Phase synchronization of chaotic oscillators

Detection of coupling in Duffing oscillator systems

In complex dynamical systems, the detection of coupling and its direction from observed time series is a challenging task. We study coupling in coupled Duffing oscillator systems in regular and chaotic dynamical regimes. By observing the conditional mutual information (CMI) based on the Shannon entropy, we successfully infer the direction of coupling for different system regimes. Moreover, we show that, in the weak coupling limit, the values of CMI can be used to infer the coupling parameters by computing the derivative of the conditional mutual information with respect to the coupling strength, called the information susceptibility. The complete numerical implementation is available at https://repo.ijs.si/mbresar/duffing-cmi.

Synchronization in populations of electrochemical bursting oscillators with chaotic slow dynamics

We investigate the synchronization of coupled electrochemical bursting oscillators using the electrodissolution of iron in sulfuric acid. The dynamics of a single oscillator consisted of slow chaotic oscillations interrupted by a burst of fast spiking, generating a multiple time-scale dynamical system. A wavelet analysis first decomposed the time series data from each oscillator into a fast and a slow component, and the corresponding phases were also obtained. The phase synchronization of the fast and slow dynamics was analyzed as a function of electrical coupling imposed by an external coupling resistance. For two oscillators, a progressive transition was observed: With increasing coupling strength, first, the fast bursting intervals overlapped, which was followed by synchronization of the fast spiking, and finally, the slow chaotic oscillations synchronized. With a population of globally coupled 25 oscillators, the coupling eliminated the fast dynamics, and only the synchronization of the slow dynamics can be observed. The results demonstrated the complexities of synchronization with bursting oscillations that could be useful in other systems with multiple time-scale dynamics, in particular, in neuronal networks.

Source-level EEG and graph theory reveal widespread functional network alterations in focal epilepsy

OBJECTIVE

The hypersynchronous neuronal activity associated with epilepsy causes widespread functional network disruptions extending beyond the epileptogenic zone. This altered network topology is considered a mediator for non-seizure symptoms, such as cognitive impairment. The aim of this study was to investigate functional network alterations in focal epilepsy patients with good seizure control and high quality of life.

METHODS

We compared twenty-two focal epilepsy patients and sixteen healthy controls on graph metrics derived from functional connectivity of source-level resting-state EEG. Graph metrics were calculated over a range of network densities in five frequency bands.

RESULTS

We observed a significantly increased small world index in patients relative to controls. On the local level, two left-hemisphere regions displayed a shift towards greater alpha band "hubness". The findings were not mediated by age, sex or education, nor by age of epilepsy onset, duration or focus lateralisation.

CONCLUSIONS

Widespread functional network alterations are evident in focal epilepsy, even in a cohort characterised by successful anti-seizure medication therapy and high quality of life. These findings might support the position that functional network analysis could hold clinical relevance for epilepsy.

SIGNIFICANCE

Focal epilepsy is accompanied by global and local functional network aberrancies which might be implied in the sustenance of non-seizure symptoms.

Synchronizing Chaos with Imperfections

Previous research on nonlinear oscillator networks has shown that chaos synchronization is attainable for identical oscillators but deteriorates in the presence of parameter mismatches. Here, we identify regimes for which the opposite occurs and show that oscillator heterogeneity can synchronize chaos for conditions under which identical oscillators cannot. This effect is not limited to small mismatches and is observed for random oscillator heterogeneity on both homogeneous and heterogeneous network structures. The results are demonstrated experimentally using networks of Chua's oscillators and are further supported by numerical simulations and theoretical analysis. In particular, we propose a general mechanism based on heterogeneity-induced mode mixing that provides insights into the observed phenomenon. Since individual differences are ubiquitous and often unavoidable in real systems, it follows that such imperfections can be an unexpected source of synchronization stability.

The Study for Synchronization between Two Coupled FitzHugh-Nagumo Neurons Based on the Laplace Transform and the Adomian Decomposition Method

The synchronization between two coupled FitzHugh-Nagumo (FHN) neurons with or without external current is studied by using the Laplace transform and the Adomian decomposition method. Different from other researches, the synchronization error system is expressed as sets of Volterra integral equations based on the convolution theorem in the Laplace transform. Then, it is easy to analytically obtain the conditions that synchronization errors disappear based on the successive approximation method in integral equation theorem, the correctness of which is verified by numerical simulations. Furthermore, the synchronous dynamics of the two coupled FHN neurons also can be written in the form of Volterra integral equations, which is more convenient to analytically solve by using the Adomian decomposition method. It is found that the occurrence of synchronization between the two FHN neurons only depends on the coupling strength and is irrelevant to the external current. Only synchronous rest state in the two FHN neurons without external current can be achieved, while synchronous spikes appear if the external current is not zero.

