The Shannon entropy of a time series is a standard measure to assess the complexity of a dynamical process and can be used to quantify transitions between different dynamical regimes. An alternative way of quantifying complexity is based on state recurrences, such as those available in recurrence quantification analysis. Although varying definitions for recurrence-based entropies have been suggested so far, for some cases they reveal inconsistent results. Here we suggest a method based on weighted recurrence plots and show that the associated Shannon entropy is positively correlated with the largest Lyapunov exponent. We demonstrate the potential on a prototypical example as well as on experimental data of a chemical experiment.
This paper is concerned with the problem of pinning synchronization of nonlinear dynamical networks with multiple stochastic disturbances. Two kinds of pinning schemes are considered: 1) pinned nodes are fixed along the time evolution and 2) pinned nodes are switched from time to time according to a set of Bernoulli stochastic variables. Using Lyapunov function methods and stochastic analysis techniques, several easily verifiable criteria are derived for the problem of pinning distributed synchronization. For the case of fixed pinned nodes, a novel mixed optimization method is developed to select the pinned nodes and find feasible solutions, which is composed of a traditional convex optimization method and a constraint optimization evolutionary algorithm. For the case of switching pinning scheme, upper bounds of the convergence rate and the mean control gain are obtained theoretically. Simulation examples are provided to show the advantages of our proposed optimization method over previous ones and verify the effectiveness of the obtained results.
Mathematical modeling approaches are becoming ever more established in clinical neuroscience. They provide insight that is key to understanding complex interactions of network phenomena, in general, and interactions within the migraine-generator network, in particular.
We deduce rigorous conditions for the onset of amplitude death (AD) and oscillation death (OD) in a system of identical coupled paradigmatic Stuart-Landau oscillators. A nonscalar coupling and high frequency are beneficial for the onset of AD. In strong contrast, scalar diffusive coupling and low intrinsic frequency are in favor of the emergence of OD. Our finding contributes to clearly distinguish intrinsic geneses for AD and OD, and further substantially corroborates that AD and OD are indeed two dynamically distinct oscillation quenching phenomena due to distinctly different mechanisms.
This paper presents an analytical study of synchronization in an array of output-coupled temporal Boolean networks. A temporal Boolean network (TBN) is a logical dynamic system developed to model Boolean networks with regulatory delays. Both state delay and output delay are considered, and these two delays are assumed to be different. By referring to the algebraic representations of logical dynamics and using the semi-tensor product of matrices, the output-coupled TBNs are firstly converted into a discrete-time algebraic evolution system, and then the relationship between the states of coupled TBNs and the initial state sequence is obtained. Then, some necessary and sufficient conditions are derived for the synchronization of an array of TBNs with an arbitrary given initial state sequence. Two numerical examples including one epigenetic model are finally given to illustrate the obtained results.
In this paper, we develop an approach to achieve either frequency or amplitude modulation of an oscillator merely through feedback control. We present and implement a unified theory of our approach for any finite-dimensional continuous dynamical system that exhibits oscillatory behavior. The approach is illustrated not only for the normal forms of dynamical systems but also for representative biological models, such as the isolated and coupled FitzHugh-Nagumo model. We demonstrate the potential usefulness of our approach to uncover the mechanisms of frequency and amplitude modulations experimentally observed in a wide range of real systems.
A method to identify distinct dynamical regimes and transitions between those regimes in a short univariate time series was recently introduced [N. Malik et al., Europhys. Lett. 97, 40009 (2012)], employing the computation of fluctuations in a measure of nonlinear similarity based on local recurrence properties. In this work, we describe the details of the analytical relationships between this newly introduced measure and the well-known concepts of attractor dimensions and Lyapunov exponents. We show that the new measure has linear dependence on the effective dimension of the attractor and it measures the variations in the sum of the Lyapunov spectrum. To illustrate the practical usefulness of the method, we identify various types of dynamical transitions in different nonlinear models. We present testbed examples for the new method's robustness against noise and missing values in the time series. We also use this method to analyze time series of social dynamics, specifically an analysis of the US crime record time series from 1975 to 1993. Using this method, we find that dynamical complexity in robberies was influenced by the unemployment rate until the late 1980s. We have also observed a dynamical transition in homicide and robbery rates in the late 1980s and early 1990s, leading to increase in the dynamical complexity of these rates.
The vulnerability to real-life networks against small initial attacks has been one of outstanding challenges in the study of interrelated networks. We study cascading failures in two interrelated networks S and B composed from dependency chains and connectivity links respectively. This work proposes a realistic model for cascading failures based on the redistribution of traffic flow. We study the Barabási-Albert networks (BA) and Erd?s-Rényi graphs (ER) with such structure, and found that the efficiency sharply decreases with increasing percentages of the dependency nodes for removing a node randomly. Furthermore, we study the robustness of interrelated traffic networks, especially the subway and bus network in Beijing. By analyzing different attacking strategies, we uncover that the efficiency of the city traffic system has a non-equilibrium phase transition at low capacity of the networks. This explains why the pressure of the traffic overload is relaxed by singly increasing the number of small buses during rush hours. We also found that the increment of some buses may release traffic jam caused by removing a node of the bus network randomly if the damage is limited. However, the efficiencies to transfer people flow will sharper increase when the capacity of the subway network ?(S) > ?0.
The study of the foraging behavior of group animals (especially ants) is of practical ecological importance, but it also contributes to the development of widely applicable optimization problem-solving techniques. Biologists have discovered that single ants exhibit low-dimensional deterministic-chaotic activities. However, the influences of the nest, ants' physical abilities, and ants' knowledge (or experience) on foraging behavior have received relatively little attention in studies of the collective behavior of ants. This paper provides new insights into basic mechanisms of effective foraging for social insects or group animals that have a home. We propose that the whole foraging process of ants is controlled by three successive strategies: hunting, homing, and path building. A mathematical model is developed to study this complex scheme. We show that the transition from chaotic to periodic regimes observed in our model results from an optimization scheme for group animals with a home. According to our investigation, the behavior of such insects is not represented by random but rather deterministic walks (as generated by deterministic dynamical systems, e.g., by maps) in a random environment: the animals use their intelligence and experience to guide them. The more knowledge an ant has, the higher its foraging efficiency is. When young insects join the collective to forage with old and middle-aged ants, it benefits the whole colony in the long run. The resulting strategy can even be optimal.
