Fractal Structure of the Nested Actions of Keeping the Beat

Fractal Structure of the Nested Actions in Keeping the Beat Charles A. Coey (coeyca@mail.uc.edu) Center for Cognition, Action and Perception, Department of Psychology, University of Cincinnati, ML 0376, 4150 Edwards Cl., University of Cincinnati, Cincinnati, OH 45221-0376 USA Justin Hassebrock (hassebja@mail.uc.edu) Center for Cognition, Action and Perception, Department of Psychology, University of Cincinnati, ML 0376, 4150 Edwards Cl., University of Cincinnati, Cincinnati, OH 45221-0376 USA Heidi Kloos (heidi.kloos@uc.edu) Center for Cognition, Action and Perception, Department of Psychology, University of Cincinnati, ML 0376, 4150 Edwards Cl., University of Cincinnati, Cincinnati, OH 45221-0376 USA Michael J. Richardson (richamo@ucmail.uc.edu) Center for Cognition, Action and Perception, Department of Psychology, University of Cincinnati, ML 0376, 4150 Edwards Cl., University of Cincinnati, Cincinnati, OH 45221-0376 USA Abstract The current experiments investigated the fractal structure in the nested actions of tapping behavior. The results revealed that task constraints (e.g., tapping to a metronome) alter the fractal structure of a given aspect of the behavior (e.g., inter- tap interval) and decouple its long-term interactions with other aspects of the behavior (e.g., key-press duration). These results support the idea that fractal structure reflects the dynamical organization of complex systems. Keywords: complex systems, fractal scaling, finger tapping Introduction There is certainly no shortage of complexity for the student of human mind and behavior. The human system is at once physical, chemical, biological, psychological, and social. We cognitive scientists carefully design experimental tasks and manipulations to gain an empirical purchase on the many forces that shape human behavior. Traditionally, the field has relied on classical techniques of linear statistics. We take different averages or degrees of variability in reaction time to be indicative of the cognitive processes underlying performance in our experimental tasks. Recently, however, researchers have turned to examining more subtle and complex statistical facets of data to understand the processes involved in the organization of human behavior; namely, fractal structure. Comprehensive review of this statistical property, the available mathematical techniques for its assessment, and the potential implications for theories of cognitive science is not possible in the limited space provided here (see Brown & Liebovitch, 2010; Delignieres & Marmelat, 2013; Holden, 2005; Van Orden Kloos, & Wallot, 2010). Nonetheless, a brief introduction to the topic is warranted. The term “fractal structure” is here being used loosely to refer to patterns of variability in repeated measurements of human behavior. Most traditional, linear statistical techniques operate on the assumptions that deviations from mean performance will obey a Gaussian distribution and that these “errors” will be uncorrelated with one another. Data displaying fractal structure violates these assumptions. That is, fluctuations in repeated performances exhibit “long- term dependencies” such that errors in early observations are correlated with errors in much later observations. Fractal data obey power-law scaling such that size of a given error is inversely proportional to how often errors of that size occur. Thus, like geometric fractals, these data are said to be “self-similar” and “scale-invariant”. Fractal data entail nested patterns of variability wherein small variations in measurement have the same structure as large variations. Such structure in repeated measurements is often referred to as “pink noise”, as contrasted against the random variation entailed in “white noise” (Holden, 2005). In part, these patterns are important to researchers in cognitive science as they have been discovered in a plethora of human data from the simplest of reaction time tasks taking place over the course of minutes (Van Orden, Holden, & Turvey, 2003) to measurements of self-esteem over a many months (Delignieres, Fortes, Ninot, 2004). More importantly, experimental manipulations of the type typically employed by cognitive scientists have been shown to affect fractal structure. For instance, Kello et al., (2007) demonstrated that reaction times to unpredictable cues were not only slower, but also closer to white noise (i.e., random) variation, than reaction times to predictable cues. Despite their widespread occurrence, there is not yet a unified account of how these fractal patterns get into the data or what they imply for theories of cognitive science (e.g., Van Orden, Holden, & Turvey, 2005; Wagenmakers, Farrell, & Ratcliff, 2005). The current experiments are intended to contribute to the on-going discussion by examining the fractal structure in the nested actions in a tapping task, their dynamical interaction with one another, and the impact of employing different task constraints.