Predicting Transitions in Complex Systems

Here we propose a method for controlling the coupled non-uniform second-order Kuramoto oscillators in complex network and find the resilience pattern of such complex system. By control, we mean how to drive a given complete incoherent state of the system to full synchronized ones using few number of nodes, which are known as driver nodes. For a given topology, we identify the driver nodes and determine that how much extra dissipation needed to add in these nodes for pushing the system to the perfect synchronization. Finally by mapping N coupled Kuramoto oscillators in given network to an effective one-dimensional dynamical system, where its effective state variable Xeff is a function of time and a global control parameter, $\beta_{eff}$, we find its resilience pattern of the system in a two dimensional $X_{eff} -- \beta_{eff}$ plane. The control parameter can be constructed by solely underlying network properties of coupled oscillators. We find an interval of control parameter $\beta_{eff}$, where the system has a chance to synchronize.

Natural systems are complex. Transition in complex system may occur huge effects in nature. Prediction of such transitions is thus of utmost importance. The complex systems may show diverse complex dynamics. Understanding of which is the basic need to anticipate the transitions which may occur in the system. The multistability is one of the salient and potent features of the complex systems. Multistability appears in diverse forms, and their study is an exciting topic of research in science and engineering. A particular form of multistability is bistability: it shows many variants, such as the coexistence of two stable steady states (SSS), one stable steady state and one stable limit cycle (LC), two stable limit cycles, or two chaotic attractors. Birhythmicity is the phenomenon of coexistence of two stable limit cycles separated by an unstable limit cycle with different amplitudes and frequencies. In many physical systems, birhythmicity is undesirable as in energy harvesting systems, but in most biological systems, e.g., enzymatic oscillations, it is desirable. Therefore, control of birhythmicity is of utmost importance. Although the control of multistability is a well studied topic, the control of birhythmicity has not been explored to that extent. We propose two control schemes, namely, conjugate self-feedback and self-feedback, that can effectively control (predict) and, whenever required, can eliminate birhythmicity. We theoretically explore and numerically establish the technique of control of birhythmicity and transitions to any desired attractor. Also we experimentally study the efficacy of the conjugate self-feedback control scheme in electronic circuit by controlling the initial conditions in the circuit. A number of engineering and biological systems, such as, a birhythmic van der Pol oscillator, an energy harvesting system, the p53-Mdm2 network for protein genesis (the OAK model), and a glycolysis model (modified Decroly-Goldbeter model) are investigated with the proposed control schemes to establish the efficacy and generality of the schemes. The main essence of these control schemes lies in the fact that these are easily realizable and offer an efficient mean to control birhythmicity.

In this work, we study the time scales of diffusion processes occurring on single and multiplex networks. We propose here a dissimilarity metric for weighted graphs, and a measure based on the distance to a complete graph. We show that this measure has a strong correlation with previous results based on the spectrum of the supra-Laplacian. This novel measure has the advantage of its simple computation, even when considering several layers.

We present evidence of extreme events in two Hindmarsh-Rose (HR) bursting neurons mutually interacting via two different coupling congurations: 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 (APS) of the coupled systems via two different routes, intermittency and quasiperiodicity for purely excitatory and inhibitory synaptic or repulsive diffusive coupling, respectively. A simple electronic experiment using two repulsively coupled analog circuits of the HR neuron model confirms occurrence of the dragon-king-like extreme events.

We solve an adaptive search model where a random walker or Lévy flight stochastically resets to previously visited sites on a d-dimensional lattice containing one trapping site. Because of reinforcement, a phase transition occurs when the resetting rate crosses a threshold above which nondiffusive stationary states emerge, localized around the inhomogeneity. The threshold depends on the trapping strength and on the walker’s return probability in the memoryless case. The transition belongs to the same class as the self-consistent theory of Anderson localization. These results show that similarly to many living organisms and unlike the well-studied Markovian walks, non-Markov movement processes can allow agents to learn about their environment and promise to bring adaptive solutions in search tasks.

