Recurrent chaotic clustering and slow chaos in adaptive networks.

Adaptive dynamical networks are network systems in which the structure co-evolves and interacts with the dynamical state of the nodes. We study an adaptive dynamical network in which the structure changes on a slower time scale relative to the fast dynamics of the nodes. We identify a phenomenon we refer to as recurrent adaptive chaotic clustering (RACC), in which chaos is observed on a slow time scale, while the fast time scale exhibits regular dynamics. Such slow chaos is further characterized by long (relative to the fast time scale) regimes of frequency clusters or frequency-synchronized dynamics, interrupted by fast jumps between these regimes. We also determine parameter values where the time intervals between jumps are chaotic and show that such a state is robust to changes in parameters and initial conditions.

Stickiness and recurrence plots: An entropy-based approach.

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RPs), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient. We obtain a multi-modal distribution of the finite-time RTE and find that each mode corresponds to the motion around islands of different hierarchical levels.

How heterogeneity in connections and cycles matter for synchronization of complex networks.

We analyze how the structure of complex networks of non-identical oscillators influences synchronization in the context of the Kuramoto model. The complex network metrics assortativity and clustering coefficient are used in order to generate network topologies of Erdös-Rényi, Watts-Strogatz, and Barabási-Albert types that present high, intermediate, and low values of these metrics. We also employ the total dissonance metric for neighborhood similarity, which generalizes to networks the standard concept of dissonance between two non-identical coupled oscillators. Based on this quantifier and using an optimization algorithm, we generate Similar, Dissimilar, and Neutral natural frequency patterns, which correspond to small, large, and intermediate values of total dissonance, respectively. The emergency of synchronization is numerically studied by considering these three types of dissonance patterns along with the network topologies generated by high, intermediate, and low values of the metrics assortativity and clustering coefficient. We find that, in general, low values of these metrics appear to favor phase locking, especially for the Similar dissonance pattern.

Ubiquity of ring structures in the control space of complex oscillators.

We report the discovery of two types of stability rings in the control parameter space of a vertical-cavity surface-emitting semiconductor laser. Stability rings are closed parameter paths in the laser control space. Inside such rings, laser stability thrives even in the presence of small parameter fluctuations. Stability rings were also found in rather distinct contexts, namely, in the way that cancerous, normal, and effector cells interact under ionizing radiation and in oscillations of an electronic circuit with a junction-gate field-effect transistor (JFET) diode. We argue that stability-enhancing rings are generic structures present in the control parameter space of many complex systems.

Scientometric analysis of the Chaos journal (1991-2019): From descriptive statistics to complex networks viewpoints.

We performed a scientometric analysis of Chaos papers from 1991 to 2019, applying a careful disambiguation process for identifying the authors correctly. First, we used standard scientometric tools based on descriptive statistics. This analysis enabled us to compute productivity and the degree of collaboration. The evolution in the number of authors, countries, and topics per article has an increasing trend. An analysis of the citations considering their temporal mean number exhibits a growing tendency in time. Second, we dealt with Lotka-Zipf's law considering the rank distributions of 15 datasets. We found that the sum of Crossref citations by country was the only dataset for which the power-law was the only plausible distribution. Next, we examined the networks of authors, countries, and topics, going from the simplest case of undirected and unweighted networks to the general case of weighted and directed networks and assigning a weight to the individual nodes. Based on the networks' topology and features, we introduced diversity, collaboration, influence, and productivity measures and found a significant increase in the diversity of all the considered networks (authors, countries, and topics) but manifesting a very different network structure. The computation of typical network quantities combined with the communities' identification reveals the presence of several hubs and the existence of various communities that encompass nodes of all the continents in the case of countries. Finally, using the most general networks, it was possible to compute influence and productivity indexes to find the USA, China, and Germany's leadership inside the network.

Emergence of Neuronal Synchronisation in Coupled Areas.

