Quickly cool a piece of super-heated, liquid glass and a strange thing happens. The glass becomes hard, but very brittle—so much so that it may abruptly and startlingly shatter without warning. This is because the bonds between the molecules are under strain, and the cool temperature and low velocity means they cannot escape, as if caught in a negative equity trap with the neighbors from hell. And, like warring neighbors, eventually something gives way and the strain relieves itself catastrophically. Glass-makers avoid such catastrophes by “annealing” the glass, which means holding it for a long time at a high enough temperature that the molecules can move past each other but not too fast—in this way, the glass can find its way into a minimum-energy state where each molecule has had a chance to settle itself comfortably next to its neighbors with as little strain as possible, after which it can be completely cooled without problems.
Systems in which elements interact with their neighbors and settle into stable states are called attractors, and the stable states they settle into are called attractor states, or local minima. The term “attractor” arises from the property that if the system finds itself near one of these states it will tend to be attracted towards it, like a marble rolling downhill into a hollow. If there are multiple hollows—multiple local minima—then the marble may settle into a nearby one that is not necessarily the lowest point it can reach. To find the “global minimum” the whole thing may need to be shaken up so that the marble can jiggle itself out of its suboptimal local minimum and try and find a better one, including eventually (hopefully) the global one. This jiggling, or injection of energy, is what annealing accomplishes, and the process of moving into progressively lower energy states is called gradient descent.
Many natural systems show attractor-like dynamics. A murmuration of starlings, for example, produces aerial performances of such extraordinary, balletic synchrony that it seems like a vast, amorphous, purposeful organism, and yet the synchronized movements arise simply from the interactions between each bird and its nearest neighbors. Each flow of the flock in a given direction is a transient stable state, and periodic perturbations cause the flock to ruffle up and re-form in a new state, swooping and swirling across the sky. At a finer scale, brain scientists frequently recruit attractor dynamics to explain stable states in brain activity, such as the persistent firing of the neurons that signal which way you are facing, or where you are. Unlike glass particles or starlings, neurons do not physically move, but they express states of activity that influence the activity of their “neighbors” (neurons they are connected to) such that the activity of the whole network eventually stabilizes. Some theoreticians even think that memories might be attractor states—presenting a reminder of a memory is akin to placing the network near a local minimum, and the evolution of the system’s activity towards that minimum, via gradient descent, is analogous to retrieving the memory.
Attractors also characterize aspects of human social organization. The problem of pairing everybody off so that the species can reproduce successfully is a problem of annealing. Each individual is trying to optimize constraints—they want the most attractive, productive partner but so do all their competitors, and so compromises need to be made —bonds are made and broken, made again and broken again, until each person (approximately speaking) has found a mate. Matching people to jobs is another annealing problem, and one that we haven’t solved yet—how to find a low-strain social organization in which each individual is matched to their ideal job? If this is done badly, and society settles into a strained local minimum in which some people are happy but large numbers of people are trapped in jobs they dislike with little chance of escape, then the only solution may be an annealing one—to inject energy into the system and shake it up so that it can find a better local minimum. This need to de-stabilize a system in order to obtain a more stable one might be why populations sometimes vote for seemingly destructive social change. The alternative is to maintain a strained status quo in which tensions fail to dissipate and society eventually ruptures, like shattered glass.
Attractors are all around us, and we should pay more attention to them.