Predicting Transitions in Complex Systems

We consider critical transitions in dynamical systems perturbed by bounded noise. Dynamical systems perturbed by bounded noise can be modeled either as random dynamical systems (describing the dynamics for each noise realisation) or set-valued dynamical systems (describing only the collection of all possible behaviour). We focus in this talk on set-valued dynamical systems and demonstrate that under reasonable assumptions, there are only two different types of critical transitions in set-valued dynamical systems, and we characterise these critical transitions by means of collisions of attractor-repeller pair. Analogous questions are studied in the nonautonomous case. Joint work with Jochen Bröcker, Giulia Carigi, Jeroen Lamb, and Christian Rodrigues.

Stock networks have become a valid tool to characterize extreme events in stock markets such as financial crisis. To enable this characterization a careful construction of such a network should be adopted taking into account various properties of corresponding multivariate datasets. Recent studies have shown the presence of nonlinearity in these datasets as well as indicated possible effects of these contributions on further use of stock networks. Motivated by these studies we explore in more detail particular effects of nonlinearity within the whole process of complex network analysis applied on stock market data. We design a particular pipeline to mitigate any alleged effects and localize remaining nonlinear contributions. This pipeline is in form of suggested recipe to perform a careful construction of a stock network and enables thus a valid characterization of corresponding extreme events. This work is joint cooperation with Jaroslav Hlinka.

The brain is a complex system that is known change its state on a range of temporal scales, both as part of healthy dynamics and in disease. We shall discuss several approaches to characterize such states and to conceptualize their transitions. These will include the use of network theory to characterize brain states and some pitfalls of this methodology, detecting transitions in brain states using temporal clustering techniques, and a model of dynamics of switching between ictal and inter-ictal state in epilepsy, where we shall focus on its counter-intuitive behaviour under external perturbations. These examples will help us demonstrate how taking a closer look at some mathematical conceptualization of brain states and their transitions can help us see beyond the initial intuitions.

An emerging idea in theoretical neuroscience is that complex behaviors and cognitive functions rely on an optimal balance between functional integration and functional differentiation in cortical circuits, otherwise defined as brain complexity (Tononi and Edelman 1998; Sporns et al. 2000; Bassett and Bullmore 009; Deco et al. 2015). Neurophysiologically, this balance is contingent on the ability of multiple, functionally specialized groups of cortical neurons (differentiation) to engage in rapid causal interactions (integration) and produce complex patterns of activity.Empirically, it has been demonstrated that the complexity of cortical responses reliably discriminates between consciousness and unconsciousness in healthy humans during sleep and anesthesia as well as in brain-injured patients. In this presentation I will discuss our experiments measuring the complexity of the cortical networks under different conditions to explore how complexity is modulated, the underlying neuronal mechanisms, computational consequences and implications on consciousness levels.

Connectivity networks are formed from multivariate time series, such as multi-channel electroencephalograms, to study the causality structure of the underlying system. Here, the problem of estimating the connectivity network in the presence of prominent unobserved variables is addressed. The causality measure of partial mutual information from mixed embedding (PMIME), designed to estimate direct causality in the presence of many observed variables, is adapted to estimate also instantaneous (zero-lag) causality, termed PMIME0. The estimation of zero-lag causality by PMIME0 is a signature of the presence of hidden source in the observed system, as demonstrated analytically on a toy system. The PMIME0 is found to identify the true instantaneous and other causality effects with great accuracy in a variety of multivariate dynamical systems (linear, chaotic, discrete, continuous). The method is applied to electroencephalogram (EEG) data with epileptiform discharges (ED), where the results imply a strong impact of unobserved confounders during the EDs. This finding comes as a possible explanation for the increased levels of causality during epileptic seizures estimated by some measures affected by the presence of a common source.

The detection of structural changes is a popular research topic and in the recent years the study of changes in the dynamics of a coupled system has gained much attention and has been applied in various fields for early warning and prediction. Granger causality measures have been extensively used to capture the connectivity structure of coupled systems and form causality networks from multivariate time series. A large number of network indices are available to quantify characteristics of the causality networks. The aim of this study is to provide a methodology for the identification of changes in the connectivity structure of a high-dimensional dynamical system observed from a multivariate time series, using an appropriate connectivity measure, network indices and data mining techniques. In particular, the partial mutual information from mixed embedding (PMIME) is used to estimate direct causal effects among many observed variables, and a large number of network indices are computed on the network formed by PMIME. Feature selection is applied to derive an optimal subset of features (network indices), which is then fed into the cumulative sum control chart (CUSUM) and multivariate cumulative sum control chart (MCUSUM) algorithms for the detection of structural change points. In terms of a simulation study, this setting is applied for two different scenarios, i.e. sharp and smooth changes of the coupling structure of a high-dimensional system (Mackey-Glass of regimes of different complexity). The same setting is also applied to electroencephalogram (EEG) recordings to detect connectivity changes before, during and after epileptic seizures.

The organization of biological form and function is a classic problem cut-crossing disciplines which include a variety of transitions between complex spatiotemporal patterns. Historically, work focussed first into understand self-organization, and more recently the attention shifted to scale-free collective fluctuations, many of them shown to correspond to critical phenomena. In this lecture we will review our recent work across several scales uncovering novel understanding of proteins, mitocondria and brain function and dynamics.