One of the most fundamental questions in the field of neuroscience is the emergence of synchronous behaviour in the brain, such as phase, anti-phase, and shift-phase synchronisation. In this work, we investigate how the connectivity between brain areas can influence the phase angle and the neuronal synchronisation. To do this, we consider brain areas connected by means of excitatory and inhibitory synapses, in which the neuron dynamics is given by the adaptive exponential integrate-and-fire model. Our simulations suggest that excitatory and inhibitory connections from one area to another play a crucial role in the emergence of these types of synchronisation. Thus, in the case of unidirectional interaction, we observe that the phase angles of the neurons in the receiver area depend on the excitatory and inhibitory synapses which arrive from the sender area. Moreover, when the neurons in the sender area are synchronised, the phase angle variability of the receiver area can be reduced for some conductance values between the areas. For bidirectional interactions, we find that phase and anti-phase synchronisation can emerge due to excitatory and inhibitory connections. We also verify, for a strong inhibitory-to-excitatory interaction, the existence of silent neuronal activities, namely a large number of excitatory neurons that remain in silence for a long time.

Influence of Autapses on Synchronization in Neural Networks With Chemical Synapses.

A great deal of research has been devoted on the investigation of neural dynamics in various network topologies. However, only a few studies have focused on the influence of autapses, synapses from a neuron onto itself via closed loops, on neural synchronization. Here, we build a random network with adaptive exponential integrate-and-fire neurons coupled with chemical synapses, equipped with autapses, to study the effect of the latter on synchronous behavior. We consider time delay in the conductance of the pre-synaptic neuron for excitatory and inhibitory connections. Interestingly, in neural networks consisting of both excitatory and inhibitory neurons, we uncover that synchronous behavior depends on their synapse type. Our results provide evidence on the synchronous and desynchronous activities that emerge in random neural networks with chemical, inhibitory and excitatory synapses where neurons are equipped with autapses.

Influence of Delayed Conductance on Neuronal Synchronization.

In the brain, the excitation-inhibition balance prevents abnormal synchronous behavior. However, known synaptic conductance intensity can be insufficient to account for the undesired synchronization. Due to this fact, we consider time delay in excitatory and inhibitory conductances and study its effect on the neuronal synchronization. In this work, we build a neuronal network composed of adaptive integrate-and-fire neurons coupled by means of delayed conductances. We observe that the time delay in the excitatory and inhibitory conductivities can alter both the state of the collective behavior (synchronous or desynchronous) and its type (spike or burst). For the weak coupling regime, we find that synchronization appears associated with neurons behaving with extremes highest and lowest mean firing frequency, in contrast to when desynchronization is present when neurons do not exhibit extreme values for the firing frequency. Synchronization can also be characterized by neurons presenting either the highest or the lowest levels in the mean synaptic current. For the strong coupling, synchronous burst activities can occur for delays in the inhibitory conductivity. For approximately equal-length delays in the excitatory and inhibitory conductances, desynchronous spikes activities are identified for both weak and strong coupling regimes. Therefore, our results show that not only the conductance intensity, but also short delays in the inhibitory conductance are relevant to avoid abnormal neuronal synchronization.

Bistable Firing Pattern in a Neural Network Model.

Excessively high, neural synchronization has been associated with epileptic seizures, one of the most common brain diseases worldwide. A better understanding of neural synchronization mechanisms can thus help control or even treat epilepsy. In this paper, we study neural synchronization in a random network where nodes are neurons with excitatory and inhibitory synapses, and neural activity for each node is provided by the adaptive exponential integrate-and-fire model. In this framework, we verify that the decrease in the influence of inhibition can generate synchronization originating from a pattern of desynchronized spikes. The transition from desynchronous spikes to synchronous bursts of activity, induced by varying the synaptic coupling, emerges in a hysteresis loop due to bistability where abnormal (excessively high synchronous) regimes exist. We verify that, for parameters in the bistability regime, a square current pulse can trigger excessively high (abnormal) synchronization, a process that can reproduce features of epileptic seizures. Then, we show that it is possible to suppress such abnormal synchronization by applying a small-amplitude external current on > 10% of the neurons in the network. Our results demonstrate that external electrical stimulation not only can trigger synchronous behavior, but more importantly, it can be used as a means to reduce abnormal synchronization and thus, control or treat effectively epileptic seizures.

Neuron dynamics variability and anomalous phase synchronization of neural networks.