The cheapest and thus widespread way to add new generators to a high-voltage power grid is by a simple tree-like connection scheme. However, it is not entirely clear how such locally cost-minimizing connection schemes affect overall system performance, in particular the stability against blackouts. Here we investigate how local patterns in the network topology influence a power grid's ability to withstand blackout-prone large perturbations. Employing basin stability, a nonlinear concept, we find in numerical simulations of artificially generated power grids that tree-like connection schemes--so-called dead ends and dead trees--strongly diminish stability. A case study of the Northern European power system confirms this result and demonstrates that the inverse is also true: repairing dead ends by addition of a few transmission lines substantially enhances stability. This may indicate a topological design principle for future power grids: avoid dead ends.
Control gains play an important role in the control of a natural or a technical system since they reflect how much resource is required to optimize a certain control objective. This paper is concerned with the controllability of neuronal networks with constraints on the average value of the control gains injected in driver nodes, which are in accordance with engineering and biological backgrounds. In order to deal with the constraints on control gains, the controllability problem is transformed into a constrained optimization problem (COP). The introduction of the constraints on the control gains unavoidably leads to substantial difficulty in finding feasible as well as refining solutions. As such, a modified dynamic hybrid framework (MDyHF) is developed to solve this COP, based on an adaptive differential evolution and the concept of Pareto dominance. By comparing with statistical methods and several recently reported constrained optimization evolutionary algorithms (COEAs), we show that our proposed MDyHF is competitive and promising in studying the controllability of neuronal networks. Based on the MDyHF, we proceed to show the controlling regions under different levels of constraints. It is revealed that we should allocate the control gains economically when strong constraints are considered. In addition, it is found that as the constraints become more restrictive, the driver nodes are more likely to be selected from the nodes with a large degree. The results and methods presented in this paper will provide useful insights into developing new techniques to control a realistic complex network efficiently.
Low-dimensional behavior of large systems of globally coupled oscillators has been intensively investigated since the introduction of the Ott-Antonsen ansatz. In this report, we generalize the Ott-Antonsen ansatz to second-order Kuramoto models in complex networks. With an additional inertia term, we find a low-dimensional behavior similar to the first-order Kuramoto model, derive a self-consistent equation and seek the time-dependent derivation of the order parameter. Numerical simulations are also conducted to verify our analytical results.
Spontaneous explosive emergent behavior takes place in heterogeneous networks when the frequencies of the nodes are positively correlated to the node degree. A central feature of such explosive transitions is a hysteretic behavior at the transition to synchronization. We unravel the underlying mechanisms and show that the dynamical origin of the hysteresis is a change of basin of attraction of the synchronization state. Our findings hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, reveal how microscopic network parameters such as node degree and frequency affect the global network properties and can be used for network design and control.
In this paper a new learning rule for the coupling weights tuning of Hopfield like chaotic neural networks is developed in such a way that all neurons behave in a synchronous manner, while the desirable structure of the network is preserved during the learning process. The proposed learning rule is based on sufficient synchronization criteria, on the eigenvalues of the weight matrix belonging to the neural network and the idea of Structured Inverse Eigenvalue Problem. Our developed learning rule not only synchronizes all neuron's outputs with each other in a desirable topology, but also enables us to enhance the synchronizability of the networks by choosing the appropriate set of weight matrix eigenvalues. Specifically, this method is evaluated by performing simulations on the scale-free topology.
Complex network approaches have recently been applied to continuous spatial dynamical systems, like climate, successfully uncovering the system's interaction structure. However the relationship between the underlying atmospheric or oceanic flow's dynamics and the estimated network measures have remained largely unclear. We bridge this crucial gap in a bottom-up approach and define a continuous analytical analogue of Pearson correlation networks for advection-diffusion dynamics on a background flow. Analysing complex networks of prototypical flows and from time series data of the equatorial Pacific, we find that our analytical model reproduces the most salient features of these networks and thus provides a general foundation of climate networks. The relationships we obtain between velocity field and network measures show that line-like structures of high betweenness mark transition zones in the flow rather than, as previously thought, the propagation of dynamical information.
Event-related potentials provide strong evidence for a face-specific process that peaks at about 170 ms following stimulus onset--the N170 effect. The N170 has been shown to be sensitive to adaptation, reflected in an amplitude reduction by repeated face presentation, which is usually considered to be driven by bottom-up processes. Here we investigated whether the N170 adaptation profile can be modulated by top-down factors aiming at holistic or feature-based processing. Adaptor stimuli were Mooney faces, isolated facial features (eyes or mouths), or houses. Target faces required either a gender decision (holistic task), or a decision on a facial feature (detail task). We observed an intricate crossover interaction pattern, reflected in opposite effects on adaptation to Mooney faces and eyes as compared to mouth conditions. These findings provide evidence that adaptation effects can be modulated by top-down processes.
Sleep disorders are a major risk factor for cardiovascular diseases. Sleep apnea is the most common sleep disturbance and its detection relies on a polysomnography, i.e., a combination of several medical examinations performed during a monitored sleep night. In order to detect occurrences of sleep apnea without the need of combined recordings, we focus our efforts on extracting a quantifier related to the events of sleep apnea from a cardiovascular time series, namely systolic blood pressure (SBP). Physiologic time series are generally highly nonstationary and entrap the application of conventional tools that require a stationary condition. In our study, data nonstationarities are uncovered by a segmentation procedure which splits the signal into stationary patches, providing local quantities such as mean and variance of the SBP signal in each stationary patch, as well as its duration L. We analysed the data of 26 apneic diagnosed individuals, divided into hypertensive and normotensive groups, and compared the results with those of a control group. From the segmentation procedure, we identified that the average duration , as well as the average variance 2>, are correlated to the apnea-hypoapnea index (AHI), previously obtained by polysomnographic exams. Moreover, our results unveil an oscillatory pattern in apneic subjects, whose amplitude S* is also correlated with AHI. All these quantities allow to separate apneic individuals, with an accuracy of at least 79%. Therefore, they provide alternative criteria to detect sleep apnea based on a single time series, the systolic blood pressure.