Complex dynamical processes show sudden transitions to extremely large amplitude events when a control parameter is varied. Crisis-induced intermittency is one important route to such transition. A search for other possible reasons is necessary that might provide clues on the prediction of such events. We investigated two different dynamical systems, one forced Liénard system and a coupled bursting neuron model that showed three different routes to such sudden occurrence of large events. In the Lienard system, we observed two different parameter regimes where, (1) a period-doubling route to chaos with a slow growth of the chaotic attractor and then a sudden large expansion of the attractor, (2) period to intermittency route to chaos and then a sudden large expansion of the attractor. The amplitude of the large events crosses a significant height that is taken as a qualifier of extreme events. In both the cases, we observed long-tail non-Gaussian probability of the events indicating emergence of rare, but occasional large amplitude events. In the second model, we coupled two bursting Hindmarsh-Rose neurons and apply chemical synaptic interactions, when we noted (3) a third kind of transition, a quasiperiodic route to chaos with an expansion of attractor for a type of chemical synaptic interactions, while in another regime, we noted the intermittency route once again. Here the occasional very large amplitude events are outliers to a power-law followed by small to moderate size events. So far we guess from our observations that presence of saddle orbit or a saddle causes such large amplitude transitions. We are yet to find the exact cause of such transitions.

We study the dynamical regimes demonstrated by a pair of identical 3-element ring oscillators (reduced version of synthetic 3-gene genetic Repressilator) coupled using the design of the ‘quorum sensing (QS)’ process natural for interbacterial communications. In this work QS is implemented as an additional network incorporating elements of the ring as both the source and the activation target of the fast diffusion QS signal. This version of indirect nonlinear coupling, in cooperation with the reasonable extension of the parameters which control properties of the isolated oscillators, exhibits the formation of a very rich array of attractors. Using a parameter-space defined by the individual oscillator amplitude and the coupling strength, we found the extended area of parameter-space where the identical oscillators demonstrate quasiperiodicity, which evolves to chaos via the period doubling of either resonant limit cycles or complex antiphase symmetric limit cycles with five winding numbers. The symmetric chaos extends over large parameter areas up to its loss of stability, followed by a system transition to an unexpected mode: an asymmetric limit cycle with a winding number of 1:2. In turn, after long evolution across the parameter-space, this cycle demonstrates a period doubling cascade which restores the symmetry of dynamics by formation of symmetric chaos, which nevertheless preserves the memory of the asymmetric limit cycles in the form of stochastic alternating “polarization” of the time series. All stable attractors coexist with some others, forming remarkable and complex multistability including the coexistence of torus and limit cycles, chaos and regular attractors, symmetric and asymmetric regimes. We traced the paths and bifurcations leading to all areas of chaos, and presented a detailed map of all transformations of the dynamics. Accepted manuscript: 10.1016/j.cnsns.2018.03.006

Electro-mechanical excitation waves in the heart may exhibit different spatio-temporal dynamics ranging from repeated plane waves to scroll waves or spatio-temporal chaos, resulting in life threatening arrhythmias. This kind of chaotic dynamics in excitable cardiac media is often characterised by significant complexity fluctuations (including laminar phases) and can be non-persistent exhibiting supertransients, with lifetimes of the chaotic phases increasing exponentially with the system size. Terminating or at least shortening chaotic transients can be life saving in the medical context of cardiac arrhythmias. Therefore, we study the impact of perturbations on the duration of transients and features of the terminal phase of chaotic transients. Practically, such perturbations can be applied via so-called virtual electrodes where electrical heterogeneities of the cardiac muscle act as local excitation sites when subjected to a global electric field. In the talk we shall present novel results on the nonlinear dynamics of the heart including features of the terminal phase of transient chaos, parameter and state estimation, as well as experimental studies and modalities.

Schädliche Algenblüten treten dann auf, wenn bestimmte Algenarten, die potentiell schädlich für ihre Konkurrenten oder ihre Fraßfeinde sind, explosionsartig wachsen und die gesamte Dynamik des Ökosystems verändern. Einige Arten können auch Gifte bilden, die über die Nahrungskette oder durch Hautkontakt beim Baden gesundheitsschädlich für den Menschen sein können. Wir untersuchen welche biotischen und abiotischen Faktoren zu solchen Blüten führen können und wie sie sich im Meer ausbreiten können.

We analyse climate dynamics from a complex network approach. This leads to an inverse problem: Is there a backbone-like structure underlying the climate system? For this we propose a method to reconstruct and analyze a complex network from data generated by a spatio-temporal dynamical system. This approach enables us to uncover relations to global circulation patterns in oceans and atmosphere. This concept is then applied to Monsoon data; in particular, we develop a general framework to predict extreme events by combining a non-linear synchronization technique with complex networks. Applying this method, we uncover a new mechanism of extreme floods in the eastern Central Andes which could be used for operational forecasts. Moreover, we analyze the Indian Summer Monsoon (ISM) and identify two regions of high importance. By estimating an underlying critical point, this leads to a substantially improved prediction of the onset of the ISM. References Runge, J. , J. Heitzig, V. Petoukhov, J. Kurths, Phys. Rev. Lett. 108, 258701 (2012) Boers, N., B. Bookhagen, N. Marwan, J. Kurths, and J. Marengo, Geophys. Res. Lett. 40, 4386 (2013) N. Boers, B. Bookhagen, H.M.J. Barbosa, N. Marwan, J. Kurths, and J.A. Marengo, Nature Communications 5, 5199 (2014) N. Boers, R. Donner, B. Bookhagen, and J. Kurths, Climate Dynamics 45, 619 (2015) J. Runge et al., Nature Communications 6, 8502 (2015) V. Stolbova, E. Surovyatkina, B. Bookhagen, and J. Kurths, Geophys. Res. Lett. (2016) D. Eroglu, F. McRobies, I. Ozken, T. Stemler, K. Wyrwoll, S. Breitenbach, N. Marwan, J. Kurths, Nature Communications 7, 12929 (2016) B. Goswami, N. Boers, A. Rheinwalt, N. Marwan, J. Heitzig, S. Breitenbach, J. Kurths, Nature Communications 9, 48(2018)

