Birds of a feather flock together, without knowing anything about the mathematics of pattern formation.
Or maybe they do. Who knows what goes on in bird brains? A more interesting question, though, is not whether birds understand the math behind their flocking but whether physicists do.
Physicists have long sought formulas to describe the flying patterns of bird flocks. A flock in flight offers a spectacular example of collective biological behavior: Dozens or hundreds of birds assemble into a blob that flies off as a unit in a specific direction. Physicists would like to see if they can describe such behavior with the same math that describes flocks of atoms and molecules.
That math, known as statistical mechanics, successfully quantifies the large-scale behavior of tiny molecules interacting among themselves. Individual gas molecules fly around in all sorts of directions and speeds that statistical math can average in a way that allows precise knowledge of temperature and pressure, for instance. Imagine the fun that scientists could have if those methods similarly quantified the behavior of living things, such as stockbrokers or politicians. “Physicists have long hoped that … collective behaviors in biological systems could be understood in the same way as we understand collective behavior in physics,” physicist William Bialek of Princeton University and collaborators write in a paper recently posted online at arXiv.org.
Bialek and colleagues note that in previous work describing bird flight with math, a gap has persisted between theory and experiment. In their new paper, they try to bridge that gap using flight data from flocks of starlings. (What kind of starlings do you mean? European!) Without assuming anything special about bird flocking, the scientists let the data build the mathematical model, with the aid of a trick called the maximum entropy approach.
“Maximum entropy” (“maxent” for short) is a concept proposed in 1957 by Edwin Jaynes, a maverick physicist fond of the Bayesian approach to statistics. Maxent describes the condition in which all the possible arrangements of a system are equally probable. In information theory, that’s just another way of saying you don’t really know anything about the system you’re studying. Jaynes showed how you could combine information theory and statistical mechanics to compute probabilities for a system’s various possible behaviors without knowing anything specific, or making any assumptions, about the details of that system.
Many physicists long ignored or even ridiculed the Jaynes approach, although it has a number of loyal followers. Yet while its usefulness has been disputed, the maximum entropy approach does seem to help explain bird behavior, Bialek’s team finds.
Applied to flocks of starlings, the maxent approach assumes nothing other than that for the flock to retain its form, what any one bird does must depend on what at least some of the other birds are doing. Finding the right math to describe those bird-bird interactions isn’t so easy. Many possible formulas for how birds interact could be consistent with the overall flow of the flock. Maximum entropy reasoning allowed the physicists to find the simplest of the many possible bird-interaction formulas.
There’s no obvious reason why this simplest approach should match observations of actual flocks. Many combinations of changing speeds and directions can be consistent with the flock’s overall flow. Each bird might compute its path by gathering data on numerous other birds and combining all that input in a complicated way. On the other hand, in physics “simple” often works well, as with math describing the spinning particles in a magnet. One particle’s spin depends on its neighbors’, and neighborly interactions lead to all the spins aligning in a single direction.
Something similar apparently happens with the starlings. Equations based on the maxent method show that each bird’s flight depends on what the nearest birds are doing. More specifically, the birds base their behavior on the flight of a fixed number of neighboring birds, not on all the birds within a given distance. In other words, the density of the flock does not change the rules of flying.
These findings support the idea that “social forces” — interactions between individual birds — are driving collective behavior in a group of living organisms. And most intriguing, the maximum entropy formulation of statistical physics seems capable of capturing something about those behavioral dynamics.
“Our approach can be seen as part of a larger effort using maximum entropy methods to link the collective behavior of real biological systems to theories grounded in statistical mechanics,” Bialek and collaborators write. “We view the success of our theory as an encouraging first step.”
So someday, maybe, statistical mechanical methods may graduate from birds to human brains.