Science since antiquity has asked whether mathematical relationships among acoustic frequencies govern musical relationships. Psychophysics rejected frequency ratio theories, focusing on sensory phenomena predicted by linear analysis of sound. Cognitive psychologists have since focused on long-term exposure to the music of one’s culture and short-term sensitivity to statistical regularities. Today evidence is rapidly mounting that oscillatory neurodynamics is an important source of nonlinear auditory responses. This leads us to reevaluate the significance of frequency relationships in the perception of music. Here, we present a dynamical systems analysis of mode-locked neural oscillation that predicts cross-cultural invariances in music perception and cognition. We show that this theoretical framework combines with short- and long-term learning to explain the perception of Hindustani rāgas, not only by encultured Indian listeners but also by Western listeners unfamiliar with the style. These findings demonstrate that intrinsic neurodynamics contribute significantly to the perception of musical structure.

Oscillatory instability at the Hopf bifurcation is a dynamical phenomenon that has been suggested to characterize active nonlinear processes observed in the auditory system. Networks of oscillators poised near Hopf bifurcation points and tuned to tonotopically distributed frequencies have been used as models of auditory processing at various levels, but systematic investigation of the dynamical properties of such oscillatory networks is still lacking. Here we provide a dynamical systems analysis of a canonical model for gradient frequency neural networks driven by a periodic signal. We use linear stability analysis to identify various driven behaviors of canonical oscillators for all possible ranges of model and forcing parameters. The analysis shows that canonical oscillators exhibit qualitatively different sets of driven states and transitions for different regimes of model parameters. We classify the parameter regimes into four main categories based on their distinct signal processing capabilities. This analysis will lead to deeper understanding of the diverse behaviors of neural systems under periodic forcing and can inform the design of oscillatory network models of auditory signal processing.

We study mode locking in a canonical model of gradient frequency neural networks under periodic forcing. The canonical model is a generic mathematical model for a network of nonlinear oscillators tuned to a range of distinct frequencies. It is mathematically more tractable than biological neuron models and allows close analysis of mode-locking behaviors. Here we analyze individual modes of synchronization for a periodically forced canonical model and present a complete set of driven behaviors for all parameter regimes available in the model. Using a closed-form approximation, we show that the Arnold tongue (i.e., locking region) for $k:m$ synchronization gets narrower as $k$ and $m$ increase. We find that numerical simulations of the canonical model closely follow the analysis of individual modes when forcing is weak, but they deviate at high forcing amplitudes for which oscillator dynamics are simultaneously influenced by multiple modes of synchronization.