Mathematical models of rhythm synchronization and anticipation
Keywords: Music.

When humans synchronize with a periodic stimulus, endogenous processes like neural delays and spontaneous rates are related to the systematic asynchronies observed between human movements and stimulus onsets. This dissertation presents two different models that capture how neural delays and spontaneous rates of movement affect human synchronization. Additionally, these models have been implemented with tensorflow 2, allowing for parameter optimization and general applications in signal processing algorithms. The first model captures how, in metronome synchronization tasks, people tend to tap slightly before the metronome clicks. This anticipation tendency increases with longer stimulus periods of up to 3500ms, but is less pronounced in trained individuals like musicians compared to non-musicians. In non-biological systems, anticipation is observed between delayed-coupled systems. Therefore, the human anticipation tendency could be explained with such a system because delayed communication is inherent to the sensorimotor system during perception-action coordination. This dissertation tests this hypothesis with a dynamical systems model consisting of an oscillator receiving its own delayed activity as input. Simulation experiments were conducted using previously published behavioral data from human studies with either synchronization to a metronome or interpersonal synchronization. Our new model is validated by its ability to simulate real human synchronization and anticipation data. As a result, our model informs theories of adaptive human synchronization. The second model captures how interpersonal synchronization is affected by an individual's spontaneous rates of movement. Specifically, the greater discrepancy between two synchronizing musicians' spontaneous rates, the greater asynchronies observed during joint duet performance. Interestingly, a musician's spontaneous rate remains stable after experiencing a joint performance, suggesting short-term tempo adaptation during joint performance and spontaneous rate restoration afterwards. This dissertation tests whether an oscillatory dynamical system with frequency elasticity and Hebbian learning could explain how spontaneous rates affect interpersonal synchronization. The model consists of an oscillator with a natural frequency that emulates the human spontaneous rate. The oscillator's frequency term is elastic to allow for short-term frequency changes during synchronization with a periodic stimulus of an arbitrary frequency. However, elasticity makes the oscillator return to its original natural frequency when it is no longer stimulated. Our model is validated by its ability to simulate human synchronization data using its adaptive and elastic frequency learning mechanism. This model can simulate duet musical performance and capture how asynchronies between performers are systematically influenced by the difference between two performers' spontaneous rates. Finally, this dissertation presents a novel implementation of these oscillatory models in tensorflow 2. This toolbox is written with a broad user-base in mind, and it includes general numerical methods for integration of ordinary differential equations. Besides allowing users to simulate different types of neural oscillators in multi-scale networks, the toolbox also allows oscillatory models to be combined with deep learning networks for the first time. Oscillatory networks are a new alternative for time-frequency analysis in deep learning algorithms, and can improve performance in common signal processing tasks