Enhanced multiresolution wavelet analysis of complex dynamics in nonlinear systems

Multiresolution wavelet analysis (MWA) is a powerful data processing tool that provides a characterization of complex signals over multiple time scales. Typically, the standard deviations of wavelet coefficients are computed depending on the resolution level and such quantities are used as measures for diagnosing different types of system behavior. To enhance the capabilities of this tool, we propose a combination of MWA with detrended fluctuation analysis (DFA) of detail wavelet coefficients. We find that such an MWA&DFA approach is capable of revealing the correlation features of wavelet coefficients in independent ranges of scales, which provide more information about the complex organization of datasets compared to variances or similar statistical measures of the standard MWA. Using this approach, we consider changes in the dynamics of coupled chaotic systems caused by transitions between different types of complex oscillations. We also demonstrate the potential of the MWA&DFA method for characterizing different physiological conditions by analyzing the electrical brain activity in mice.

Asymmetric synchronization in lattices of pinned spiral waves

Networks of coupled oscillators show a wealth of fascinating dynamics and are capable of storing and processing information. In biological and social networks, the coupling is often asymmetric. We use the chirality of rotating spiral waves to introduce this asymmetry in an excitable reaction-diffusion model. The individual vortices are pinned to unexcitable disks and arranged at a constant spacing L along straight lines or simple geometric patterns. In the case of periodic boundaries or pinning disks arranged along the edge of a closed shape, small L values lead to synchronization via repeated wave collisions. The rate of synchronization as a function of L shows a single maximum and is determined by the dispersion behavior of a continuous wave train traveling along the system boundary. For finite systems, spirals are affected by their upstream neighbor, and a single dominant spiral exists along each chain. Specific initial conditions can decouple neighboring vortices even for small L values. We also present a time-delay differential equation that reproduces the phase dynamics in periodic systems.

Coupled phase-amplitude dynamics in heterogeneous metacommunities

Spatial synchrony of population fluctuations is an important tool for predicting regional stability. Its application to natural systems is still limited by the complexity of ecological time series displaying great variation in the frequency and amplitude of their fluctuations, which are not fully resolved by current ecological theories of spatial synchrony. In particular, while environmental fluctuations and limited dispersal can each control the dynamics of frequency and amplitude of population fluctuations, ecological theories of spatial synchrony still need to resolve their role on synchrony and stability in heterogeneous metacommunities. Here, we adopt a heterogeneous predator-prey metacommunity model and study the response of dispersal-driven phase locking and frequency modulation to among-patch heterogeneity in carrying capacity. We find that frequency modulation occurs at intermediate values of dispersal and habitat heterogeneity. We also show how frequency modulation can emerge in metacommunities of autonomously oscillating populations as well as through the forcing of local communities at equilibrium. Frequency modulation was further found to produce temporal variation in population amplitudes, promoting local and regional stability through cyclic patterns of local and regional variability. Our results highlight the importance of approaching spatial synchrony as a non-stationary phenomenon, with implications for the assessment and interpretation of spatial synchrony observed in experimental and natural systems.

Performance comparison of coupling-evaluation methods in discriminating between pregnancy and labor EHG signals

BACKGROUND

Recent years have seen an increased interest in electrohysterogram (EHG) signals as a means to evaluate the synchronization of uterine contractions. Several studies have pointed out that the quality of signal processing - and hence the interpretation of measurement results - is affected significantly by the choice of measurement technique and the presence of non-stationary frequency content in EHG signals. To our knowledge, the effect of time variance on the quality of EHG signal processing has never been fully investigated. How best to process EHG signals with the goal of distinguishing labor-induced contractions from their harmless, pre-labor cousins, remains an open question.

METHOD

Our methodology is based on three pillars. The first consists of a new method for EHG preprocessing in which we apply a second-order Butterworth filter to retain only the EHG fast-wave, low-frequency band (FWL), then use a bivariate piecewise stationary pre-segmentation (bPSP) algorithm to segment the EHG signal into stationary parts. The second pillar addresses the estimation of connectivity and directionality using three methods: nonlinear correlation coefficient (h²), general synchronization (H), and Granger causality (GC). The third pillar is related to signal classification and discrimination between pregnancy and labor using receiver operating curves (ROC) and connectivity and direction maps. For this purpose, we analyze the impact of four factors on data processing efficiency: i) method of connectivity detection, ii) effect of piecewise stationary segmentation preprocessing, iii) retained frequency content and iv) electrode configuration used for EHG recording (bipolar vs. unipolar).

RESULTS

Our results show that piecewise signal segmentation and filtering considerably improves classification performance and statistical significance for some connectivity methods, in particular the h². To this end we propose a new approach (detailed below) for h² called Filtered-Windowed (FW) h² that better highlights the differences between pregnancy and labor in the connectivity matrix and directionality maps.