The existence of persistent midlatitude atmospheric flow regimes with time-scales larger than 5-10 days and indications of preferred transitions between them motivates to develop early warning indicators for such regime transitions. In this talk, I will use this problem as a motivating example to develop more general indicators of geophysical flow transitions (on many different time scales), such as the collapse of the ocean's meridional overturning circulation, using operator type techniques.

Approaches to modelling critical transitions from time series data are discussed. A non-stationary low-order stochastic dynamical system of appropriate complexity to capture the transition mechanism under consideration is estimated from data. The technique is generic, not requiring detailed a priori knowledge about the underlying dynamics of the system. A hierarchy of methods is considered. First, linear models are fitted and the time evolution of their eigenvalues studied and extrapolated beyond the learning data window. Second, nonlinear modelling is employed. In the simplest case, the model is a one-dimensional effective Langevin equation, but also higher-dimensional dynamical reconstructions based on time-delay embedding and local modelling are considered. Integrations with the non-stationary models are performed beyond the learning data window to predict the nature and timing of critical transitions. Also spatially extended systems are addressed; they are projected onto their essential modes before applying the techniques described above. Example systems from the field of weather and climate science are discussed.

Forecasts of local scale weather are based on ensemble simulations using a mesoscale limited area numerical weather prediction (NWP) model. Particularly on the atmospheric mesoscale, uncertainties are large and predictions are probabilistic in nature. Thus verification and post-processing of local weather is an important part in forecasting local scale and extreme weather. We will present and discuss approaches for verification and ensemble post-processing of local weather forecasts, introducing the concept of proper scoring rules and the application thereof.

Any scientific discipline strives to explain causes of observed phenomena. Quantitative, mathematical description of causality is possible when studying phenomena evolving in time and providing measurable quantities which can be registered in consecutive instants of time and stored in datasets called time series. As examples we can mention long-term recordings of air temperature, or recordings of the electrical activity of human brain, known as the electroencephalogram. In this talk we will follow ideas of the father of cybernetics, Norbert Wiener, and Nobel prize winner Sir C.W.J. Granger. We will explain how to detect causality using probability distribution functionals from information theory and the interpretation of causal relations as information transfer. We will study the information transfer in chaotic systems on the route to synchronization. The time and the the arrow of time play a natural role in the definition of causality: the cause precedes the effect. We will investigate whether this principle is obeyed by chaotic dynamical systems and how causality estimators behave near transitions/bifurcations. Another role of time can be seen in complex systems evolving on multiple time scales. We will show how to measure the information transfer across time scales. In an application we will demonstrate a causal influence of climate oscillations with a period about 7-8 years on the amplitude of the annual temperature cycle and the inter-annual variability of the mean winter temperature in central Europe.

We study daily surface air temperature (SAT) reanalysis in a grid over the Earth surface and identify and quantify changes in SAT variability patterns during the period 1979–2016. By analysing Hilbert amplitude and frequency we identify the regions where relative variations are most pronounced (larger than $\pm$50% for the amplitude and $\pm$100% for the frequency). Amplitude variations are interpreted as due to changes in precipitation or ice melting; frequency variations as due to a northward shift of the inter-tropical convergence zone (ITCZ) and a widening of the rainfall band in the western Pacific Ocean. The ITCZ is the ascending branch of the Hadley cell and thus, by affecting the tropical atmospheric circulation, ITCZ migration has far reaching climatic consequences. As the methodology proposed here can be applied to many other geophysical time series, our work will stimulate new research that will advance the understanding of climate change impacts.

Earth’s orbit and axial tilt imprint a strong seasonal cycle on climatological data. Climate variability is typically viewed in terms of fluctuations in the seasonal cycle induced by higher frequency processes. We can interpret this as a competition between the orbitally enforced monthly stability and the fluctuations/noise induced by weather. Here we introduce a new time-series method that determines these contributions from monthly-averaged data. We find that the spatio-temporal distribution of the monthly stability and the magnitude of the noise reveal key fingerprints of several important climate phenomena, including the evolution of the Arctic sea ice cover, the El-Nino Southern Oscillation (ENSO), the Atlantic-Nino and the Indian Dipole Mode. In analogy with the classical destabilising influence of the ice-albedo feedback on summertime sea ice, we find that during some time interval of the season a destabilising process operates in all of these climate phenomena. The interaction between the destabilisation and the accumulation of noise, which we term the memory effect, underlies phase locking to the seasonal cycle and the statistical nature of seasonal predictability.

