Extreme events in excitable systems and mechanisms of their generation

Extreme events in gene regulatory networks with time-delays

This work explores distinct complex dynamics of simplified two nodes of coupled gene regulatory networks with multiple delays in two self-inhibitory and mutually activated genes. We have identified the emergence of extreme events within a specific range of system parameter values. A detailed analysis of the time delay-induced emergence of extreme events is illustrated using bifurcation analysis, two-parameter phase diagrams, return maps, temporal plots, and probability density functions. The reasons behind the advent of extreme events are discussed in detail, with possible analogies to simplified two nodes of gene regulatory networks. The occasional large-amplitude bursting originated in the system via interior crisis-induced intermittency, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic intermittency routes. Additionally, we have used various recurrence quantification statistical measures, such as mean recurrence time, determinism, and recurrence time entropy, to describe the transition from periodic or chaotic to unforeseen large deviations. Our approach shows that the sudden surge of variance and mean recurrence time at the transition points can be used as a new metric to detect the critical transitions of distinct extreme bursting events. The comprehensive overview of the interaction between gene regulatory networks, with insights into the formation of unusual dynamics, is beneficial to grasping different neuronal diseases.

Extreme events in two coupled chaotic oscillators

Since 1970, the Rössler system has remained as a considerably simpler and minimal-dimensional chaos serving system. Unveiling the dynamics of a system of two coupled chaotic oscillators that lead to the emergence of extreme events in the system is an engrossing and crucial scientific research area. Our present study focuses on the emergence of extreme events in a system of diffusively and bidirectionally two coupled Rössler oscillators and unraveling the mechanism behind the genesis of extreme events. We find the appearance of extreme events in three different observables: average velocity, synchronization error, and one transverse directional variable to the synchronization manifold. The emergence of extreme events in average velocity variables happens due to the occasional in-phase synchronization. The on-off intermittency plays a crucial role in the genesis of extreme events in the synchronization error dynamics and in the transverse directional variable to the synchronization manifold. The bubble transition of the chaotic attractor due to the on-off intermittency is illustrated for the transverse directional variable. We use generalized extreme value theory to study the statistics of extremes. The extreme events data sets concerning the average velocity variable follow a generalized extreme value distribution. The inter-event intervals of the extreme events in the average velocity variable spread well exponentially. The upshot of the interplay between the coupling strength and the frequency mismatch between the oscillators in the genesis of extreme events in the coupled system is depicted numerically.

Intermittent cluster synchronization in a unidirectional ring of bursting neurons

We report a new mechanism through which extreme events with a dragon-king-like distribution emerge in a network of unidirectional ring of Hindmarsh-Rose bursting neurons interacting through chemical synapses. We establish and substantiate the fact that depending on the choice of initial conditions, the neurons are divided into different clusters. These clusters transit from a phase-locked state (antiphase) to phase synchronized regime with increasing value of the coupling strength. Before attaining phase synchronization, there exist some regions of the coupling strength where these clusters are phase synchronized intermittently. During such intermittent phase synchronization, extreme events originate in the mean field of the membrane potential. This mechanism, which we name as intermittent cluster synchronization, is proposed as the new precursor for the generation of emergent extreme events in this system. These results are also true for diffusive coupling (gap junctions). The distribution of the local maxima of the collective observable shows a long-tailed non-Gaussian while the interevent interval follows the Weibull distribution. The goodness of fit is corroborated using probability-probability plot and quantile-quantile plot. This intermittent phase synchronization becomes rarer and rarer with an increase in the number of clusters of initial conditions.

Investigation of transient extreme events in a mutually coupled star network of theoretical Brusselator system

In this article, we present evidence of a distinct class of extreme events that occur during the transient chaotic state within network modeling using the Brusselator with a mutually coupled star network. We analyze the phenomenon of transient extreme events in the network by focusing on the lifetimes of chaotic states. These events are identified through the finite-time Lyapunov exponent and quantified using threshold and statistical methods, including the probability distribution function (PDF), generalized extreme value (GEV) distribution, and return period plots. We also evaluate the transitions of these extreme events by examining the average synchronization error and the system's energy function. Our findings, validated across networks of various sizes, demonstrate consistent patterns and behaviors, contributing to a deeper understanding of transient extreme events in complex networks.

Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems

Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems' intended or desired functionality, it is crucial to understand the underlying mechanisms, to identify precursors of such transitions, and to reliably detect them in time series of suitable system observables to enable forecasts. This review critically assesses the various steps of investigation involved in time-series-analysis-based detection of critical transitions in real-world non-autonomous systems: from the data recording to evaluating the reliability of offline and online detections. It will highlight pros and cons to stimulate further developments, which would be necessary to advance understanding and forecasting nonlinear behavior such as critical transitions in complex systems.

Impact of time varying interaction: Formation and annihilation of extreme events in dynamical systems

This study investigates the emergence of extreme events in two different coupled systems: the FitzHugh-Nagumo neuron model and the forced Liénard system, both based on time-varying interactions. The time-varying coupling function between the systems determines the duration and frequency of their interaction. Extreme events in the coupled system arise as a result of the influence of time-varying interactions within various parameter regions. We specifically focus on elucidating how the transition point between extreme events and regular events shifts in response to the duration of interaction time between the systems. By selecting the appropriate interaction time, we can effectively mitigate extreme events, which is highly advantageous for controlling undesired fluctuations in engineering applications. Furthermore, we extend our investigation to networks of oscillators, where the interactions among network elements are also time dependent. The proposed approach for coupled systems holds wide applicability to oscillator networks.

Extreme rotational events in a forced-damped nonlinear pendulum

Since Galileo's time, the pendulum has evolved into one of the most exciting physical objects in mathematical modeling due to its vast range of applications for studying various oscillatory dynamics, including bifurcations and chaos, under various interests. This well-deserved focus aids in comprehending various oscillatory physical phenomena that can be reduced to the equations of the pendulum. The present article focuses on the rotational dynamics of the two-dimensional forced-damped pendulum under the influence of the ac and dc torque. Interestingly, we are able to detect a range of the pendulum's length for which the angular velocity exhibits a few intermittent extreme rotational events that deviate significantly from a certain well-defined threshold. The statistics of the return intervals between these extreme rotational events are supported by our data to be spread exponentially at a specific pendulum's length beyond which the external dc and ac torque are no longer sufficient for a full rotation around the pivot. The numerical results show a sudden increase in the size of the chaotic attractor due to interior crisis, which is the source of instability that is responsible for triggering large amplitude events in our system. We also notice the occurrence of phase slips with the appearance of extreme rotational events when the phase difference between the instantaneous phase of the system and the externally applied ac torque is observed.

Adaptive synapse-based neuron model with heterogeneous multistability and riddled basins

Biological neurons can exhibit complex coexisting multiple firing patterns dependent on initial conditions. To this end, this paper presents a novel adaptive synapse-based neuron (ASN) model with sine activation function. The ASN model has time-varying equilibria with the variation of externally applied current and its equilibrium stability involves transitions between stable and unstable points through fold and Hopf bifurcations, resulting in complex distributions of attractive regions with heterogeneous multi-stability. Globally coexisting heterogeneous behaviors are studied by bifurcation diagram, phase portrait, dynamical distribution, and basin of attraction. The results show that the number of coexisting heterogeneous attractors can be up to 12, but for a simple neuron model, such a large number of coexisting heterogeneous attractors has not been reported in the relevant literature. Most interestingly, the ASN model also has riddled-like complex basins of attraction and four illustrative examples are depicted by the phase portraits with small changes of the initial conditions. Besides, the ASN model is implemented using a simple microcontroller platform, and various heterogeneous coexisting attractors are acquired experimentally to validate the numerical results.

Extreme events in a complex network: Interplay between degree distribution and repulsive interaction

The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and, thereby, confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator associated with a node.

Regulating dynamics through intermittent interactions

In this article we experimentally demonstrate an efficient scheme to regulate the behavior of coupled nonlinear oscillators through dynamic control of their interaction. It is observed that introducing intermittency in the interaction term as a function of time or the system state predictably alters the dynamics of the constituent oscillators. Choosing the nature of the interaction, attractive or repulsive, allows for either suppression of oscillations or stimulation of activity. Two parameters Δ and τ, that reign the extent of interaction among subsystems, are introduced. They serve as a harness to access the entire range of possible behaviors from fixed points to chaos. For fixed values of system parameters and coupling strength, changing Δ and τ offers fine control over the dynamics of coupled subsystems. We show this experimentally using coupled Chua's circuits and elucidate their behavior for a range of coupling parameters through detailed numerical simulations.