CONCLUSIONS

This is the first comparative study of the effects of multiple processing factors on connectivity measurement efficiency. Our results indicate that appropriate preprocessing can improve the differentiation of pregnancy and labor-induced contraction signals and may lead to innovative applications in the prevention of preterm labor.

Effect of Stress on Cardiorespiratory Synchronization of Ironman Athletes

The aim of this paper is to investigate the cardiorespiratory synchronization in athletes subjected to extreme physical stress combined with a cognitive stress tasks. ECG and respiration were measured in 14 athletes before and after the Ironman competition. Stroop test was applied between the measurements before and after the Ironman competition to induce cognitive stress. Synchrogram and empirical mode decomposition analysis were used for the first time to investigate the effects of physical stress, induced by the Ironman competition, on the phase synchronization of the cardiac and respiratory systems of Ironman athletes before and after the competition. A cognitive stress task (Stroop test) was performed both pre- and post-Ironman event in order to prevent the athletes from cognitively controlling their breathing rates. Our analysis showed that cardiorespiratory synchronization increased post-Ironman race compared to pre-Ironman. The results suggest that the amount of stress the athletes are recovering from post-competition is greater than the effects of the Stroop test. This indicates that the recovery phase after the competition is more important for restoring and maintaining homeostasis, which could be another reason for stronger synchronization.

Evaluating phase synchronization methods in fMRI: A comparison study and new approaches

In recent years there has been growing interest in measuring time-varying functional connectivity between different brain regions using resting-state functional magnetic resonance imaging (rs-fMRI) data. One way to assess the relationship between signals from different brain regions is to measure their phase synchronization (PS) across time. There are several ways to perform such analyses, and we compare methods that utilize a PS metric together with a sliding window, referred to here as windowed phase synchronization (WPS), with those that directly measure the instantaneous phase synchronization (IPS). In particular, IPS has recently gained popularity as it offers single time-point resolution of time-resolved fMRI connectivity. In this paper, we discuss the underlying assumptions required for performing PS analyses and emphasize the importance of band-pass filtering the data to obtain valid results. Further, we contrast this approach with the use of Empirical Mode Decomposition (EMD) to achieve similar goals. We review various methods for evaluating PS and introduce a new approach within the IPS framework denoted the cosine of the relative phase (CRP). We contrast methods through a series of simulations and application to rs-fMRI data. Our results indicate that CRP outperforms other tested methods and overcomes issues related to undetected temporal transitions from positive to negative associations common in IPS analysis. Further, in contrast to phase coherence, CRP unfolds the distribution of PS measures, which benefits subsequent clustering of PS matrices into recurring brain states.

Lag synchronization of coupled time-delayed FitzHugh-Nagumo neural networks via feedback control

Synchronization plays a significant role in information transfer and decision-making by neurons and brain neural networks. The development of control strategies for synchronizing a network of chaotic neurons with time delays, different direction-dependent coupling (unidirectional and bidirectional), and noise, particularly under external disturbances, is an essential and very challenging task. Researchers have extensively studied the synchronization mechanism of two coupled time-delayed neurons with bidirectional coupling and without incorporating the effect of noise, but not for time-delayed neural networks. To overcome these limitations, this study investigates the synchronization problem in a network of coupled FitzHugh-Nagumo (FHN) neurons by incorporating time delays, different direction-dependent coupling (unidirectional and bidirectional), noise, and ionic and external disturbances in the mathematical models. More specifically, this study investigates the synchronization of time-delayed unidirectional and bidirectional ring-structured FHN neuronal systems with and without external noise. Different gap junctions and delay parameters are used to incorporate time-delay dynamics in both neuronal networks. We also investigate the influence of the time delays between connected neurons on synchronization conditions. Further, to ensure the synchronization of the time-delayed FHN neuronal networks, different adaptive control laws are proposed for both unidirectional and bidirectional neuronal networks. In addition, necessary and sufficient conditions to achieve synchronization are provided by employing the Lyapunov stability theory. The results of numerical simulations conducted for different-sized multiple networks of time-delayed FHN neurons verify the effectiveness of the proposed adaptive control schemes.

Capability of processed EEG parameters to monitor conscious sedation in endoscopy is similar to general anaesthesia

Background

Reliable and safe sedation is a prerequisite for endoscopic interventions. The current standard is rather safe, yet, an objective device to measure sedation depth is missing. To date, anaesthesia monitors based on processed electroencephalogram (EEG) have not been utilised in conscious sedation.

Objective

To investigate EEG parameters to differentiate consciousness in endoscopic propofol sedation.

Methods

In total, 171 patients aged 21–83 years (ASA I–III) undergoing gastrointestinal and bronchial endoscopy were enrolled. Standard monitoring and a frontotemporal two‐channel EEG were recorded. The state of consciousness was identified by repeated requests to squeeze the investigator's hand.

