A Network of Relaxation Oscillators that Finds Downbeats in Rhythms

A network of relaxation oscillators is used to find downbeats in rhythmical patterns. In this study, a novel model is described in detail. Its behavior is tested by exposing it to patterns having various levels of rhythmic complexity. We analyze the performance of the model and relate its success to previous work dealing with fast synchrony in coupled oscillators. 1 Downbeat Induction The term beats refers to sounds that are perceived as being equally spaced in time. Downbeats are particularly salient beats that usually occur at a comfortable tapping rate. When you tap your feet to the radio you are finding downbeats, a skill called beat induction. Downbeats act as a unifying force, lending music the feeling of movement by allowing the listener to predict the onset of important musical events. The process of beat induction is influenced by many aspects of music including harmony, melody and rhythm [2, 9]. Because interactions are not always simple, it can be difficult to predict the locations of downbeats. For example, in rock and roll music, even though chord changes (harmonic components) usually occur on the first note of a musical bar, downbeats are usually aligned with the second note due to syncopated drumming style. Furthermore, although beats are perceived as being equally-spaced in time, perfect timing is rarely if ever found on the radio. Both motor noise and deviations due to expressive timing are found in performed music. The task of beat induction can be simplified by considering only patterns of equal-amplitude beeps or clicks. This removes the influence of melody and harmony (and in addition amplitude and timbre variations). However, even in this “rhythm-only” domain, two important influences compete to decide downbeat location. First, the grouping of events in the pattern is important. For example, when three or more events are presented in rapid succession, the first and last events are more perceptually salient than the middle one. Second, the meter of the pattern is important. Meter is the sense of strong and weak beats that arises from interactions among hierarchical levels in a pattern having nested periodic components. Such a hierarchy is implied in Western music notation, where different levels are indicated by kinds of notes (whole notes, half notes, quarter notes, etc.) and where bars establish measures of an equal number of beats. For an overview see [5]. For these reasons, downbeat induction can be seen as a seemingly simple cognitive skill (tapping along with music) that is in fact challenging to model. We address this challenge by presenting a network of coupled oscillators that performs downbeat induction. The network consists of Fitzhugh-Nagumo oscillators [3, 13] designed to model the dynamics of neural action potential. We show in this paper that they can also “tap along” when forced by rhythmical pulses of voltage. Computational simulations using a popular set of rhythmical patterns reveal that our model succeeds at finding the most likely downbeat in almost all cases. In the paper we analyze model performance and provide a possible explanation of why a small network of simulated neurons would do so well at the task. In the literature, many have addressed the question of whether a simple model can find downbeats in patterns. Longuet-Higgins & Lee [10] and Povel & Essens [15] offer rules based symbolic models that find downbeats in symbolic patterns. Parncutt [14] considers the importance of presentation rate in beat induction. Large & Kolen [8] and McAuley [12] provide oscillator models that work online to find downbeats in temporal signals.