Mode locking in periodically forced gradient frequency neural networks

Musical neurodynamics

A great deal of research in the neuroscience of music suggests that neural oscillations synchronize with musical stimuli. Although neural synchronization is a well-studied mechanism underpinning expectation, it has even more far-reaching implications for music. In this Perspective, we survey the literature on the neuroscience of music, including pitch, harmony, melody, tonality, rhythm, metre, groove and affect. We describe how fundamental dynamical principles based on known neural mechanisms can explain basic aspects of music perception and performance, as summarized in neural resonance theory. Building on principles such as resonance, stability, attunement and strong anticipation, we propose that people anticipate musical events not through predictive neural models, but because brain-body dynamics physically embody musical structure. The interaction of certain kinds of sounds with ongoing pattern-forming dynamics results in patterns of perception, action and coordination that we collectively experience as music. Statistically universal structures may have arisen in music because they correspond to stable states of complex, pattern-forming dynamical systems. This analysis of empirical findings from the perspective of neurodynamic principles sheds new light on the neuroscience of music and what makes music powerful.

Neural dynamics of predictive timing and motor engagement in music listening

Why do humans spontaneously dance to music? To test the hypothesis that motor dynamics reflect predictive timing during music listening, we created melodies with varying degrees of rhythmic predictability (syncopation) and asked participants to rate their wanting-to-move (groove) experience. Degree of syncopation and groove ratings are quadratically correlated. Magnetoencephalography data showed that, while auditory regions track the rhythm of melodies, beat-related 2-hertz activity and neural dynamics at delta (1.4 hertz) and beta (20 to 30 hertz) rates in the dorsal auditory pathway code for the experience of groove. Critically, the left sensorimotor cortex coordinates these groove-related delta and beta activities. These findings align with the predictions of a neurodynamic model, suggesting that oscillatory motor engagement during music listening reflects predictive timing and is effected by interaction of neural dynamics along the dorsal auditory pathway.

Exploring the dynamics of intentional sensorimotor desynchronization using phasing performance in music

Humans tend to synchronize spontaneously to rhythmic stimuli or with other humans, but they can also desynchronize intentionally in certain situations. In this study, we investigate the dynamics of intentional sensorimotor desynchronization using phasing performance in music as an experimental paradigm. Phasing is a compositional technique in modern music that requires musicians to desynchronize from each other in a controlled manner. A previous case study found systematic nonlinear trajectories in the phasing performance between two expert musicians, which were explained by coordination dynamics arising from the interaction between the intrinsic tendency of synchronization and the intention of desynchronization. A recent exploratory study further examined the dynamics of phasing performance using a simplified task of phasing against a metronome. Here we present a further analysis and modeling of the data from the exploratory study, focusing on the various types of phasing behavior found in non-expert participants. Participants were instructed to perform one phasing lap, and individual trials were classified as successful (1 lap), unsuccessful (> 1 laps), or incomplete (0 lap) based on the number of laps made. It was found that successful phasing required a gradual increment of relative phase and that different types of failure (unsuccessful vs. incomplete) were prevalent at slow vs. fast metronome tempi. The results are explained from a dynamical systems perspective, and a dynamical model of phasing performance is proposed which captures the interaction of intrinsic dynamics and intentional control in an adaptive-frequency oscillator coupled to a periodic external stimulus. It is shown that the model can replicate the multiple types of phasing behavior as well as the effect of tempo observed in the human experiment. This study provides further evidence that phasing performance is governed by the nonlinear dynamics of rhythmic coordination. It also demonstrates that the musical technique of phasing provides a unique experimental paradigm for investigating human rhythmic behavior.

Infants show enhanced neural responses to musical meter frequencies beyond low-level features

Music listening often entails spontaneous perception and body movement to a periodic pulse-like meter. There is increasing evidence that this cross-cultural ability relates to neural processes that selectively enhance metric periodicities, even when these periodicities are not prominent in the acoustic stimulus. However, whether these neural processes emerge early in development remains largely unknown. Here, we recorded the electroencephalogram (EEG) of 20 healthy 5- to 6-month-old infants, while they were exposed to two rhythms known to induce the perception of meter consistently across Western adults. One rhythm contained prominent acoustic periodicities corresponding to the meter, whereas the other rhythm did not. Infants showed significantly enhanced representations of meter periodicities in their EEG responses to both rhythms. This effect is unlikely to reflect the tracking of salient acoustic features in the stimulus, as it was observed irrespective of the prominence of meter periodicities in the audio signals. Moreover, as previously observed in adults, the neural enhancement of meter was greater when the rhythm was delivered by low-pitched sounds. Together, these findings indicate that the endogenous enhancement of metric periodicities beyond low-level acoustic features is a neural property that is already present soon after birth. These high-level neural processes could set the stage for internal representations of musical meter that are critical for human movement coordination during rhythmic musical behavior. RESEARCH HIGHLIGHTS: 5- to 6-month-old infants were presented with auditory rhythms that induce the perception of a periodic pulse-like meter in adults. Infants showed selective enhancement of EEG activity at meter-related frequencies irrespective of the prominence of these frequencies in the stimulus. Responses at meter-related frequencies were boosted when the rhythm was conveyed by bass sounds. High-level neural processes that transform rhythmic auditory stimuli into internal meter templates emerge early after birth.