Computational methods have complemented experimental and clinical neurosciences and led to improvements in our understanding of the nervous systems in health and disease. In parallel, neuromodulation in form of electric and magnetic stimulation is gaining increasing acceptance in chronic and intractable diseases. In this paper, we firstly explore the relevant state of the art in fusion of both developments towards translational computational neuroscience. Then, we propose a strategy to employ the new theoretical concept of dynamical network biomarkers (DNB) in episodic manifestations of chronic disorders. In particular, as a first example, we introduce the use of computational models in migraine and illustrate on the basis of this example the potential of DNB as early-warning signals for neuromodulation in episodic migraine.
Sleep is a physiological process with an internal program of a number of well defined sleep stages and intermediate wakefulness periods. The sleep stages modulate the autonomous nervous system and thereby the sleep stages are accompanied by different regulation regimes for the cardiovascular and respiratory system. The differences in regulation can be distinguished by new techniques of cardiovascular physics. The number of patients suffering from sleep disorders increases unproportionally with the increase of the human population and aging, leading to very high expenses in the public health system. Therefore, the challenge of cardiovascular physics is to develop highly-sophisticated methods which are able to, on the one hand, supplement and replace expensive medical devices and, on the other hand, improve the medical diagnostics with decreasing the patients risk. Methods of cardiovascular physics are used to analyze heart rate, blood pressure and respiration to detect changes of the autonomous nervous system in different diseases. Data driven modeling analysis, synchronization and coupling analysis and their applications to biosignals in healthy subjects and patients with different sleep disorders are presented. Newly derived methods of cardiovascular physics can help to find indicators for these health risks.
The analysis of symbolic dynamics applied to physiological time series is able to retrieve information about dynamical properties of the underlying system that cannot be gained with standard methods like e.g. spectral analysis. Different approaches for the transformation of the original time series to the symbolic time series have been proposed. Yet the differences between the approaches are unknown. In this study three different transformation methods are investigated: (1) symbolization according to the deviation from the average time series, (2) symbolization according to several equidistant levels between the minimum and maximum of the time series, (3) binary symbolization of the first derivative of the time series. Each method was applied to the cardiac interbeat interval series RRi and its difference ?RRI of 17 healthy subjects obtained during head-up tilt testing. The symbolic dynamics of each method is analyzed by means of the occurrence of short sequences (words) of length 3. The occurrence of words is grouped according to words without variations of the symbols (0V%), words with one variation (1V%), two like variations (2LV%) and two unlike variations (2UV%). Linear regression analysis showed that for method 1 0V%, 1V%, 2LV% and 2UV% changed with increasing tilt angle. For method 2 0V%, 2LV% and 2UV% changed with increasing tilt angle and method 3 showed changes for 0V% and 1V%. In conclusion, all methods are capable of reflecting changes of the cardiac autonomic nervous system during head-up tilt. All methods show that even the analysis of very short symbolic sequences is capable of tracking changes of the cardiac autonomic regulation during head-up tilt testing.
In this paper, we investigate the multiobjective identification of controlling areas in the neuronal network of a cats brain by considering two measures of controllability simultaneously. By utilizing nondominated sorting mechanisms and composite differential evolution (CoDE), a reference-point-based nondominated sorting composite differential evolution (RP-NSCDE) is developed to tackle the multiobjective identification of controlling areas in the neuronal network. The proposed RP-NSCDE shows its promising performance in terms of accuracy and convergence speed, in comparison to nondominated sorting genetic algorithms II. The proposed method is also compared with other representative statistical methods in the complex network theory, single objective, and constraint optimization methods to illustrate its effectiveness and reliability. It is shown that there exists a tradeoff between minimizing two objectives, and therefore pareto fronts (PFs) can be plotted. The developed approaches and findings can also be applied to coordination control of various kinds of real-world complex networks including biological networks and social networks, and so on.
In this paper, we study the controllability of networks with different numbers of communities and various strengths of community structure. By means of simulations, we show that the degree descending pinning scheme performs best among several considered pinning schemes under a small number of pinned nodes, while the degree ascending pinning scheme is becoming more powerful by increasing the number of pinned nodes. It is found that increasing the number of communities or reducing the strength of community structure is beneficial for the enhancement of the controllability. Moreover, it is revealed that the pinning scheme with evenly distributed pinned nodes among communities outperforms other kinds of considered pinning schemes.
Amplitude death (AD) and oscillation death (OD) are two structurally different oscillation quenching types in coupled nonlinear oscillators. The transition from AD to OD has been recently realized due to the interplay between heterogeneity and coupling strength [A. Koseska et al., Phys. Rev. Lett. 111, 024103 (2013)]. We identify here the transition from AD to OD in nonlinear oscillators with couplings of distinct natures. It is demonstrated that the presence of time delay in the coupling cannot induce such a transition in identical oscillators, but it can indeed facilitate its occurrence with a low degree of heterogeneity. Moreover, it is further shown that the AD to OD transition is reliably observed in identical oscillators with dynamic and conjugate couplings. The coexistence of AD and OD and rich stable OD configurations after the transition are revealed, which are of great significance for potential applications in physics, biology, and control studies.
In this paper, we investigate the multiobjective identification of controlling areas in the neuronal network of cats’ brain by considering two measures of controllability simultaneously. By utilizing nondominated sorting mechanisms and composite differential evolution (CoDE), a reference point based nondominated sorting composite differential evolution (RP-NSCDE) is developed to tackle the multiobjective identification of controlling areas in the neuronal network. The proposed RP-NSCDE shows its promising performance in terms of accuracy and convergence speed, in comparison to nondominated sorting genetic algorithms II. The proposed method is also compared with other representative statistical methods in complex network theory, single objective and constraint optimization methods to illustrate its effectiveness and reliability. It is shown that there exists a trade-off between minimizing two objectives, and therefore pareto fronts (PFs) can be plotted. The developed approaches and findings can also be applied to coordination control of various kinds of real-world complex networks including biological networks and social networks, etc.