The sudden and apparently unpredictable nature of seizures is one of the most disabling aspects of the disease epilepsy that affects about 1% of the world population. Identifying seizure precursors from brain dynamics could drastically improve therapeutic possibilities and thus the quality of life of people with epilepsy. Over the last two decades, an improved characterization of the complex spatial-temporal dynamics of the epileptic brain could be achieved with tools from nonlinear dynamics, statistical physics, synchronization and network theory. These tools appear to be capable of defining seizure precursors from electroencephalographic recordings. In this talk, I will provide an overview of the progress that has been made in the field: from preliminary descriptions of pre-seizure phenomena and proof of principle studies via controlled studies to the recent development of implantable seizure prediction and prevention systems.

We present a data-driven analysis approach for the identification of a critical transitional phase and for estimating its resilience. We map the complex structure of time-resolved similarity matrices of some multivariate dynamics onto a finite number of states, and from the "distance" between successive states we derive an estimate for the resilience of a given state. Using our approach, we investigate long-term (covering days to weeks), multichannel electroencephalographic (EEG) recordings from up to now more than 40 epilepsy patients that captured more than 60 seizures. We provide evidence that our analysis approach allows for a detection of pre-seizure states in the majority of patients. More importantly, the corresponding changes in resilience provide important clues for a possible controlability of a critical transitional pre-seizure phase.

Complex spatiotemporal patterns, called chimera states, consist of coexisting coherent and incoherent domains and can be observed in networks of coupled oscillators. The interplay of synchrony and asynchrony in complex brain networks is an important aspect in studies of brain functions and dysfunctions. We analyse the collective dynamics of FitzHugh-Nagumo neurons in complex networks motivated by neuroscience. We compare two topologies: an empirical structural neural connectivity derived from weighted magnetic resonance imaging and a mathematically constructed network with modular fractal connectivity. We analyse the properties of chimeras and partially synchronized states, and obtain regions of their stability in the parameter planes. Furthermore, we study the influence of the removal of nodes on the network synchronizability, which can be useful for applications to epileptic seizures.

In this talk, I shall report on recent progress on several frontiers in the analysis and application of early-warning signs. Classical techniques such as using slowing down and variance/autocorrelation in the context of stochastic ordinary differential equations are by-now well explored. Yet, many open questions remain for warning signs in the following areas: (a) statistical quantification, (b) stochastic partial differential equations, and (c) network dynamics. For each area, I am going to describe the current state of mathematical theory as well as examples from applications. I am also going to propose some potential future directions for the theory.

When concepts like the bulk free energy and the interface tension are available, it is the competition between both that determines the type of phase conversion in a phase transition. The conversion can proceed via nucleation or spinodal decomposition, for example. Here we study a transition between collective fixed-point and collective synchronized oscillatory behavior, where these concepts are not applicable. The system consists of coupled dynamical units, which individually can be in an excitable or oscillatory state. The conversion is triggered by the change of a single bifurcation parameter. Of particular interest is the arrest of oscillations. We identify the criterion that determines the seeds of arrest and the propagation of arrest fronts in terms of the vicinity to the future attractor. Due to a high degree of multistability we observe versatile patterns of phase locked motion in the oscillatory regime. Quenching the system into the regime, where oscillatory states are no longer stable, we observe qualitatively distinct approaches of the fixed-point attractor, depending on the initial seeds. If the seeds of arrest are isolated single sites of the lattice, the arrest propagates via bubble formation, visually similar to nucleation processes; if the seed is extended along a line of lowest amplitudes, the freezing follows the spatial patterns of phase-locked motion with long interfaces between arrested and oscillating units [1]. In an outlook we consider a transition in the collective behavior of Kuramoto oscillators between a unique collective fixed-point solution for positive couplings to multistable solutions of partial synchronization, for which the multistability depends on the strength of negative couplings [2]. References: [1] D. Labavic and H. Meyer-Ortmanns, On the arrest of synchronized oscillations, Europhys.Lett. 109, 10 001-p1-p6 (2015). [2] S. Esmaeili, D. Labavic, M. Pleimling and H. Meyer-Ortmanns, Breaking of time-translation invariance in Kuramoto dynamics with multiple time scales, Europhys. Lett.118, 40006 (2017).

Superstatistics is a widely employed tool of non-equilibrium statistical physics which plays an important role in analysis of hierarchical complex dynamical systems. Yet, its canonical'' formulation in terms of a single nuisance parameter is often too restrictive when applied to complex empirical data. Here we show that a multi-scale generalization of the superstatistics paradigm is more versatile, allowing to address such pertinent issues as transmutation of statistics or inter-scale stochastic behavior. To put some flesh on the bare bones, we provide a numerical evidence for a transition between two superstatistics regimes, by analyzing high-frequency (minute-tick) data for share-price returns of seven selected companies. Salient issues, such as breakdown of superstatistics in fractional diffusion processes or connection with Brownian subordination are also briefly discussed. Related article: Petr Jizba, Jan Korbel, Hynek Lavička, Martin Prokš, Václav Svoboda, Christian Beck, Physica A 493 (2018), 29-46.

Understanding the mechanism of large scale outages is the precondition for designing of an effective prevention scheme. Traditional methods face the combinatorial explosion problems and have to sacrifices the modeling accuracy for large-scale power systems. Complex system method looks at the problem in a “top-down” way, and the move from a ‘‘local’’ to a ‘‘global’’ view at power system critical failures could increase our capacity to gain a better understanding of root causes, such as the conditions triggering instabilities, alternative system designs to avoid or mitigate crises, and side effects of policy measures. A review on recent publications related to the application of complex system method on topics including self-organized criticality (SOC) and critical transitions is given first. Our work on explaining the mechanism of system entering SOC by critical transition is introduced next in detail. The temporal and spatial statistics of time series of percentage load shedding and transmission capacity are analyzed with respect to different local reliability of the transmission system. The performance of early warning based on them are discussed. Moreover, the fact that our research is conducted from a long-term point of view enables it to serve as a planning-assistant tool for a future power system with higher resilience.

