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A bee-swarm model of eye movements

Right now I'm reading Philip Ball's trilogy on "nature's patterns". In three books (Shapes, Flow and Branches), Ball describes all kinds of patterns, from the black and white stripes on a zebra's coat to the shape of a milk splash. These are quite possibly the best (not too) popular science books that I've read in years, but more about that some other time, perhaps. Right now I want to focus on one thing that I found particularly intriguing: Ball's description of the organized motion of a swarm.

A milk splash has a surprisingly regular shape (Source: [url=]Robbie[/url])By definition, a swarm consists of a group of individuals. In the case of bees, and arguably in the case of humans as well, these individuals are driven by simple impulses. They react to other individuals in their immediate vicinity, but there is no master plan, and very little in the way of group coordination. And yet, somehow, from this teeming, chaotic mess, organized behavior does arise.

Consider, for example, what happens when a few bees have discovered a potential nest site, or some other place-to-bee. You might expect the few "informed" bees to separate from the swarm, being unable to communicate their find and thus unable to convince the other members of the swarm to follow their lead. Or you might expect the informed bees to be re-absorbed into the chaos of the swarm, quickly forgetting their find. But, surprisingly enough, this doesn't need to happen: A few informed bees can cause an entire swarm to migrate towards a newly discovered site.

A swarm of bees (source: [url=]Wikimedia Commons[/url])So how do you get from individual bees to coordinated swarm behavior? Couzin and colleagues have proposed a model that explains many aspects of swarm behavior, using only a few simple and biologically plausible rules for the behavior of individual bees. First and foremost, bees don't want to bump into each other, for obvious reasons. So, in the model, each bee is surrounded by a small region of avoidance. If there are other bees within this region, a bee will do its best to avoid a collision. Avoidance has priority, but if the coast is clear, a bee will scan a slightly larger region. And it will fly towards others bees in this region and align its direction. Simply put, a bee will follow its neighbors unless there is a chance of collision.

Finally, if a bee is informed (i.e., it wants to fly in a particular direction), its actual direction will be a weighted average of its preferred direction and the direction that results from the interactions with other bees. As it turns out, these few rules are enough for coordinated swarm behavior to emerge: The slight bias exerted by a few informed bees is sufficient to guide an entire swarm with a surprising degree of accuracy.

Now, you might ask, what does all of this have to do with eye movements? And here's were things get interesting, I think. If you take a look at typical swarm trajectories, you notice a few patterns (which is why Philip Ball is interested in them). For example, if a swarm has conflicting information, so that, say, 20 bees want to go to site A, whereas 10 bees want to go to site B, the swarm will generally end up at site A. As it should, of course, because the majority vote wins. But the minority is not completely powerless: It will cause the trajectory of the swarm to swerve slightly in the direction of site B.

I've recreated Couzin's model (a slight variation on it, actually) and made a few demonstration videos. In the video below you can see the curved trajectory. The majority of the bees wants to go to the green dot, but there is nevertheless a slight curve towards the red dot.

If you are familiar with eye movements, you might recognize this type of trajectory. It is analogous to what happens when humans make an eye movement to some "target" object (the green dot, in this case), while they are distracted by a "distractor" object (the red dot). In this case, the eyes will typically curve towards the distractor on their way to the target (or away from, but this can be modeled quite easily as well).

It's also well known that the eyes sometimes fail to reach a target, particularly if there is a very salient (e.g., very bright) distractor present. In that case, the eyes are sometimes "captured" by the distractor. This too, occasionally happens to the simulated bees, as you can see in the video below.

Of course, all of this would be a mere curiosity if the similarity stopped here. But it doesn't. Let's consider, once more, what happens if there is conflicting information in the swarm. But now the two conflicting groups are equally large: Let's say, 20 bees want to go to site A, and 20 bees want to go to site B.

In this case, two things can happen. If sites A and B are far apart, the swarm will reach a consensus and randomly select one of the sites. But more interestingly, if A and B are close together, the swarm will often fly to some intermediate location, halfway between A and B, as shown in the video below.

Again, this is analogous to eye movements. If people make an eye movement to one of two objects, of approximately equal saliency, the eyes will frequently land somewhere in between the two objects. But, as with the simulated swarm, this phenomenon, which is called the "global effect", occurs only if two objects are equally salient and positioned close together.

There's also an element of chance involved. The global effect does not always occur, and sometimes the eyes accurately land on the target object, as you can see in the video below. In this case, the characteristic curvature is still observed.

I will probably run some more simulations, to investigate the extent to which results from eye movement studies can be simulated with this model. But my suspicion is that the bee-swarm model will go a long way. And this, I think, is interesting.

The brain does not consist of bees, obviously, but there are parallels. The brain is like a swarm of neurons: Individual neurons do very little, but coordinated behavior arises from interactions between many neurons. And there is usually lots of conflicting information (lots of objects to look at, for example), which needs to be resolved.

To understand these type of swarm-like systems, perhaps, in the spirit of Philip Ball's trilogy, it's useful to focus on patterns, rather than on the specifics of the system. Bees are not neurons, but what is it about bees and neurons that makes them act so similarly when aggregated into a swarm?


Ball, P. (2009). Flow. Oxford, UK: Oxford University Press.

Couzin, I. D., Krause, J., Franks, N. R., & Levin, S. A. (2005). Effective leadership and decision-making in animal groups on the move. Nature, 433, 513-516.