Results

In total, 1132 state‐of‐consciousness (SOC) transitions were recorded in procedures ranging from 5 to 69 min. Thirty‐four EEG parameters from the frequency domain, time‐frequency domain and complexity measures were calculated. Area under the curve ranged from 0.51 to 0.82 with complexity and optimised frequency domain parameters yielding the best results.

Conclusion

Prediction of the SOC with processed EEG parameters is feasible, and the results for sedation in endoscopic procedures are similar to those reported from general anaesthesia. These results are insufficient for a clinical application, but prediction capability may be increased with optimisation and modelling.

Electroencephalogram (EEG)‐based anaesthesia monitors, like the Bispectral Index, have been investigated as an adjunct to monitor propofol sedation in the endoscopy ward. These studies showed very limited benefit.

Capability of processed EEG parameters to differentiate the state of consciousness (SOC) in endoscopy is similar to general anaesthesia.

However, artefacts arising from the less controlled endoscopy environment as compared to anaesthesia limit their use in sedation monitoring.

The Bispectral Index and its underlying parameters are ineffective in the determination of the SOC in sedation during endoscopic procedures.

Bubbling transition as a mechanism of destruction of synchronous oscillations of identical microbubble contrast agents

We study the process of the destruction of synchronous oscillations in a model of two interacting microbubble contrast agents exposed to an external ultrasound field. Completely synchronous oscillations in this model are possible in the case of identical bubbles when the governing system of equations possess a symmetry leading to the existence of a synchronization manifold. Such synchronous oscillations can be destructed without breaking the corresponding symmetry of the governing dynamical system. Here, we describe the phenomenological mechanism responsible for such destruction of synchronization and demonstrate its implementation in the studied model. We show that the appearance and expansion of transversally unstable areas in the synchronization manifold leads to the transformation of a synchronous chaotic attractor into a hyperchaotic one. We also demonstrate that this bifurcation sequence is stable with respect to symmetry breaking perturbations.

Reduction of the synchronization time in random logistic maps

We report on the effects of additive noises in a nonchaotic logistic map. In this system, the Lyapunov exponent changes from negative to positive as the noise intensity is increased. When the Lyapunov exponent is negative, the synchronization of orbits with different initial conditions occurs. We find that the synchronization time cannot be determined solely by the Lyapunov exponent when the noise intensity is greater than a point at which the Lyapunov exponent is minimum. We show that this reduction of the synchronization time is attributed to initial nonstationary behaviors, where the critical point of the logistic map plays an important role.

Natural ELF fields in the atmosphere and in living organisms

Most electrical activity in vertebrates and invertebrates occurs at extremely low frequencies (ELF), with characteristic maxima below 50 Hz. The origin of these frequency maxima is unknown and remains a mystery. We propose that over billions of years during the evolutionary history of living organisms on Earth, the natural electromagnetic resonant frequencies in the atmosphere, continuously generated by global lightning activity, provided the background electric fields for the development of cellular electrical activity. In some animals, the electrical spectrum is difficult to differentiate from the natural background atmospheric electric field produced by lightning. In this paper, we present evidence for the link between the natural ELF fields and those found in many living organisms, including humans.

Decreased connection density and modularity of functional brain networks during n-back working memory paradigm

INTRODUCTION

Investigating how the brain adapts to increased mental workload through large-scale functional reorganization appears as an important research question. Functional connectivity (FC) aims at capturing how disparate regions of the brain dynamically interact, while graph theory provides tools for the topological characterization of the reconstructed functional networks. Although numerous studies investigated how FC is altered in response to increased working memory (WM) demand, current results are still contradictory as few studies confirmed the robustness of these findings in a low-density setting.

METHODS

In this study, we utilized the n-back WM paradigm, in which subjects were presented stimuli (single digits) sequentially, and their task was to decide for each given stimulus if it matched the one presented n-times earlier. Electroencephalography recordings were performed under a control (0-back) and two task conditions of varying difficulty (2- and 3-back). We captured the characteristic connectivity patterns for each difficulty level by performing FC analysis and described the reconstructed functional networks with various graph theoretical measures.

RESULTS

We found a substantial decrease in FC when transitioning from the 0- to the 2- or 3-back conditions, however, no differences relating to task difficulty were identified. The observed changes in brain network topology could be attributed to the dissociation of two (frontal and occipitotemporal) functional modules that were only present during the control condition. Furthermore, behavioral and performance measures showed both positive and negative correlations to connectivity indices, although only in the higher frequency bands.