Anomalous phase synchronization describes a synchronization phenomenon occurring even for the weakly coupled network and characterized by a non-monotonous dependence of the synchronization strength on the coupling strength. Its existence may support a theoretical framework to some neurological diseases, such as Parkinson's and some episodes of seizure behavior generated by epilepsy. Despite the success of controlling or suppressing the anomalous phase synchronization in neural networks applying external perturbations or inducing ambient changes, the origin of the anomalous phase synchronization as well as the mechanisms behind the suppression is not completely known. Here, we consider networks composed of N = 2000 coupled neurons in a small-world topology for two well known neuron models, namely, the Hodgkin-Huxley-like and the Hindmarsh-Rose models, both displaying the anomalous phase synchronization regime. We show that the anomalous phase synchronization may be related to the individual behavior of the coupled neurons; particularly, we identify a strong correlation between the behavior of the inter-bursting-intervals of the neurons, what we call neuron variability, to the ability of the network to depict anomalous phase synchronization. We corroborate the ideas showing that external perturbations or ambient parameter changes that eliminate anomalous phase synchronization and at the same time promote small changes in the individual dynamics of the neurons, such that an increasing individual variability of neurons implies a decrease of anomalous phase synchronization. Finally, we demonstrate that this effect can be quantified using a well known recurrence quantifier, the "determinism." Moreover, the results obtained by the determinism are based on only the mean field potential of the network, turning these measures more suitable to be used in experimental situations.

Quantifying entropy using recurrence matrix microstates.

We conceive a new recurrence quantifier for time series based on the concept of information entropy, in which the probabilities are associated with the presence of microstates defined on the recurrence matrix as small binary submatrices. The new methodology to compute the entropy of a time series has advantages compared to the traditional entropies defined in the literature, namely, a good correlation with the maximum Lyapunov exponent of the system and a weak dependence on the vicinity threshold parameter. Furthermore, the new method works adequately even for small segments of data, bringing consistent results for short and long time series. In a case where long time series are available, the new methodology can be employed to obtain high precision results since it does not demand large computational times related to the analysis of the entire time series or recurrence matrices, as is the case of other traditional entropy quantifiers. The method is applied to discrete and continuous systems.

Optimizing the detection of nonstationary signals by using recurrence analysis.

Recurrence analysis and its quantifiers are strongly dependent on the evaluation of the vicinity threshold parameter, i.e., the threshold to regard two points close enough in phase space to be considered as just one. We develop a new way to optimize the evaluation of the vicinity threshold in order to assure a higher level of sensitivity to recurrence quantifiers to allow the detection of even small changes in the dynamics. It is used to promote recurrence analysis as a tool to detect nonstationary behavior of time signals or space profiles. We show that the ability to detect small changes provides information about the present status of the physical process responsible to generate the signal and offers mechanisms to predict future states. Here, a higher sensitive recurrence analysis is proposed as a precursor, a tool to predict near future states of a particular system, based on just (experimentally) obtained signals of some available variables of the system. Comparisons with traditional methods of recurrence analysis show that the optimization method developed here is more sensitive to small variations occurring in a signal. The method is applied to numerically generated time series as well as experimental data from physiology.

Recurrence measure of conditional dependence and applications.

Identifying causal relations from observational data sets has posed great challenges in data-driven causality inference studies. One of the successful approaches to detect direct coupling in the information theory framework is transfer entropy. However, the core of entropy-based tools lies on the probability estimation of the underlying variables. Here we propose a data-driven approach for causality inference that incorporates recurrence plot features into the framework of information theory. We define it as the recurrence measure of conditional dependence (RMCD), and we present some applications. The RMCD quantifies the causal dependence between two processes based on joint recurrence patterns between the past of the possible driver and present of the potentially driven, excepting the contribution of the contemporaneous past of the driven variable. Finally, it can unveil the time scale of the influence of the sea-surface temperature of the Pacific Ocean on the precipitation in the Amazonia during recent major droughts.

A deforestation-induced tipping point for the South American monsoon system.

The Amazon rainforest has been proposed as a tipping element of the earth system, with the possibility of a dieback of the entire ecosystem due to deforestation only of parts of the rainforest. Possible physical mechanisms behind such a transition are still subject to ongoing debates. Here, we use a specifically designed model to analyse the nonlinear couplings between the Amazon rainforest and the atmospheric moisture transport from the Atlantic to the South American continent. These couplings are associated with a westward cascade of precipitation and evapotranspiration across the Amazon. We investigate impacts of deforestation on the South American monsoonal circulation with particular focus on a previously neglected positive feedback related to condensational latent heating over the rainforest, which strongly enhances atmospheric moisture inflow from the Atlantic. Our results indicate the existence of a tipping point. In our model setup, crossing the tipping point causes precipitation reductions of up to 40% in non-deforested parts of the western Amazon and regions further downstream. The responsible mechanism is the breakdown of the aforementioned feedback, which occurs when deforestation reduces transpiration to a point where the available atmospheric moisture does not suffice anymore to release the latent heat needed to maintain the feedback.

