Collective behavior: flocks, magnets, neurons and mathematics

The analysis of collective behavior is quickly becoming cross-disciplinary.  I wrote a few years ago about a study that analyzed the coordination of starling flocks. That post was based on the work of Thierry Mora and William Bialek, presented in their paper Are Biological Systems Poised at Criticality. The paper was published in the Journal of Statistical Physics in 2011.

The mathematics of critical transitions describes systems that reach a state from which they are almost instantly transformed. Such a transformation could be liquid turning to gas or metals becoming magnetized. The authors of this paper found that the birds in a flock were connected in such a way that a flock turning in unison could be described, mathematically, as a phase transition.

In the past few years, new, larger scale experiments have made it possible to construct statistical mechanics models of biological systems directly from real data. We review the surprising successes of this “inverse” approach, using examples form families of proteins, networks of neurons, and flocks of birds. Remarkably, in all these cases the models that emerge from the data are poised at a very special point in their parameter space–a critical point. This suggests there may be some deeper theoretical principle behind the behavior of these diverse systems.

I hope that the last point, to which I added emphasis, will grow in relevance.

Also in 2011, Mora and Bialek were among the seven coauthors of the paper: Statistical Mechanics for Natural Flocks of Birds. This study focused on the alignment of flight direction in a flock.

Rather than affecting every other flock member, orientation changes caused only a bird’s seven closest neighbors to alter their flight. That number stayed consistent regardless of flock density, making the equations “topological” rather than critical in nature.

“The orientations are not at a critical point,” said Giardina. Even without criticality, however, changes rippled quickly through flocks — from one starling to seven neighbors, each of which affected seven more neighbors, and so on.

The closest statistical fit for this behavior comes from the physics of magnetism, and describes how the electron spins of particles align with their neighbors as metals become magnetized.

The paper’s abstract tells us that these models are mathematically equivalent to the quantum-mechanical Heisenberg model of magnetism.

An interesting observation here is that the interaction among birds is defined by a number of neighboring birds, not by the number of birds in a neighboring area. In other words, if it was a metric distance that governed their interaction, when the flock was more dense, the number of birds neighboring an individual bird would increases and so the number of birds interacting would also increase. But this seems not to be the case. The number of birds interacting is the quantity that stays constant. There is a very nice description of how this observation came about, and what it might mean here. From the paper:

The collective behaviour of large groups of animals is an imposing natural phenomenon, very hard to cast into a systematic theory [1]. Physicists have long hoped that such collective behaviours in biological systems could be understood in the same way as we understand collective behaviour in physics, where statistical mechanics provides a bridge between microscopic rules and macroscopic phenomena [2, 3]. A natural test case for this approach is the emergence of order in a flock of birds: out of a network of distributed interactions among the individuals, the entire flock spontaneously chooses a unique direction in which to fly [4], much as local interactions among individual spins in a ferromagnet lead to a spontaneous magnetization of the system as a whole [5]. Despite detailed development of these ideas [6{9], there still is a gap between theory and experiment. Here we show how to bridge this gap, by constructing a maximum entropy model [10] based on field data of large flocks of starlings [11{13]. We use this framework to show that the effective interactions among birds are local, and that the number of interacting neighbors is independent of flock density, confirming that interactions are ruled by topological rather than metric distance.

In a synopsis of a more recent paper, Michael Schirber explains a new refinement in the study of flocks.

Andrea Cavagna from the National Research Council (CNR) in Rome, Italy, and his colleagues have now explored this spin wave model in the continuous limit, where the birds can be thought of as fluid elements in a large hydrodynamic system. Both spin waves  and density waves  can occur, but in some cases they damp out before traveling very far. The researchers show that only spin waves propagate in small flocks, whereas density waves dominate for large flocks. In the intermediate region, no waves can propagate, which would make flocks of this size unsustainable. The results may have implications for other animal groups, such as fish schools and mammal herds.

In a New Scientist piece on the same study this point is made:

“I think it is interesting because it identifies purely physical mechanisms for the propagation of information across the flock,” says Cristina Marchetti of Syracuse University in New York. “More importantly, it imposes constraints on such a propagation, which imply constraints on the size of the flock.”

The theme that runs through all of these studies is the recognition that the behavior of all kinds of systems (physical, behavioral, biological) can look the same when viewed from certain perspectives. This ‘sameness’ is most often brought to light with mathematics. There are many things suggested by this, about nature and about mathematics. But, today, my inclination is to say this:  Mathematics itself looks like one of the many faces of nature when we imagine that mathematics itself is an evolving organization of things related to each other (in our own heads, if you will). Like the biological systems that produce organisms or the matter and energy systems that produce galaxies, mathematics produces something. Our conscious experience of mathematics, denoted and investigated by mathematicians, is as difficult to pin to the physical as consciousness itself. But mathematics, like delicate new tissue that runs through us and around us, does consistently provide the mechanism for seeing and understanding. And so, what we call ‘seeing’ and ‘understanding’ must also be some aspect of nature organizing itself.

Here’s a couple of more links:

Wired 11.08.11

Wired 03.13.12

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