Fractal models for event-based and dynamical timers

Some recent papers proposed to distinguish between event-based and emergent timing. Event-based timing is conceived as prescribed by events produced by a central clock, and seems to be used in discrete tasks (e.g., finger tapping). Emergent or dynamical timing refers to the exploitation of the dynamical properties of effectors, and is typically used in continuous tasks (e.g., circle drawing). The analysis of period series suggested that both timing control processes possess fractal properties, characterized by self-similarity and long-range dependence. The aim of this article is to present two models that produce period series presenting the statistical properties previously evidenced in discrete and continuous rhythmic tasks. The first one is an adaptation of the classical activation/threshold models, including a plateau-like evolution of the threshold over time. The second one is a hybrid limit-cycle model, including a time-dependent linear stiffness parameter. Both models reproduced satisfactorily the spectral signatures of event-based and dynamical timing processes, respectively. The models also produced auto-correlation functions similar to those experimentally observed. Using ARFIMA modeling we show that these simulated series possess fractal properties. We suggest in conclusion some possible extensions of this modeling approach, to account for the effects of metronomic pacing, or to analyze bimanual coordination.