The anticipation of critical transitions in complex systems is a field of active research in such diverse disciplines as ecology, climate research or engineering. In (Cotilla-Sanchez et.al, IEEE Transactions on Smart Grid,2012) large scale power outages are considered as critical transitions in the operation of power grids. It was shown for the large scale power outage on August 10 in 1996 in the USA that the event could be anticipated from critical fluctuations of the system frequency. In this talk we present results from the analysis of two dif- ferent data sets of the system frequency from the same event that have been measured at two different positions in the grid. We show that critical fluctu- ations seem not to be a reliable indicator for a critical transition in the power grid. In contrast to that, analyzing the data from the viewpoint of Langevin equations by estimating the drift and diffusion coefficients shows to be more reliable to anticipate the outage.

The transition to synchrony in complex networks has been extensively studied since already long ago. In the vast majority of cases, dynamics and topologies, this transition is of the second-order type, i.e. continuous and reversible. Recently, however, explosive synchronization, that is, the abrupt, discontinuous and irreversible emergence of a collective coherence state has been reported, and experimentally observed in power grids, circuits and chemical systems. Very recently, the interest has extended also to neuroscience, since some studies have related this phenomena to the onset of epilepsy seizures and the transition to and from consciousness. Various substantial contributions have been made which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations. This knowledge has not only given tools for predicting the explosive transitions, but have greatly enhanced our general understanding of phase transitions in networked systems.

The synchronization of nonlinear dynamical systems is a topic of interest in many fields of science. Particular attention has been paid to the synchronization of chaotic systems, both in unidirectional or bidirectional coupling configurations. An interesting type of synchronization, so-called anticipated synchronization, was proposed by H.U. Voss [Phys. Rev. E 61, 5115 (2000)]. This author showed that, for particular parameter values, two identical chaotic systems unidirectionally coupled can synchronize in such a manner that the trajectories of the “slave” (the response system) anticipate (i.e., predict) those of the “master” (the sender system). Noticeably, the anticipation regime can be achieved without perturbing at all the dynamics of the master. In this talk I will explain how this peculiar synchronization scheme can be used to predict, and eventually control, unwanted features of the dynamical trajectory and explain recent extensions to extreme events and spatially extended systems.

The entrainment phenomenon, by which an oscillator adapts its natural rhythm to an external signal, has been observed in physics, chemistry, biology, etc. Recent attention has focused on the optimal conditions for entraining an oscillator to a weak external periodic signal. Here we use an experimental setup consisting of a semiconductor laser with optical feedback, as a testbed to study entrainment phenomena. The laser operates in the so-called low frequency fluctuations (LFFs) regime, in which the laser intensity displays apparently random spikes \cite{tiana-alsina:2010}. Our goal is to experimentally identify and characterize the transitions between different locking regimes, when a weak periodic signal modulates the laser current. We investigate the transition between 1:1 locking (the frequency of the optical spikes is equal to the modulation frequency) and 2:1 locking (the spike frequency is half the modulation frequency), as well as higher order, noisier locking regimes. The experimental control parameters are the amplitude of the weak signal, its frequency, and the control parameter of the laser (the dc value of the pump current, which controls the natural frequency of the spikes). Three measures are used to quantify the entrainment quality and to identify the boundaries of the different locking regimes. We also discuss how the laser responds to a weak non-periodic signal, when it operates at the boundary between locking regions. \bibitem{tiana-alsina:2010} J. Tiana-Alsina, M.C. Torrent, O.A. Rosso, C. Masoller, J. Garc'{\i}a-Ojalvo, Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback'', Phys. Rev. A \textbf{82}(1), 013819 (2010).

Semiconductor lasers with time-delayed optical feedback display a wide range of dynamical regimes, which have found various practical applications. They also provide excellent testbeds for data analysis tools for characterizing complex signals. Recently, several of us have identified the onset of low-frequency fluctuations (LFFs), where the laser intensity displays abrupt dropouts, and the onset of coherence collapse (CC), where the intensity fluctuations are highly irregular. Here we map these regimes when both, the laser current and the feedback strength vary. We show that the shape of the distribution of intensity fluctuations (characterized by the standard deviation, the skewness, and the kurtosis) allows to distinguish among noise, LFFs, and CC, and to quantitatively determine (in spite of the gradual nature of the transitions) the boundaries of the three regimes. Ordinal analysis of the inter-dropout time intervals consistently identifies the three regimes occurring in the same parameter regions as the analysis of the intensity distribution. Simulations of the well-known time-delayed Lang–Kobayashi model are in good qualitative agreement with the observations.