Model-assisted deep learning of rare extreme events from partial observations

To predict rare extreme events using deep neural networks, one encounters the so-called small data problem because even long-term observations often contain few extreme events. Here, we investigate a model-assisted framework where the training data are obtained from numerical simulations, as opposed to observations, with adequate samples from extreme events. However, to ensure the trained networks are applicable in practice, the training is not performed on the full simulation data; instead, we only use a small subset of observable quantities, which can be measured in practice. We investigate the feasibility of this model-assisted framework on three different dynamical systems (Rössler attractor, FitzHugh-Nagumo model, and a turbulent fluid flow) and three different deep neural network architectures (feedforward, long short-term memory, and reservoir computing). In each case, we study the prediction accuracy, robustness to noise, reproducibility under repeated training, and sensitivity to the type of input data. In particular, we find long short-term memory networks to be most robust to noise and to yield relatively accurate predictions, while requiring minimal fine-tuning of the hyperparameters.

Towards a Data-Driven Estimation of Resilience in Networked Dynamical Systems: Designing a Versatile Testbed

Estimating resilience of adaptive, networked dynamical systems remains a challenge. Resilience refers to a system's capacity "to absorb exogenous and/or endogenous disturbances and to reorganize while undergoing change so as to still retain essentially the same functioning, structure, and feedbacks." The majority of approaches to estimate resilience requires exact knowledge of the underlying equations of motion; the few data-driven approaches so far either lack appropriate strategies to verify their suitability or remain subject of considerable debate. We develop a testbed that allows one to modify resilience of a multistable networked dynamical system in a controlled manner. The testbed also enables generation of multivariate time series of system observables to evaluate the suitability of data-driven estimators of resilience. We report first findings for such an estimator.

Investigating climate tipping points under various emission reduction and carbon capture scenarios with a stochastic climate model

We study the mitigation of climate tipping point transitions using an energy balance model. The evolution of the global mean surface temperature is coupled with the CO 2 concentration through the green-house effect. We model the CO 2 concentration with a stochastic delay differential equation (SDDE), accounting for various carbon emission and capture scenarios. The resulting coupled system of SDDEs exhibits a tipping point phenomena: if CO 2 concentration exceeds a critical threshold (around 478   ppm ), the temperature experiences an abrupt increase of about six degrees Celsius. We show that the CO 2 concentration exhibits a transient growth which may cause a climate tipping point, even if the concentration decays asymptotically. We derive a rigorous upper bound for the CO 2 evolution which quantifies its transient and asymptotic growths, and provides sufficient conditions for evading the climate tipping point. Combining this upper bound with Monte Carlo simulations of the stochastic climate model, we investigate the emission reduction and carbon capture scenarios that would avert the tipping point.

Optimized ensemble deep learning framework for scalable forecasting of dynamics containing extreme events

The remarkable flexibility and adaptability of both deep learning models and ensemble methods have led to the proliferation for their application in understanding many physical phenomena. Traditionally, these two techniques have largely been treated as independent methodologies in practical applications. This study develops an optimized ensemble deep learning framework wherein these two machine learning techniques are jointly used to achieve synergistic improvements in model accuracy, stability, scalability, and reproducibility, prompting a new wave of applications in the forecasting of dynamics. Unpredictability is considered one of the key features of chaotic dynamics; therefore, forecasting such dynamics of nonlinear systems is a relevant issue in the scientific community. It becomes more challenging when the prediction of extreme events is the focus issue for us. In this circumstance, the proposed optimized ensemble deep learning (OEDL) model based on a best convex combination of feed-forward neural networks, reservoir computing, and long short-term memory can play a key role in advancing predictions of dynamics consisting of extreme events. The combined framework can generate the best out-of-sample performance than the individual deep learners and standard ensemble framework for both numerically simulated and real-world data sets. We exhibit the outstanding performance of the OEDL framework for forecasting extreme events generated from a Liénard-type system, prediction of COVID-19 cases in Brazil, dengue cases in San Juan, and sea surface temperature in the Niño 3.4 region.

Enhancement of extreme events through the Allee effect and its mitigation through noise in a three species system

We consider the dynamics of a three-species system incorporating the Allee Effect, focussing on its influence on the emergence of extreme events in the system. First we find that under Allee effect the regular periodic dynamics changes to chaotic. Further, we find that the system exhibits unbounded growth in the vegetation population after a critical value of the Allee parameter. The most significant finding is the observation of a critical Allee parameter beyond which the probability of obtaining extreme events becomes non-zero for all three population densities. Though the emergence of extreme events in the predator population is not affected much by the Allee effect, the prey population shows a sharp increase in the probability of obtaining extreme events after a threshold value of the Allee parameter, and the vegetation population also yields extreme events for sufficiently strong Allee effect. Lastly we consider the influence of additive noise on extreme events. First, we find that noise tames the unbounded vegetation growth induced by Allee effect. More interestingly, we demonstrate that stochasticity drastically diminishes the probability of extreme events in all three populations. In fact for sufficiently high noise, we do not observe any more extreme events in the system. This suggests that noise can mitigate extreme events, and has potentially important bearing on the observability of extreme events in naturally occurring systems.