Inferring functional network connectivity from the timing of events at different nodes has gained fundamental attention in diverse fields like neurosciences and climatology. However, a deep theoretical understanding of the reconstruction skills of such approaches has so far been missing to a large extent. In this work, our interest in reconstructing network topology based upon the observed timing of specific events is motivated by the need for predicting the emergence of such events at a given node in a network under the hypothesis that the processes controlling the timing of these events at all nodes are mutually interrelated. We approach this problem by numerically studying a simple stochastic event propagation model. Specifically, we take some network with a prescribed topology and seed events that are randomly triggered (i.e., follow Poisson processes) independently at each node. In addition, all events at a given node are propagated to its neighbors along the links of the network with a certain probability and time delay. The mutual simultaneity of events among the different per-node event sequences is quantified by means of event coincidence analysis based on possible precursor events. In this framework, one event sequence is taken as a reference, and the fraction of events in this series is computed which has been preceded by at least one event in the second series within a given time window. Since by construction, events at a given node are with a certain probability followed by events at all of its neighbors, an elevated event coincidence rate between two nodes should point to an existing link in the network. We perform ensemble simulations of the described process on different types of standard network architectures (Erdös-Renyi, Barabasi-Albert, etc.) and characterize the obtained reconstruction accuracy by means of ROC analysis. By varying the intrinsic rate of seed events at each node as well as the propagation probabilities, we study a wide variety of settings that could mimic real-world cases like extreme events in some climate variable. Finally, we propose a refinement of the aforementioned reconstruction approach for finding causal relationships between the dynamics at different nodes, which assumes a directed network topology and explicitly accounts for the asymmetry of event coincidence rates between two series depending on which of the two is taken as reference data. By applying the PC algorithm in conjunction with conditional event coincidence rates, we iteratively unveil the original network links by testing for conditional independence between all possible pairs of nodes.

Background Sudden transitions are an established pattern of change toward recovery in depression research. However, it is unknown if these transitions are preceded by early-warning signals (EWS), and thus whether recovery from depression functions according to the principals of complex dynamic systems. This research project examines the use of EWS in detecting oncoming shifts in depressive symptoms within individuals that are expected to improve in the near future. We aim to map micro-level changes in symptom dynamics using frequent self-ratings as well as physiological measurements. Methods/Design We recruit 45 people with depressive symptoms who are due to start psychological treatment. All participants will report on their mood, activities, and cognitions in a brief questionnaire five times a day, for a period of four months. They also wear an actigraph continuously, and measure their heart rate for 5 minutes, twice a day. For six months, they report on their depressive symptoms weekly, and monthly for six months thereafter. Personalized analyses will be conducted, testing the presence of EWS such as rising autocorrelation, rising variance and rising connection strength in the daily measures before transitions toward remission of depression. Discussion With 600 observations per person, this study is the first to monitor the process of recovery in depression in this much detail. We expect that it will be possible to predict a transition in depressive symptoms using EWS within individuals. This study addresses a gap in our knowledge of change processes in depression, and could lead to the development of early interventions to promote remission of depression.

The adaptive SIS model is one of the most elementary models in which the dynamics on the network is coupled to the dynamics of the network. We investigate the critical behavior of the network topology by looking at the densities of motives, the clustering, the compactness, the degree distribution and assortativity, the effective branching ratio and spectral distribution. The proximity to the persistence threshold can be sensed in some of these quantities long before the transition via the appearance of local maxima due to a crossover phenomenon. This is also reconcilable with the pair approximation. A good signal-to-noise ratio makes them novel candidates for precursors of the transition. We explain the mechanism that gives rise to this behavior and some caveats. We also discuss the issues that arise when fitting power laws to these quantities.

Earth climate, in general, varies on many temporal and spatial scales. In particular, air temperature exhibits recurring patterns and quasi-oscillatory phenom- ena with different periods. Although these oscillations are usually weak in amplitude, they might have non-negligible influence on temperature variability on shorter time- scales due to cross-scale interactions, recently observed by Paluš [1]. In this letter, we show how to discern possible cross-scale interactions and, if present, how to quantify their effect on air temperature in Europe. M. Paluš, “Multiscale atmospheric dynamics: Cross-frequency phase-amplitude coupling in the air temperature,” Physical Review Letters, vol. 112, no. 7, pp. 1–5, 2014.

