Rhythmic structure enables precise temporal expectations that are essential to human communication, including speech and music. Computational models have been developed to account for how humans perceive, produce, and learn rhythmic sequences. However, it is unclear how different types of models relate to each other and how they can be evaluated. In this review and perspective, we discuss how three major classes of models-entrainment, probabilistic, and timekeeper models-have been used to study rhythmic expectations. We critically assess each model class in terms of its level of explanation, the rhythmic behaviors it captures, its ability to account for learning and enculturation, and its ability to integrate other features, such as pitch. We show that entrainment, probabilistic, and timekeeper models differ substantially in the aspects of rhythmic expectations they can capture. To move the field forward, we propose that model comparison and integration are crucial. We identify key challenges to this effort, such as the varying nature of the input and output signals and divergent modeling goals. To address these challenges, we arrive at several practical recommendations: to equate input and output signals when comparing models, to consider several model outcomes beyond goodness-of-fit measures in model evaluation, to use model-integration efforts to inform theory building, and to make code and data openly accessible. Ultimately, understanding how models of rhythmic expectations relate, and how features in these models account for behavioral, neural, and cognitive aspects of rhythmic expectations, will deepen our understanding of a core aspect of human behavior.

A musician's spontaneous rate of movement, called spontaneous motor tempo (SMT), can be measured while spontaneously playing a simple melody. Data shows that the SMT influences the musician's tempo and synchronization. In this study we present a model that captures these phenomena. We review the results from three previously-published studies: solo musical performance with a pacing metronome tempo that is different from the SMT, solo musical performance without a metronome at a tempo that is faster or slower than the SMT, and duet musical performance between musicians with matching or mismatching SMTs. These studies showed, respectively, that the asynchrony between the pacing metronome and the musician's tempo grew as a function of the difference between the metronome tempo and the musician's SMT, musicians drifted away from the initial tempo toward the SMT, and the absolute asynchronies were smaller if musicians had matching SMTs. We hypothesize that the SMT constantly acts as a pulling force affecting musical actions at a tempo different from a musician's SMT. To test our hypothesis, we developed a model consisting of a non-linear oscillator with Hebbian tempo learning and a pulling force to the model's spontaneous frequency. While the model's spontaneous frequency emulates the SMT, elastic Hebbian learning allows for frequency learning to match a stimulus' frequency. To test our hypothesis, we first fit model parameters to match the data in the first of the three studies and asked whether this same model would explain the data the remaining two studies without further tuning. Results showed that the model's dynamics allowed it to explain all three experiments with the same set of parameters. Our theory offers a dynamical-systems explanation of how an individual's SMT affects synchronization in realistic music performance settings, and the model also enables predictions about performance settings not yet tested.

Dancing and playing music require people to coordinate actions with auditory rhythms. In laboratory perception-action coordination tasks, people are asked to synchronize taps with a metronome. When synchronizing with a metronome, people tend to anticipate stimulus onsets, tapping slightly before the stimulus. The anticipation tendency increases with longer stimulus periods of up to 3500ms, but is less pronounced in trained individuals like musicians compared to non-musicians. Furthermore, external factors influence the timing of tapping. These factors include the presence of auditory feedback from one's own taps, the presence of a partner performing coordinated joint tapping, and transmission latencies (TLs) between coordinating partners. Phenomena like the anticipation tendency can be explained by delay-coupled systems, which may be inherent to the sensorimotor system during perception-action coordination. Here we tested whether a dynamical systems model based on this hypothesis reproduces observed patterns of human synchronization. We simulated behavior with a model consisting of an oscillator receiving its own delayed activity as input. Three simulation experiments were conducted using previously-published behavioral data from 1) simple tapping, 2) two-person alternating beat-tapping, and 3) two-person alternating rhythm-clapping in the presence of a range of constant auditory TLs. In Experiment 1, our model replicated the larger anticipation observed for longer stimulus intervals and adjusting the amplitude of the delayed feedback reproduced the difference between musicians and non-musicians. In Experiment 2, by connecting two models we replicated the smaller anticipation observed in human joint tapping with bi-directional auditory feedback compared to joint tapping without feedback. In Experiment 3, we varied TLs between two models alternately receiving signals from one another. Results showed reciprocal lags at points of alternation, consistent with behavioral patterns. Overall, our model explains various anticipatory behaviors, and has potential to inform theories of adaptive human synchronization.

Presents a method for the generation of real-time dynamic autonomous agents in game environments. The method is based on the use of dynamical systems theory which allows us to express intelligent behaviors using systems of differential equations. These differential equations operate at two distinct levels. At one level, the differential equations governing movement comprise a carefully designed set of attractor-repeller fields. At a second level, behavior selection is modeled using a competition dynamics that determines the relevant contributions to the movement dynamics at each time instant. Thus, our autonomous agents are capable of exhibiting useful behaviors in complex environments. Using this approach, we are able to demonstrate, in real time, behaviors such as single and multiple target tracking with both stationary and moving obstacles.

We address scaling of the "dynamic systems" approach for robot planning to multi-agent cooperation. To accommodate this extension it is necessary to carefully consider how individual behaviors contribute to the vector field. To avoid spurious minima and related problems a competition dynamics is introduced and its stability is analyzed. A system of two cooperating agents is designed, and examples are presented to illustrate the utility of this approach.

Many connectionist approaches to musical expectancy and music composition let the question of ‘What next?’ overshadow the equally important question of ‘When next?’. One cannot escape the latter question, one of temporal structure, when considering the perception of musical meter. We view the perception of metrical structure as a dynamic process where the temporal organization of external musical events synchronizes, or entrains, a listener's internal processing mechanisms. This article introduces a novel connectionist unit, based upon a mathematical model of entrainment, capable of phase- and frequency-locking to periodic components of incoming rhythmic patterns. Networks of these units can self-organize temporally structured responses to rhythmic patterns. The resulting network behavior embodies the perception of metrical structure. The article concludes with a discussion of the implications of our approach for theories of metrical structure and musical expectancy.