Gene conversion is a ubiquitous phenomenon that leads to the exchange of genetic information between homologous DNA regions and maintains co-evolving multi-gene families in most pro- and eukaryotic organisms. In this talk, I will consider its implications for the evolution of a single functional gene with a silenced duplicate, using two different mathematical models of evolution on rugged fitness landscapes. Our analytical and numerical results show that, by helping to circumvent valleys of low fitness, gene conversion with an inactive duplicate gene can cause a significant speedup of adaptation which depends non-trivially on the frequency of gene conversion and the structure of the landscape. Stochastic effects due to finite population sizes further increase the likeliness of exploiting this evolutionary pathway. Our results reveal the potential for duplicate genes to act as a scratch paper'' that frees evolution from being limited to strictly beneficial mutations in strongly selective environments.

Natural or technical systems possess often several possible stable states of operation. Linear stability theory is the appropriate tool to study the stability properties of such states with respect to small perturbations. However, in nature perturbations are not necessarily small but are finite in size. We discuss two different methods how to investigate the stability with respect to large perturbations such as single shocks. Both methods aim to determine the distance to the boundary of the basin of attraction or the edge of chaos, respectively. The first method determines the minimal destabilizing perturbation for large dynamical systems such as networks. Besides the size of this perturbations the method allows also to obtain the direction of this perturbation. We illustrate this method using pollinator networks in ecology and energy networks and identify relations between the topology of a network and its stability properties. The second method measures return times to a stable state at the edge of chaos. This is demonstrated for the transition from laminar to turbulent motion in a shear flow.

The collective behavior of living organisms in an heterogeneous environment is a central issue in population ecology. In particular, it is relevant to know whether a population will survive long term or go extinct. In a recent article, we address this question for a single species population in a bounded one-dimensional habitat of length $L$. We assume that the evolution of the population density distribution $\rho$ is governed by elementary processes such as growth and dispersal. In standard models, these processes are typically described by a constant per capita growth rate and normal diffusion, respectively, however, feedbacks in the regulatory mechanisms and external factors can produce density-dependent rates. The role of these plausible nonlinearities has been overlooked in previous studies. Then, we analyze a generalization of the standard evolution equation, with diffusion coefficient $\rho^{\nu-1}$ and per capita growth rate $\rho^{\mu-1}$, which depend on $\rho$ when $\nu,\mu\neq 1$. This equation is complemented by absorbing boundaries, mimicking a nonviable neighborhood. For this nonlinear problem, we obtain, analytically, exact expressions of the critical habitat size $L_c$ for population survival, as a function of the exponents and initial conditions. We find that depending on the kind of nonlinearities present, population survival occurs for $L \ge L_c$, $L\le L_c$ or for any $L$. This result generalizes the usual statement that $L_c$ represents the minimum habitat size. Additionally, nonlinearities introduce sensitivity to the initial conditions, affecting $L_c$. Ref.: E Colombo, C Anteneodo, J Theor Biol(2018) https://doi.org/10.1016/j.jtbi.2018.02.030

Complex dynamics can be studied as resulting either from high-dimensional deterministic systems, or from random, stochastic systems, where the dimensionality is so high that it does not usually matter. But then the details of the relationship between cause and consequence are buried in the probabilistic description and cannot be revealed. To date, the deterministic approaches have been limited mainly to autonomous systems, while nonautonomous systems have either been deemed as being easily expressed as autonomous, or treated as stochastic. In this talk we will discuss the dynamics of oscillatory systems subject to non-autonomous perturbation. We will consider oscillators subject to driving at a slowly varying frequency, and discuss both their short-term and long-term stability properties. We will argue that, in general, deterministic non-autonomous perturbation is a very good candidate for stabilising complex dynamics, and as such is useful for modelling living systems. Cell energy metabolism will be considered as an example. Nonautonomous time-varying oscillatory dynamics, observed in experiments, will be used to model the continuous process of matching fluctuating supply and demand in energy metabolism. We will discuss our hypothesis that altered cellular energy metabolism is a hallmark of many diseases, including cancer and dementia, and illustrate the hypothesis with models of coupled phase oscillators that have variable frequencies.

Ecosystems are usually faced with limiting amount of resources, and need to develop strategies to survive under those conditions. We have recently shown that dynamics, in the form of growth oscillations, can help cellular populations (specifically bacterial biofilms) to cope with limiting resources. In this talk I will focus on the transition leading to such oscillatory behavior, which involves a subcritical Hopf bifurcation induced by delay. A minimal delay differential model is proposed that accounts for a variety of experimental observations, and makes non-trivial predictions that are validated experimentally.

Ventricular fibrillation (VF) is a life-threatening cardiac arrhythmia which leads to a ceased cardiac pumping functionality. In the electrocardiogram it manifests itself as a highly irregular time-series with varying amplitude and complexity. We investigate the variations in complexity using permutation entropy applied to short windows of the ECG and demonstrate that the phenomenon can be found during VF in ECG time series of different species. Using 3D Fenton-Karma simulations of rabbit hearts we show that the complexity variations are caused by cardiac excitation waves at different levels of spatio-temporal complexity which can be quantified by spatio-temporal permutation entropies. Using cross correlation and nearest neighbor prediction we demonstrate that it is possible to link cardiac signals measured using ECG electrodes to the spatio-temporal complexity on the heart muscle in simulations and in data from optical mapping experiments.