Self-organized criticality (SOC) with conservative systems is by now a very understood subject, fully integrated to the Statistical Mechanics of non-equilibrium systems. But for dissipative systems, the situation is not clear and several open problems remain. Models of dissipative systems (earthquakes, forest fires, neuronal avalanches) have been proposed since the 90´s with the claim that they are examples of true SOC systems. The present consensus is that they are not: even if one can propose slow drive mechanisms to counterbalance dissipation, so that the system is at least conservative in average, this needs some fine tuning in the hyperparameters of these adaptive mechanisms. Without this fine tuning, such models present stochastic oscillations that do not disappear in the thermodynamic limit, not true convergence to the critical point, what has been called self-organized quasi-criticality (SOqC). Here we explain the origin of these stochastic oscillations as an outcome of a different criticality scenario: instead of the presence of the standard transcritical transition point, usually found in SOC models, we have a Neimark-Sacker bifurcation from a stable spiral to an indiferent one. In the critical region, demographic noise excites and maintain the stochastic oscillations. By using simple stochastic neurons, we obtain transparent analytic results.

Large-scale neuronal dynamics models, i.e. whole-brain models, have attracted great attention of the scientific community to unveil key mechanisms about the relationship between brain function and brain structure. Despite the vast amount of whole-brain models found in literature with different degrees of complexity, some empirical signatures of brain at rest, such as the functional connections (FC) between brain regions and the emergence of resting state networks (RSNs) are only partially understood and still poorly reproduced. The increase in the predictive power of whole-brain models is a fundamental step that can trigger many fundamental applications, such as quantitative discriminations between healthy and lesioned subjects. It has been shown [1] that brain at rest may be poised near a critical state, defined as the special point in the space of parameters where the system displays the maximum susceptibility/complexity, i.e. it optimally and collectively responds to external inputs [2]. Exploiting this notion of criticality and following seminal work of Haimovici et al. [2], we propose a biological meaningful yet simple stochastic model which is able to predict the brain organization into resting state networks through only few independent parameters. In this study we developed biological meaningful parametrization of the structural connectivity matrix (SC) (i.e., the human connectome) that increases the match between simulated and empirical data. In addition, using our modeling approach we are able to distinguish between simulated healthy and lesioned brains. [1] Beggs, J. M., & Plenz, D. (2003). Neuronal avalanches in neocortical circuits. Journal of neuroscience, 23(35), 11167-11177. [2] Haimovici, A., Tagliazucchi, E., Balenzuela, P., & Chialvo, D. R. (2013). Brain organization into resting state networks emerges at criticality on a model of the human connectome. Physical review letters, 110(17), 178101.

Detection of causal relations between elements is a commonly used approach in the description of complex systems in many fields of current science. While in some cases (computer or social networks) the existence of connections can be naturally defined, in other systems (neuroscience, climatology) it is often problematic to determine this structure by direct observation. For this reason, methods for inference of causal structure using only knowledge of time series have been developed. However, these methods in general require the computation of high-dimensional information functionals which is invoking the curse of dimensionality with increasing network size. Recently, several methods based on iterative procedures for assessment of conditional (in)dependencies have been developed to mitigate this problem. Based on theoretical analysis and numerical simulations (including random and realistic complex coupling patterns), we bring a comparison of accuracy and computational demands of selected algorithms. Numerical simulations suggest that accuracy of studied algorithms is (under suitable parameter choice) very similar, however computational demands of the algorithms differ in a manner depending on the density and size of estimated network.

Usually when investigating noisy bifurcations in dynamical systems the noise is assumed to be white in time. However, this is always an idealisation; in many applications there is a memory time scale associated with the noise. Here, we study the influence of red noise on fold bifurcations. We use as examples a model of monsoon transitions and a low-order model of the meridional overturning circulation. The likelihood of a transition is calculated as a function of time in a non-stationary model integration. It is found that with red noise the transition on average happens earlier than in the white noise case. Moreover, the uncertainty about the timing of the transition is larger in the red noise case.

Chimera states, characterized by coexisting regions of phase coherence and incoherence, are stable patterns even in a system with translational and rotational symmetry. It has been experimentally demonstrated in chemical, mechanical, electronic, and opto-electronic systems. Here, we show that the formation of chimera patterns can also be observed in conservative Hamiltonian systems with long-range nonlocal hopping in which both energy and particle number are conserved. In particular, we find that such patterns should be observable in ultracold atoms and can be implemented in a two-component Bose-Einstein condensate with a spin-dependent optical lattice. The calculated parameter regime are well within the current technology. The present work highlights the connections between chimera patterns, nonlinear dynamics, condensed matter, and ultracold atoms.

