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.

Phase reduction and phase-based optimal control for biological systems: a tutorial.

A powerful technique for the analysis of nonlinear oscillators is the rigorous reduction to phase models, with a single variable describing the phase of the oscillation with respect to some reference state. An analog to phase reduction has recently been proposed for systems with a stable fixed point, and phase reduction for periodic orbits has recently been extended to take into account transverse directions and higher-order terms. This tutorial gives a unified treatment of such phase reduction techniques and illustrates their use through mathematical and biological examples. It also covers the use of phase reduction for designing control algorithms which optimally change properties of the system, such as the phase of the oscillation. The control techniques are illustrated for example neural and cardiac systems.

Entrainment and synchronization in networks of Rayleigh-van der Pol oscillators with diffusive and Haken-Kelso-Bunz couplings.

We analyze a network of non-identical Rayleigh-van der Pol (RvdP) oscillators interconnected through either diffusive or nonlinear coupling functions. The work presented here extends existing results on the case of two nonlinearly coupled RvdP oscillators to the problem of considering a network of three or more of them. Specifically, we study synchronization and entrainment in networks of heterogeneous RvdP oscillators and contrast the effects of diffusive linear coupling strategies with the nonlinear Haken-Kelso-Bunz coupling, originally introduced to study human bimanual experiments. We show how convergence of the error among the nodes' trajectories toward a bounded region is possible with both linear and nonlinear coupling functions. Under the assumption that the network is connected, simple, and undirected, analytical results are obtained to prove boundedness of the error when the oscillators are coupled diffusively. All results are illustrated by way of numerical examples and compared with the experimental findings available in the literature on synchronization of people rocking chairs, confirming the effectiveness of the model we propose to capture some of the features of human group synchronization observed experimentally in the previous literature.

Analyzing nonlinear oscillations with He's frequency-amplitude method and numerical comparison in jet engine vibration system.

In this study, we describe and successfully solve a jet engine vibration equation using a straightforward tool known as the He's frequency-amplitude Method (HFAM). The jet engine vibration system demonstrates diverse applications across aerospace, power generation, industrial machinery, transportation, marine propulsion, energy optimization, defense, and aviation training. Utilizing HFAM, we derive periodic solutions in a general form for this system, considering various cases dependent on damping and driving forces. The obtained results highlight the effectiveness of HFAM as a distinct and straightforward technique for nonlinear equations. By comparing the solutions with numerical results obtained using the fourth-order Runge-Kutta method, we demonstrate the excellent accuracy of our solutions.

Engineering spatiotemporal patterns: information encoding, processing, and controllability in oscillator ensembles.

The ability to finely manipulate spatiotemporal patterns displayed in neuronal populations is critical for understanding and influencing brain functions, sleep cycles, and neurological pathologies. However, such control tasks are challenged not only by the immense scale but also by the lack of real-time state measurements of neurons in the population, which deteriorates the control performance. In this paper, we formulate the control of dynamic structures in an ensemble of neuron oscillators as a tracking problem and propose a principled control technique for designing optimal stimuli that produce desired spatiotemporal patterns in a network of interacting neurons without requiring feedback information. We further reveal an interesting presentation of information encoding and processing in a neuron ensemble in terms of its controllability property. The performance of the presented technique in creating complex spatiotemporal spiking patterns is demonstrated on neural populations described by mathematically ideal and biophysical models, including the Kuramoto and Hodgkin-Huxley models, as well as real-time experiments on Wein bridge oscillators.

Data-Efficient Inference of Nonlinear Oscillator Networks.

Decoding the connectivity structure of a network of nonlinear oscillators from measurement data is a difficult yet essential task for understanding and controlling network functionality. Several data-driven network inference algorithms have been presented, but the commonly considered premise of ample measurement data is often difficult to satisfy in practice. In this paper, we propose a data-efficient network inference technique by combining correlation statistics with the model-fitting procedure. The proposed approach can identify the network structure reliably in the case of limited measurement data. We compare the proposed method with existing techniques on a network of Stuart-Landau oscillators, oscillators describing circadian gene expression, and noisy experimental data obtained from Rössler Electronic Oscillator network.

