The research of COBBS through seven short stories
We collect experimental data on the three-dimensional motion of large biological groups in their natural environment and use statistical field theory methods to analyse the data, thus building new theory as directly as possible inspired by the data. At the theoretical level our main interest is understanding what kind of interaction rules these groups and, more broadly, what are the effective dynamical equations regulating their collective behaviour. We started our research on collective animal behaviour by studying flocks of starlings and later turned to insect swarms, in order to have experimental systems as far as possible from each other along the line order-disorder. The overarching concept crossing all our research is that of correlation, both at the static and at the dynamical level. Here below you can find a brief summary of some of our most relevant results.
It took us more than 3 years to produce the first generation of 3D data (2005-2007), but after that we immediately stumbled into something interesting. By analyzing the proximity structure of birds within a flock we detected a strong anisotropy in the nearest neighbours (NN) spatial distribution: given a bird, its NN are found with higher probability at its sides, rather than along the longitudinal (front/rear) direction. We used such anisotropy as a proxy of the inter-individual interaction and found that it decays with the distance: the 1st NN has a significantly more anisotropic distribution than the 10th NN. No surprise there, except for one fact: by analyzing flocks with different densities, we found that the rate of decay of the anisotropy was a constant number when measured in units of neighbours, rather than in units of meters. We concluded that interaction in flocks is not based on metric distance as in physical systems (and as assumed by all models up to then), but on a density-invariant topological distance, which is more robust against ‘evaporation’ of the group under the violent density fluctuations typical of biological systems under predation. This work has now more than a thousands citations.
|Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study|
The interaction range that we found in flocks was approximately of 7 neighbours, which is quite short. Given the startling coordination of flocks, it was clear that something had to make up for such short-ranged interaction at the level of the correlation. In 2010 we thus measured the equal-time spatial correlations of velocity fluctuations and found them to be long-ranged, namely with a correlation length, ξ, that scales with the system’s size. From a field-theory standpoint this result was not surprising: flocks have a spontaneously-broken continuous symmetry (rotation), hence we should expect some massless Goldstone mode, namely an infinite correlation length. Definitely less trivial was the fact that also the supposedly massive mode, namely the modulus of the velocity (i.e. the speed), was long-range correlated. This was evidence that correlations are strong even when they are not compelled to be so, hence hinting at criticality. The difference between massless Goldstone modes and massive longitudinal modes was not fully appreciated by the biological community, which was more impressed than it should have been by the general scale-free scenario. This case illustrates that theoretical physics (Goldstone theorem and massless modes, in this case) is not an optional, but an essential tool to make genuine progress in biology.
|Scale-free correlations in starling flocks|
Modeling is a tricky business. Even when we declare that we “build models directly from the data”, it is not that clear what this means by that and even less how to do it. Quite often, when modeling a phenomenon starting from the data, we anyway put into the model many of our a priori assumptions, be it in an explicit or implicit form. It is in general very hard to build models by letting the data speak for themselves, with no assumptions on our part. The maximum entropy method tries to do exactly this: given a set of experimental observations, it builds the least structured probability distribution compatible with the given data. In other words, maximum entropy keeps to the bare minimum the number of assumptions (and therefore of parameters). We applied this method to experimental data about the velocity correlations in bird flocks and the results are actually quite neat: you succeed in fitting the data and in predicting new experimental quantities, by means of a maximum entropy model which has a surprisingly low number of parameters.
|Statistical mechanics for natural flocks of birds|
Information transfer and spin
All our analysis up to 2010 had been static, whereas collective behaviour is primarily a dynamical phenomenon and cross-group information transfer is a major biological issue. Between 2010 and 2013 we run a new data-taking campaign to measure information transfer across turning flocks. Data clearly indicated the existence of linear phase waves (the phase φi being the angle between the group direction of motion and that of individual i), that could not be reproduced by previous theories of collective motion. By comparing data to active matter models, we realized in 2014 that there was a key missing ingredient in the latter, namely the coupling between the order parameter and the generator of the rotations of the order parameter, i.e. the spin. Such coupling gives rise to a non-dissipative second-order inertial term in the equation of motion, which produces linear spin waves. Our new theory made a quantitative prediction about how the speed of propagation of the wave depends on the polarization of the system. Experimental data show that this prediction is very accurately verified.
|Information transfer and behavioural inertia in starling flocks|
Finite-size scaling and near-criticality
After working for several years on flocks, in 2011 we started collecting data on natural swarms of midges in the field. The idea was to explore the opposite extreme of the order-disorder spectrum and check whether something interesting might be going on in the disordered (swarm) phase as well. Most past studies argued that it couldn’t, as swarming was supposed to be just an epiphenomenon of the confined random walk of each insect around an environmental marker (light, water, etc). We thought differently, and proceeded to build an entirely new experimental setup to investigate this issue. The data immediately showed that, despite the low value of the order parameter, velocity correlation was very large. In the symmetry-unbroken phase of swarms (no net motion), no Goldstone mode could explain the near-critical finite-size scaling behaviour that we found, hence we could not help suggesting in 2014 that criticality might in fact play a role in the collective behaviour of swarms.
|Collective behaviour without collective order in wild swarms of midges|
Dynamic tracking in 3D
We build theories about collective behaviour in biological systems starting directly from the empirical data. But where is the data from? If all of our work were an iceberg, we can safely say that the theory-building part would be just the tip of the iceberg; the biggest, submerged part of the iceberg is producing the data. In the case of flocks and swarms, this is tracking. Our experiments consist of collecting synchronized triplets of images at a fast rate. In order to transform these raw images into actual 3D spatio-temporal trajectories, one needs to match different images of the same individual across different cameras and to follow it in time, which is not an easy business if the individual belongs to a large, crowded group. The breakthrough which allowed us to gather all of our data was a new tracking method which combines computer vision and optimization theory, together with idea from statistical physics and renormalization, into one single algorithm – GRETA.
|GReTA – A Novel Global and Recursive Tracking Algorithm in Three Dimensions|
Having established in a robust experimental way the existence of unusually strong correlation in swarms, where no Goldstone mode could justify it, we turned our attention to dynamical scaling. From a statistical physics viewpoint, this was the natural next step along the path of building a comprehensive theory of collective behaviour. The dynamic correlation function, C(k,t), is in principle a complicated function not only of space (or momentum, k) and time, but also of all possible biological parameters, like density, noise, species, etc. To our great surprise (and delight), we found that by adapting to natural swarms the classic prescription of Halperin and Hohenberg for dynamic scaling, we could collapse all space-time correlation functions of many different real swarms onto one single master curve. This discovery of dynamic scaling in 2017 was the last piece of a decade-long composing puzzle about correlation, dynamics and universality.
|Dynamic scaling in natural swarms|