Characterizing complex patterns arising from horizontal oil-water two-phase flows is a contemporary and challenging problem of paramount importance. We design a new multisector conductance sensor and systematically carry out horizontal oil-water two-phase flow experiments for measuring multivariate signals of different flow patterns. We then infer multivariate recurrence networks from these experimental data and investigate local cross-network properties for each constructed network. Our results demonstrate that a cross-clustering coefficient from a multivariate recurrence network is very sensitive to transitions among different flow patterns and recovers quantitative insights into the flow behavior underlying horizontal oil-water flows. These properties render multivariate recurrence networks particularly powerful for investigating a horizontal oil-water two-phase flow system and its complex interacting components from a network perspective.
We study the phase reduction of two coupled van der Pol oscillators with asymmetric repulsive coupling under an external harmonic force. We show that the system of two phase oscillators undergoes a Hopf bifurcation and possesses multistability on a 2?-periodic phase plane. We describe the bifurcation mechanisms of formation of multistability in the phase-reduced system and show that the Andronov-Hopf bifurcation in the phase-reduced system is not an artifact of the reduction approach but, indeed, has its prototype in the nonreduced system. The bifurcational mechanisms presented in the paper enable one to describe synchronization effects in a wide class of interacting systems with repulsive coupling e.g., genetic oscillators.
By introducing a processing delay in the coupling, we find that it can effectively annihilate the quenching of oscillation, amplitude death (AD), in a network of coupled oscillators by switching the stability of AD. It revives the oscillation in the AD regime to retain sustained rhythmic functioning of the networks, which is in sharp contrast to the propagation delay with the tendency to induce AD. This processing delay-induced phenomenon occurs both with and without the propagation delay. Further this effect is rather general from two coupled to networks of oscillators in all known scenarios that can exhibit AD, and it has a wide range of applications where sustained oscillations should be retained for proper functioning of the systems.
Coupled oscillators are shown to experience two structurally different oscillation quenching types: amplitude death (AD) and oscillation death (OD). We demonstrate that both AD and OD can occur in one system and find that the transition between them underlies a classical, Turing-type bifurcation, providing a clear classification of these significantly different dynamical regimes. The implications of obtaining a homogeneous (AD) or inhomogeneous (OD) steady state, as well as their significance for physical and biological applications and control studies, are also pointed out.
Amplitude death (AD) is an emergent phenomenon whereby two or more autonomously oscillating systems completely lose their oscillations due to coupling. In this work, we study AD in nonlinear oscillators with mixed time-delayed coupling, which is a combination of instantaneous and time-delayed couplings. We find that the mixed time-delayed coupling favors the onset of AD for a larger set of parameters than in the limiting cases of purely instantaneous or completely time-delayed coupling. Coupled identical oscillators experience AD under instantaneous coupling mixed with a small proportion of time-delayed coupling. Our work gives a deeper understanding of delay-induced AD in coupled nonlinear oscillators.
The analysis of effects from coupling in and between systems is important in data-driven investigations as practiced in many scientific fields. It allows deeper insights into the mechanisms of interaction emerging among individual smaller systems when forming complex systems as in the human circulatory system. For systems featuring various regimes, usually only the epochs before and after a transition between different regimes are analyzed, although relevant information might be hidden within these transitions. Transient behavior of cardiovascular variables may emerge, on the one hand, from the recovery of the system after a severe disturbance or, on the other hand, from adaptive behavior throughout changes of states. It contains important information about the processes involved and the relations between state variables such as heart rate, blood pressure, and respiration. Therefore, we apply an ensemble approach to extend the method of symbolic coupling traces to time-variant coupling analysis. These new ensemble symbolic coupling traces are capable of determining coupling direction, strength, and time offset ? from transient dynamics in multivariate cardiovascular data. We use this method to analyze data recorded during an orthostatic test to reveal a transient structure that cannot be detected by classic methods.
The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter we demonstrate that the nodes in a second-order Kuramoto model perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call cluster explosive synchronization. We provide a rigorous analytical treatment using a mean-field analysis in uncorrelated networks. Our findings are in good agreement with numerical simulations and fundamentally deepen the understanding of microscopic mechanisms toward synchronization.
This paper proposes the concept of pinning noise and then investigates the phenomenon of stochastic resonance of coupled complex systems driven by pinning noise, where the noise has an ?-stable distribution. Two kinds of pinning noise are taken into account: partial noise and switching noise. In particular, we establish a connection between switching noise and global noise when Gaussian noise is considered. It is shown that switching noise can not only achieve a stronger resonance effect, but it is also more robust to induce the resonance effect than partial noise.
We report the experimental verification of noise-enhanced logic behaviour in an electronic analog of a synthetic genetic network, composed of two repressors and two constitutive promoters. We observe good agreement between circuit measurements and numerical prediction, with the circuit allowing for robust logic operations in an optimal window of noise. Namely, the input-output characteristics of a logic gate is reproduced faithfully under moderate noise, which is a manifestation of the phenomenon known as Logical Stochastic Resonance. The two dynamical variables in the system yield complementary logic behaviour simultaneously. The system is easily morphed from AND/NAND to OR/NOR logic.
The asymmetry of coupling between complex systems can be studied by conditional probabilities of recurrence, which can be estimated by joint recurrence plots. This approach is applied for the first time on experimental data: time series of the human cardiorespiratory system in order to investigate the couplings between heart rate, mean arterial blood pressure and respiration. We find that the respiratory system couples towards the heart rate, and the heart rate towards the mean arterial blood pressure. However, our analysis could not detect a clear coupling direction between the mean arterial blood pressure and respiration.
We analyse cardiovascular time series with the aim of performing early prediction of preeclampsia (PE), a pregnancy-specific disorder causing maternal and foetal morbidity and mortality. The analysis is made using a novel approach, namely the ?-recurrence networks applied to a phase space constructed by means of the time series of the variabilities of the heart rate and the blood pressure (systolic and diastolic). All the possible coupling structures among these variables are considered for the analysis. Network measures such as average path length, mean coreness, global clustering coefficient and scale-local transitivity dimension are computed and constitute the parameters for the subsequent quadratic discriminant analysis. This allows us to predict PE with a sensitivity of 91.7 per cent and a specificity of 68.1 per cent, thus validating the use of this method for classifying healthy and preeclamptic patients.