Noise amplification precedes extreme epileptic events on human EEG

Extreme events are rare and sudden abnormal deviations of the system's behavior from a typical state. Statistical analysis reveals that if the time series contains extreme events, its distribution has a heavy tail. In dynamical systems, extreme events often occur due to developing instability preceded by noise amplification. Here, we apply this theory to analyze generalized epileptic seizures in the human brain. First, we demonstrate that the time series of electroencephalogram (EEG) spectral power in a frequency band of 1-5 Hz obeys a heavy-tailed distribution, confirming the presence of extreme events. Second, we report that noise on EEG signals gradually increases before the seizure onset. Thus, we hypothesize that generalized epileptic seizures in humans are the extreme events emerging from instability accompanied by preictal noise amplification similar to other dynamical systems.

Optical rogue waves in coupled fiber Raman lasers

For coupled linear cavity-random fiber Raman lasers, for the first time, to the best of our knowledge, we demonstrate a new mechanism of emergence of the random pulses, with the anomalous statistics satisfying optical rogue waves' criteria experimentally. The rogue waves appear as a result of the coupling of two Raman cascades, namely, a linear cavity laser with a wavelength of 1.55 µm and a random laser with a wavelength nearly 1.67 µm, along with coupling of the orthogonal states of polarization (SOPs). The coherent coupling of SOPs causes localization of the trajectories in the vicinity of these states, whereas polarization instability drives escape taking the form of chaotic oscillations. Antiphase dynamics in two cascades result in the suppression of low amplitude chaotic oscillations and enable the anomalous spikes, satisfying rogue waves criteria.

Parametric excitation induced extreme events in MEMS and Liénard oscillator

Two paradigmatic nonlinear oscillatory models with parametric excitation are studied. The authors provide theoretical evidence for the appearance of extreme events (EEs) in those systems. First, the authors consider a well-known Liénard type oscillator that shows the emergence of EEs via two bifurcation routes: intermittency and period-doubling routes for two different critical values of the excitation frequency. The authors also calculate the return time of two successive EEs, defined as inter-event intervals that follow Poisson-like distribution, confirming the rarity of the events. Further, the total energy of the Liénard oscillator is estimated to explain the mechanism for the development of EEs. Next, the authors confirmed the emergence of EEs in a parametrically excited microelectromechanical system. In this model, EEs occur due to the appearance of a stick-slip bifurcation near the discontinuous boundary of the system. Since the parametric excitation is encountered in several real-world engineering models, like macro- and micromechanical oscillators, the implications of the results presented in this paper are perhaps beneficial to understand the development of EEs in such oscillatory systems.

Understanding the origin of extreme events in El Niño southern oscillation

We investigate a low-dimensional slow-fast model to understand the dynamical origin of El Niño southern oscillation. A close inspection of the system dynamics using several bifurcation plots reveals that a sudden large expansion of the attractor occurs at a critical system parameter via a type of interior crisis. This interior crisis evolves through merging of a cascade of period-doubling and period-adding bifurcations that leads to the origin of occasional amplitude-modulated extremely large events. More categorically, a situation similar to homoclinic chaos arises near the critical point; however, atypical global instability evolves as a channellike structure in phase space of the system that modulates variability of amplitude and return time of the occasional large events and makes a difference from the homoclinic chaos. The slow-fast timescale of the low-dimensional model plays an important role on the onset of occasional extremely large events. Such extreme events are characterized by their heights when they exceed a threshold level measured by a mean-excess function. The probability density of events' height displays multimodal distribution with an upper-bounded tail. We identify the dependence structure of interevent intervals to understand the predictability of return time of such extreme events using autoregressive integrated moving average model and box-plot analysis.

