Aperiodic Dynamics Research Articles

Resting-state aperiodic neural dynamics predict individual differences in visuomotor performance and learning.

An emerging body of work has demonstrated that resting-state non-oscillatory, or aperiodic, 1/f neural activity is a functional and behaviorally relevant marker of cognitive function capacity. In the motor domain, previous work has only applied 1/f analyses to investigations of motor coordination and performance measures. The value of aperiodic resting-state neural dynamics as a marker of individual visuomotor performance capacity remains unknown. Accordingly, the aim of this work was to investigate if individual 1/f intercept and slope parameters of aperiodic resting-state neural activity predict reaction time and perceptual sensitivity in an immersive virtual reality marksmanship task. The marksmanship task required speeded selection of target stimuli and avoidance of selecting non-target stimuli. Motor and perceptual demands were incrementally increased across task blocks and participants performed the task across three training sessions spanning one week. When motor demands were high, steeper individual 1/f slope predicted shorter reaction time. This relationship did not change with practice. Increased 1/f intercept and a steeper 1/f slope were associated with higher perceptual sensitivity, measured as d'. However, this association was only observed under the highest levels of perceptual demand and only in the initial exposure to these conditions. Individuals with a lower 1/f intercept and a shallower 1/f slope demonstrated the greatest gains in perceptual sensitivity from task practice. These findings demonstrate that individual differences in motor and perceptual performance can be accounted for with resting-state aperiodic neural dynamics. The 1/f aperiodic parameters are most informative in predicting visuomotor performance under complex and demanding task conditions. In addition to predicting capacity for high visuomotor performance with a novel task, 1/f aperiodic parameters might also be useful in predicting which individuals might derive the most improvements from practice.

Hidden Strange Nonchaotic Attractors

In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed.

Is the PPG Signal Chaotic?

This paper shows how the dynamics of the PhotoPlethysmoGraphic (PPG) signal, an easily accessible biological signal from which valuable diagnostic information can be extracted, of young and healthy individuals performs at different timescales. On a small timescale, the dynamic behavior of the PPG signal is predominantly quasi-periodic. On a large timescale, a more complex dynamic diversity emerges, but never a chaotic behavior as earlier studies had reported. The procedure by which the dynamics of the PPG signal is determined consists of contrasting the dynamics of a PPG signal with well-known dynamics---named reference signals in this study---, mostly present in physical systems, such as periodic, quasi-periodic, aperiodic, chaotic or random dynamics. For this purpose, this paper provides two methods of analysis based on Deep Neural Network (DNN) architectures. The former uses a Convolutional Neural Network (CNN) architecture model. Upon training with reference signals, the CNN model identifies the dynamics present in the PPG signal at different timescales, assigning, according to a classification process, an occurrence probability to each of them. The latter uses a Recurrent Neural Network (RNN) based on a Long Short-Term Memory (LSTM) architecture. With each of the signals, whether reference signals or PPG signals, the RNN model infers an evolution function (nonlinear regression model) based on training data, and considers its predictive capability over a relatively short time horizon.

Impact of Subharmonic and Aperiodic Laryngeal Dynamics on the Phonatory Process Analyzed in Ex Vivo Rabbit Models.

Normal voice is characterized by periodic oscillations of the vocal folds. On the other hand, disordered voice dynamics (e.g., subharmonic and aperiodic oscillations) are often associated with voice pathologies and dysphonia. Unfortunately, not all investigations may be conducted on human subjects; hence animal laryngeal studies have been performed for many years to better understand human phonation. The rabbit larynx has been shown to be a potential model of the human larynx. Despite this fact, only a few studies regarding the phonatory parameters of rabbit larynges have been performed. Further, to the best of our knowledge, no ex vivo study has systematically investigated phonatory parameters from high-speed, audio and subglottal pressure data with irregular oscillations. To remedy this, the present study analyzes experiments with sustained phonation in 11 ex vivo rabbit larynges for 51 conditions of disordered vocal fold dynamics. (1) The results of this study support previous findings on non-disordered data, that the stronger the glottal closure insufficiency is during phonation, the worse the phonatory characteristics are; (2) aperiodic oscillations showed worse phonatory results than subharmonic oscillations; (3) in the presence of both types of irregular vibrations, the voice quality (i.e., cepstral peak prominence) of the audio and subglottal signal greatly deteriorated compared to normal/periodic vibrations. In summary, our results suggest that the presence of both types of irregular vibration have a major impact on voice quality and should be considered along with glottal closure measures in medical diagnosis and treatment.

Entrainment of aperiodic and periodic oscillations in the Mercury Beating Heart system using external periodic forcing.