CONCLUSION

The marked decrease in FC may be due to temporarily abandoned connections that are redundant or irrelevant in solving the specific task. Our results indicate that FC analysis is a robust tool for investigating the response of the brain to increased cognitive workload.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

Uncovering Interaction Between The Loops Of Autonomic Regulation Of Blood Circulation From Long Time Series

The purpose of this work is to study the interaction between the autonomic regulatory loops of blood circulation from long time series. Methods ― We simultaneously recorded four-hour signals of electrocardiogram and photoplethysmogram from the ear and finger of ten healthy adults. We determined the intervals of phase synchronization of the studied regulatory loops and analyzed the dependence of their length on the recording time. The deviations of the total percentage of phase synchronization (index S) from its mean value were estimated in moving non-overlapping windows. Results ― For studied signals we found no significant correlation between the length of synchronization epoch and the time of its beginning. A sharp increase in the deviation of the index S from its mean was shown at the end of the experiment. Conclusion ― The increase in the deviation from the mean at the end of our records is most likely associated more with psychosomatic influences than with hormonal regulation or immobilization stress.

Koopman analysis in oscillator synchronization

Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems, while a global statistical approach is overly cursory. Koopman operator technique seems to balance well the two approaches. In this paper we extend Koopman analysis to the study of synchronization of coupled oscillators by extracting important eigenvalues and eigenfunctions from the observed time series. A renormalization group analysis is designed to derive an analytic approximation of the eigenfunction in the case of weak coupling that dominates the oscillation. For moderate or strong couplings, numerical computation further confirms the importance of the average frequencies and the associated eigenfunctions. The synchronization transition points could be located with quite high accuracy by checking the correlation of neighboring eigenfunctions at different coupling strengths, which is readily applied to other nonlinear systems.

Asymptotic and Finite-Time Cluster Synchronization of Coupled Fractional-Order Neural Networks With Time Delay

This article is devoted to the cluster synchronization issue of coupled fractional-order neural networks. By introducing the stability theory of fractional-order differential systems and the framework of Filippov regularization, some sufficient conditions are derived for ascertaining the asymptotic and finite-time cluster synchronization of coupled fractional-order neural networks, respectively. In addition, the upper bound of the settling time for finite-time cluster synchronization is estimated. Compared with the existing works, the results herein are applicable for fractional-order systems, which could be regarded as an extension of integer-order ones. A numerical example with different cases is presented to illustrate the validity of theoretical results.

Explosive synchronization in temporal networks: A comparative study

We present a comparative study on Explosive Synchronization (ES) in temporal networks consisting of phase oscillators. The temporal nature of the networks is modeled with two configurations: (1) oscillators are allowed to move in a closed two-dimensional box such that they couple with their neighbors and (2) oscillators are static and they randomly switch their coupling partners. Configuration (1) is further studied under two possible scenarios: in the first case, oscillators couple to fixed numbers of neighbors, while, in the other case, they couple to all oscillators lying in their circle of vision. Under these circumstances, we monitor the degrees of temporal networks, velocities, and radius of circle of vision of the oscillators and the probability of forming connections in order to study and compare the critical values of the coupling required to induce ES in the population of phase oscillators.

A Novel Strategy for Complete and Phase Robust Synchronizations of Chaotic Nonlinear Systems

Our work here is to propose a novel technique by which chaos complete and phase synchronizations can be accomplished via a low-cost scheme. We call the proposed technique a “single-state feedback track synchronization control algorithm”. A single-state feedback track synchronization control algorithm is designed so that both complete and phase synchronizations can be accomplished using the same controller. Complete synchronization between two chaotic systems means complete symmetry between them, but phase synchronization means complete symmetry with a phase shift. In addition, the proposed method is applied to the synchronization of two identical chaotic Lorenz models. The effectiveness and robustness of the proposed algorithm is well illustrated via exhaustive numerical simulation experiments based on the Matlab toolbox of the powerful genetic algorithm. The robustness of the proposed algorithm motivated us to apply this method of synchronization in a secure communication application.

Synchronization within synchronization: transients and intermittency in ecological networks

Transients are fundamental to ecological systems with significant implications to management, conservation and biological control. We uncover a type of transient synchronization behavior in spatial ecological networks whose local dynamics are of the chaotic, predator–prey type. In the parameter regime where there is phase synchronization among all the patches, complete synchronization (i.e. synchronization in both phase and amplitude) can arise in certain pairs of patches as determined by the network symmetry—henceforth the phenomenon of  ‘synchronization within synchronization.’ Distinct patterns of complete synchronization coexist but, due to intrinsic instability or noise, each pattern is a transient and there is random, intermittent switching among the patterns in the course of time evolution. The probability distribution of the transient time is found to follow an algebraic scaling law with a divergent average transient lifetime. Based on symmetry considerations, we develop a stability analysis to understand these phenomena. The general principle of symmetry can also be exploited to explain previously discovered, counterintuitive synchronization behaviors in ecological networks.

