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Reservoir computing is a machine learning method that is closely linked to dynamical systems theory. This connection is highlighted in a brief introduction to the general concept of reservoir computing. We then address a recently suggested approach to improve the performance of reservoir systems by incorporating past values of the input signal or of the reservoir state variables into the readout used to forecast the input or cross-predict other variables of interest. The efficiency of this extension is illustrated by a minimal example in which a three-dimensional reservoir system based on the Lorenz-63 model is used to predict the variables of a chaotic Rössler system.

Abstract 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.

Although other methods exist to store and manage data in modern information technology, the standard solution is file systems. Therefore, keeping well-organized file structures and file system layouts can be key to a sustainable research data management infrastructure. However, file structures alone lack several important capabilities for FAIR data management: the two most significant being insufficient visualization of data and inadequate possibilities for searching and obtaining an overview. Research data management systems (RDMSs) can fill this gap, but many do not support the simultaneous use of the file system and RDMS. This simultaneous use can have many benefits, but keeping data in RDMS in synchrony with the file structure is challenging. Here, we present concepts that allow for keeping file structures and semantic data models (in RDMS) synchronous. Furthermore, we propose a specification in yaml format that allows for a structured and extensible declaration and implementation of a mapping between the file system and data models used in semantic research data management. Implementing these concepts will facilitate the re-use of specifications for multiple use cases. Furthermore, the specification can serve as a machine-readable and, at the same time, human-readable documentation of specific file system structures. We demonstrate our work using the Open Source RDMS LinkAhead (previously named “CaosDB”).

The mechanical contraction of the pumping heart is driven by electrical excitation waves running across the heart muscle due to the excitable electrophysiology of heart cells. With cardiac arrhythmias these waves turn into stable or chaotic spiral waves (also called rotors) whose observation in the heart is very challenging. While mechanical motion can be measured in 3D using ultrasound, electrical activity can (so far) not be measured directly within the muscle and with limited resolution on the heart surface, only. To bridge the gap between measurable and not measurable quantities we use two approaches from machine learning, echo state networks and convolutional autoencoders, to solve two relevant data modelling tasks in cardiac dynamics: Recovering excitation patterns from noisy, blurred or undersampled observations and reconstructing complex electrical excitation waves from mechanical deformation. For the synthetic data sets used to evaluate both methods we obtained satisfying solutions with echo state networks and good results with convolutional autoencoders, both clearly indicating that the data reconstruction tasks can in principle be solved by means of machine learning.

In this study, ordinal pattern analysis and classical frequency-based EEG analysis methods are used to differentiate between EEGs of different age groups as well as individuals. As characteristic features, functional connectivity as well as single-channel measures in both the time and frequency domain are considered. We compare the separation power of each feature set after nonlinear dimensionality reduction using t-distributed stochastic neighbor embedding and demonstrate that ordinal pattern-based measures yield results comparable to frequency-based measures applied to preprocessed data, and outperform them if applied to raw data. Our analysis yields no significant differences in performance between single-channel features and functional connectivity features regarding the question of age group separation.

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.

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 vs. Blebbistatin-arrested Langendorff-perfused rabbit hearts (N = 10). We found action potential durations (APD) to be, on average, 25 ± 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 ± 2Hz with Blebbistatin and 15 ± 4Hz without Blebbistatin, and maintained a broader frequency spectrum. In contracting hearts, the average number of phase singularities was NPS = 11 ± 4 compared to NPS = 6 ± 3 with Blebbistatin during VF on the anterior 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.

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.

Storing scientific data on the filesystem in a meaningful and transparent way is no trivial task. In particular, when the data have to be accessed after their originator has left the lab, the importance of a standardized filesystem layout cannot be underestimated. It is desirable to have a structure that allows for the unique categorization of all kinds of data from experimental results to publications. They have to be accessible to a broad variety of workflows, e.g., via graphical user interface as well as via command line, in order to find widespread acceptance. Furthermore, the inclusion of already existing data has to be as simple as possible. We propose a three-level layout to organize and store scientific data that incorporates the full chain of scientific data management from data acquisition to analysis to publications. Metadata are saved in a standardized way and connect original data to analyses and publications as well as to their originators. A simple software tool to check a file structure for compliance with the proposed structure is presented.

Spatio-temporal chaotic dynamics in a two-dimensional excitable medium is (cross-) estimated using a machine learning method based on a convolutional neural network combined with a conditional random field. The performance of this approach is demonstrated using the four variables of the Bueno-Orovio-Fenton-Cherry model describing electrical excitation waves in cardiac tissue. Using temporal sequences of two-dimensional fields representing the values of one or more of the model variables as input the network successfully cross-estimates all variables and provides excellent forecasts when applied iteratively.

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.