Studies on heart rate variability (HRV) have become popular and the possibility of diagnosis based on non-invasive techniques compels us to overcome the difficulties originated on the environmental changes that can affect the signal. We perform a non-parametric segmentation which consists of locating the points where the signal can be split into stationary segments. By finding stationary segments we are able to analyze the size of these segments and evaluate how the signal changes from one segment to another, looking at the statistical moments given in each patch, for example, mean and variance. We analyze HRV data for 15 patients with congestive heart failure (CHF; 11 males, 4 females, age 56±11 years), 18 elderly healthy subjects (EH; 11 males, 7 females, age 50±7 years), and 15 young healthy subjects (YH; 11 females, 4 males, age 31±6 years). Our results confirm higher variance for YH, and EH, while CHF displays diminished variance with p-values <0.01, when compared to the healthy groups, presenting higher HRV in healthy subjects. Moreover, it is possible to distinguish between YH and EH with p < 0.05 through the segmentation outcomes. We found high correlations between the results of segmentation and standard measures of HRV analysis and a connection to results of detrended fluctuation analysis (DFA). The segmentation applied to HRV studies detects aging and pathological conditions effects on the non-stationary behavior of the analyzed groups, promising to contribute in complexity analysis and providing risk stratification measures.
The equivalent system for a multiple-rational-order (MRO) fractional differential system is studied, where the fractional derivative is in the sense of Caputo or Riemann-Liouville. With the relationship between the Caputo derivative and the generalized fractional derivative, we can change the MRO fractional differential system with a Caputo derivative into a higher-dimensional system with the same Caputo derivative order lying in (0,1). The stability of the zero solution to the original system is studied through the analysis of its equivalent system. For the Riemann-Liouville case, we transform the MRO fractional differential system into a new one with the same order lying in (0,1), where the properties of the Riemann-Liouville derivative operator and the fractional integral operator are used. The corresponding stability is also studied. Finally, several numerical examples are provided to illustrate the derived results.
This paper presents a brief overview of recent developments in chaos synchronization in coupled fractional differential systems, where the original viewpoints are retained. In addition to complete synchronization, several other extended concepts of synchronization, such as projective synchronization, hybrid projective synchronization, function projective synchronization, generalized synchronization and generalized projective synchronization in fractional differential systems, are reviewed.
Potential paleoclimatic driving mechanisms acting on human evolution present an open problem of cross-disciplinary scientific interest. The analysis of paleoclimate archives encoding the environmental variability in East Africa during the past 5 Ma has triggered an ongoing debate about possible candidate processes and evolutionary mechanisms. In this work, we apply a nonlinear statistical technique, recurrence network analysis, to three distinct marine records of terrigenous dust flux. Our method enables us to identify three epochs with transitions between qualitatively different types of environmental variability in North and East Africa during the (i) Middle Pliocene (3.35-3.15 Ma B.P.), (ii) Early Pleistocene (2.25-1.6 Ma B.P.), and (iii) Middle Pleistocene (1.1-0.7 Ma B.P.). A deeper examination of these transition periods reveals potential climatic drivers, including (i) large-scale changes in ocean currents due to a spatial shift of the Indonesian throughflow in combination with an intensification of Northern Hemisphere glaciation, (ii) a global reorganization of the atmospheric Walker circulation induced in the tropical Pacific and Indian Ocean, and (iii) shifts in the dominating temporal variability pattern of glacial activity during the Middle Pleistocene, respectively. A reexamination of the available fossil record demonstrates statistically significant coincidences between the detected transition periods and major steps in hominin evolution. This result suggests that the observed shifts between more regular and more erratic environmental variability may have acted as a trigger for rapid change in the development of humankind in Africa.
Recent studies of brain connectivity and language with methods of complex networks have revealed common features of organization. These observations open a window to better understand the intrinsic relationship between the brain and the mind by studying how information is either physically stored or mentally represented. In this paper, we review some of the results in both brain and linguistic networks, and we illustrate how modelling approaches can serve to comprehend the relationship between the structure of the brain and its function. On the one hand, we show that brain and neural networks display dynamical behaviour with optimal complexity in terms of a balance between their capacity to simultaneously segregate and integrate information. On the other hand, we show how principles of neural organization can be implemented into models of memory storage and recognition to reproduce spontaneous transitions between memories, resembling phenomena of memory association studied in psycholinguistic experiments.
Sleep is a physiological process with an internal program of a number of well defined sleep stages and intermediate wakefulness periods. The sleep stages do modulate the autonomous nervous system and thereby the sleep stages are accompanied by different regulation regimes for the cardiovascular and respiratory system. The differences in regulation can be distinguished by new analysis techniques on the recorded signals. In addition to normal sleep regulation some sleep disorders affect the cardiovascular and respiratory regulation. The most prevalent disorder linked to sleep and changes in the autonomous system is obstructive sleep apnea. In patients with obstructive sleep apnea marked short term changes in cardiovascular and respiratory regulation are observed during sleep. These abnormalities in regulation are further differentiated between the sleep stages. For long term changes obstructive sleep apnea is recognized as a major risk factor for arterial hypertension. Treatment of obstructive sleep apnea lowers blood pressure during the night and over time also lowers blood pressure during daytime. In this study we investigated 18 patients with sleep apnea and normal blood pressure, 10 patients with sleep apnea and arterial hypertension and 10 normal subjects as controls. Both patient groups were tested with cardiorespiratory polysomnography before and under CPAP therapy. The effects on cardiovascular and respiratory regulation during sleep and daytime are investigated in the three groups.
Heart rate and blood pressure variability analysis as well as baroreflex sensitivity have been proven to be powerful tools for the assessment of autonomic control in clinical practice. Their ability to detect systematic changes caused by different states, diseases and treatments shall be shown for sleep disorders. Therefore, we consider 18 normotensive and 10 hypertensive patients suffering from obstructive sleep apnea syndrome (OSAS) before and after a three-month continuous positive airway pressure (CPAP) therapy. Additionally, an age and sex matched control group of 10 healthy subjects is examined. Linear and nonlinear parameters of heart rate and blood pressure fluctuation as well as the baroreflex sensitivity are used to answer the question whether there are differences in cardiovascular regulation between the different sleep stages and groups. Moreover, the therapeutic effect of CPAP therapy in OSAS patients shall be investigated. Kruskal-Wallis tests between the sleep stages for each group show significant differences in the very low spectral component of heart rate (VLF/P: 0.0033-0.04 Hz, p<0.01) which indicates differences in metabolic activity during the night. Furthermore, the decrease of Shannon entropy of word distribution as a parameter of systolic blood pressure during non-REM sleep reflects the local dominance of the vagal system (p<0.05). The increased sympathetic activation of the patients leads to clear differences of cardiovascular regulation in different sleep stages between controls and patients. We found a significant reduction of baroreflex sensitivity in slow wave sleep in the OSAS patients (Mann-Whitney test, p<0.05) compared to controls, which disappeared after three months of CPAP therapy. Hence, our results demonstrate the ability of cardiovascular analyzes to separate between healthy and pathological regulation as well as between different severities of OSAS in this retrospective study.