Identifying edges that facilitate the generation of extreme events in networked dynamical systems

The collective dynamics of complex networks of FitzHugh-Nagumo units exhibits rare and recurrent events of high amplitude (extreme events) that are preceded by so-called proto-events during which a certain fraction of the units become excited. Although it is well known that a sufficiently large fraction of excited units is required to turn a proto-event into an extreme event, it is not yet clear how the other units are being recruited into the final generation of an extreme event. Addressing this question and mimicking typical experimental situations, we investigate the centrality of edges in time-dependent interaction networks. We derived these networks from time series of the units' dynamics employing a widely used bivariate analysis technique. Using our recently proposed edge-centrality concepts together with an edge-based network decomposition technique, we observe that the recruitment is primarily facilitated by sets of certain edges that have no equivalent in the underlying topology. Our finding might aid to improve the understanding of generation of extreme events in natural networked dynamical systems.

Advent of extreme events in predator populations

We study the dynamics of a ring of patches with vegetation–prey–predator populations, coupled through interactions of the Lotka–Volterra type. We find that the system yields aperiodic, recurrent and rare explosive bursts of predator density in a few isolated spatial patches from time to time. Further, the global predator biomass also exhibits sudden uncorrelated occurrences of large deviations from the mean as the coupled system evolves. The maximum value of the predator population in a patch, as well as the maximum value of the predator biomass, increases with coupling strength. These trends are further corroborated by fits to Generalized Extreme Value distributions, where the location and scale factor of the distribution increases markedly with coupling strength, indicating the crucial role of coupling interactions in the generation of extreme events. These results indicate how occurrences of extremely large predator populations can emerge in coupled population dynamics, and in a more general context they suggest a generic class of deterministic nonlinear systems that can naturally exhibit extreme events

Routes to extreme events in dynamical systems: Dynamical and statistical characteristics

Intermittent large amplitude events are seen in the temporal evolution of a state variable of many dynamical systems. Such intermittent large events suddenly start appearing in dynamical systems at a critical value of a system parameter and continues for a range of parameter values. Three important processes of instabilities, namely, interior crisis, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic motion, are most common as observed in many systems that lead to such occasional and rare transitions to large amplitude spiking events. We characterize these occasional large events as extreme events if they are larger than a statistically defined significant height. We present two exemplary systems, a single system and a coupled system, to illustrate how the instabilities work to originate extreme events and they manifest as non-trivial dynamical events. We illustrate the dynamical and statistical properties of such events.

Extreme events in a network of heterogeneous Josephson junctions

We report intermittent large spiking events in a heterogeneous network of forced Josephson junctions under the influence of repulsive interaction. The response of the individual junctions has been inspected instead of the collective response of the ensemble, which reveals the large spiking events in a subpopulation with characteristic features of extreme events (EE). The network splits into three clusters of junctions, one in coherent libration, one in incoherent rotational motion, and another subpopulation originating EE, which resembles a chimeralike pattern. EE migrates spatially from one to another subpopulation of junctions with the repulsive strength. The origin of EE in a subpopulation and chimera pattern is a generic effect of distributed damping parameter and repulsive interaction, which we verify with another network of the Liénard system. EE originates in the subpopulation via a local riddling of in-phase synchronization. The probability density function of event heights confirms the rare occurrence of large events and the return time of EE as expressed by interevent intervals in the subgroup follows a Poisson distribution. The mechanism of the origin of such a unique clustering is explained qualitatively.

Unstructured network topology begets order-based representation by privileged neurons

How spiking activity reverberates through neuronal networks, how evoked and spontaneous activity interacts and blends, and how the combined activities represent external stimulation are pivotal questions in neuroscience. We simulated minimal models of unstructured spiking networks in silico, asking whether and how gentle external stimulation might be subsequently reflected in spontaneous activity fluctuations. Consistent with earlier findings in silico and in vitro, we observe a privileged subpopulation of 'pioneer neurons' that, by their firing order, reliably encode previous external stimulation. We also confirm that pioneer neurons are 'sensitive' in that they are recruited by small fluctuations of population activity. We show that order-based representations rely on a 'chain' of pioneer neurons with different degrees of sensitivity and thus constitute an emergent property of collective dynamics. The forming of such representations is greatly favoured by a broadly heterogeneous connection topology-a broad 'middle class' in degree of connectedness. In conclusion, we offer a minimal model for the representational role of pioneer neurons, as observed experimentally in vitro. In addition, we show that broadly heterogeneous connectivity enhances the representational capacity of unstructured networks.

