Hebbian learning with elasticity explains how the spontaneous motor tempo affects music performance synchronization
Authors: Roman Iran R. , Roman Adrian S. , Kim Ji Chul , Large Edward W. 
Published: October 2020

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: (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 musicians with matching or mismatching SMTs. These studies showed, respectively, that (1) 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, (2) musicians drifted away from the initial tempo toward the SMT, and (3) 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 published in (1) and asked whether this same model would explain the data in (2) and (3) 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.

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 adaptive frequency learning during synchronization tasks to adapt the rate of their movements and match another individual’s actions or metronome tempo. Adaptive frequency learning also explains why an individual’s spontaneous rate persists after carrying out a musical synchronization task. We define a new model with adaptive 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.