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The conventional termination technique of life threatening cardiac arrhythmia like ventricular fibrillation is the application of a high-energy electrical defibrillation shock, coming along with severe side-effects. In order to improve the current treatment reducing these side-effects, the application of pulse sequences of lower energy instead of a single high-energy pulse are promising candidates. In this study, we show that in numerical simulations the dose-response function of pulse sequences applied to two-dimensional spiral wave chaos is not necessarily monotonously increasing, but exhibits a non-trivial frequency dependence. This insight into crucial phenomena appearing during termination attempts provides a deeper understanding of the governing termination mechanisms in general, and therefore may open up the path towards an efficient termination of cardiac arrhythmia in the future.

The main aim of this paper is to demonstrate the benefit of the application of high-performance computing techniques in the field of non-linear science through two kinds of dynamical systems as test models. It is shown that high-resolution, multi-dimensional parameter scans (in the order of millions of parameter combinations) via an initial value problem solver are an efficient tool to discover new features of dynamical systems that are hard to find by other means. The employed initial value problem solver is an in-house code written in C++ and CUDA C software environments, which can exploit the high processing power of professional graphics cards (GPUs). The first test model is the Keller–Miksis equation, a non-linear oscillator describing the dynamics of a driven single spherical gas bubble placed in an infinite domain of liquid. This equation is important in the field of cavitation and sonochemistry. Here, the high-resolution parameter scans gave us the opportunity to lay down the basis of a non-feedback technique to control multi-stability in which direct selection of the desired attractor is possible. The second test model is related to a pressure relief valve that can exhibit a special kind of impact dynamics called grazing impact. A fine scan of the initial conditions revealed a second focal point of the grazing lines in the initial-condition space that was hidden in previous studies.

CaosDB is a flexible semantic research data management system, released as open source software. Its versatile data model and data integration toolkit allows for applications in complex and very heterogeneous scientific workflows and different scientific domains. Successful implementations include biomedical physics [1] and glaciology [2]. Here, we present a recent implementation of a use case in marine biogeochemistry which has a special focus on bridging between experimental work and numerical ocean modelling. CaosDB is used to store, index and link data during different stages of research on the marine carbon cycle: Data from experiments and field campaigns is integrated and mapped onto semantic data structures. This data is then linked to data from numerical ocean simulations. The ocean model, here with a specific focus on natural marine carbon sequestration of dissolved organic carbon (DOC), uses the georeferenced data to evaluate model performance. By simultaneously linking empirical data and the sampled model parameter space together with the model output, CaosDB enhances the efficiency of model development. In the current implementation simulated data is linked to georeferenced DOC concentration data. We plan to expand it to complex data sets including thousands of dissolved organic matter molecular formulae and metagenomes of pelagic microbial communities. The combined management of these heterogeneous data structures with semantic models allows us to perform complex searches and seamlessly connect to automated data analysis pipelines.[1] Fitschen, T.; Schlemmer, A.; Hornung, D.; tom Wörden, H.; Parlitz, U.; Luther, S. CaosDB—Research Data Management for Complex, Changing, and Automated Research Workflows. Data 2019, 4, 83. https://doi.org/10.3390/data4020083[2] Schlemmer, A.; tom Wörden, H.; Freitag, J.; Fitschen, T.; Kerch, J.; Schlomann, Y.; ... & Luther, S. Evaluation of the semantic research data management system CaosDB in glaciology. deRSE 2019. https://doi.org/10.5446/42478

Like most machine learning algorithms, Echo State Networks possess several hyperparameters that have to be carefully tuned for achieving best performance. For minimizing the error on a specific task, we present a gradient based optimization algorithm, for the input scaling, the spectral radius, the leaking rate, and the regularization parameter.