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

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: solo musical performance with a pacing metronome tempo that is different from the SMT, solo musical performance without a metronome at a tempo that is faster or slower than the SMT, and duet musical performance between musicians with matching or mismatching SMTs. These studies showed, respectively, that 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, musicians drifted away from the initial tempo toward the SMT, and 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 in the first of the three studies and asked whether this same model would explain the data the remaining two studies 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.

Dynamic models for musical rhythm perception and coordination

Rhythmicity permeates large parts of human experience. Humans generate various motor and brain rhythms spanning a range of frequencies. We also experience and synchronize to externally imposed rhythmicity, for example from music and song or from the 24-h light-dark cycles of the sun. In the context of music, humans have the ability to perceive, generate, and anticipate rhythmic structures, for example, "the beat." Experimental and behavioral studies offer clues about the biophysical and neural mechanisms that underlie our rhythmic abilities, and about different brain areas that are involved but many open questions remain. In this paper, we review several theoretical and computational approaches, each centered at different levels of description, that address specific aspects of musical rhythmic generation, perception, attention, perception-action coordination, and learning. We survey methods and results from applications of dynamical systems theory, neuro-mechanistic modeling, and Bayesian inference. Some frameworks rely on synchronization of intrinsic brain rhythms that span the relevant frequency range; some formulations involve real-time adaptation schemes for error-correction to align the phase and frequency of a dedicated circuit; others involve learning and dynamically adjusting expectations to make rhythm tracking predictions. Each of the approaches, while initially designed to answer specific questions, offers the possibility of being integrated into a larger framework that provides insights into our ability to perceive and generate rhythmic patterns.

Why art? The role of arts in arts and health

This article is an answer to a report called "What is the evidence on the role of the arts in improving health and well-being?" The authors conclude that the arts have an impact on mental and physical health. Yet, the question of the role of the arts remains unanswered. What is and what is not an art effect? Recently, embodied theory has inspired articles on the perception of art. These articles have not yet received attention in the field of Arts and Health. Scholars in psychosomatic medicine have argued for an approach based on recent work in enactive embodied theory to investigate the connection between the body and the mind. The present article examines how key concepts in this theory relate to art. This leads to a discussion of art in terms of empathy-the relation between the internal state of the artist and the internal state of the beholder. I exemplify with a conceptual framework of musical empathy. Implications for health are addressed.

Neural Entrainment to Musical Pulse in Naturalistic Music Is Preserved in Aging: Implications for Music-Based Interventions

Neural entrainment to musical rhythm is thought to underlie the perception and production of music. In aging populations, the strength of neural entrainment to rhythm has been found to be attenuated, particularly during attentive listening to auditory streams. However, previous studies on neural entrainment to rhythm and aging have often employed artificial auditory rhythms or limited pieces of recorded, naturalistic music, failing to account for the diversity of rhythmic structures found in natural music. As part of larger project assessing a novel music-based intervention for healthy aging, we investigated neural entrainment to musical rhythms in the electroencephalogram (EEG) while participants listened to self-selected musical recordings across a sample of younger and older adults. We specifically measured neural entrainment to the level of musical pulse-quantified here as the phase-locking value (PLV)-after normalizing the PLVs to each musical recording's detected pulse frequency. As predicted, we observed strong neural phase-locking to musical pulse, and to the sub-harmonic and harmonic levels of musical meter. Overall, PLVs were not significantly different between older and younger adults. This preserved neural entrainment to musical pulse and rhythm could support the design of music-based interventions that aim to modulate endogenous brain activity via self-selected music for healthy cognitive aging.