We report experimental results indicating entrainment of aperiodic and periodic oscillatory dynamics in the Mercury Beating Heart (MBH) system under the influence of superimposed periodic forcing. Aperiodic oscillations in MBH were controlled to generate stable topological modes, namely, circle, ellipse, and triangle, evolving in a periodic fashion at different parameters of the forcing signal. These periodic dynamics show 1:1 entrainment for circular and elliptical modes, and additionally the controlled system exhibits 1:2 entrainment for elliptical and triangular modes at a different set of parameters. The external periodic forcing of the periodic MBH system reveals the existence of domains of entrainment (1:1, 1:2, 1:3, and 1:4) represented in the Arnold tongue structures. Moreover, Devil's staircase is obtained when the amplitude-frequency space of parameters of the applied signal is scanned.

Topological classification of quasiperiodically driven quantum systems

Few level quantum systems driven by $n_\mathrm{f}$ incommensurate fundamental frequencies exhibit temporal analogues of non-interacting phenomena in $n_\mathrm{f}$ spatial dimensions, a consequence of the generalisation of Floquet theory in frequency space. We organise the fundamental solutions of the frequency lattice model for $n_\mathrm{f}=2$ into a quasi-energy band structure and show that every band is classified by an integer Chern number. In the trivial class, all bands have zero Chern number and the quasi-periodic dynamics is qualitatively similar to Floquet dynamics. The topological class with non-zero Chern bands has dramatic dynamical signatures, including the pumping of energy from one drive to the other, chaotic sensitivity to initial conditions, and aperiodic time dynamics of expectation values. The topological class is however unstable to generic perturbations due to exact level crossings in the quasi-energy spectrum. Nevertheless, using the case study of a spin in a quasi-periodically varying magnetic field, we show that topological class can be realised at low frequencies as a pre-thermal phase, and at finite frequencies using counter-diabatic tools.

Applicability of strange nonchaotic Wien-bridge oscillators for secure communication

In this paper, experimental realisation of synchronisation of strange nonchaotic attractor in unidirectionally coupled Wien-bridge oscillator is presented. Preliminary work of chaotic and strange nonchaotic dynamics with diode as the nonlinear element is published in J. Nonlinear Dyn. 2015, 1 (2015). In this work, two Wien-bridge oscillators are coupled at the aperiodic dynamics for application in secure communication. The dynamics of the coupled system is explored by experimental and numerical studies. Employing the simple communication masking method with aperiodic dynamics offers some advantages rather than using chaotic attractors, i.e., the range of control parameter value is smaller than the range for which the system exhibits chaotic behaviour. The performance of the system is evaluated and it is found that experimental result agrees with numerical results.

Control, synchronization, and enhanced reliability of aperiodic oscillations in the Mercury Beating Heart system.

Experiments involving the Mercury Beating Heart (MBH) oscillator, exhibiting irregular (aperiodic) dynamics, are performed. In the first set of experiments, control over irregular dynamics of the MBH oscillator was obtained via a superimposed periodic voltage signal. These irregular (aperiodic) dynamics were recovered once the control was switched off. Subsequently, two MBH oscillators were coupled to attain synchronization of their aperiodic oscillations. Finally, two uncoupled MBH oscillators were subjected, repeatedly, to a common stochastic forcing, resulting in an enhancement of their mutual phase correlation.

Boundary-induced pattern formation from uniform temporal oscillation

Pattern dynamics triggered by fixing a boundary is investigated. By considering a reaction-diffusion equation that has a unique spatially uniform and limit cycle attractor under a periodic or Neumann boundary condition, and then by choosing a fixed boundary condition, we found three novel phases depending on the ratio of diffusion constants of activator to inhibitor: transformation of temporally periodic oscillation into a spatially periodic fixed pattern, travelling wave emitted from the boundary, and aperiodic spatiotemporal dynamics. The transformation into a fixed, periodic pattern is analyzed by crossing of local nullclines at each spatial point, shifted by diffusion terms, as is analyzed by using recursive equations, to obtain the spatial pattern as an attractor. The generality of the boundary-induced pattern formation as well as its relevance to biological morphogenesis is discussed.

Periodicity, chaos and multiple coexisting attractors in a generalized Moore–Spiegel system

Introduced in 1966 by Moore and Spiegel, the so called Moore–Spiegel system displays aperiodic dynamics that describes the irregular variability of the luminosity of the stars. The Moore–Spiegel system is defined by a jerk system with a single cubic nonlinearity that is responsible for the chaotic dynamics of the whole system. In this contribution, the dynamics of the generalized Moore-Spiegel system recently investigated by [Letellier and Malasoma, Chaos, Solitons & Fractals 69 (2014) 40–49] is considered. Some fundamental dynamical properties of the model such as fixed points, phase portraits, basins of attraction, bifurcation diagrams, and Lyapunov exponents are investigated. Analysis shows that chaos is achieved via period-doubling and symmetry restoring crisis scenarios. One of the major results of this work is the finding of various windows in the parameters’ space in which two, three, four or six different attractors coexist, depending solely on the choice of initial conditions. This unusual and striking phenomenon has not yet been reported previously in the Moore–Spiegel system, and thus deserves dissemination. Compared to some lower dimensional systems capable of six disconnected coexisting periodic and chaotic attractors reported to date, the Moore–Spiegel system represents one of the simplest and the most ‘elegant’ paradigms. Some PSIM based simulations of the nonlinear dynamics of the system are carried out to validate the results of theoretical analyzes.