The characterization of spatiotemporal complexity remains a challenging task. This holds in particular for the analysis of data from fluorescence imaging (optical mapping), which allows for the measurement of membrane potential and intracellular calcium at high spatial and temporal resolutions and, therefore, allows for an investigation of cardiac dynamics. Dominant frequency maps and the analysis of phase singularities are frequently used for this type of excitable media. These methods address some important aspects of cardiac dynamics; however, they only consider very specific properties of excitable media. To extend the scope of the analysis, we present a measure based on entropy rates for determining spatiotemporal complexity patterns of excitable media. Simulated data generated by the Aliev–Panfilov model and the cubic Barkley model are used to validate this method. Then, we apply it to optical mapping data from monolayers of cardiac cells from chicken embryos and compare our findings with dominant frequency maps and the analysis of phase singularities. The studies indicate that entropy rate maps provide additional information about local complexity, the origins of wave breakup and the development of patterns governing unstable wave propagation.

We present CaosDB, a Research Data Management System (RDMS) designed to ensure seamless integration of inhomogeneous data sources and repositories of legacy data in a FAIR way. Its primary purpose is the management of data from biomedical sciences, both from simulations and experiments during the complete research data lifecycle. An RDMS for this domain faces particular challenges: research data arise in huge amounts, from a wide variety of sources, and traverse a highly branched path of further processing. To be accepted by its users, an RDMS must be built around workflows of the scientists and practices and thus support changes in workflow and data structure. Nevertheless, it should encourage and support the development and observation of standards and furthermore facilitate the automation of data acquisition and processing with specialized software. The storage data model of an RDMS must reflect these complexities with appropriate semantics and ontologies while offering simple methods for finding, retrieving, and understanding relevant data. We show how CaosDB responds to these challenges and give an overview of its data model, the CaosDB Server and its easy-to-learn CaosDB Query Language. We briefly discuss the status of the implementation, how we currently use CaosDB, and how we plan to use and extend it.

We investigate a system of two viscoelastically coupled, modified Van der Pol oscillators to compare their synchronization properties due to elastic and viscoelastic coupling. We show that viscoelastic coupling leads to in-phase synchronization while elastic coupling favours anti-phase synchronization. To study the impact of symmetry and nonlinearity, the restoring forces in the Van der Pol oscillators are extended to include nonlinear and asymmetric components. If the asymmetry, or rather the nonlinearity, of the restoring forces exceeds a certain threshold, only in-phase synchronized motion is found to be stable. Another important finding is that chaotic solutions can only be found if the restoring forces are asymmetric and the coupling incorporates viscosity.

The Anthropocene is characterized by close interdependencies between the natural Earth system and the global human society, posing novel challenges to model development. Here we present a conceptual model describing the long-term co-evolution of natural and socio-economic subsystems of Earth. While the climate is represented via a global carbon cycle, we use economic concepts to model socio-metabolic flows of biomass and fossil fuels between nature and society. A well-being-dependent parametrization of fertility and mortality governs human population dynamics. Our analysis focuses on assessing possible asymptotic states of the Earth system for a qualitative understanding of its complex dynamics rather than quantitative predictions. Low dimension and simple equations enable a parameter-space analysis allowing us to identify preconditions of several asymptotic states and hence fates of humanity and planet. These include a sustainable co-evolution of nature and society, a global collapse and everlasting oscillations. We consider different scenarios corresponding to different socio-cultural stages of human history. The necessity of accounting for the ‘human factor’ in Earth system models is highlighted by the finding that carbon stocks during the past centuries evolved opposing to what would ‘naturally’ be expected on a planet without humans. The intensity of biomass use and the contribution of ecosystem services to human well-being are found to be crucial determinants of the asymptotic state in a (pre-industrial) biomass-only scenario without capital accumulation. The capitalistic, fossil-based scenario reveals that trajectories with fundamentally different asymptotic states might still be almost indistinguishable during even a centuries-long transient phase. Given current human population levels, our study also supports the claim that besides reducing the global demand for energy, only the extensive use of renewable energies may pave the way into a sustainable future.

The amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems. This quantity, known as the mutual information rate (MIR), is calculated from the mutual information, which is rigorously defined only for random systems. Moreover, the definition of mutual information is based on probabilities of significant events. This work offers a simple alternative way to calculate the MIR in dynamical (deterministic) networks or between two time series (not fully deterministic), and to calculate its upper and lower bounds without having to calculate probabilities, but rather in terms of well known and well defined quantities in dynamical systems. As possible applications of our bounds, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators.

We suggest a control based approach to topology estimation of networks with N elements. This method first drives the network to steady states by a delayed feedback control; then performs structural perturbations for shifting the steady states M times; and finally infers the connection topology from the steady states' shifts by matrix inverse algorithm (M = N) or l(1)-norm convex optimization strategy applicable to estimate the topology of sparse networks from M << N perturbations. We discuss as well some aspects important for applications, such as the topology reconstruction quality and error sources, advantages and disadvantages of the suggested method, and the influence of (control) perturbations, inhomegenity, sparsity, coupling functions, and measurement noise. Some examples of networks with Chua's oscillators are presented to illustrate the reliability of the suggested technique.