Universal generation of devil's staircases near Hopf bifurcations via modulated forcing of nonlinear systems

The discrete circle map is the archetypical example of a driven periodic system, showing a complex resonance structure under a change of the forcing frequency known as the devil's staircase. Adler's equation can be seen as the direct continuous equivalent of the circle map, describing locking effects in periodic systems with continuous forcing. This type of locking produces a single fundamental resonance tongue without higher-order resonances, and a devil's staircase is not observed. We show that, with harmonically modulated forcing, nonlinear oscillations close to a Hopf bifurcation generically reproduce the devil's staircase even in the continuous case. Experimental results on a semiconductor laser driven by a modulated optical signal show excellent agreement with our theoretical predictions. The locking appears as a modulation of the oscillation amplitude as well as the angular oscillation frequency. Our results show that by proper implementation of an external drive, additional regions of stable frequency locking can be introduced in systems which originally show only a single Adler-type resonance tongue. The induced resonances can be precisely controlled via the modulation parameters.

Fingerprint of volcanic forcing on the ENSO-Indian monsoon coupling

Coupling of the El Niño-Southern Oscillation (ENSO) and Indian monsoon (IM) is central to seasonal summer monsoon rainfall predictions over the Indian subcontinent, although a nonstationary relationship between the two nonlinear phenomena can limit seasonal predictability. Radiative effects of volcanic aerosols injected into the stratosphere during large volcanic eruptions (LVEs) tend to alter ENSO evolution; however, their impact on ENSO-IM coupling remains unclear. Here, we investigate how LVEs influence the nonlinear behavior of the ENSO and IM dynamical systems using historical data, 25 paleoclimate reconstructions, last-millennium climate simulations, large-ensemble targeted climate sensitivity experiments, and advanced analysis techniques. Our findings show that LVEs promote a significantly enhanced phase-synchronization of the ENSO and IM oscillations, due to an increase in the angular frequency of ENSO. The results also shed innovative insights into the physical mechanism underlying the LVE-induced enhancement of ENSO-IM coupling and strengthen the prospects for improved seasonal monsoon predictions.

Metastable neural dynamics underlies cognitive performance across multiple behavioural paradigms

Despite resting state networks being associated with a variety of cognitive abilities, it remains unclear how these local areas act in concert to express particular cognitive operations. Theoretical and empirical accounts indicate that large-scale resting state networks reconcile dual tendencies towards integration and segregation by operating in a metastable regime of their coordination dynamics. Metastability may confer important behavioural qualities by binding distributed local areas into large-scale neurocognitive networks. We tested this hypothesis by analysing fMRI data in a large cohort of healthy individuals (N = 566) and comparing the metastability of the brain's large-scale resting network architecture at rest and during the performance of several tasks. Metastability was estimated using a well-defined collective variable capturing the level of 'phase-locking' between large-scale networks over time. Task-based reasoning was principally characterised by high metastability in cognitive control networks and low metastability in sensory processing areas. Although metastability between resting state networks increased during task performance, cognitive ability was more closely linked to spontaneous activity. High metastability in the intrinsic connectivity of cognitive control networks was linked to novel problem solving or fluid intelligence, but was less important in tasks relying on previous experience or crystallised intelligence. Crucially, subjects with resting architectures similar or 'pre-configured' to a task-general arrangement demonstrated superior cognitive performance. Taken together, our findings support a key linkage between the spontaneous metastability of large-scale networks in the cerebral cortex and cognition.

Basin of attraction for chimera states in a network of Rössler oscillators

Chimera states are spatiotemporal patterns in which coherent and incoherent dynamics coexist simultaneously. These patterns were observed in both locally and nonlocally coupled oscillators. We study the existence of chimera states in networks of coupled Rössler oscillators. The Rössler oscillator can exhibit periodic or chaotic behavior depending on the control parameters. In this work, we show that the existence of coherent, incoherent, and chimera states depends not only on the coupling strength, but also on the initial state of the network. The initial states can belong to complex basins of attraction that are not homogeneously distributed. Due to this fact, we characterize the basins by means of the uncertainty exponent and basin stability. In our simulations, we find basin boundaries with smooth, fractal, and riddled structures.

Predicting phase and sensing phase coherence in chaotic systems with machine learning

Recent interest in exploiting machine learning for model-free prediction of chaotic systems focused on the time evolution of the dynamical variables of the system as a whole, which include both amplitude and phase. In particular, in the framework based on reservoir computing, the prediction horizon as determined by the largest Lyapunov exponent is often short, typically about five or six Lyapunov times that contain approximately equal number of oscillation cycles of the system. There are situations in the real world where the phase information is important, such as the ups and downs of species populations in ecology, the polarity of a voltage variable in an electronic circuit, and the concentration of certain chemical above or below the average. Using classic chaotic oscillators and a chaotic food-web system from ecology as examples, we demonstrate that reservoir computing can be exploited for long-term prediction of the phase of chaotic oscillators. The typical prediction horizon can be orders of magnitude longer than that with predicting the entire variable, for which we provide a physical understanding. We also demonstrate that a properly designed reservoir computing machine can reliably sense phase synchronization between a pair of coupled chaotic oscillators with implications to the design of the parallel reservoir scheme for predicting large chaotic systems.