Heart rate and blood pressure variability as well as baroreflex sensitivity (BRS) lead to additional insights on the patients prognosis after cardiovascular events. The following study was performed to assess the differences in the postoperative recovery of the autonomic regulation after transcatheter aortic valve implantation (TAVI) and surgical aortic valve replacement (SAVR). Fifty-eight consecutive patients were enrolled in a prospective study; 24 underwent TAVI and 34 SAVR. BRS was calculated according to the Dual Sequence Method, heart rate variability (HRV) was evaluated using standard linear as well as nonlinear parameters. HRV and BRS parameters were reduced after surgery in patients with SAVR only (meanNN: p<0.001, sdNN: p<0.05, Shannon: p<0.01, BRS: p<0.01), while these indexes were preserved in patients after TAVI. Simultaneously, an increased complexity of blood pressure (BP) in SAVR patients (fwShannon: p<0.001, fwRenyi4: p<0.001), but not in TAVI patients was recorded. In this study we were able to demonstrate for the first time that, in contrast to patients undergoing conventional open surgery, there are fewer alterations of the cardiovascular autonomic system in patients with TAVI.
A coupling phase is deemed to be crucial in stabilizing behavior in nonlinear systems. In this paper, we study how the coupling phase influences the delay-induced oscillation death (OD) in coupled oscillators. The OD boundaries are identified analytically even in the presence of the coupling phase. We find that OD only occurs for a coupling phase belonging to a certain interval. The optimal coupling phase, under which the largest OD island forms, is characterized well by a power law scaling with respect to the frequency. The coupling phase turns out to be a key parameter that determines a delay-induced OD. Furthermore, the controlling role of the coupling phase generally is proved to hold fairly for networked delay-coupled oscillators.
In this paper, we present an efficient opinion control strategy for complex networks, in particular, for social networks. The proposed adaptive bridge control (ABC) strategy calls for controlling a special kind of nodes named bridge and requires no knowledge of the node degrees or any other global or local knowledge, which are necessary for some other immunization strategies including targeted immunization and acquaintance immunization. We study the efficiency of the proposed ABC strategy on random networks, small-world networks, scale-free networks, and the random networks adjusted by the edge exchanging method. Our results show that the proposed ABC strategy is efficient for all of these four kinds of networks. Through an adjusting clustering coefficient by the edge exchanging method, it is found out that the efficiency of our ABC strategy is closely related with the clustering coefficient. The main contributions of this paper can be listed as follows: (1) A new high-order social network is proposed to describe opinion dynamic. (2) An algorithm, which does not require the knowledge of the nodes degree and other global?local network structure information, is proposed to control the "bridges" more accurately and further control the opinion dynamics of the social networks. The efficiency of our ABC strategy is illustrated by numerical examples. (3) The numerical results indicate that our ABC strategy is more efficient for networks with higher clustering coefficient.
In this paper, multiobjective synchronization of chaotic systems is investigated by especially simultaneously minimizing optimization of control cost and convergence speed. The coupling form and coupling strength are optimized by an improved multiobjective evolutionary approach that includes a hybrid chromosome representation. The hybrid encoding scheme combines binary representation with real number representation. The constraints on the coupling form are also considered by converting the multiobjective synchronization into a multiobjective constraint problem. In addition, the performances of the adaptive learning method and non-dominated sorting genetic algorithm-II as well as the effectiveness and contributions of the proposed approach are analyzed and validated through the Ro?ssler system in a chaotic or hyperchaotic regime and delayed chaotic neural networks.
An electronic analog of a synthetic genetic network known as the repressilator is proposed. The repressilator is a synthetic biological clock consisting of a cyclic inhibitory network of three negative regulatory genes which produces oscillations in the expressed protein concentrations. Compared to previous circuit analogs of the repressilator, the circuit here takes into account more accurately the kinetics of gene expression, inhibition, and protein degradation. A good agreement between circuit measurements and numerical prediction is observed. The circuit allows for easy control of the kinetic parameters thereby aiding investigations of large varieties of potential dynamics.
The intrinsic relationship between the architecture of the brain and the range of sensory and behavioral phenomena it produces is a relevant question in neuroscience. Here, we review recent knowledge gained on the architecture of the anatomical connectivity by means of complex network analysis. It has been found that cortico-cortical networks display a few prominent characteristics: (i) modular organization, (ii) abundant alternative processing paths, and (iii) the presence of highly connected hubs. Additionally, we present a novel classification of cortical areas of the cat according to the role they play in multisensory connectivity. All these properties represent an ideal anatomical substrate supporting rich dynamical behaviors, facilitating the capacity of the brain to process sensory information of different modalities segregated and to integrate them toward a comprehensive perception of the real world. The results here exposed are mainly based on anatomical data of cats brain, but further observations suggest that, from worms to humans, the nervous system of all animals might share these fundamental principles of organization.
Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience.
Frank Moss was a leading figure in the study of nonlinear and stochastic processes in biological systems. His work, particularly in the area of stochastic resonance, has been highly influential to the interdisciplinary scientific community. This Focus Issue pays tribute to Moss with articles that describe the most recent advances in the field he helped to create. In this Introduction, we review Mosss seminal scientific contributions and introduce the articles that make up this Focus Issue.
The dynamical structure of genetic networks determines the occurrence of various biological mechanisms, such as cellular differentiation. However, the question of how cellular diversity evolves in relation to the inherent stochasticity and intercellular communication remains still to be understood. Here, we define a concept of stochastic bifurcations suitable to investigate the dynamical structure of genetic networks, and show that under stochastic influence, the expression of given proteins of interest is defined via the probability distribution of the phase variable, representing one of the genes constituting the system. Moreover, we show that under changing stochastic conditions, the probabilities of expressing certain concentration values are different, leading to different functionality of the cells, and thus to differentiation of the cells in the various types.