Modeling the tonotopic map using a two-dimensional array of neural oscillators

We present a model of a tonotopic map known as the Oscillatory Tonotopic Self-Organizing Map (OTSOM). It is a 2-dimensional, self-organizing array of Hopf oscillators, capable of performing a Fourier-like decomposition of the input signal. While the rows in the map encode the input phase, the columns encode frequency. Although Hopf oscillators exhibit resonance to a sinusoidal signal when there is a frequency match, there is no obvious way to also achieve phase tuning. We propose a simple method by which a pair of Hopf oscillators, unilaterally coupled through a coupling scheme termed as modified power coupling, can exhibit tuning to the phase offset of sinusoidal forcing input. The training of OTSOM is performed in 2 stages: while the frequency tuning is adapted in Stage 1, phase tuning is adapted in Stage 2. Earlier tonotopic map models have modeled frequency as an abstract parameter unconnected to any oscillation. By contrast, in OTSOM, frequency tuning emerges as a natural outcome of an underlying resonant process. The OTSOM model can possibly be regarded as an approximation of the tonotopic map found in the primary auditory cortices of mammals, particularly exemplified in the studies of echolocating bats.

Disordered quenching in arrays of coupled Bautin oscillators

In this work, we study the phenomenon of disordered quenching in arrays of coupled Bautin oscillators, which are the normal form for bifurcation in the vicinity of the equilibrium point when the first Lyapunov coefficient vanishes and the second one is nonzero. For particular parameter values, the Bautin oscillator is in a bistable regime with two attractors-the equilibrium and the limit cycle-whose basins are separated by the unstable limit cycle. We consider arrays of coupled Bautin oscillators and study how they become quenched with increasing coupling strength. We analytically show the existence and stability of the dynamical regimes with amplitude disorder in a ring of coupled Bautin oscillators with identical natural frequencies. Next, we numerically provide evidence that disordered oscillation quenching holds for rings as well as chains with nonidentical natural frequencies and study the characteristics of this effect.

The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.

A Dynamical, Radically Embodied, and Ecological Theory of Rhythm Development

Musical rhythm abilities-the perception of and coordinated action to the rhythmic structure of music-undergo remarkable change over human development. In the current paper, we introduce a theoretical framework for modeling the development of musical rhythm. The framework, based on Neural Resonance Theory (NRT), explains rhythm development in terms of resonance and attunement, which are formalized using a general theory that includes non-linear resonance and Hebbian plasticity. First, we review the developmental literature on musical rhythm, highlighting several developmental processes related to rhythm perception and action. Next, we offer an exposition of Neural Resonance Theory and argue that elements of the theory are consistent with dynamical, radically embodied (i.e., non-representational) and ecological approaches to cognition and development. We then discuss how dynamical models, implemented as self-organizing networks of neural oscillations with Hebbian plasticity, predict key features of music development. We conclude by illustrating how the notions of dynamical embodiment, resonance, and attunement provide a conceptual language for characterizing musical rhythm development, and, when formalized in physiologically informed dynamical models, provide a theoretical framework for generating testable empirical predictions about musical rhythm development, such as the kinds of native and non-native rhythmic structures infants and children can learn, steady-state evoked potentials to native and non-native musical rhythms, and the effects of short-term (e.g., infant bouncing, infant music classes), long-term (e.g., perceptual narrowing to musical rhythm), and very-long term (e.g., music enculturation, musical training) learning on music perception-action.

Mapping between sound, brain and behaviour: four-level framework for understanding rhythm processing in humans and non-human primates

Humans perceive and spontaneously move to one or several levels of periodic pulses (a meter, for short) when listening to musical rhythm, even when the sensory input does not provide prominent periodic cues to their temporal location. Here, we review a multi-levelled framework to understanding how external rhythmic inputs are mapped onto internally represented metric pulses. This mapping is studied using an approach to quantify and directly compare representations of metric pulses in signals corresponding to sensory inputs, neural activity and behaviour (typically body movement). Based on this approach, recent empirical evidence can be drawn together into a conceptual framework that unpacks the phenomenon of meter into four levels. Each level highlights specific functional processes that critically enable and shape the mapping from sensory input to internal meter. We discuss the nature, constraints and neural substrates of these processes, starting with fundamental mechanisms investigated in macaque monkeys that enable basic forms of mapping between simple rhythmic stimuli and internally represented metric pulse. We propose that human evolution has gradually built a robust and flexible system upon these fundamental processes, allowing more complex levels of mapping to emerge in musical behaviours. This approach opens promising avenues to understand the many facets of rhythmic behaviours across individuals and species. This article is part of the theme issue 'Synchrony and rhythm interaction: from the brain to behavioural ecology'.