Cardiomyocyte and stroma cell cross-talk is essential for the formation of collagen-based engineered heart muscle, including engineered human myocardium (EHM). Fibroblasts are a main component of the myocardial stroma. We hypothesize that fibroblasts, by compacting the surrounding collagen network, support the self-organization of cardiomyocytes into a functional syncytium. With a focus on early self-organization processes in EHM, we studied the molecular and biophysical adaptations mediated by defined populations of fibroblasts and embryonic stem cell-derived cardiomyocytes in a collagen type I hydrogel. After a short phase of cell-independent collagen gelation (30 min), tissue compaction was progressively mediated by fibroblasts. Fibroblast-mediated tissue stiffening was attenuated in the presence of cardiomyocytes allowing for the assembly of stably contracting, force-generating EHM within 4 weeks. Comparative RNA-sequencing data corroborated that fibroblasts are particularly sensitive to the tissue compaction process, resulting in the fast activation of transcription profiles, supporting heart muscle development and extracellular matrix synthesis. Large amplitude oscillatory shear (LAOS) measurements revealed nonlinear strain stiffening at physiological strain amplitudes (>2%), which was reduced in the presence of cells. The nonlinear stress-strain response could be characterized by a mathematical model. Collectively, our study defines the interplay between fibroblasts and cardiomyocytes during human heart muscle self-organization in vitro and underscores the relevance of fibroblasts in the biological engineering of a cardiomyogenesis-supporting viscoelastic stroma. We anticipate that the established mathematical model will facilitate future attempts to optimize EHM for in vitro (disease modelling) and in vivo applications (heart repair).

International audience; Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to “reconstruct” some supposed influences. In the 1970s, a new way of performing science under the name “chaos” emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements—which were never published—illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.

Chaotic dynamics of a dynamical system is not necessarily persistent. If there is (without any active intervention from outside) a transition towards a (possibly nonchaotic) attractor, this phenomenon is called transient chaos, which can be observed in a variety of systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. Also, chimera states, which show coherent and incoherent dynamics in spatially distinct regions of the system, are often chaotic transients. In many practical cases, the control of the chaotic dynamics (either the termination or the preservation of the chaotic dynamics) is desired. Although the self-termination typically occurs quite abruptly and can so far in general not be properly predicted, previous studies showed that in many systems a 'terminal transient phase" (TTP) prior to the self-termination existed, where the system was less susceptible against small but finite perturbations in different directions in state space. In this study, we show that, in the specific case of chimera states, these susceptible directions can be related to the structure of the chimera, which we divide into the coherent part, the incoherent part and the boundary in between. That means, in practice, if self-termination is close we can identify the direction of perturbation which is likely to maintain the chaotic dynamics (the chimera state). This finding improves the general understanding of the state space structure during the TTP, and could contribute also to practical applications like future control strategies of epileptic seizures which have been recently related to the collapse of chimera states.

A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.

The development of new approaches to control cardiac arrhythmias requires a deep understanding of spiral wave dynamics. Optogenetics offers new possibilities for this. Preliminary experiments show that sub-threshold illumination affects electrical wave propagation in the mouse heart. However, a systematic exploration of these effects is technically challenging. Here, we use state-of-the-art computer models to study the dynamic control of spiral waves in a two-dimensional model of the adult mouse ventricle, using stationary and non-stationary patterns of sub-threshold illumination. Our results indicate a light intensity-dependent increase in cellular resting membrane potentials, which together with diffusive cell-cell coupling leads to the development of spatial voltage gradients over differently illuminated areas. A spiral wave drifts along the positive gradient. These gradients can be strategically applied to ensure drift-induced termination of a spiral wave, both in optogenetics and in conventional methods of electrical defibrillation.

In many real-life systems, transient chaotic dynamics plays a major role. For instance, the chaotic spiral or scroll wave dynamics of electrical excitation waves during life-threatening cardiac arrhythmias can terminate by itself. Epileptic seizures have recently been related to the collapse of transient chimera states. Controlling chaotic transients, either by maintaining the chaotic dynamics or by terminating it as quickly as possible, is often desired and sometimes even vital (as in the case of cardiac arrhythmias). We discuss in this study that the difference of the underlying structures in state space between a chaotic attractor (persistent chaos) and a chaotic saddle (transient chaos) may have significant implications for efficient control strategies in real life systems. In particular, we demonstrate that in the latter case, chaotic dynamics in spatially extended systems can be terminated via a relatively low number of (spatially and temporally) localized perturbations. We demonstrate as a proof of principle that control and targeting of high-dimensional systems exhibiting transient chaos can be achieved with exceptionally small interactions with the system. This insight may impact future control strategies in real-life systems like cardiac arrhythmias.