Seizure prediction - ready for a new era

Epilepsy is a common disorder characterized by recurrent seizures. An overwhelming majority of people with epilepsy regard the unpredictability of seizures as a major issue. More than 30 years of international effort have been devoted to the prediction of seizures, aiming to remove the burden of unpredictability and to couple novel, time-specific treatment to seizure prediction technology. A highly influential review published in 2007 concluded that insufficient evidence indicated that seizures could be predicted. Since then, several advances have been made, including successful prospective seizure prediction using intracranial EEG in a small number of people in a trial of a real-time seizure prediction device. In this Review, we examine advances in the field, including EEG databases, seizure prediction competitions, the prospective trial mentioned and advances in our understanding of the mechanisms of seizures. We argue that these advances, together with statistical evaluations, set the stage for a resurgence in efforts towards the development of seizure prediction methodologies. We propose new avenues of investigation involving a synergy between mechanisms, models, data, devices and algorithms and refine the existing guidelines for the development of seizure prediction technology to instigate development of a solution that removes the burden of the unpredictability of seizures.

Intermittent large deviation of chaotic trajectory in Ikeda map: Signature of extreme events

We notice signatures of extreme eventslike behavior in a laser based Ikeda map. The trajectory of the system occasionally travels a large distance away from the bounded chaotic region, which appears as intermittent spiking events in the temporal dynamics. The large spiking events satisfy the conditions of extreme events as usually observed in dynamical systems. The probability density function of the large spiking events shows a long-tail distribution consistent with the characteristics of rare events. The interevent intervals obey a Poissonlike distribution. We locate the parameter regions of extreme events in phase diagrams. Furthermore, we study two Ikeda maps to explore how and when extreme events terminate via mutual interaction. A pure diffusion of information exchange is unable to terminate extreme events where synchronous occurrence of extreme events is only possible even for large interaction. On the other hand, a threshold-activated coupling can terminate extreme events above a critical value of mutual interaction.

Traceability and dynamical resistance of precursor of extreme events

Extreme events occur in a variety of natural, technical, and societal systems and often have catastrophic consequences. Their low-probability, high-impact nature has recently triggered research into improving our understanding of generating mechanisms, providing early warnings as well as developing control strategies. For the latter to be effective, knowledge about dynamical resistance of a system prior to an extreme event is of utmost importance. Here we introduce a novel time-series-based and non-perturbative approach to efficiently monitor dynamical resistance and apply it to high-resolution observations of brain activities from 43 subjects with uncontrollable epileptic seizures. We gain surprising insights into pre-seizure dynamical resistance of brains that also provide important clues for success or failure of measures for seizure prevention. The novel resistance monitoring perspective advances our understanding of precursor dynamics in complex spatio-temporal systems with potential applications in refining control strategies.

Emergence of extreme events in networks of parametrically coupled chaotic populations

We consider a collection of populations modelled by the prototypical chaotic Ricker map, relevant to the population growth of species with non-overlapping generations. The growth parameter of each population patch is influenced by the local mean field of its neighbourhood, and we explore the emergent patterns in such a parametrically coupled network. In particular, we examine the dynamics and distribution of the local populations, as well as the total biomass. Our significant finding is the following: When the range of coupling is sufficiently large, namely, when enough neighbouring populations influence the growth rate of a population, the system yields remarkably large biomass values that are very far from the mean. These extreme events are relatively rare and uncorrelated in time. We also find that at any point in time, exceedingly large population densities emerge in a few patches, analogous to an extreme event in space. Thus, we suggest a new mechanism in coupled chaotic systems that naturally yield extreme events in both time and space.

Complexity and irreducibility of dynamics on networks of networks

We study numerically the dynamics of a network of all-to-all-coupled, identical sub-networks consisting of diffusively coupled, non-identical FitzHugh-Nagumo oscillators. For a large range of within- and between-network couplings, the network exhibits a variety of dynamical behaviors, previously described for single, uncoupled networks. We identify a region in parameter space in which the interplay of within- and between-network couplings allows for a richer dynamical behavior than can be observed for a single sub-network. Adjoining this atypical region, our network of networks exhibits transitions to multistability. We elucidate bifurcations governing the transitions between the various dynamics when crossing this region and discuss how varying the couplings affects the effective structure of our network of networks. Our findings indicate that reducing a network of networks to a single (but bigger) network might not be accurate enough to properly understand the complexity of its dynamics.