Spiral and scroll waves determine the complex spatio-temporal dynamics in excitable media (e.g. during cardiac arrhythmias). In numerical simulations, it can be shown that the chaotic dynamics in these systems is often transient, rather than permanent. The underlying chaotic set governing the dynamics is thus a chaotic saddle and not a chaotic attractor. This distinction has significant implications for the structure of the state space. Using a control method for networks developed by S. P. Cornelius et al. [1] we demonstrate that the investigated chaotic transients can be terminated via small and spatially localized perturbations. This approach exploits the specific structure of the state space near a chaotic saddle. The simulations show that the chaotic dynamics of the investigated high-dimensional systems can be terminated with a small number of (minimal invasive) local perturbations. This proof of concept might be relevant for developing novel defibrillation strategies. [1] S. P. Cornelius et al. Realistic control of network dynamics. Nat. Commun. 4:1942 (2013).

Obtaining early warnings of approaching abrupt transitions in the dynamics of complex spatially extended systems is relevant in many contexts such as climate, ecology or physiology. Here we show that a functional network constructed from spatial correlations experiences a percolation transition way before the actual system reaches a bifurcation point. Tools from percolation theory are then used to introduce early warning precursors anticipating the tipping point. We illustrate the generality and versatility of our percolation-based framework with model systems experiencing different types of bifurcations and with sea-surface temperatures associated to El Nino phenomenon. Work done in collaborations with V. Rodriguez-Mendez, V.M. Eguiluz and J.J. Ramasco

Transient phenomena dominate our world and may cause severe qualitative and often long term transitions in complex dynamical systems. Yet transients are far from fully understood conceptually. We here work out aspects about why understanding even basic questions about transient deterministic dynamics still remain unanswered after a century of research. As the first example, we consider simple models of network dynamical systems to highlight that it is typically mathematically impossible to compute peak times and arrival amplitudes of a spreading signal originating from one network unit. Although it might appear odd for deterministic systems, we present an ansatz from probability theory to quantify arrival times and time-dependent network-wide signal strengths. As a second example, we provide hints about how to identify sources of spreading signals in a network and discuss ways to dynamically shield units from transient spreading signals. https://arxiv.org/abs/1710.09687

Unravelling the interrelationship between structure and dynamic stability of ecological networks is an important issue in theoretical biology. We contribute to this subject by analyzing the stability of plant-animal mutualistic systems against non-small perturbations. This non-local approach seems valid as we assume a mutualistic network to be a multistable system in which one ’desired’ dynamical regime, enabling the largest number of coexisting species, competes with several 'undesired' regimes of partial extinction. As a measure for this nonlinear stability, we introduce the Minimal Fatal Shock (MiFaS) which is defined as the smallest (Minimal) instantaneous perturbation (Shock) of state variables which is capable of inducing a (Fatal) transition from the desired to an undesired state. We then use the MiFaS approach to locate weak points in various mutualistic networks and derive topological features which adversely affect a networks’s dynamic stability.

Transition network are constructed from time series data by associating nodes on the network with dynamical states (appropriately coarse-grained) of the underlying system. There are several ways to achieve this representation and we will review our favourites. We provide a theoretical justification for these structure providing an adequate representation of the underlying dynamics and show that this can be used to detect transitions in the original dynamics in two ways: (1) by constructing distinct networks with distinct topological properties from distinct regimes, and (2) by indicating that an incoming trajectory will deviate from the historical network as a dynamical transition is approached.

Probabilistic time series forecasting is instrumental in several fields of science and technology, including meteorology, seismology, climatology and, more recently, renewable energies. Therefore, it is useful, both theoretically and practically, to know its limits. To this end, we assume a perfect model scenario, that is to say, the system producing the data is supposed to be perfectly known, while the uncertainty results solely from the finite precision of the observations. Two further qualifications are added to this general setting to connect with nonlinear time series analysis. First, the system is supposed to be a discrete-time dynamical system. Second, the uncertainty is due to the quantization of the state space rather than being introduced via additive noise (as in the conventional statistical approach). In this nonlinear perfect model scenario, the quality of probabilistic predictions n time steps ahead will be measured by the so-called ignorance score Ig(n), which is a conditional entropy. It turns out that Ig(n) is bounded on the n vs Ig(n) plane, the boundaries of its admissible domain depending on the entropies of the dynamic and state space quantization. Other properties of Ig(n) will be derived in two further scenarios: (i) the imperfect model scenario, corresponding to the real situation in which the underlying system is imperfectly known, and (ii) the differential perfect and imperfect model scenarios, where the probability mass functions are replaced by probability densities, and the discrete entropies by differential entropies (as usually in practice). It follows that the slope of Ig(n) in an imperfect scenario can be used to compare the performance of probabilistic forecasting algorithms, as we will show with numerical examples.

We analyze the final state sensitivity of nonlocal networks with respect to initial conditions of their units. By changing the initial conditions of the network units, we perturb an initially synchronized state. Depending on the perturbation strength, we observe the existence of two possible network long-term states: (i) The network assimilates the perturbation effects and returns to its initial synchronized configuration. (ii) The perturbation leads the network to an alternative desynchronized state. By computing uncertainty exponents of a two-dimensional chart of such perturbations, we find the synchronized state to be sensitive dependent on the initial conditions of networks units in two regimes: one less uncertain regime corresponding to the fractal boundaries of a continuous region of initial conditions for which the network synchronizes; and another extremely sensitive corresponding to a riddled-like combination of initial conditions for which the network synchronizes or not.

