Hebbian learning with elasticity explains how the spontaneous motor tempo affects music performance synchronization

Music has a tempo (or frequency of the underlying beat) that musicians maintain throughout a performance. Musicians can maintain this tempo on their own or paced by a metronome. Behavioral studies have found that each musician shows a spontaneous rate of movement, called spontaneous motor tempo (SMT), which can be measured when a musician spontaneously plays a simple melody. Data shows that a musician’s SMT systematically influences how actions affect tempo and synchronization. In this study we present a model that captures this phenomenon. To develop our model, we review the results from three musical performance settings that have been previously published: (1) solo musical performance with a pacing metronome tempo that is different from the SMT, (2) solo musical performance without a metronome at a tempo that is faster or slower than the SMT, and (3) duet musical performance between musician pairs with matching or mismatching SMTs. In the first setting, 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. In the second setting, musicians drifted away from the initial spontaneous tempo toward the SMT. And in the third setting, the absolute asynchronies between performing musicians were smaller if their SMTs matched compared to when they did not. Based on these observations, we hypothesize that, while musicians can perform musical actions at a tempo different from their SMT, the SMT constantly acts as a pulling force. We developed a model to test our hypothesis. The model is an oscillatory dynamical system with Hebbian and elastic tempo learning that simulates music performance. We model SMT as the dynamical system’s natural frequency. Hebbian learning lets the system’s frequency adapt to match the stimulus frequency. The pulling force is modeled as an elasticity term that pulls the learned frequency toward the system’s natural frequency. We used this model to simulate the three music performance settings, replicating behavioral results. Our model also lets us make predictions about performance settings not yet tested. The present study offers a dynamical explanation of how an individual’s SMT affects adaptive synchronization in realistic musical performance. Author summary Individuals can keep a musical tempo on their own or timed by another individual or a metronome. Experiments show that individuals show a specific spontaneous rate of periodic action, for example walking, blinking, or singing. Moreover, in a simple metronome synchronization task, an individual’s spontaneous rate determines that the individual will tend to anticipate a metronome that is slower, and lag a metronome that is faster. Researchers have hypothesized the mechanisms explaining how spontaneous rates affect synchronization, but no hypothesis can account for all observations yet. Our hypothesis is that individuals rely on elastic Hebbian frequency learning during synchronization tasks to adapt the rate of their movements and match another individual’s actions or metronome tempo. Elastic Hebbian frequency learning also explains why an individual’s spontaneous rate persists after carrying out a musical synchronization task. We define a new model with elastic Hebbian frequency learning and use it to simulate existing empirical data. Not only can our model explain the empirical data, but it can also make testable predictions. Our results support the theory that the brain’s endogenous rhythms give rise to spontaneous rates of movement, and that learning dynamics interact with such brain rhythms to allow for flexible synchronization. ### Competing Interest Statement Iran R. Roman, Adrian S. Roman declare that no competing financial interest exist. Edward W. Large declares a competing financial interest as CEO of Oscilloscape, LLC.