The sheep are loose, and the sheepdogs—two players in a psychology experiment developed by researchers Michael Richardson and Patrick Nalepka—must get them back into the herd! How they solve this problem appears to be governed by a relatively simple mathematical model representing a few different state variables.
When people start playing the game, Richardson explained at a talk on dynamical methods this morning at the Dynamic Systems and Computational Modeling Preconference at the SPSP Annual Convention, they tend to begin with a search and recovery strategy. When a sheep gets too far away from the center, they chase it back in.
But as they get more experienced, they tend to hit upon an optimal strategy: cycle back and forth around the herd in the center, as a pair of oscillators encircling the herd in an even way.
Then Richardson deftly steps through a series of simple mathematical representations of first the search and recovery strategy—the distance of the furthest sheep, the angle of the sheep, the radius of the “home base”—and the oscillating containment strategy. Finally, he includes a parameter that governs how people switch between strategies.
When he demonstrated his model, there were a few spontaneous bursts of surprised laughter from the audience. The behavior of the mathematical representation closely mimicked the play of humans, chasing sheep until they had them rounded up and then running oscillating containment routes.
But this was just one of the many approaches Richardson demoed to examine how behavior unfolds over time. His research group at the University of Cincinnati offers a week-long workshop on Nonlinear Methods for Psychological Science every summer, and his presentation today touched on many of these—from cross-recurrence quantification analysis to explicit mathematical models of coupled oscillators to extracting summary measures of complexity like the fractal dimension.
These models have allowed him to capture patterns of variability across a variety of situations—such as individuals trying to avoid collision following crossed paths or jazz pianists improvising with each other.
In work with Ashley Walton, he has found, for example, that when jazz pianists are riffing off of a simple “drone” versus a standard swing track, they tend to exhibit more coordinated patterns of playing. They believe the simplicity of the drone background doesn’t create enough regularity to allow for more diverse improvisatory moves—while the regular pattern of the swing track allows for these moves. This finding came naturally from an approach focused on variability over time.
Underlying this dynamical perspective is a conviction that psychologists don’t need to just model minds, and they don’t need to just pay attention to summary statistics from experiments. Instead, we should be thinking more about specific task dynamics and how behavior changes over time. When we explore dynamics, we open up a whole new frontier for description and explanation.
Alex Danvers is a PhD student in social psychology studying emotions in social interactions. He uses dynamical systems and evolutionary perspectives, and is interested in new methods for exploring psychological phenomena.
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