Genetic algorithm shape optimization to manipulate the nonlinear response of a clamped-clamped beam.

Dynamical systems, which are described by differential equations, can have an enhanced response because of their nonlinearity. As one example, the Duffing oscillator can exhibit multiple stable vibratory states for some external forcing frequencies. Although discrete systems that are described by ordinary differential equations have helped to build fundamental groundwork, further efforts are needed in order to tailor nonlinearity into distributed parameter, continuous systems, which are described by partial differential equations. To modify the nonlinear response of continuous systems, topology optimization can be used to change the shape of the mechanical system. While topology optimization is well-developed for linear systems, less work has been pursued to optimize the nonlinear vibratory response of continuous systems. In this paper, a genetic algorithm implementation of shape optimization for continuous systems is described. The method is very general, with flexible objective functions and very few assumptions; it is applicable to any continuous system. As a case study, a clamped-clamped beam is optimized to have a more nonlinear or less nonlinear vibratory response. This genetic algorithm implementation of shape optimization could provide a tool to improve the performance of many continuous structures, including MEMS sensors, actuators, and macroscale civil structures.

Infochemistry and the Future of Chemical Information Processing.

Nowadays, information processing is based on semiconductor (e.g., silicon) devices. Unfortunately, the performance of such devices has natural limitations owing to the physics of semiconductors. Therefore, the problem of finding new strategies for storing and processing an ever-increasing amount of diverse data is very urgent. To solve this problem, scientists have found inspiration in nature, because living organisms have developed uniquely productive and efficient mechanisms for processing and storing information. We address several biological aspects of information and artificial models mimicking corresponding bioprocesses. For instance, we review the formation of synchronization patterns and the emergence of order out of chaos in model chemical systems. We also consider molecular logic and ion fluxes as information carriers. Finally, we consider recent progress in infochemistry, a new direction at the interface of chemistry, biology, and computer science, considering unconventional methods of information processing.

Blinking Networks of Memristor Oscillatory Circuits in the Flux-Charge Domain.

Multistability phenomena and complex nonlinear dynamics in memristor oscillators pave the way to obtain efficient solutions to optimization problems by means of novel computational architectures based on the interconnection of single-device oscillators. It is well-known that topological properties of interconnections permit to control synchronization and spatio-temporal patterns in oscillatory networks. When the interconnections can change in time with a given probability to connect two oscillators, the whole network acts as a complex network with blinking couplings. The work of has shown that a particular class of blinking complex networks are able to completely synchronize in a faster fashion with respect to other coupling strategies. This work focuses on the specific class of blinking complex networks made of Memristor-based Oscillatory Circuits (MOCs). By exploiting the recent Flux-Charge Analysis Method, we make clear that synchronization phenomena in blinking networks of memristor oscillators having stochastic couplings, i.e., Blinking Memristor Oscillatory Networks (BMONs), correspond to global periodic oscillations on invariant manifolds and the effect of a blinking link is to shift the nonlinear dynamics through the infinite (invariant) manifolds. Numerical simulations performed on MOCs prove that synchronization phenomena can be controlled just by changing the coupling amongst them.

Simultaneous normal form transformation and model-order reduction for systems of coupled nonlinear oscillators.

In this paper, we describe a direct normal form decomposition for systems of coupled nonlinear oscillators. We demonstrate how the order of the system can be reduced during this type of normal form transformation process. Two specific examples are considered to demonstrate particular challenges that can occur in this type of analysis. The first is a 2 d.f. system with both quadratic and cubic nonlinearities, where there is no internal resonance, but the nonlinear terms are not necessarily ε 1-order small. To obtain an accurate solution, the direct normal form expansion is extended to ε 2-order to capture the nonlinear dynamic behaviour, while simultaneously reducing the order of the system from 2 to 1 d.f. The second example is a thin plate with nonlinearities that are ε 1-order small, but with an internal resonance in the set of ordinary differential equations used to model the low-frequency vibration response of the system. In this case, we show how a direct normal form transformation can be applied to further reduce the order of the system while simultaneously obtaining the normal form, which is used as a model for the internal resonance. The results are verified by comparison with numerically computed results using a continuation software.