Optical mapping is a high-resolution fluorescence imaging technique, that uses voltage- or calcium-sensitive dyes to visualize electrical excitation waves on the heart surface. However, optical mapping is very susceptible to the motion of cardiac tissue, which results in so-called motion artifacts in the fluorescence signal. To avoid motion artifacts, contractions of the heart muscle are typically suppressed using pharmacological excitation-contraction uncoupling agents, such as Blebbistatin. The use of pharmacological agents, however, may influence cardiac electrophysiology. Recently, it has been shown that numerical motion tracking can significantly reduce motion-related artifacts in optical mapping, enabling the simultaneous optical measurement of cardiac electrophysiology and mechanics. Here, we combine ratiometric optical mapping with numerical motion tracking to further enhance the robustness and accuracy of these measurements. We evaluate the method's performance by imaging and comparing cardiac restitution and ventricular fibrillation (VF) dynamics in contracting, non-working versus Blebbistatin-arrested Langendorff-perfused rabbit hearts ($N=10$). We found action potential durations (APD) to be, on average, $25 \pm 5 \%$ shorter in contracting hearts compared to hearts uncoupled with Blebbistatin. The relative shortening of the APD was found to be larger at higher frequencies. VF was found to be significantly accelerated in contracting hearts, i.e., $9 \pm 2 Hz$ with Blebbistatin and $15 \pm 4 Hz$ without Blebbistatin ($N=10$ hearts), and maintained a broader frequency spectrum. In contracting hearts, the average number of phase singularities was $N_{PS} = 11 \pm 4$ compared to $N_{PS} = 6 \pm 3$ with Blebbistatin during VF on the anterior left ventricular surface. VF inducibility was reduced with Blebbistatin. We found the effect of Blebbistatin to be concentration-dependent and reversible by washout. Aside from the electrophysiological characterization, we also measured and analyzed cardiac motion. Our findings may have implications for the interpretation of optical mapping data, and highlight that physiological conditions, such as oxygenation and metabolic demand, must be carefully considered in ex vivo imaging experiments.

Genetic circuits that control specific cellular functions are never fully insulated against influences of other parts of the cell. For example, they are subject to periodic modulation by the cell cycle through volume growth and gene doubling. To investigate possible effects of the cell cycle on oscillatory gene circuits dynamics, we modelled a simple synthetic genetic oscillator, the repressilator, and studied hallmarks of the resulting nonlinear dynamics. We found that the repressilator coupled to the cell cycle shows typical quasiperiodic motion with discrete Fourier spectra and windows in parameter space with synchronization of the two oscillators, with a devil’s stair case indicating the Arnold tongues of synchronization. In the case of identical parameters for the three genes of the repressilator and simultaneous gene duplication, we identify two classes of synchronization windows, symmetric and asymmetric, depending on whether the trajectories satisfy a discrete three-fold rotation symmetry, corresponding to cyclic permutation of the three genes. Unexpectedly changing the gene doubling time revealed that the width of the Arnold tongues is connected to that three-fold symmetry of the synchronization trajectories: non-simultaneous gene duplication increases the width of asymmetric synchronization regions, for some of them by an order of magnitude. By contrast, there is only a small or even a negative effect on the window size for symmetric synchronization. This observation points to a control mechanism of synchronization via the location of the genes on the chromosome.

A novel method to control multistability of nonlinear oscillators by applying dual-frequency driving is presented. The test model is the Keller–Miksis equation describing the oscillation of a bubble in a liquid. It is solved by an in-house initial-value problem solver capable to exploit the high computational resources of professional graphics cards. During the simulations, the control parameters are the two amplitudes of the acoustic driving at fixed, commensurate frequency pairs. The high-resolution bi-parametric scans in the control parameter plane show that a period-2 attractor can be continuously transformed into a period-3 one (and vice versa) by proper selection of the frequency combination and by proper tuning of the driving amplitudes. This phenomenon has opened a new way to drive the system to a desired, pre-selected attractor directly via a non-feedback control technique without the need of the annihilation of other attractors. Moreover, the residence in transient chaotic regimes can also be avoided. The results are supplemented with simulations obtained by the boundary-value problem solver AUTO, which is capable to compute periodic orbits directly regardless of their stability, and trace them as a function of a control parameter with the pseudo-arclength continuation technique.