The dynamics of complex systems, which involve a large number of degrees of freedom are generally non-stationary. Using the mutual information similarity matrix, as a nonlinear measure, we map the complex structure of similarity matrix of multivariate electroencephalographic (iEEG) time series onto a finite number of states. The states are the clusters of mutual information matrices with similar correlation structures. The tree diagram of epileptic brain states close and far from the seizures show different state dynamics. We find that the number of states can be increasing or decreasing by approaching seizure.

We develop a perturbation theory for a class of rst order nonlinear non-autonomous stochastic ordinary dierential equations that arise in climate physics. The perturbative procedure produces moments in terms of integral delay equations, whose order by order decay is characterized in a Floquet-like sense. Both additive and multiplicative noise are discussed and the question of how the nature of the noise influences the method of calculus, and hence the results, is addressed. Namely, whether the noise is white or continuous in the sense of $L^2$ convergence, determines whether Ito or Stratonovich calculus should be used. The analysis is tested by comparison with numerical solutions. The particular climate dynamics problem upon which we focus involves a low-order model for the evolution of Arctic sea ice under the influence of increasing greenhouse gas forcing $\Delta
F_0$. The deterministic model, developed by Eisenman and Wettlaufer (2009) exhibits several transitions as
$\Delta
F_0$ increases and the stochastic analysis is used to understand the manner in which noise changes these transitions and the stability of the system.

Pearson correlation matrices have been long used in biology or perhaps most notably in finance. Together with the backing of random matrix theory (RMT) they are fit to denoise and offer insight into collective behaviour of multichannel signals such as, for instance, recordings of electrical brain activity. In this respect electroencephalography (EEG) has the advantage of large temporal resolution, resulting in a small N/T ratio (number of channels/length of time series) and hence also enhanced eigen-inference. We explore the information apparent in eigenspectra of these correlation matrices (among others using recent advances in RMT on lagged correlations [1]) and find indications of some common measurement artefacts and brain rhythms (usually in the largest eigenvalues and corresponding eigenvectors), as well as overall highly skewed distributions depending on the task the subjects performed. We numerically compare these characteristics with autoregressive family time series models on the one hand and with Kuramoto and 2-D Ising models on the other. The last one has only recently been studied [2] in terms of equal-time correlation matrix eigenspectra, but not from the perspective of lagged correlation matrices. [1] MA Nowak and W Tarnowski. Spectra of large time-lagged correlation matrices from random matrix theory. J. Stat. Mech. (2017) 063405. [2] T Vinayak et al. Spectral analysis of finite-time correlation matrices near equilibrium phase transitions. EPL 108 (2014) 20006.

authors: Aleksandra Pidde and Aneta Stefanovska Dynamical systems abound in nature. Yet, the question about the nature of the coupling between them still remains highly non-trivial to answer. In recent years, the investigation of coupling functions has developed into a very active and rapidly evolving field [1]. Bispectral analysis [2, 3] belongs to a group of techniques based on high-order statistics that may be used to analyse non-Gaussian signals, to obtain phase information, to suppress Gaussian noise of unknown spectral form, to detect and characterize signal nonlinearities, or to study couplings between interacting oscillatory processes [4]. Unlike the various methods for investigation of the coupling, the bispectral analysis does not require prior phase extraction, or phase response curve estimation, which can bring additional problems. The method is robust against the noise, does not require any prior assumptions, and can be applied regardless from the level of the synchrony between oscillators. We investigate time variable model of forced oscillator and elaborate the procedure involving bispectral analysis and statistical testing to reliably determine the existence of intermittent coupling. We will show that wavelet bispectral analysis is particularly useful when coupling changes with time. [1] T. Stankovski, T. Pereira, P. V. E. McClintock, and A. Stefanovska. Rev. Mod. Phys., 89:045001, 2017 [2] B. P. van Milligen, C. Hidalgo, and E. Sánchez. Phys. Rev. Lett., 74:395–398, 1995. [3] J. Jamšek, A. Stefanovska, P. V. E. McClintock, and I. A. Khovanov. Phys. Rev. E, 68(1):016201, 2003. [4] J. Jamšek, M. Paluš, and A. Stefanovska. Phys. Rev. E, 81(3):036207, 2010.