Dragon-king-like extreme events in coupled bursting neurons

We present evidence of extreme events in two Hindmarsh-Rose (HR) bursting neurons mutually interacting via two different coupling configurations: chemical synaptic- and gap junctional-type diffusive coupling. A dragon-king-like probability distribution of the extreme events is seen for combinations of synaptic coupling where small- to medium-size events obey a power law and the larger events that cross an extreme limit are outliers. The extreme events originate due to instability in antiphase synchronization of the coupled systems via two different routes, intermittency and quasiperiodicity routes to complex dynamics for purely excitatory and inhibitory chemical synaptic coupling, respectively. For a mixed type of inhibitory and excitatory chemical synaptic interactions, the intermittency route to extreme events is only seen. Extreme events with our suggested distribution is also seen for gap junctional-type diffusive, but repulsive, coupling where the intermittency route to complexity is found. A simple electronic experiment using two diffusively coupled analog circuits of the HR neuron model, but interacting in a repulsive way, confirms occurrence of the dragon-king-like extreme events.

Riddled basins of attraction in systems exhibiting extreme events

Using a system of two FitzHugh-Nagumo units, we demonstrate the occurrence of riddled basins of attraction in delay-coupled systems as the coupling between the units is increased. We characterize riddled basins using the uncertainty exponent which is a measure of the dimensions of the basin boundary. Additionally, we show that the phase space can be partitioned into pure and mixed regions, where initial conditions in the pure regions certainly avoid the generation of extreme events, while initial conditions in the mixed region may or may not exhibit such events. This implies that any tiny perturbation of initial conditions in the mixed region could yield the emergence of extreme events because the latter state possesses a riddled basin of attraction.

Extreme events in the forced Liénard system

We observe extremely large amplitude intermittent spikings in a dynamical variable of a periodically forced Liénard-type oscillator and characterize them as extreme events, which are rare, but recurrent and larger in amplitude than a threshold. The extreme events occur via two processes, an interior crisis and intermittency. The probability of occurrence of the events shows a long-tail distribution in both the cases. We provide evidence of the extreme events in an experiment using an electronic analog circuit of the Liénard oscillator that shows good agreement with our numerical results.

Early warning signal for interior crises in excitable systems

The ability to reliably predict critical transitions in dynamical systems is a long-standing goal of diverse scientific communities. Previous work focused on early warning signals related to local bifurcations (critical slowing down) and nonbifurcation-type transitions. We extend this toolbox and report on a characteristic scaling behavior (critical attractor growth) which is indicative of an impending global bifurcation, an interior crisis in excitable systems. We demonstrate our early warning signal in a conceptual climate model as well as in a model of coupled neurons known to exhibit extreme events. We observed critical attractor growth prior to interior crises of chaotic as well as strange-nonchaotic attractors. These observations promise to extend the classes of transitions that can be predicted via early warning signals.

Extreme events in FitzHugh-Nagumo oscillators coupled with two time delays

We study two identical FitzHugh-Nagumo oscillators which are coupled with one or two different time delays. If only a single-delay coupling is used, the length of the delay determines whether the synchronization manifold is transversally stable or unstable, exhibiting mixed-mode or chaotic oscillations in which the small amplitude oscillations are always in phase but the large amplitude oscillations are in phase or out of phase, respectively. For two delays we find an intricate dynamics which comprises an irregular alteration of small amplitude oscillations, in-phase and out-of-phase large amplitude oscillations, also called extreme events. This transient chaotic dynamics is sandwiched between a bubbling transition and a blowout bifurcation.

Reflex seizures, traits, and epilepsies: from physiology to pathology

Epileptic seizures are generally unpredictable and arise spontaneously. Patients often report non-specific triggers such as stress or sleep deprivation, but only rarely do seizures occur as a reflex event, in which they are objectively and consistently modulated, precipitated, or inhibited by external sensory stimuli or specific cognitive processes. The seizures triggered by such stimuli and processes in susceptible individuals can have different latencies. Once seizure-suppressing mechanisms fail and a critical mass (the so-called tipping point) of cortical activation is reached, reflex seizures stereotypically manifest with common motor features independent of the physiological network involved. The complexity of stimuli increases from simple sensory to complex cognitive-emotional with increasing age of onset. The topography of physiological networks involved follows the posterior-to-anterior trajectory of brain development, reflecting age-related changes in brain excitability. Reflex seizures and traits probably represent the extremes of a continuum, and understanding of their underlying mechanisms might help to elucidate the transition of normal physiological function to paroxysmal epileptic activity.