Synchronization based state estimation tries to synchronize a model with the true evolution of a system via the observations. In practice, an extra term is added to the model equations which hampers growth of instabilities transversal to the synchronization manifold. Therefore, there is a very close connection between synchronization and data assimilation. Recently, synchronization with time-delayed observations has been proposed, in which observations at future times are used to help synchronize a system that does not synchronize using only present observations, with remarkable successes. Unfortunately, these schemes are limited to small-dimensional problems. In this article, we lift that restriction by proposing an ensemble-based synchronization scheme. Tests were performed using the Lorenz’96 model for 20-, 100- and 1000-dimension systems. Results show global synchronization errors stabilizing at values of at least an order of magnitude lower than the observation errors, suggesting that the scheme is a promising tool to steer model states to the truth. While this framework is not a complete data assimilation method, we develop this methodology as a potential choice for a proposal density in a more comprehensive data assimilation method, like a fully nonlinear particle filter.

We present a dynamical observer for two dimensional partial differential equation models describing excitable media, where the required cross prediction from observed time series to not measured state variables is provided by Echo State Networks receiving input from local regions in space, only. The efficacy of this approach is demonstrated for (noisy) data from a (cubic) Barkley model and the Bueno-Orovio-Cherry-Fenton model describing chaotic electrical wave propagation in cardiac tissue.

We present an approach for data-driven prediction of high-dimensional chaotic time series generated by spatially-extended systems. The algorithm employs a convolutional autoencoder for dimension reduction and feature extraction combined with a probabilistic prediction scheme operating in the feature space, which consists of a conditional random field. The future evolution of the spatially-extended system is predicted using a feedback loop and iterated predictions. The excellent performance of this method is illustrated and evaluated using Lorenz-96 systems and Kuramoto-Sivashinsky equations of different size generating time series of different dimensionality and complexity.

The active cavitation threshold of a dual-frequency driven single spherical gas bubble is studied numerically. This threshold is defined as the minimum intensity required to generate a given relative expansion (Rmax-RE)/RE, where RE is the equilibrium size of the bubble and Rmax is the maximum bubble radius during its oscillation. The model employed is the Keller-Miksis equation that is a second order ordinary differential equation. The parameter space investigated is composed by the pressure amplitudes, excitation frequencies, phase shift between the two harmonic components and by the equilibrium bubble radius (bubble size). Due to the large 6-dimensional parameter space, the number of the parameter combinations investigated is approximately two billion. Therefore, the high performance of graphics processing units is exploited; our in-house code is written in C++ and CUDA C software environments. The results show that for (Rmax-RE)/RE=2, the best choice of the frequency pairs depends on the bubble size. For small bubbles, below 3μm, the best option is to use just a single frequency of a low value in the giant response region. For medium sized bubbles, between 3μm and 6μm, the optimal choice is the mixture of low frequency (giant response) and main resonance frequency. For large bubbles, above 6μm, the main resonance dominates the active cavitation threshold. Increasing the prescribed relative expansion value to (Rmax-RE)/RE=3, the optimal choice is always single frequency driving with the lowest value (20kHz here). Thus, in this case, the giant response always dominates the active cavitation threshold. The phase shift between the harmonic components of the dual-frequency driving (different frequency values) has no effect on the threshold.

Viscoelastically coupled excitable oscillators are used to model individually beating spatially separated cardiomyocytes surrounded by an extracellular matrix (ECM). We investigate how mechanical coupling via the ECM can synchronize two such oscillators with excitation contraction coupling and electromechanical feedback and how this synchronization depends on the rheological properties of the ECM. Extending our study to a linear chain of coupled oscillators we find a transition to synchronization as the ECM becomes stiffer. In the case of purely elastic coupling we observe antiphase chimera states.

