The mechanisms that support sensorimotor synchronization — that is, the temporal coordination of movement with an external rhythm — are often investigated using linear computational models. The main method used for estimating the parameters of this type of model was established in the seminal work of Vorberg and Schulze (2002), and is based on fitting the model to the observed auto-covariance function of asynchronies between movements and pacing events. Vorberg and Schulze also identified the problem of parameter interdependence, namely, that different sets of parameters might yield almost identical fits, and therefore the estimation method cannot determine the parameters uniquely. This problem results in a large estimation error and bias, thereby limiting the explanatory power of existing linear models of sensorimotor synchronization. We present a mathematical analysis of the parameter interdependence problem. By applying the Cramér–Rao lower bound, a general lower bound limiting the accuracy of any parameter estimation procedure, we prove that the mathematical structure of the linear models used in the literature determines that this problem cannot be resolved by any unbiased estimation method without adopting further assumptions. We then show that adding a simple and empirically justified constraint on the parameter space — assuming a relationship between the variances of the noise terms in the model — resolves the problem. In a follow-up paper in this volume, we present a novel estimation technique that uses this constraint in conjunction with matrix algebra to reliably estimate the parameters of almost all linear models used in the literature.
Linear models have been used in several contexts to study the mechanisms that underpin sensorimotor synchronization. Given that their parameters are often linked to psychological processes such as phase correction and period correction, the fit of the parameters to experimental data is an important practical question. We present a unified method for parameter estimation of linear sensorimotor synchronization models that extends available techniques and enhances their usability. This method enables reliable and efficient analysis of experimental data for single subject and multi-person synchronization. In a previous paper (Jacoby et al., 2015), we showed how to significantly reduce the estimation error and eliminate the bias of parameter estimation methods by adding a simple and empirically justified constraint on the parameter space. By applying this constraint in conjunction with the tools of matrix algebra, we here develop a novel method for estimating the parameters of most linear models described in the literature. Through extensive simulations, we demonstrate that our method reliably and efficiently recovers the parameters of two influential linear models: Vorberg and Wing (1996), and Schulze et al. (2005), together with their multi-person generalization to ensemble synchronization. We discuss how our method can be applied to include the study of individual differences in sensorimotor synchronization ability, for example, in clinical populations and ensemble musicians.
The production of speech and music are two human behaviors that involve complex hierarchical structures with implications for timing. Timing constraints may arise from a human proclivity to form ‘self-organized’ metrical structures for perceived and produced event sequences, especially those that involve repetition. To test whether the propensity to organize events in time arises even for simple motor behaviors, we developed a novel experimental tapping paradigm investigating whether participants use the beat structure of a tapped pattern to determine the interval between repetitions. Participants listened to target patterns of 3, 4, or 5 events, occurring at one of four periodic rates, and tapped out the pattern 11 times, creating 10 inter-pattern intervals (IPIs), which participants chose freely. The ratio between mean IPI and mean inter-tap interval (ITI) was used to measure the beat-relatedness of the overall timing pattern; the closer this ratio is to an integer, the more likely the participant was timing the IPI to match a multiple of the target pattern beat. Results show that a beat-based strategy contributes prominently, although not universally, to IPI duration. Moreover, participants preferred interval cycles with even numbers of beats, especially cycles with four beats. Finally, the IPI/ITI ratio was affected by rate, with more beats of silence for the IPI at faster rates. These findings support the idea that people can generate a larger global timing structure when engaging in the repetition of simple periodic motor patterns, and use that structure to govern the timing of those motor events.