Online flow cytometry reveals microbial dynamics influenced by concurrent natural and operational events in groundwater used for drinking water treatment.

Detailed measurements of physical, chemical and biological dynamics in groundwater are key to understanding the important processes in place and their influence on water quality – particularly when used for drinking water. Measuring temporal bacterial dynamics at high frequency is challenging due to the limitations in automation of sampling and detection of the conventional, cultivation-based microbial methods. In this study, fully automated online flow cytometry was applied in a groundwater system for the first time in order to monitor microbial dynamics in a groundwater extraction well. Measurements of bacterial concentrations every 15 minutes during 14 days revealed both aperiodic and periodic dynamics that could not be detected previously, resulting in total cell concentration (TCC) fluctuations between 120 and 280 cells μL−1. The aperiodic dynamic was linked to river water contamination following precipitation events, while the (diurnal) periodic dynamic was attributed to changes in hydrological conditions as a consequence of intermittent groundwater extraction. Based on the high number of measurements, the two patterns could be disentangled and quantified separately. This study i) increases the understanding of system performance, ii) helps to optimize monitoring strategies, and iii) opens the possibility for more sophisticated (quantitative) microbial risk assessment of drinking water treatment systems.

Probing and exploiting the chaotic dynamics of a hydrodynamic photochemical oscillator to implement all the basic binary logic functions.

The convective motions within a solution of a photochromic spiro-oxazine being irradiated by UV only on the bottom part of its volume, give rise to aperiodic spectrophotometric dynamics. In this paper, we study three nonlinear properties of the aperiodic time series: permutation entropy, short-term predictability and long-term unpredictability, and degree distribution of the visibility graph networks. After ascertaining the extracted chaotic features, we show how the aperiodic time series can be exploited to implement all the fundamental two-inputs binary logic functions (AND, OR, NAND, NOR, XOR, and XNOR) and some basic arithmetic operations (half-adder, full-adder, half-subtractor). This is possible due to the wide range of states a nonlinear system accesses in the course of its evolution. Therefore, the solution of the convective photochemical oscillator results in hardware for chaos-computing alternative to conventional complementary metal-oxide semiconductor-based integrated circuits.

Supercritical dynamics of domain walls in magnetic films with in-plane anisotropy

Based on the numerical solution of Landau-Lifshitz equations, the nonlinear dynamic behavior of vortex-like domain walls in films with in-plane anisotropy has been investigated in external magnetic fields H significantly exceeding critical fields Hc, above which the stationary motion of domain walls is replaced by nonstationary (periodic or aperiodic) motion. A method has been proposed for the detection of complex aperiodic dynamics of structural rearrangements of domain walls, which is based on the construction of diagrams of the dependences of the tilt angle of the magnetization M with respect to the plane of the domain wall in some of its points on the corresponding angle in other points. It has been found that these diagrams significantly change with variations in the external magnetic field applied along the easy axis of magnetization in the range H > Hc. It has been shown that the pattern of these changes is similar to the scenario of the Feigenbaum transition to dynamic chaos.

Nonlinear Cournot duopoly with implementation delays

We study the effects of two delays on the local as well as on global stability of nonlinear Cournot duopoly dynamics. The two major findings are an analytical construction of the stability switching curve on which stability is lost and the numerical confirmation of the birth of aperiodic global dynamics when the stationary state is locally unstable. The delays matters and can generate various dynamics ranging from simple to complicated dynamics.

A model of development of a spontaneous outbreak of an insect with aperiodic dynamics

Modeling of expansion of alien species in a new environment has become an important issue in view of the recent environmental conditions. Extensive effects of introduction of invasive species for the food chains of ecosystems are described quite regularly. Rapid changes in the form of a “population outbreak” are characteristic of certain insects, and their implications are important for agriculture. Monophagous insects are not only agricultural pests but also potential agents of biological weed control. This paper considers the development of a dynamic system for simulating the scenario of population outbreaks of Cardiaspina albitextura Taylor, 1962, described in the ecological literature. We propose a continuous-event driven computing structure of the new model which takes into account the mortality rate variation at different development stages and the factor of abrupt exhaustion of food resources at the peak of the outbreak.