Traveling phase waves in asymmetric networks of noisy chaotic attractors.

We explore identical Rössler systems organized into two equally sized groups, among which differing positive and negative in- and out-coupling strengths are allowed. With this asymmetric coupling, we analyze patterns in the phase dynamics that coexist with chaotic amplitudes. We specifically investigate traveling phase waves where the oscillators settle on a new rhythm different from their own. We show that these waves are possible even without coherence in the phase angles. It is further demonstrated that the emergence of these incoherent traveling waves depends on the type of coupling, not on the individual dynamics of the Rössler systems. Together with the study of noise effects, our results suggest a promising new avenue toward the interplay of chaotic, noisy, coherent, and incoherent collective dynamics.

Suppression of phase synchronisation in network based on cat's brain.

We have studied the effects of perturbations on the cat's cerebral cortex. According to the literature, this cortex structure can be described by a clustered network. This way, we construct a clustered network with the same number of areas as in the cat matrix, where each area is described as a sub-network with a small-world property. We focus on the suppression of neuronal phase synchronisation considering different kinds of perturbations. Among the various controlling interventions, we choose three methods: delayed feedback control, external time-periodic driving, and activation of selected neurons. We simulate these interventions to provide a procedure to suppress undesired and pathological abnormal rhythms that can be associated with many forms of synchronisation. In our simulations, we have verified that the efficiency of synchronisation suppression by delayed feedback control is higher than external time-periodic driving and activation of selected neurons of the cat's cerebral cortex with the same coupling strengths.

Effects of assortative mixing in the second-order Kuramoto model.

In this paper we analyze the second-order Kuramoto model in the presence of a positive correlation between the heterogeneity of the connections and the natural frequencies in scale-free networks. We numerically show that discontinuous transitions emerge not just in disassortative but also in strongly assortative networks, in contrast with the first-order model. We also find that the effect of assortativity on network synchronization can be compensated by adjusting the phase damping. Our results show that it is possible to control collective behavior of damped Kuramoto oscillators by tuning the network structure or by adjusting the dissipation related to the phases' movement.

Sleep apnea-hypopnea quantification by cardiovascular data analysis.

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 <σ2>, 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.

Diminished heart beat non-stationarities in congestive heart failure.

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 role of extracellular potassium dynamics in the different stages of ictal bursting and spreading depression: a computational study.

Experimental evidences point out the participation of nonsynaptic mechanisms (e.g., fluctuations in extracellular ions) in epileptiform bursting and spreading depression (SD). During these abnormal oscillatory patterns, it is observed an increase of extracellular potassium concentration [K(+)](o) and a decrease of extracellular calcium concentration [Ca(2+)](o) which raises the neuronal excitability. However, whether the high [K(+)](o) triggers and propagates these abnormal neuronal activities or plays a secondary role into this process is unclear. To better understand the influence of extracellular potassium dynamics in these oscillatory patterns, the experimental conditions of high [K(+)](o) and zero [Ca(2+)](o) were replicated in an extended Golomb model where we added important regulatory mechanisms of ion concentration as Na(+)-K(+) pump, ion diffusion and glial buffering. Within these conditions, simulations of the cell model exhibit seizure-like discharges (ictal bursting). The SD was elicited by the interruption of the Na(+)-K(+) pump activity, mimicking the effect of cellular hypoxia (an experimental protocol to elicit SD, the hypoxia-induced SD). We used the bifurcation theory and the fast-slow method to analyze the interference of K(+) dynamics in the cellular excitability. This analysis indicates that the system loses its stability at a high [K(+)](o), transiting to an elevated state of neuronal excitability. Effects of high [K(+)](o) are observed in different stages of ictal bursting and SD. In the initial stage, the increase of [K(+)](o) creates favorable conditions to trigger both oscillatory patterns. During the neuronal activity, a continuous growth of [K(+)](o) by outward K(+) flow depresses K(+) currents in a positive feedback way. At the last stage, due to the depression of K(+) currents, the Na(+)-K(+) pump is the main mechanism in the end of neuronal activity. Thus, this work suggests that [K(+)](o) dynamics may play a fundamental role in these abnormal oscillatory patterns.