In our project, we apply vector autoregressive models and sparse linear regression methods for detection the links in multivariate time series. Several selection criteria for sparse methods LASSO and Dantzig Selector are discussed. The models are applied on epilepsy data.

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 ver- satile, 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.

A model for a network of networks for Fitzhugh-Nagumo oscillators is proposed as a prototype to show both the complexity of the system's dynamics, as well as the irreducibility of the concept of networks of networks to single networks. Under specified conditions, a set of novel dynamical regimes emerges, not present in previous constructions of single networks. These novel dynamics, arising in an anomalous region of our parameter space, thwart the notion of reducibility of networks of networks into single networks. The new dynamical regimes cannot be described by the previous single network description, thus begging a renewed acceptance of the concept of irreducibility of networks of networks.

The robust operation of power grids depends on the stability of the combined functional electric generators at play. Mechanical rotating generators can be described by third- or fourth-order oscillators, embodying both the rotor-angle and the voltage dynamics. We tackle the stability analysis of third- and fourth-order models of synchronous generators, aiming at uncovering routes to instability, via both sufficient and necessary stability criteria. Rigorous analytic results are obtained, including an outlook on graph-theoretical connectivity measures, unveiling bounds to both the network structure and the physical characterising parameters of the generator machines.

By the introduction of the resilience concept in 1973, Holling induced a paradigm shift from the constancy-perspective of ecological stability to the perspective of dynamical persistence within a domain of attraction. Today, the concept of resilience is accessible with a growing number of mathematical tools via the architecture of nonlinear systems. While this paradigm shift has appeared to be very fruitful for both experimental und theoretical research, it still discloses the ap- plication of important ideas such as adaptation and transformation. We argue that basing the idea of ecological resilience strictly on basins of attraction is a logical fallacy. In this work, we attempt to build a new resilience framework which explains resilience from a dynamical systems perspective, but based on context-dependent ecological reasoning. Thereby, we open up the resilience concept to ecologically meaningful system properties like system functioning, which are pre- sumably based on (potentially multiple) attractors instead of single domains of attraction.

In this work two models based on the Hodgkin-Huxley formalism, describing dynamics of neurons, with bistability between silent state and bursting state are considered. For the case of such kind bistability we observe, that basin of attraction of silent state is very small, and the probability to reach silent state for randomly distributed initial conditions is less then 5%. In the work we show that in dependence on the structure of phase space and localization of the coexisting attractors we can induce swithching, or not by noise.

In modern era of time series analysis one faces a problem of extraction of information from a large number of measured components, while the number of consecutive measurements is limited by the time resolution of the apparatus (fMRI), stationarity of time series (EEG) or internal dynamics of a system (stock markets), which limits the usability of classical probabilistic tools. A way to overcome this difficulty is by reducing the dimensionality of the system. One of possibilities is the diagonalization of the correlation matrix. In the limit when both the number of components and the number of measurements are large its spectrum is described by the famous Marcenko-Pastur distribution from Random Matrix Theory (RMT). This approach gives knowledge only about equal-time correlations. The desired auto-cross-correlations are in general asymmetric, impeding their analysis. In the poster, we propose a RMT setting for calculation of the complex spectra of time-lagged correlation matrices, which we suggest to be a powerful tool for studying collective autocorrelations in complex systems. [1] MA Nowak, W Tarnowski, Spectra of large time-lagged correlation matrices from Random Matrix Theory J. Stat. Mech. 2017, 063405; Arxiv [1612.06552]

We introduce a heterogeneous continuous time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatio-temporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.

Environmental variations due to climate change or anthropogenic influences might give rise to unexpected responses of ecosystems, particularly their rapid decline. We study a paradigmatic predator-prey system, the Rosenzweig-MacArthur model, and demonstrate that a fast decline of nutrients can lead to a collapse of the ecosystem. We assume that nutrients decrease with a ramping parameter linear in time. As a result, the stable equilibrium also changes in time. When nutrients decrease too fast, the system is not able to track the moving stable equilibrium and the system will undergo a rate-induced transition. In addition, we use the method described in Wieczorek, [2011] to compute the threshold which separates initial conditions that undergo rate-induced transitions from those initial conditions which track the moving equilibrium for a given rate of change. Further we demonstrate that predator-prey systems which undergo rate-induced transitions are at high risk of extinction because they are determined by low population densities after the transitions. Because small populations often collapse when they are exposed to fast environmental variations and diseases, the analysis of rate-induced transitions is needed to prevent endangered populations.