Cardiovascular disease is often related to defects of subcellular components in cardiac myocytes, specifically in the dyadic cleft, which include changes in cleft geometry and channel placement. Modeling of these pathological changes requires both spatially resolved cleft as well as whole cell level descriptions. We use a multiscale model to create dyadic structure-function relationships to explore the impact of molecular changes on whole cell electrophysiology and calcium cycling. This multiscale model incorporates stochastic simulation of individual L-type calcium channels and ryanodine receptor channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca2+ and Ca2+-binding molecules in the bulk of the cell. We found action potential duration, systolic, and diastolic [Ca2+] to respond most sensitively to changes in L-type calcium channel current. The ryanodine receptor channel cluster structure inside dyadic clefts was found to affect all biomarkers investigated. The shape of clusters observed in experiments by Jayasinghe et al. and channel density within the cluster (characterized by mean occupancy) showed the strongest correlation to the effects on biomarkers. peerReviewed

The spatiotemporal dynamics of excitable media may exhibit chaotic transients. We investigate this transient chaos in the 2D Fenton–Karma model describing the propagation of electrical excitation waves in cardiac tissue and compute the average duration of chaotic transients in dependence on model parameter values. Furthermore, other characteristics like the dominant frequency, the size of the excitable gap, pseudo ECGs, the number of phase singularities and parameters characterizing the action potential duration restitution curve are determined and it is shown that these quantities can be used to predict the average transient time using polynomial regression.

We propose an image processing method for ordering anterior chamber optical coherence tomography (OCT) images in a fully unsupervised manner. The method consists of three steps: Firstly we preprocess the images (filtering the noise, aligning and normalizing the resolution); secondly, a distance measure between images is computed for every pair of images; thirdly we apply a machine learning algorithm that exploits the distance measure to order the images in a two-dimensional plane. The method is applied to a large (~1000) database of anterior chamber OCT images of healthy subjects and patients with angle-closure and the resulting unsupervised ordering and classification is validated by two ophthalmologists.

Chaotic spiral or scroll wave dynamics can be found in diverse systems. In cardiac dynamics, spiral or scroll waves of electrical excitation determine the dynamics during life-threatening arrhythmias like ventricular fibrillation. In numerical studies it was found that chaotic episodes of spiral and scroll waves can be transient, thus they terminate spontaneously. We show in this study that this behavior can also be observed using models which describe the ion channel dynamics of human cardiomyocytes (Bueno-Orovio-Cherry-Fenton model and the Ten Tusscher-Noble-Noble-Panfilov model). For both models we find that the average lifetime of the chaotic transients grows exponentially with the system size. With this behavior, we classify the systems into the group of type-II supertransients. We observe a significant difference of the breakup behavior between the models, which results in a distinct dynamics during the final phase just before the termination. The observation of a (temporally) stable single-spiral state affects the prevailing description of the dynamics of type-II supertransients as being "quasi-stationary" and also the feasibility of predicting the spontaneous termination of the spiral wave dynamics. In the long term, the relation between the breakup behavior of spiral waves and properties of chaotic transients like predictability or average transient lifetime may contribute to an improved understanding and classification of cardiac arrhythmias. peerReviewed

There are many examples of excitable media, such as the heart, that can show complex dynamics and where control is a challenging task. Heavy means like a strong electric shock are nowadays still necessary to control and terminate ventricular fibrillation (VF). It is known that heterogeneities in an excitable medium can stabilize the activity, e.g., spiral waves can pin to such obstacles. This might also be a reason for the persistence of VF and the difficulty to control it. Previous studies investigated systems with a single pinned spiral wave and demonstrated how the spiral can be unpinned. In this article, we extend this case and investigate a generic excitable system with multiple pinned spiral waves. We describe a control technique that allows the simultaneous unpinning of pinned spiral waves. Apart from theoretical considerations, we provide numerical evidence that the proposed technique is superior to the underdrive pacing method that has reportedly high success rates when applied to a single pinned spiral.

Transient chaos can emerge in a variety of diverse systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. The end of the chaotic episode can either be desired or not, depending on the specific system and application. In both cases, however, a prediction of the end of the chaotic dynamics is required. Despite the general challenges of reliably predicting chaotic dynamics for a long time period, the recent observation of a "terminal transient phase" of chaotic transients provides new insights into the transition from chaos to the subsequent (nonchaotic) regime. In spatially extended systems and also low-dimensional maps it was shown that the structure of the state space changes already a significant amount of time before the actual end of the chaotic dynamics. In this way, the terminal transient phase provides the conceptual foundation for a possible prediction of the upcoming end of the chaotic episode a significant amount of time in advance. In this study, we strengthen the general validity of the terminal transient phase by verifying its existence in another spatially extended model (Gray-Scott model) and the Henon map, where in the latter case the underlying mechanisms can be understood in an intuitive way. Furthermore, we show that the temporal length of the terminal transient phase remains approximately constant, when changing the system size (Gray-Scott) or parameters (Henon map) of the investigated models, although the average lifetime of the observed chaotic transients sensitively depends on these variations. Since the timescale of the terminal transient phase is in this sense relatively robust, this insight might be essential for possible applications, where the ratio between the length of the terminal transient phase and the relevant timescale of the dynamics may probably be crucial when a reasonable prediction (thus a sufficient time before) the end of the chaotic episode is required.