Linear Augmentation for Stabilizing Stationary Solutions: Potential Pitfalls and Their Application

Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

A highly specific test for periodicity

We present a method that allows to distinguish between nearly periodic and strictly periodic time series. To this purpose, we employ a conservative criterion for periodicity, namely, that the time series can be interpolated by a periodic function whose local extrema are also present in the time series. Our method is intended for the analysis of time series generated by deterministic time-continuous dynamical systems, where it can help telling periodic dynamics from chaotic or transient ones. We empirically investigate our method's performance and compare it to an approach based on marker events (or Poincaré sections). We demonstrate that our method is capable of detecting small deviations from periodicity and outperforms the marker-event-based approach in typical situations. Our method requires no adjustment of parameters to the individual time series, yields the period length with a precision that exceeds the sampling rate, and its runtime grows asymptotically linear with the length of the time series.

Data-driven prediction and prevention of extreme events in a spatially extended excitable system

Extreme events occur in many spatially extended dynamical systems, often devastatingly affecting human life, which makes their reliable prediction and efficient prevention highly desirable. We study the prediction and prevention of extreme events in a spatially extended system, a system of coupled FitzHugh-Nagumo units, in which extreme events occur in a spatially and temporally irregular way. Mimicking typical constraints faced in field studies, we assume not to know the governing equations of motion and to be able to observe only a subset of all phase-space variables for a limited period of time. Based on reconstructing the local dynamics from data and despite being challenged by the rareness of events, we are able to predict extreme events remarkably well. With small, rare, and spatiotemporally localized perturbations which are guided by our predictions, we are able to completely suppress extreme events in this system.

Transitions between dynamical behaviors of oscillator networks induced by diversity of nodes and edges

We study the impact of dynamical and structural heterogeneity on the collective dynamics of large small-world networks of pulse-coupled integrate-and-fire oscillators endowed with refractory periods and time delay. Depending on the choice of homogeneous control parameters (here, refractoriness and coupling strength), these networks exhibit a large spectrum of dynamical behaviors, including asynchronous, partially synchronous, and fully synchronous states. Networks exhibit transitions between these dynamical behaviors upon introducing heterogeneity. We show that the probability for a network to exhibit a certain dynamical behavior (network susceptibility) is affected differently by dynamical and structural heterogeneity and depends on the respective homogeneous dynamics.

Measuring predictability of autonomous network transitions into bursting dynamics

Understanding spontaneous transitions between dynamical modes in a network is of significant importance. These transitions may separate pathological and normal functions of the brain. In this paper, we develop a set of measures that, based on spatio-temporal features of network activity, predict autonomous network transitions from asynchronous to synchronous dynamics under various conditions. These metrics quantify spike-timing distributions within a narrow time window as a function of the relative location of the active neurons. We applied these metrics to investigate the properties of these transitions in excitatory-only and excitatory-and-inhibitory networks and elucidate how network topology, noise level, and cellular heterogeneity affect both the reliability and the timeliness of the predictions. The developed measures can be calculated in real time and therefore potentially applied in clinical situations.

Actin and PIP3 waves in giant cells reveal the inherent length scale of an excited state

The membrane and actin cortex of a motile cell can autonomously differentiate into two states, one typical of the front, the other of the tail. On the substrate-attached surface of Dictyostelium discoideum cells, dynamic patterns of front-like and tail-like states are generated that are well suited to monitor transitions between these states. To image large-scale pattern dynamics independently of boundary effects, we produced giant cells by electric-pulse-induced cell fusion. In these cells, actin waves are coupled to the front and back of phosphatidylinositol (3,4,5)-trisphosphate (PIP3)-rich bands that have a finite width. These composite waves propagate across the plasma membrane of the giant cells with undiminished velocity. After any disturbance, the bands of PIP3 return to their intrinsic width. Upon collision, the waves locally annihilate each other and change direction; at the cell border they are either extinguished or reflected. Accordingly, expanding areas of progressing PIP3 synthesis become unstable beyond a critical radius, their center switching from a front-like to a tail-like state. Our data suggest that PIP3 patterns in normal-sized cells are segments of the self-organizing patterns that evolve in giant cells.

Route to extreme events in excitable systems

Systems of FitzHugh-Nagumo units with different coupling topologies are capable of self-generating and -terminating strong deviations from their regular dynamics that can be regarded as extreme events due to their rareness and recurrent occurrence. Here we demonstrate the crucial role of an interior crisis in the emergence of extreme events. In parameter space we identify this interior crisis as the organizing center of the dynamics by employing concepts of mixed-mode oscillations and of leaking chaotic systems. We find that extreme events occur in certain regions in parameter space, and we show the robustness of this phenomenon with respect to the system size.