Gait coordination in overground walking with a virtual reality avatar

Little information is currently available on interpersonal gait synchronization in overground walking. This is caused by difficulties in continuous gait monitoring over many steps while ensuring repeatability of experimental conditions. These challenges could be overcome by using immersive virtual reality (VR), assuming it offers ecological validity. To this end, this study provides some of the first evidence of gait coordination patterns for overground walking dyads in VR. Six subjects covered the total distance of 27 km while walking with a pacer. The pacer was either a real human subject or their anatomically and biomechanically representative VR avatar driven by an artificial intelligence algorithm. Side-by-side and front-to-back arrangements were tested without and with the instruction to synchronize steps. Little evidence of spontaneous gait coordination was found in both visual conditions, but persistent gait coordination patterns were found in the case of intentional synchronization. Front-to-back rather than side-by-side arrangement consistently yielded in the latter case higher mean synchronization strength index. Although the mean magnitude of synchronization strength index was overall comparable in both visual conditions when walking under the instruction to synchronize steps, quantitative and qualitative differences were found which might be associated with common limitations of VR solutions.

Intermittent theta burst stimulation of the right dorsolateral prefrontal cortex accelerates visuomotor adaptation with delayed feedback

Implicit adaptation to visual rotations during fast reaching is a well-recognized function of the cerebellum. However, there is still no well-established understanding of the neural underpinnings that support explicit processes during visuomotor adaptation. We tested the causative involvement of dorsolateral prefrontal cortex (DLPFC) in an adaptive reaching task by employing excitatory intermittent theta burst stimulation (iTBS) to left or right DLPFC during learning to adapt to a sudden large visual rotation with delayed terminal feedback. Spontaneous resting-state electroencephalography (EEG) signals were recorded before and immediately after the administration of iTBS. iTBS to right DLPFC, compared to left DLPFC or control, induced faster adaptation to the rotation and had a greater adjustment of aiming directions in early adaptation trials. Moreover, resting-state functional connectivity of EEG of the frontal cortex after iTBS predicted subsequent adaptation rate. These results suggest a critical role of right DLPFC in supporting explicit learning in the adaptive reaching task.

Introduction to Focus Issue: Symmetry and optimization in the synchronization and collective behavior of complex systems

Synchronization phenomena and collective behavior are commonplace in complex systems with applications ranging from biological processes such as coordinated neuron firings and cell cycles to the stability of alternating current power grids. A fundamental pursuit is the study of how various types of symmetry-e.g., as manifest in network structure or coupling dynamics-impact a system's collective behavior. Understanding the intricate relations between structural and dynamical symmetry/asymmetry also provides new paths to develop strategies that enhance or inhibit synchronization. Previous research has revealed symmetry as a key factor in identifying optimization mechanisms, but the particular ways that symmetry/asymmetry influence collective behavior can generally depend on the type of dynamics, networks, and form of synchronization (e.g., phase synchronization, group synchronization, and chimera states). Other factors, such as time delay, noise, time-varying structure, multilayer connections, basin stability, and transient dynamics, also play important roles, and many of these remain underexplored. This Focus Issue brings together a survey of theoretical and applied research articles that push forward this important line of questioning.

Can we detect clusters of chaotic dynamical networks via causation entropy?

It is known that chaotic dynamical systems in the coupled networks can synchronize, and they can even form clusters. Our study addresses the issue of determining the membership information of continuous-time dynamical networks forming clusters. We observe the output vectors of individual systems in the networks and reconstruct the state space according to Takens' embedding theorem. Afterward, we estimate the information-theoretic measures in the reconstructed state space. We propose the average integrated causation entropy as a model-free distinctive measure to distinguish the clusters in the network using the k-means clustering algorithm. We have demonstrated the proposed procedure on three networks that contain Chua systems. The results indicate that we can determine the members of clusters and the membership information from the data, conclusively.

Cross-frequency phase coupling of brain oscillations and relevance attribution as saliency detection in abstract reasoning

reasoning is associated with the ability to detect relations among objects, ideas, events. It underlies the understanding of other individuals' thoughts and intentions. In natural settings, individuals have to infer relevant associations that have proven to be reliable or precise predictors. Salience theory suggests that the attribution of meaning to stimulus depends on their contingency, saliency, and relevance to adaptation. So far, subjective estimates of relevance have mostly been explored in motivation and implicit learning. Mechanisms underlying formation of associations in abstract thinking with regard to their subjective relevance, or salience, are not clear. Applying novel computational methods, we investigated relevance detection in categorization tasks in 17 healthy individuals. Two models of relevance detection were developed: a conventional one with nouns from the same semantic category, an aberrant one based on an insignificant common feature. Control condition introduced non-related words. The participants were to detect either a relevant principle or an insignificant feature to group presented words. In control condition they inferred that the stimuli were irrelevant to any grouping idea. Cross-frequency phase coupling analysis revealed statistically distinct patterns of synchronization representing search and decision in the models of normal and aberrant relevance detection. Significantly distinct frontotemporal functional networks with central and parietal components in the theta and alpha frequency bands may reflect differences in relevance detection.

