the border of the group. By means of practical examples, we demonstrate that neglecting border effects gives rise to artefacts when studying the 3D structure of a group. Moreover, we show that mathematical rigour is essential to distinguish important biological properties from trivial geometric features of animal groups., Comment: To be published in Animal Behaviour

The most startling examples of collective animal behaviour are provided by very large and cohesive groups moving in three dimensions. Paradigmatic examples are bird flocks, fish schools and insect swarms. However, because of the sheer technical difficulty of obtaining 3D data, empirical studies conducted to date have only considered loose groups of a few tens of animals. Moreover, these studies were very seldom conducted in the field. Recently the STARFLAG project achieved the 3D reconstruction of thousands of birds under field conditions, thus opening the way to a new generation of quantitative studies of collective animal behaviour. Here, we review the main technical problems in 3D data collection of large animal groups and we outline some of the methodological solutions adopted by the STARFLAG project. In particular, we explain how to solve the stereoscopic correspondence - or matching - problem, which was the major bottleneck of all 3D studies in the past., Comment: To be published in Animal Behaviour

We study analytically the dynamics of a generalized p-spin model, starting with a thermalized initial condition. The model presents birth and death of states, hence the dynamics (even starting at equilibrium) may go out of equilibrium when the temperature is varied. We give a full description of this constrained out of equilibrium behavior and we clarify the connection to the thermodynamics by computing (sub-dominant) TAP states, constrained to the starting equilibrium configuration., Comment: 10 pages, 3 figures; longer version with appendix