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# Toward a “Thermodynamics” of Collective Behavior

Wheeling flocks of birds, swirling schools of fish, stampeding herds of ungulates: these are some of the most spectacular sights in the natural world. Animal aggregations like these seemingly represent examples of “collective behavior,” in which group form and dynamics arise spontaneously without one or more animals in charge or possessing global information. This paradigm’s dissimilarity to the way people tend to approach design and control has led to significant interest in the aggregations. How, for example, is the large-scale motion of a flock of birds coordinated, and how does it maintain cohesion? Why do groups often outperform individuals in accomplishing tasks, and how can one predict this solely from information about group makeup and individual interactions? The answers to these and related questions are not only of fundamental interest, but also have potential application to the engineering of distributed systems ranging from swarms of robots to fleets of self-driving cars.

As with any question in physics or applied math, we can both substantiate our understanding and pave the way for applications by building and validating models. In the case of collective behavior, agent-based modeling is the dominant paradigm. Each “animal” is assigned a simple behavior that it performs in isolation (such as moving in a straight line at a constant speed), in addition to a few interaction rules expected to produce the desired group behavior. These rules generally consist of a combination of a long-range attraction (to encourage aggregation), a short-range repulsion (to prevent collapse), and a tendency to align one’s direction of motion with that of nearby agents (to promote group order).

Such models can successfully produce spontaneous group order, and different group morphologies—all of which are reminiscent of examples found in nature—are obtained as the relative strength and range of these rules is varied [1]. But do they actually describe real animal groups? That is a difficult question — in part because it is not clear exactly how to validate the models. Traditionally, models are evaluated based on whether the group morphology they produce resembles what is found in nature. Group morphology on its own, however, is almost certainly not sufficient to accurately describe animal aggregations — pattern alone does not imply function [8].

Moving beyond group structure for evaluating models has proved surprisingly challenging. This reflects in part (until recently) the scarcity of quantitative empirical data for animal groups. But deciding what questions to ask of the data is the more fundamental dilemma. One recent line of research attempts to go from observations of animal movement to the rules that generated the motion [4, 5], thus providing support for modeling choices. However, this is a very challenging inverse problem, and not necessarily one that is well-posed, given that animal interactions are almost certainly nonlinear. One can surely try to fit model parameters to the data. But since these agent-based models are not derived from any conservation laws or other first principles, fitting the data to the model neither validates the model per se nor indicates that the chosen framework correctly describes the real biology.

Figure 1. Positions and velocities of the swarm center of mass for a free, undriven swarm (blue) and an acoustically-driven swarm (black). The motion of the center of mass is normally stochastic, but becomes coherent when driven. Adapted from [6].

A more tractable approach would be to match the output of a model to the observational data. Yet that requires the selection of nontrivial quantities to compute for both the real animals and the model output; as argued above, these quantities should not simply be structural or morphological. In a nutshell, the entire concept of collective behavior is predicated on the idea that the group has an identity distinct from its constituent members. We seek a precise and sufficiently discriminatory way of characterizing the group identity that is unlikely to appear in a model by chance or for the wrong reasons.

There are many ways to approach this problem. One appealing recent direction involves recognizing that physics has faced a similar question in the realm of thermodynamics and materials characterization. In those cases, one seeks to precisely describe the properties of large groups of molecules (namely, a piece of a material) in a way that does not directly reference the molecular properties. Although we cannot indiscriminately apply the prescriptions of classical thermodynamics to animal groups, given the absence of conservation laws or well-described physical interactions, we can approach the problem with a similar mindset. In particular, we can recognize that one cannot typically determine a material’s properties by passive observation alone; one must interact with it, either by applying external perturbations or putting it into contact with another material.

Of course, applying controlled perturbations to animal groups in the wild is very difficult. On the other hand, working in the laboratory places practical restrictions on the types of animals that can be used. Thus, most of the work in determining material-like properties of animal aggregations has used insects as model organisms. For example, Michael Tennenbaum, Zhongyang Liu, David Hu, and Alberto Fernandez-Nieves [10] studied the properties of fire ant aggregations by placing them in an oscillatory rheometer. Fire ants are known to form macroscopic structures by linking their bodies together. The authors used their ant rheometer to show that these aggregations behave similarly to viscoelastic fluids, becoming stiffer and more purely elastic as the density of ants increases.