Several tipping elements are known to exist in the climate system (Lenton et al., 2008). Some of them might be at risk of undergoing a critical transition into a qualitatively different state within a global warming of 1.5°C to 2°C compared to pre-industrial levels (Schellnhuber et al., 2016). Among these are crucial components of the cryosphere: the West Antarctic Ice Sheet, the Greenland Ice Sheet as well as the Arctic summer sea ice and mountain glaciers. Their disintegration could possibly lead to severe changes in the climate system itself, through positive nonlinear feedbacks such as the ice albedo feedback which act back on temperature. For the quantification and evaluation of the climate dynamics as well as feedbacks we use the CLIMBER-2 model (Pethoukov et al., 2000, Ganopolski et al., 2001), an Earth-system model of intermediate complexity on a coarse spatial resolution, that includes an atmosphere, ocean and sea ice model as well as a dynamic vegetation model and the global carbon cycle. With conceptual calculations, backed up by CLIMBER-2 simulations, we here assess the effect of critical transitions of (some or all of) the cryo-elements on global mean temperature. Furthermore, we separate the total model response into contributions from the different fast climate feedbacks including the albedo, water vapor, clouds and lapse rate feedbacks.

Modern acceleration devices allow the long-term measurement of three-axial acceleration on various points of the human body over long durations ($>$ 24h) with a high resolution ($>$100 Hz). Apparently the recorded values depend on the placement of the accelerometer. Here we studied 24h acceleration data from hip and wrist as well as simultaneously recorded ECG data in 600 subjects. The data was acquired in the ongoing German National Cohort study. We find that sleep and weak motion activity can be well distinguished by wrist-worn devices. However, data from the hip rather allows studying and characterizing transitions between more intensive activities including exercising. This leads to the idea to combine the two acceleration signals in order to classify activity levels and examine regime transitions. The resulting curves of hip versus wrist activity are surprisingly similar for most subjects. For a detailed look we added the pulse as derived from 6-second ECG segments. The corresponding three dimensional plots allow a fast interpretation of technical and medical issues like asynchronicity, non-wear-time, tachycardia and bradycardia. In addition, they facilitate studies of the relationship between activity regime transitions and corresponding changes of heart rate. Furthermore, we observe that the high-resolution acceleration signals contain more information than just movement and locomotion. Time-series analyses for sleep episodes reveal the respiration signal and a 'physiological tremor' peak, which we want to exploit to further study and characterize physiological transitions during sleep.

Gait disturbances are among the most disabling motor symptoms of Parkinson’s disease (PD). Freezing of gait (FOG), a paroxysmal inability to generate effective stepping, is experienced by about half of all individuals with advanced PD. We have recorded 32-channel electroencephalogram (EEG) data and simultaneous 4-channel leg electromyogram (EMG) data in 15 patients with PD and 8 healthy elderly controls during various gait tasks. Altogether, data from 123 FOG episodes has been collected. The main goals of the project are (i) the identification of neuronal control patterns associated with FOG and (ii) the detection and prediction of FOG episodes. In contrast to our initial expectation, we find that phase synchronization of neural oscillations ('brainwave' sin several EEG bands) across brain hemispheres is significantly {\it increased} in subjects with PD as compared with healthy controls. Hence, the gait problems do not seem to be related with a deteriorating coordination between the brain hemispheres (or body sides). There is no systematic further increase (or decrease) of EEG phase synchronization before or during actual FOG episodes, and there also seem to be no systematic changes in EEG amplitudes in several bands. In this respect, we cannot confirm corresponding previous reports by an Australian group, that used EEG amplitudes for predicting FOG episodes in a machine learning approach. EEG multiplex recurrence networks’ properties do also not seem to yield a clear prediction for FOG episodes. Further, we find that the amplitudes of EMG oscillations are significantly reduced in PD patients as compared with healthy controls, while EMG frequencies are increased in PD patients without FOG only. During FOG episodes we observe no systematic changes of EMG amplitudes and a slight ($\sim$ 25\%) rise in EMG frequencies. It is unlikely that this feature can be exploited for predicting FOG episodes. However, when studying cross-correlations between EEG amplitudes and EMG amplitudes, we observe that such correlations do not exist during normal gait but appear during and somewhat before FOG episodes.