In this paper, the transitions of burst synchronization are explored in a neuronal network consisting of subnetworks. The studied network is composed of electrically coupled bursting Hindmarsh-Rose neurons. Numerical results show that two types of burst synchronization transitions can be induced not only by the variations of intra- and intercoupling strengths but also by changing the probability of random links between different subnetworks and the number of subnetworks. Furthermore, we find that the underlying mechanisms for these two bursting synchronization transitions are different: one is due to the change of spike numbers per burst, while the other is caused by the change of the bursting type. Considering that changes in the coupling strengths and neuronal connections are closely interlaced with brain plasticity, the presented results could have important implications for the role of the brain plasticity in some functional behavior that are associated with synchronization.
Inferring regulatory interactions between genes from transcriptomics time-resolved data, yielding reverse engineered gene regulatory networks, is of paramount importance to systems biology and bioinformatics studies. Accurate methods to address this problem can ultimately provide a deeper insight into the complexity, behavior, and functions of the underlying biological systems. However, the large number of interacting genes coupled with short and often noisy time-resolved read-outs of the system renders the reverse engineering a challenging task. Therefore, the development and assessment of methods which are computationally efficient, robust against noise, applicable to short time series data, and preferably capable of reconstructing the directionality of the regulatory interactions remains a pressing research problem with valuable applications.
We present a model of synchronization in networks of autonomous agents where the topology changes due to agents motion. We introduce two timescales, one for the topological change and another one for local synchronization. If the former scale is much shorter, an approximation that averages out the effect of motion is available. Here we show, however, that the time required for synchronization achievement is larger than the prediction of the approximation in the opposite case, especially close to the continuum percolation transition point. The simulation results are confirmed by means of spectral analysis of the time-dependent Laplacian matrix. Our results show that the tradeoff between these two timescales, which have opposite effects on synchronization, should be taken into account for the design of mobile device networks.
The recent years have seen the emergence of graph theoretical analysis of complex, functional brain networks estimated from neurophysiological measurements. The research has mainly focused on the graph characterization of the resting-state/default network, and its potential for clinical application. Functional resting-state networks usually display the characteristics of small-world networks and their statistical properties have been observed to change due to pathological conditions or aging. In the present paper we move forward in the application of graph theoretical tools in functional connectivity by investigating high-level cognitive processing in healthy adults, in a manner similar to that used in psychological research in the framework of event-related potentials (ERPs). More specifically we aim at investigating how graph theoretical approaches can help to discover systematic and task-dependent differences in high-level cognitive processes such as language perception. We will show that such an approach is feasible and that the results coincide well with the findings from neuroimaging studies.
This brief investigates globally exponential synchronization for linearly coupled neural networks (NNs) with time-varying delay and impulsive disturbances. Since the impulsive effects discussed in this brief are regarded as disturbances, the impulses should not happen too frequently. The concept of average impulsive interval is used to formalize this phenomenon. By referring to an impulsive delay differential inequality, we investigate the globally exponential synchronization of linearly coupled NNs with impulsive disturbances. The derived sufficient condition is closely related with the time delay, impulse strengths, average impulsive interval, and coupling structure of the systems. The obtained criterion is given in terms of an algebraic inequality which is easy to be verified, and hence our result is valid for large-scale systems. The results extend and improve upon earlier work. As a numerical example, a small-world network composing of impulsive coupled chaotic delayed NN nodes is given to illustrate our theoretical result.
Interacting human activities underlie the patterns of many social, technological, and economic phenomena. Here we present clear empirical evidence from Short Message correspondence that observed human actions are the result of the interplay of three basic ingredients: Poisson initiation of tasks and decision making for task execution in individual humans as well as interaction among individuals. This interplay leads to new types of interevent time distribution, neither completely Poisson nor power-law, but a bimodal combination of them. We show that the events can be separated into independent bursts which are generated by frequent mutual interactions in short times following random initiations of communications in longer times by the individuals. We introduce a minimal model of two interacting priority queues incorporating the three basic ingredients which fits well the distributions using the parameters extracted from the empirical data. The model can also embrace a range of realistic social interacting systems such as e-mail and letter communications when taking the time scale of processing into account. Our findings provide insight into various human activities both at the individual and network level. Our analysis and modeling of bimodal activity in human communication from the viewpoint of the interplay between processes of different time scales is likely to shed light on bimodal phenomena in other complex systems, such as interevent times in earthquakes, rainfall, forest fire, and economic systems, etc.
In this paper, we examine the effects of correlated Gaussian noise on a two-dimensional neuronal network that is locally modeled by the Rulkov map. More precisely, we study the effects of the noise correlation on the variations of the mean firing rate and the correlations among neurons versus the noise intensity. Via numerical simulations, we show that the mean firing rate can always be optimized at an intermediate noise intensity, irrespective of the noise correlation. On the other hand, variations of the population coherence with respect to the noise intensity are strongly influenced by the ratio between local and global Gaussian noisy inputs. Biological implications of our findings are also discussed.
We consider a three-domain model of cardiac tissue consisting of fibroblasts, myocytes, and extracellular space. We show in the one dimensional case that the fibroblasts with different resting potentials may alter restitution properties of tissue. On this basis we demonstrated that in two dimensional slice of cardiac tissue, a spiral wave break up can be caused purely by the influence of fibroblasts and, vice-versa, initially unstable spiral can be stabilized by fibroblasts depending on the value of their resting potential.
The methods of nonlinear systems form an extensive toolbox for the study of biology, and systems biology provides a rich source of motivation for the development of new mathematical techniques and the furthering of understanding of dynamical systems. This Focus Issue collects together a large variety of work which highlights the complementary nature of these two fields, showing what each has to offer the other. While a wide range of subjects is covered, the papers often have common themes such as "rhythms and oscillations," "networks and graph theory," and "switches and decision making." There is a particular emphasis on the links between experimental data and modeling and mathematical analysis.