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.

Invariant predictions of epidemic patterns from radically different forms of seasonal forcing.

Seasonal variation in environmental variables, and in rates of contact among individuals, are fundamental drivers of infectious disease dynamics. Unlike most periodically forced physical systems, for which the precise pattern of forcing is typically known, underlying patterns of seasonal variation in transmission rates can be estimated approximately at best, and only the period of forcing is accurately known. Yet solutions of epidemic models depend strongly on the forcing function, so dynamical predictions-such as changes in epidemic patterns that can be induced by demographic transitions or mass vaccination-are always subject to the objection that the underlying patterns of seasonality are poorly specified. Here, we demonstrate that the key bifurcations of the standard epidemic model are invariant to the shape of seasonal forcing if the amplitude of forcing is appropriately adjusted. Consequently, analyses applicable to real disease dynamics can be conducted with a smooth, idealized sinusoidal forcing function, and qualitative changes in epidemic patterns can be predicted without precise knowledge of the underlying forcing pattern. We find similar invariance in a seasonally forced predator-prey model, and conjecture that this phenomenon-and the associated robustness of predictions-might be a feature of many other periodically forced dynamical systems.

Deep Brain Stimulation Frequency-A Divining Rod for New and Novel Concepts of Nervous System Function and Therapy.

The efficacy of Deep Brain Stimulation (DBS) for an expanding array of neurological and psychiatric disorders demonstrates directly that DBS affects the basic electroneurophysiological mechanisms of the brain. The increasing array of active electrode configurations, stimulation currents, pulse widths, frequencies, and pulse patterns provides valuable tools to probe electroneurophysiological mechanisms. The extension of basic electroneurophysiological and anatomical concepts using sophisticated computational modeling and simulation has provided relatively straightforward explanations of all the DBS parameters except frequency. This article summarizes current thought about frequency and relevant observations. Current methodological and conceptual errors are critically examined in the hope that future work will not replicate these errors. One possible alternative theory is presented to provide a contrast to many current theories. DBS, conceptually, is a noisy discrete oscillator interacting with the basal ganglia-thalamic-cortical system of multiple re-entrant, discrete oscillators. Implications for positive and negative resonance, stochastic resonance and coherence, noisy synchronization, and holographic memory (related to movement generation) are presented. The time course of DBS neuronal responses demonstrates evolution of the DBS response consistent with the dynamics of re-entrant mechanisms. Finally, computational modeling demonstrates identical dynamics as seen by neuronal activities recorded from human and nonhuman primates, illustrating the differences of discrete from continuous harmonic oscillators and the power of conceptualizing the nervous system as composed on interacting discrete nonlinear oscillators.

Prediction of Extreme Response of Nonlinear Oscillators Subjected to Random Loading Using the Path Integral Solution Technique.

This paper studies the applicability of the path integral solution technique for estimating extreme response of nonlinear dynamic oscillators whose equations of motion can be modelled by the use of Itô stochastic differential equations. The state vector process associated with such a model is generally a diffusion process, and the probability density function of the state vector thus satisfies the Fokker-Planck-Kolmogorov equation. It is shown that the path integral solution technique combined with an appropriate numerical scheme constitutes a powerful method for solving the Fokker-Planck Kolmogorov equation with natural boundary conditions. With the calculated probability density function of the state vector in hand, one can proceed to calculate the required quantities for estimating extreme response. The proposed method distinguishes itself by remarkably high accuracy and numerical robustness. These features are highlighted by application to example studies of nonlinear oscillators excited by white noise.

On damping in the vicinity of critical points.

The effect of damping on the behaviour of oscillations in the vicinity of bifurcations of nonlinear dynamical systems is investigated. Here, our primary focus is single degree-of-freedom conservative systems to which a small linear viscous energy dissipation has been added. Oscillators with saddle-node, pitchfork and transcritical bifurcations are shown analytically to exhibit several interesting characteristics in the free decay response near a bifurcation. A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results. A transcritical bifurcation was selected because it may be used to represent generic bifurcation behaviour. It is shown that the damping ratio can be used to predict a change in the stability with respect to changing system parameters.