Permutation entropy (PE) is a robust quantity for measuring the complexity of time series. In the cardiac community it is predominantly used in the context of electrocardiogram (ECG) signal analysis for diagnoses and predictions with a major application found in heart rate variability parameters. In this article we are combining spatial and temporal PE to form a spatiotemporal PE that captures both, complexity of spatial structures and temporal complexity at the same time. We demonstrate that the spatiotemporal PE (STPE) quantifies complexity using two datasets from simulated cardiac arrhythmia and compare it to phase singularity analysis and spatial PE (SPE). These datasets simulate ventricular fibrillation (VF) on a two-dimensional and a three-dimensional medium using the Fenton-Karma model. We show that SPE and STPE are robust against noise and demonstrate its usefulness for extracting complexity features at different spatial scales.

Transient chaos in spatially extended systems can be characterized by the length of the transient phase, which typically grows quickly with the system size (supertransients). For a large class of these systems, the chaotic phase terminates abruptly, without any obvious precursors in commonly used observables. Here we investigate transient spatiotemporal chaos in two different models of this class. By probing the state space using perturbed trajectories we show the existence of a "terminal transient phase," which occurs prior to the abrupt collapse of chaotic dynamics. During this phase the impact of perturbations is significantly different from the earlier transient and particular patterns of (non)susceptible regions in state space occur close to the chaotic trajectories. We therefore hypothesize that even without perturbations proper precursors for the collapse of chaotic transients exist, which might be highly relevant for coping with spatiotemporal chaos in cardiac arrhythmias or brain functionality, for example.

Self-organized activation patterns in excitable media such as spiral waves and spatio-temporal chaos underlie dangerous cardiac arrhythmias. While the interaction of single spiral waves with different types of heterogeneity has been studied extensively, the effect of heterogeneity on fully developed spatio-temporal chaos remains poorly understood. We investigate how the complexity and stability properties of spatio-temporal chaos in the Bar-Eiswirth model of excitable media depend on the heterogeneity of the underlying medium. We employ different measures characterizing the chaoticity of the system and find that the spatial arrangement of multiple discrete lower excitability regions has a strong impact on the complexity of the dynamics. Varying the number, shape, and spatial arrangement of the heterogeneities, we observe strong emergent effects ranging from increases in chaoticity to the complete cessation of chaos, contrasting the expectation from the homogeneous behavior. The implications of our findings for the development and treatment of arrhythmias in the heterogeneous cardiac muscle are discussed.

Objective: A frequent observation during cardiac fibrillation is a fluctuation in complexity where the irregular pattern of the fibrillation is interrupted by more regular phases of varying length. Approach: We apply different measures to sliding windows of raw ECG signals for quantifying the temporal complexity. The methods include permutation entropy, power spectral entropy, a measure for the extent of the set of reconstructed states and several wavelet measures. Main results: Using these methods, variations of fibrillation patterns over time are detected and visualized. Significance: These quantifications can be used to characterize different phases of the ECG during fibrillation and might improve diagnosis and treatment methods for heart diseases.

Transferring information from observations to models of complex systems may meet impediments when the number of observations at any observation time is not sufficient. This is especially so when chaotic behavior is expressed. We show how to use time-delay embedding, familiar from nonlinear dynamics, to provide the information required to obtain accurate state and parameter estimates. Good estimates of parameters and unobserved states are necessary for good predictions of the future state of a model system. This method may be critical in allowing the understanding of prediction in complex systems as varied as nervous systems and weather prediction where insufficient measurements are typical.