Bouncing the network: A dynamical systems model of auditory-vestibular interactions underlying infants' perception of musical rhythm

Previous work suggests that auditory-vestibular interactions, which emerge during bodily movement to music, can influence the perception of musical rhythm. In a seminal study on the ontogeny of musical rhythm, Phillips-Silver and Trainor (2005) found that bouncing infants to an unaccented rhythm influenced infants' perceptual preferences for accented rhythms that matched the rate of bouncing. In the current study, we ask whether nascent, diffuse coupling between auditory and motor systems is sufficient to bootstrap short-term Hebbian plasticity in the auditory system and explain infants' preferences for accented rhythms thought to arise from auditory-vestibular interactions. First, we specify a nonlinear, dynamical system in which two oscillatory neural networks, representing developmentally nascent auditory and motor systems, interact through weak, non-specific coupling. The auditory network was equipped with short-term Hebbian plasticity, allowing the auditory network to tune its intrinsic resonant properties. Next, we simulate the effect of vestibular input (e.g., infant bouncing) on infants' perceptual preferences for accented rhythms. We found that simultaneous auditory-vestibular training shaped the model's response to musical rhythm, enhancing vestibular-related frequencies in auditory-network activity. Moreover, simultaneous auditory-vestibular training, relative to auditory- or vestibular-only training, facilitated short-term auditory plasticity in the model, producing stronger oscillator connections in the auditory network. Finally, when tested on a musical rhythm, models which received simultaneous auditory-vestibular training, but not models that received auditory- or vestibular-only training, resonated strongly at frequencies related to their "bouncing," a finding qualitatively similar to infants' preferences for accented rhythms that matched the rate of infant bouncing.

Multifrequency Hebbian plasticity in coupled neural oscillators

We study multifrequency Hebbian plasticity by analyzing phenomenological models of weakly connected neural networks. We start with an analysis of a model for single-frequency networks previously shown to learn and memorize phase differences between component oscillators. We then study a model for gradient frequency neural networks (GrFNNs) which extends the single-frequency model by introducing frequency detuning and nonlinear coupling terms for multifrequency interactions. Our analysis focuses on models of two coupled oscillators and examines the dynamics of steady-state behaviors in multiple parameter regimes available to the models. We find that the model for two distinct frequencies shares essential dynamical properties with the single-frequency model and that Hebbian learning results in stronger connections for simple frequency ratios than for complex ratios. We then compare the analysis of the two-frequency model with numerical simulations of the GrFNN model and show that Hebbian plasticity in the latter is locally dominated by a nonlinear resonance captured by the two-frequency model.

A canonical oscillator model of cochlear dynamics

Nonlinear responses to acoustic signals arise through active processes in the cochlea, which has an exquisite sensitivity and wide dynamic range that can be explained by critical nonlinear oscillations of outer hair cells. Here we ask how the interaction of critical nonlinearities with the basilar membrane and other organ of Corti components could determine tuning properties of the mammalian cochlea. We propose a canonical oscillator model that captures the dynamics of the interaction between the basilar membrane and organ of Corti, using a pair of coupled oscillators for each place along the cochlea. We analyze two models in which a linear oscillator, representing basilar membrane dynamics, is coupled to a nonlinear oscillator poised at a Hopf instability. The coupling in the first model is unidirectional, and that of the second is bidirectional. Parameters are determined by fitting 496 auditory-nerve (AN) tuning curves of macaque monkeys. We find that the unidirectionally and bidirectionally coupled models account equally well for threshold tuning. In addition, however, the bidirectionally coupled model exhibits low-amplitude, spontaneous oscillation in the absence of stimulation, predicting that phase locking will occur before a significant increase in firing frequency, in accordance with well known empirical observations. This leads us to a canonical oscillator cochlear model based on the fundamental principles of critical nonlinear oscillation and coupling dynamics. The model is more biologically realistic than widely used linear or nonlinear filter-based models, yet parsimoniously displays key features of nonlinear mechanistic models. It is efficient enough for computational studies of auditory perception and auditory physiology.

Modeling infants' perceptual narrowing to musical rhythms: neural oscillation and Hebbian plasticity

Previous research suggests that infants' perception of musical rhythm is fine-tuned to culture-specific rhythmic structures over the first postnatal year of human life. To date, however, little is known about the neurobiological principles that may underlie this process. In the current study, we used a dynamical systems model featuring neural oscillation and Hebbian plasticity to simulate infants' perceptual learning of culture-specific musical rhythms. First, we demonstrate that oscillatory activity in an untrained network reflects the rhythmic structure of either a Western or a Balkan training rhythm in a veridical fashion. Next, during a period of unsupervised learning, we show that the network learns the rhythmic structure of either a Western or a Balkan training rhythm through the self-organization of network connections. Finally, we demonstrate that the learned connections affect the networks' response to violations to the metrical structure of native and nonnative rhythms, a pattern of findings that mirrors the behavioral data on infants' perceptual narrowing to musical rhythms.