Stability and tunneling dynamics of a dark-bright soliton pair in a harmonic trap

We consider a binary repulsive Bose-Einstein condensate in a harmonic trap in one spatial dimension and investigate particular solutions consisting of two dark-bright (DB) solitons. There are two different stationary solutions characterized by the phase difference in the bright component, in-phase and out-of-phase states. We show that above a critical particle number in the bright component, a symmetry breaking bifurcation of the pitchfork type occurs that leads to a new asymmetric solution whereas the parental branch, i.e., the out-of-phase state becomes unstable. These three different states support different small amplitude oscillations, characterized by an almost stationary density of the dark component and a tunneling of the bright component between the two dark solitons. Within a suitable effective double-well picture, these can be understood as the characteristic features of a Bosonic Josephson Junction (BJJ), and we show within a two-mode approach that all characteristic features of the BJJ phase space are recovered. For larger deviations from the stationary states, the simplifying double-well description breaks down due to the feedback of the bright component onto the dark one, causing the solitons to move. In this regime we observe intricate anharmonic and aperiodic dynamics, exhibiting remnants of the BJJ phase space.

Adiabatic approximation and aperiodic dynamics of an elliptically polarized light wave in an isotropic gyrotropic nonlinear medium

Adiabatic approximation is used to find an analytical solution to the nonintegrable propagation problem for a plane elliptically polarized light wave in an isotropic gyrotropic medium with local and nonlocal components of Kerr-type nonlinearity and second-order dispersion of group velocity. Aperiodic consistent dynamics of bound (attributable to the nonlinearity) states of two light field components with orthogonal circular polarizations is described. It is shown that this dynamics corresponds to generalization of consistent propagation of ‘soliton–multisoliton complex’ pairs to the cases with nonlinear interaction of periodic solutions—cnoidal waves.

Aperiodic dynamics in a deterministic adaptive network model of attitude formation in social groups

Adaptive network models, in which node states and network topology coevolve, arise naturally in models of social dynamics that incorporate homophily and social influence. Homophily relates the similarity between pairs of nodes’ states to their network coupling strength, whilst social influence causes coupled nodes’ states to convergence. In this paper we propose a deterministic adaptive network model of attitude formation in social groups that includes these effects, and in which the attitudinal dynamics are represented by an activator–inhibitor process. We illustrate that consensus, corresponding to all nodes adopting the same attitudinal state and being fully connected, may destabilise via Turing instability, giving rise to aperiodic dynamics with sensitive dependence on initial conditions. These aperiodic dynamics correspond to the formation and dissolution of sub-groups that adopt contrasting attitudes. We discuss our findings in the context of cultural polarisation phenomena.

Butterfly effect in a chemical oscillator

The strong sensitivity of aperiodic dynamics to initial conditions is one of the fingerprinting features of chaotic systems. While this dependence can be directly verified by means of numerical approaches, it is quite elusive and difficult to be isolated in real experimental systems. In this paper, we discuss a didactic and self-consistent method to show the divergent behaviour between two infinitesimally different solutions of the famous Belousov–Zhabotinsky oscillator simultaneously undergoing a transition to a chaotic regime. Experimental data are also used to give an intuitive visualization of the essential meaning of a Lyapunov exponent, which allows for a more quantitative characterization of the chaotic transient.

Intermittent Early Afterdepolarizations Caused by Bistability

Early afterdepolarization (EAD) is one of the main triggers for cardiac arrhythmias. The intracellular sodium concentration ([Na]i) is critical in regulating both intracellular calcium (Ca) homeostasis and membrane voltage (V). However, the role of [Na]i in the formation of early afterdepolarizations (EADs) is not completely understood. In this study, we applied mathematical modeling to show that slow [Na]i accumulation can lead to a novel form of aperiodic voltage dynamics where a sequence of action potentials (APs) with EADs occurs intermittently during regular periodic pacing. We find that these trains of EADs occur quasi-periodically and lead to intermittent EAD propagation and arrhythmias. The mechanism for these intermittent EADs can be traced to the subtle feedback between Na and Ca fluxes (especially via Na/Ca exchange (NCX) and Na/K-ATPase (IPump)) and their effect on the AP duration (APD). We analyzed the whole system by separating the slow [Na]i from the fast V-[Ca]i subsystem. We found that the fast subsystem exhibits bistability which leads to the observed EADs by forming hysteresis loops as [Na]i slowly accumulates and dissipates. That is, [Na]i can gradually decline during stable short APDs, but this gradually diminishes outward currents via IPump and INCX, which can result in abrupt APD prolongation with EADs. But that long APD gradually reverse the [Na]i decline, and shifts IPump and INCX more outward, which at some point abruptly prevents EADs and reverses APD prolongation. We argue that these intermittent EADs are robust and can occur at physiological [Na]i. Our study further provides a possible novel mechanism for the intermittency of cardiac arrhythmias.