The frequency of occurrence of ordinal patterns in an observed (or measured) times series can be used to identify equivalent dynamical system. We demonstrate this approach for system identification and parameter estimation for dynamics that can (at least approximately) be described by one-dimensional iterated maps.

In excitable media, chaotic dynamics governed by spiral or scroll waves is often not persistent but transient. Using extensive simulations employing different mathematical models we identify a specific type-II supertransient by an exponential increase of transient lifetimes with the system size in 2D and an investigation of the dynamics (number and lifetime of spiral waves, Kaplan-Yorke dimension). In 3D, simulations exhibit an increase of transient lifetimes and filament lengths only above a critical thickness. Finally, potential implications for understanding cardiac arrhythmias are discussed.

Rationale: Transverse tubules (TTs) couple electric surface signals to remote intracellular Ca 2+ release units (CRUs). Diffraction-limited imaging studies have proposed loss of TT components as disease mechanism in heart failure (HF). Objectives: Objectives were to develop quantitative super-resolution strategies for live-cell imaging of TT membranes in intact cardiomyocytes and to show that TT structures are progressively remodeled during HF development, causing early CRU dysfunction. Methods and Results: Using stimulated emission depletion (STED) microscopy, we characterized individual TTs with nanometric resolution as direct readout of local membrane morphology 4 and 8 weeks after myocardial infarction (4pMI and 8pMI). Both individual and network TT properties were investigated by quantitative image analysis. The mean area of TT cross sections increased progressively from 4pMI to 8pMI. Unexpectedly, intact TT networks showed differential changes. Longitudinal and oblique TTs were significantly increased at 4pMI, whereas transversal components appeared decreased. Expression of TT-associated proteins junctophilin-2 and caveolin-3 was significantly changed, correlating with network component remodeling. Computational modeling of spatial changes in HF through heterogeneous TT reorganization and RyR2 orphaning (5000 of 20 000 CRUs) uncovered a local mechanism of delayed subcellular Ca 2+ release and action potential prolongation. Conclusions: This study introduces STED nanoscopy for live mapping of TT membrane structures. During early HF development, the local TT morphology and associated proteins were significantly altered, leading to differential network remodeling and Ca 2+ release dyssynchrony. Our data suggest that TT remodeling during HF development involves proliferative membrane changes, early excitation-contraction uncoupling, and network fracturing.

Different cases of generalized synchronization are discussed with emphasis on methods for detecting them from time series. In particular, we shall focus on synchronization resulting in complex (non-smooth and/or non-invertible) functions or relations between drive and response.

Experimental investigations and simulations of synchronization phenomena of individual longitudinal optical modes of unidirectionally light injection coupled semiconductor lasers are presented and compared with the (anti-)synchronization of the corresponding full intensity fluctuations.

In data-driven system identification, values of parameters and not observed variables of a given model of a dynamical system are estimated from measured time series. We address the question of estimability and redundancy of parameters and variables, that is, whether unique results can be expected for the estimates or whether, for example, different combinations of parameter values would provide the same measured output. This question is answered by analyzing the null space of the linearized delay coordinates map. Examples with zero-dimensional, one-dimensional, and two-dimensional null spaces are presented employing the Hindmarsh-Rose model, the Colpitts oscillator, and the R\"ossler system.

An ensemble approach is presented for the reconstruction of a phase synchronization diagram from time series. As an example system, we analyze a forced Colpitts oscillator to show that synchronization diagram reconstructed by a single nonlinear model depends sensitively upon the model parameters, which should be estimated with a considerable amount of care. This dependence can be crucial for a precise recovery of the synchronization phenomena. To overcome this weakness, an ensemble approach is introduced. Two types of techniques, namely, (I) ensemble regression and (II) ensemble classification, are developed to show that they provide much more robust and reliable reconstruction of the synchronization diagram compared to the conventional single modeling approach.

The non-linear dependence of pre-sliding and sliding friction forces on displacement and velocity is modelled using different physics-based and black-box approaches including various Maxwell-Slip models, neural networks, non-parametric (local) models and recurrent networks. The efficiency and accuracy of these identification methods is compared for an experimental time series where the observed friction force is predicted from the measured displacement and estimated velocity. All models, although varying in their degree of accuracy, show good prediction capability of friction. Finally, it is shown that better results can be achieved by using an ensemble of the best models for prediction.