The identification of complex periodic windows in the two-dimensional parameter space of certain dynamical systems has recently attracted considerable interest. While for discrete systems, a discrimination between periodic and chaotic windows can be easily made based on the maximum Lyapunov exponent of the system, this remains a challenging task for continuous systems, especially if only short time series are available (e.g., in case of experimental data). In this work, we demonstrate that nonlinear measures based on recurrence plots obtained from such trajectories provide a practicable alternative for numerically detecting shrimps. Traditional diagonal line-based measures of recurrence quantification analysis as well as measures from complex network theory are shown to allow an excellent classification of periodic and chaotic behavior in parameter space. Using the well-studied Ro?ssler system as a benchmark example, we find that the average path length and the clustering coefficient of the resulting recurrence networks are particularly powerful discriminatory statistics for the identification of complex periodic windows.
The possibility of controlling the Calvin cycle has paramount implications for increasing the production of biomass. Multistationarity, as a dynamical feature of systems, is the first obvious candidate whose control could find biotechnological applications. Here we set out to resolve the debate on the multistationarity of the Calvin cycle. Unlike the existing simulation-based studies, our approach is based on a sound mathematical framework, chemical reaction network theory and algebraic geometry, which results in provable results for the investigated model of the Calvin cycle in which we embed a hierarchy of realistic kinetic laws. Our theoretical findings demonstrate that there is a possibility for multistationarity resulting from two sources, homogeneous and inhomogeneous instabilities, which partially settle the debate on multistability of the Calvin cycle. In addition, our tractable analytical treatment of the bifurcation parameters can be employed in the design of validation experiments.
High complexity is considered a hallmark of living systems. Here we investigate the complexity of temporal gene expression patterns using the concept of Permutation Entropy (PE) first introduced in dynamical systems theory. The analysis of gene expression data has so far focused primarily on the identification of differentially expressed genes, or on the elucidation of pathway and regulatory relationships. We aim to study gene expression time series data from the viewpoint of complexity.
Pre-eclampsia (PE), a serious pregnancy-specific disorder, causes significant neonatal and maternal morbidity and mortality. Recent studies showed that cardiovascular variability parameters as well as the baroreflex sensitivity remarkably improve its early diagnosis. For a better understanding of the dynamical changes caused by PE, in this study the coupling between respiration, systolic and diastolic blood pressure, and heart rate is investigated. Thirteen datasets of healthy pregnant women and 10 of subjects suffering from PE are included. Nonlinear additive autoregressive models with external input are used for a model-based coupling analysis following the idea of Granger causality. To overcome the problem of misdetections of standard methods in systems with a dominant driver, a heuristic ensemble approach is used here. A coupling is assumed to be real when existent in more than 80 per cent of the ensemble members, and otherwise denoted as artefacts. As the main result, we found that the coupling structure between heart rate, systolic blood pressure, diastolic blood pressure and respiration for healthy subjects and PE patients is the same and reliable. As a pathological mechanism, however, a significant increased respiratory influence on the diastolic blood pressure could be found for PE patients (p=0.003). Moreover, the nonlinear form of the respiratory influence on the heart rate is significantly different between the two groups (p=0.002). Interestingly, the influence of systolic blood pressure on the heart rate is not selected, which indicates that the baroreflex sensitivity estimation strongly demands the consideration of causal relationships between heart rate, blood pressure and respiration. Finally, our results point to a potential role of respiration for understanding the pathogenesis of PE.
The generation and synchronization of bursts are studied in intrinsically spiking neurons due to stimulation with random intracellular calcium fluctuations. It is demonstrated that sufficiently strong noise could induce qualitative change in the firing patterns of a single neuron from periodic spiking to bursting modes. The dynamical mechanism of noise-induced bursting is presented based on a global bifurcation analysis. Furthermore, it is found that a pair of uncoupled and nonidentical spiking neurons, subjected to a common noise, can exhibit synchronous firing in terms of noise-induced bursting. Furthermore, the synchronization is overall enhanced with the noise intensity increasing, and synchronization transitions are exhibited at intermediate noise levels.
Sensory stimuli entering the nervous system follow particular paths of processing, typically separated (segregated) from the paths of other modal information. However, sensory perception, awareness and cognition emerge from the combination of information (integration). The corticocortical networks of cats and macaque monkeys display three prominent characteristics: (i) modular organisation (facilitating the segregation), (ii) abundant alternative processing paths and (iii) the presence of highly connected hubs. Here, we study in detail the organisation and potential function of the cortical hubs by graph analysis and information theoretical methods. We find that the cortical hubs form a spatially delocalised, but topologically central module with the capacity to integrate multisensory information in a collaborative manner. With this, we resolve the underlying anatomical substrate that supports the simultaneous capacity of the cortex to segregate and to integrate multisensory information.
We propose a method to infer the coupling structure in networks of nonlinear oscillatory systems with multiple time scales. The method of partial phase synchronization allows us to infer the coupling structure for coupled nonlinear oscillators with one well-defined time scale. The case of oscillators with multiple time scales has remained a challenge until now. Here, we introduce partial recurrence based synchronization analysis to tackle this challenge. We successfully apply the proposed method to model systems and experimental data from coupled electrochemical oscillators. The statistical significance of the results is evaluated based on a surrogate hypothesis test.
Recently, different approaches have been proposed for studying basic properties of time series from a complex network perspective. In this work, the corresponding potentials and limitations of networks based on recurrences in phase space are investigated in some detail. We discuss the main requirements that permit a feasible system-theoretic interpretation of network topology in terms of dynamically invariant phase-space properties. Possible artifacts induced by disregarding these requirements are pointed out and systematically studied. Finally, a rigorous interpretation of the clustering coefficient and the betweenness centrality in terms of invariant objects is proposed.
This paper considers a second-order consensus problem for multiagent systems with nonlinear dynamics and directed topologies where each agent is governed by both position and velocity consensus terms with a time-varying asymptotic velocity. To describe the systems ability for reaching consensus, a new concept about the generalized algebraic connectivity is defined for strongly connected networks and then extended to the strongly connected components of the directed network containing a spanning tree. Some sufficient conditions are derived for reaching second-order consensus in multiagent systems with nonlinear dynamics based on algebraic graph theory, matrix theory, and Lyapunov control approach. Finally, simulation examples are given to verify the theoretical analysis.
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