Changes in interactions of network structures promise to disclose the underlying mechanisms and may be precursors for transitions. The direction of interaction plays a key role as it identifies causes and effects as well as loops in a network. This knowledge may then be used to determine the best target for interference with the network, for example, stimulation of a certain brain region. Reconstructing the network structure from data can be achieved using renormalised partial directed coherence. In combination with state space modelling this leads to a time-variant estimator of the underlying network. Changes in the interaction structure that precede transitions can then be identified. They promise to be useful for the prediction of transitions. As an example the proposed method is applied to neural signals.

The electroencephalogram (EEG) is a representative signal containing information about the condition of the brain. In fact, the oscillations of the EEG may contain useful information about the state of brain and how it works. We can also accept as working hypothesis that the EEG recorded under resting conditions is representative of the global state of the brain. The EEG reflect not only on characteristics of the brain but also yields clues concerning the underlying associated neuronal dynamics. The processing of information by the brain is reflected in dynamical changes in electrical activity in time, frequency and space. Therefore, the concomitant studies require methods capable of describing the quantitative and qualitative variations of the signal in both time and frequency. This objective can be reached by using the orthogonal wavelet transform, an efficient tine-frequency decomposition methods and information theory quantifiers based on its, for the EEG analysis. Relative Wavelet Energies, Wavelet Entropies, and Wavelet Statistical Complexity are used in the characterization of scalp EEG records corresponding to secondary generalized tonic-clonic epileptic seizures. In particular, we show that epileptic recruitment rhythm observed during seizure is well described in terms of Relative Wavelet Energies. In addition, during the concomitant time-period the entropy diminishes while complexity grows. This is construed as evidence supporting the conjecture that an epileptic focus, for this kind of seizures, triggers a self-organized brain state characterized by both order and maximal complexity.

Physiological time series are typically non-stationary, demanding different approaches to be analyzed in terms of standard methods. We present a non-parametric method which allow us to split a signal into stationary pieces. We explain the details of the segmentation method and how the outcomes of the segmentation can point to the complexity reduction in heart beating, due to pathological conditions and aging, and we discuss also the detection of sleep apnea through cardiovascular data. The outcomes of segmentation give us access to time characteristics of the signal that were no longer available, making possible a different approach to quantify non-stationarities in physiological time series.

Complex dynamical systems may exhibit sudden large-scale deviations from their normal behaviour, termed critical transitions. Mathematical analysis suggests that before a critical transition one should observe common dynamical signatures that can be directly measured from the signals generated by the system. In this project we tested this prediction in computational model generating different types of dynamical transitions and intracranial EEG recordings from patients with spontaneous epileptic seizures. We investigated whether typical early warning signs for critical transitions, i.e., increase of variance, AR(1) coefficient and skewness of a signal can be detected before the seizure. A systematic and thorough research was performed on 26 patients taken from European Epileptic Database. We analysed over 1000 signals from ‘origin’ depth electrodes corresponding to nearly 300 epileptic seizures of different types and occurring in different states of vigilance. In modelled time series trends in predictors before transitions were in agreement with a general theory of critical transitions. In human patients only two out of 26 (8%) exhibited a significant number of increasing variance and AR(1) coefficients and no patient exhibited all three precursor changes consistently. This study suggests that in general, seizure onset does not exhibit typical early-warning signals indicative of an impeding critical transition. The discrepancies between theoretical model and data from epileptic patients indicate that complexity may be a fundamental issue in transferring theoretical concepts to complex systems such as a human brain.

Formation of structural heterogeneities in network connectivity space is thought to underlie distributed memory formation in brain networks. We investigated dynamical signatures of memory formation in vivo and in silico. Namely, we characterized hippocampal activity patterns in mice during consolidation of fear memory and found that its in vivo dynamics is near-critical. We then measured changes in stability of functional network representations and demonstrated that the changes in representational stability, mediated by the exposure the fear conditioning paradigm, predict subsequent behavioral performance, when mouse is subjected for the second time to the same location. Next, using in silico models, we investigated the link between increased representational stability and system proximity to critical state and show that criticality mediates distributed stabilization of neuronal representations and potentially provides a dynamical substrate for system-wide memory storage. Finally, we showed that critical dynamics maybe required to preferentially promote consolidation of new memories in a network when sensory input is weak and/or sparse. Taken together we identify one of potential hallmarks of memory formation in the brain networks and elucidate dynamical states mediating its successful consolidation.

We present evidence of extreme events in two Hindmarsh-Rose bursting neurons mutually interacting via two different coupling configurations: chemical synaptic and repulsive diffusive coupling. A dragon-king-like probability distribution of the extreme events is seen for both the coupling where small to medium size events obey a power law and the extremely large events are outliers. The extreme events originate due to instability in antiphase synchronization of the coupled systems via two different routes, intermittency and quasiperiodicity for purely excitatory and inhibitory synaptic coupling respectively. The origin of extreme events follows a common mechanism as due to an emergent instability in the antiphase synchronization manifold and is independent of the route to chaos of the coupled dynamics. However, they show disparate synchronization dynamics during two different routes of transition to complex dynamics. In the quasiperiodic regime, for pure inhibitory coupling, temporal dynamics of two oscillators are in a state of antiphase burst synchronization, but the pairs of arbitrary spikes within the bursts coincide to evolve into an intermittent in-phase spiking synchrony. For pure excitatory coupling in the intermittency regime, the neurons develop a state of antiphase spiking synchronization with occasional coincidence of two pairs of arbitrary spikes. A simple electronic experiment using two repulsively coupled analog circuits of the HR neuron model confirms occurrence of the dragon-king-like extreme events.