Phase coherence is not related to topology

Phase coherence is an important measure in nonlinear science. Whereas there is no generally accepted definition for phase and therefore for phase coherence, many works associate this feature with topological aspects of the systems, such as having a well-defined rotating center. Given the relevance of this concept for synchronization problems, one aim of this paper is to argue by means of a couple of counterexamples that phase coherence is not related to the topology of the attractor. A second aim is to introduce a phase-coherence measure based on recurrence plots, for which probabilities of recurrences for two different trajectories are similar for a phase-coherent system and dissimilar for non-phase-coherent systems. The measure does not require a phase variable defined a priori.

Projective Synchronization of Delayed Neural Networks With Mismatched Parameters and Impulsive Effects

In this paper, the impulsive effects on projective synchronization between the parameter mismatched neural networks with mixed time-varying delays have been analyzed. Since complete synchronization is not possible due to the existence of parameter mismatch and projective factor, a drive has been taken to achieve the weak projective synchronization of different neural networks under impulsive control strategies. Through the use of matrix measure technique and the extended comparison principle based on the formula of variation of parameters for mixed time-varying delayed impulsive systems, sufficient criteria have been derived for exponential convergence of the networks under the effects of extensive range of impulse. Instead of upper or lower bound of the impulsive interval, the concept of the average impulsive interval is applied to estimate the number of impulsive points in an interval. The concept of calculus is applied for optimizing the synchronization error bounds which are obtained because of different ranges of impulse. Finally, the numerical simulations for various impulsive ranges for different cases are presented graphically to validate the efficiency of the theoretical results.

Origin of amplitude synchronization in coupled nonidentical oscillators

The origin of amplitude synchronization (AS), or amplitude envelope synchronization, as a peculiar form of strong correlation between amplitudes of oscillators is studied by using a model of coupled Landau-Stuart periodic oscillators. We find that the AS extensively occurs within the traditional phase drift region, and the amplitude correlation does not change with variation of the coupling strength but is dampened with increase of the frequency mismatch. The AS appears only at weak couplings and before the occurrence of phase synchronization (PS), and the oscillator amplitude is modulated by its phase. This study could build a solid foundation for AS, which has not drawn much attention in the nonlinear dynamics field before, providing a clear physical picture for synchronization including not only PS, but also AS, and arousing general interest in many interdisciplinary fields, such as neuronal systems, laser dynamics, nanomechanical resonators, and power systems, etc., where phase and amplitude are always mutually influenced and both are important.

Long-term potentiation enhancing effect of epileptic insult in the CA1 area is dependent on prior-application of primed-burst stimulation

Herein field recordings were utilized to test the effects of a transient period of pentylenetetrazol (PTZ) treatment on theta-burst long-term potentiation (LTP) at the Schaffer collateral-CA1 synapses as well as RT-PCR was used to investigate the effects of the combination of the pharmacological treatment and the theta-burst LTP induction on the expression of NMDA subunit mRNA in hippocampal slices. The slope of field excitatory postsynaptic potential (fEPSP) was unaffected while the population spike amplitude and area were increased by a transient period of PTZ treatment (3 mM, 10 min). After a theta burst, a brief PTZ exposure can lead to an enhancement of LTP as documented by fEPSP recording. The effect can be blocked by a selective NMDA receptor antagonist DL-AP5. An increase in the expression of GluN2B and GluN2A subunit mRNAs was also shown due to the combined treatment. The results indicate that the combined treatment increases the degree of NMDA-dependent LTP and are in accord with literature data on the subunit alterations of the hippocampal NMDA receptors. Moreover, our experimental paradigm can be used as a new approach to study the relevance of LTP-like phenomena and epileptic mechanisms.

Discontinuity-induced intermittent synchronization transitions in coupled non-smooth systems

The synchronization transition in coupled non-smooth systems is studied for increasing coupling strength. The average order parameter is calculated to diagnose synchronization of coupled non-smooth systems. It is found that the coupled non-smooth system exhibits an intermittent synchronization transition from the cluster synchronization state to the complete synchronization state, depending on the coupling strength and initial conditions. Detailed numerical analyses reveal that the discontinuity always plays an important role in the synchronization transition of the coupled non-smooth system. In addition, it is found that increasing the coupling strength leads to the coexistence of periodic cluster states. Detailed research illustrates that the periodic clusters consist of two or more coexisting periodic attractors. Their periodic trajectory passes from one region to another region that is divided by discontinuous boundaries in the phase space. The mutual interactions of the local nonlinearity and the spatial coupling ultimately result in a stable periodic trajectory.