Individual differences of limitation to extract beat from Kuramoto coupled oscillators: Transition from beat-based tapping to frequent tapping with weaker coupling

Abstract

Musical performers synchronize to each other despite differences in sound-onset timings which reflect each musician’s sense of the beat. A dynamical system of Kuramoto oscillators can simulate this spread of onsets at varying levels of temporal alignment with a variety of tempo and sound densities which also influence individual abilities for beat extraction. Here, we examined how people’s sense of beat emerges when tapping with Kuramoto oscillators of varying coupling strengths which distribute onsets around periodic moments in time. We hypothesized that people tap regularly close to the sound onset density peaks when coupling is strong. When weaker coupling produces multiple inter-onset intervals that are more widely spread, people may interpret their variety and distributions differently in order to form a sense of beat. Experiment 1 with a small in-person cohort indeed showed a few individuals who responded with high frequency tapping to slightly weak coupled stimuli although the rest found regular beats. Experiment 2 with a larger on-line cohort revealed three groups based on characteristics of inter-tap-intervals analyzed by k-means clustering, namely a Regular group (about 1/3 of the final sample) with the most robust beat extraction, Fast group (1/6) who maintained frequent tapping except for the strongest coupling, and Hybrid group (1/2) who maintained beats except for the weakest coupling. Furthermore, the adaptation time course of tap interval variability was slowest in Regular group. We suggest that people’s internal criterion for forming beats may involve different perceptual timescales where multiple stimulus intervals could be integrated or processed sequentially as is, and that the highly frequent tapping may reflect their approach in actively seeking synchronization. Our study provides the first documentation of the novel limits of sensorimotor synchronization and individual differences using coupled oscillator dynamics as a generative model of collective behavior.