My own lab studies mating swarms of Chironomid midges. Unlike fire ants, these flying insects do not touch each other, and their aggregations are more ephemeral. Nevertheless, we can still interact with the swarms and study their responses. For instance, Chironomids are strongly attuned to acoustic stimuli. We exploited this biological response and played the sound of a flying midge as a stimulus to the swarm, but modulated its amplitude sinusoidally in time. We observed the appearance of a coherent mode in the motion of the swarm’s center of mass (see Figure 1) whose amplitude increased linearly with that of the driving sound [6], allowing us to extract a linear susceptibility. In another set of experiments, we used swarming midges’ tendency to locate themselves over ground-based visual features to apply the equivalent of a tension test to the swarms, “pulling” one swarm into two (see Figure 2) and showing that the swarms possess an emergent elastic modulus [7].

Figure 2. Observed midge trajectories (with each midge shown in a different color) for one minute of recording time over (a) one single ground-based target (shown in black) and (b) two separated ground-based targets. A single swarm can be pulled apart into two as the targets are increasingly separated. Adapted from [7].

Most recently, we applied persistent homology [2, 3, 9, 11]—a topological data analysis technique—to study the spatial structure of swarms. Persistent homology treats a pointlike data set as discretization of some underlying unknown object, and seeks to characterize its topology. To do so, one attaches a sphere of radius $$\varepsilon/2$$ to each point and uses them to build a simplicial complex whose topology can then be analyzed; $$\varepsilon$$ is a free parameter, but topological features invariant for a range of $$\varepsilon$$ (the “persistent” in persistent homology) are assumed to be meaningful. We used this method to study the swarms’ simplest topological feature: the number of connected components, as measured by the zeroth Betti number of the simplicial complexes. This analysis revealed that swarms are composed of a dense inner core (containing one large component) and a more diffuse outer region (containing many small components). The boundary between these two regions is reminiscent of a liquid/vapor phase equilibrium, with distinct properties on either side of the interface but free exchange of individuals between the two “phases” [9].

Ultimately, we have been able to characterize properties of animal aggregations that must be reproducible by correct models and are much more stringent than simple morphology or pattern. These experiments can also help us make fundamental progress in understanding what it really means to call an aggregation collective, and why these collectives possess enhanced properties relative to individual animals. Much work remains to be done before we have a unified “thermodynamic” theory of animal aggregations, but we are well on our way.

Acknowledgments: This research was sponsored by the Army Research Laboratory and accomplished under grant numbers W911NF-12-1-0517, W911NF-13-1-0426, and W911NF-16-1-0185. The views and conclusions in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. government.

References
[1] Couzin, I.D., Krause, J., James, R., Ruxton, G.D., & Franks, N.R. (2002). “Collective memory and spatial sorting in animal groups.” J. Theor. Biol., 218, 1-11.
[2] Fasy, B.T., & Wang, B. (2016). “Exploring persistent local homology in topological data analysis.” Proceedings of the 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (6430-6434). Shanghai, China.
[3] Ghrist, R. (2008). “Barcodes: The persistent topology of data.” Bull. Am. Math. Soc., 45, 61-75.
[4] Herbert-Read, J.E., Perna, A., Mann, R.P., Schaerf, T.M., Sumpter, D.J.T., & Ward, A.J.W. (2011). “Inferring the rules of interaction of shoaling fish.” Proc. Natl. Acad. Sci. USA, 108, 18726-18731.
[5] Katz, Y., Tunstrøm, K., Ioannou, C.C., Huepe, C., & Couzin, I.D. (2011). “Inferring the structure and dynamics of interactions in schooling fish.” Proc. Natl. Acad. Sci. USA, 108, 18720-18725.
[6] Ni, R., Puckett, J.G., Dufresne, E.R., & Ouellette, N.T. (2015). “Intrinsic fluctuations and driven response of insect swarms.” Phys. Rev. Lett., 115, 118104.
[7] Ni, R., & Ouellette, N.T. (2016). “On the tensile strength of insect swarms.” Phys. Biol., 13, 045002.
[8] Parrish, J.K., & Edelstein-Keshet, L. (1999). “Complexity, pattern, and evolutionary trade-offs in animal aggregation.” Science, 284, 99-101.
[9] Sinhuber, M., & Ouellette, N.T. (2017). “Phase coexistence in insect swarms.” Phys. Rev. Lett. (in press).
[10] Tennenbaum, M., Liu, Z., Hu, D., & Fernandez-Nieves, A. (2016). “Mechanics of fire ant aggregations.” Nat. Mater., 15, 54-59.
[11] Topaz, C.M., Ziegelmeier, L., & Halverson, T. (2015). “Topological data analysis of biological aggregation models.” PLoS ONE, 10, e0126383.

Nicholas Ouellette is an associate professor in the Department of Civil and Environmental Engineering at Stanford University. His research focuses on the dynamics of nonlinear systems driven out of equilibrium, including turbulent flows, granular materials, and animal aggregations.