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We ask whether empirical finance market data (Financial Stress Index, swap and equity, emerging and developed, corporate and government, short and long maturity), with their recently observed alternations between calm periods and financial turmoil, could be described by a low-dimensional deterministic model, or whether this requests a stochastic approach. We find that a deterministic model performs at least as well as one of the best stochastic models, but may offer additional insight into the essential mechanisms that drive financial markets., Scientific Reports, 12 (1), ISSN:2045-2322

Hearing is one of the human’s foremost sensors; being able to hear again after suffering from a hearing loss is a great achievement, under all circumstances. However, in the long run, users of present-day hearing aids and cochlear implants are generally only halfway satisfied with what the commercial side offers. We demonstrate here that this is due to the failure of a full integration of these devices into the human physiological circuitry. Important parts of the hearing network that remain unestablished are the efferent connections to the cochlea, which strongly affects the faculty of listening. The latter provides the base for coping with the so-called cocktail party problem, or for a full enjoyment of multi-instrumental musical plays. While nature clearly points at how this could be remedied, to achieve this technologically will require the use of advanced high-precision electrodes and high-precision surgery, as we outline here. Corresponding efforts must be pushed forward by coordinated efforts from the side of science, as the commercial players in the field of hearing aids cannot be expected to have a substantial interest in advancements into this direction.

Many processes in nature are the result of many coupled individual subsystems (like population dynamics or neurosystems). Not always such systems exhibit simple stable behaviors that in the past science has mostly focused on. Often, these systems are characterized by bursts of seemingly stochastic activity, interrupted by quieter periods. The hypothesis is that the presence of a strong deterministic ingredient is often obscured by the stochastic features. We test this by modeling classically stochastic considered real-world data from both, the stochastic as well as the deterministic approaches to find that the deterministic approach's results level with those from the stochastic side. Moreover, the deterministic approach is shown to reveal the full dynamical systems landscape, which can be exploited for steering the dynamics into a desired regime., Scientific Reports, 12 (1), ISSN:2045-2322

In the neighborhood of critical states, distinct materials exhibit the same physical behavior, expressed by common simple laws among measurable observables, hence rendering a more detailed analysis of the individual systems obsolete. It is a widespread view that critical states are fundamental to neuroscience and directly favor computation. We argue here that from an evolutionary point of view, critical points seem indeed to be a natural phenomenon. Using mammalian hearing as our example, we show, however, explicitly that criticality does not describe the proper computational process and thus is only indirectly related to the computation in neural systems., Entropy, 24 (4), ISSN:1099-4300

We analyze empirical finance data, such as the Financial Stress Index, a number of asset classes (swaps, equity and bonds), market (emerging vs. developed), issuer (corporate vs. government bond), maturity (short vs. long) data, asking whether the recently observed alternations between calm periods and financial turmoil can be modelled in a low-dimensional deterministic manner, or whether they demand for their description a stochastic model. We find that a deterministic model performs at least as well as one of the best stochastic models, but may provide additional insight into the essential mechanisms that drive financial markets.

Various types of neural networks are currently widely used in diverse technical applications, not least because neural networks are known to be able to “generalize.” The latter property raises expectations that they should be able to handle unexpected situations with similar success than humans. Using fundamental examples, we show that in situations for which they have not been trained, artificial approaches tend to run into substantial problems, which highlights a deficit in comparisons to human abilities. For this problem–which seems to have obtained little attention so far–we provide a first analysis, based on simple examples, which exhibits some key features responsible for the difference between human and artificial intelligence.

When a system becomes unstable or noise becomes excessive, often regulations of the form of limiters (barriers obstructing excursion into undesired areas of the phase space) are imposed. It is hoped that by the influence of this element, the system can be calmed and its behaviour can be optimised. In what follows below, we consider a simple noisy nonlinear economics model that self-organises towards criticality. We demonstrate that the inherent effect of limiters is the emergence of stable cycles and that limiters need to be implemented with care in order to obtain an optimised system response. In particular, the implementation of the limiter around the maximal system response has generally a suboptimal effect, whereas the placement of the limiter in the vicinity of a period-one cycle will generally yield the optimal effect. We also show that, as a downside of the proposed approach, in order to control on period-one cycles, generally strong interventions are required. In democratic countries, a transparent control policy would therefore be an indispensable for its implementation. The discussed framework may prove helpful for this purpose.

In this chapter, we introduce the concept of the embedding dimension, as the smallest topological dimension required to ensure that an object described by simpler (often scalar) time series can be embedded in a higher topological dimension.

Biological neuronal networks excel over artificial ones in many ways, but the origin of this is still unknown. Our symbolic dynamics-based tool of excess entropies suggests that neuronal cultures naturally implement data structures of a higher level than what we expect from artificial neural networks, or from close-to-biology neural networks. This points to a new pathway for improving artificial neural networks towards a level demonstrated by biology. ISSN:0031-9007 ISSN:1079-7114

Multi-component systems often exhibit dynamics of a high degree of complexity, rendering it difficult to assess whether a proposed model’s description is adequate. For the multitude of systems that allow for a symbolic encoding, we provide a symbolic-dynamics based entropy measure that quantifies the degree of deviation obtained by a systems’s internal dynamics from random dynamics using identical average symbol probabilities. We apply this measure to several well-studied theoretical models and show its ability to characterize differences in internal dynamics, thus providing a means to accurately compare model and experiment. Data from neuronal cultures on a multi-electrode array chip validate the usefulness of our approach, revealing inadequacies of existing models and providing guidelines for their improvement. We propose our measure to be systematically used to develop future models and simulations.

Key ecological traits, like cell size, often follow scale-free or self-similar distributions. This indicates that these systems might operate near a critical (i.e. second-order) phase transition where macroscopic system behaviour is largely decoupled from microscopic system details, allowing an extremely simple, yet accurate and robust mathematical system characterisation. However, how trait-distribution scaling results from a critical transition has not yet been explicitly demonstrated. Here, we demonstrate that a generic class of cell growth and division models exhibits a critical transition from a growth-dominated to a division-dominated phase. We find experimental evidence for this transition, both in the population dynamics and in the moment scaling of chlorophyll distributions, for prokaryotic and eukaryotic phytoplankton growth under different light intensities. Our approach offers testable predictions of the response of unicellular trait-distributions to perturbations.

Key traits of unicellular species, like cell size, often follow scale-free or self-similar distributions, hinting at the possibility of an underlying critical process. However, linking such empirical scaling laws to the critical regime of realistic individual-based model classes is difficult. Here we reveal new empirical scaling evidence associated with a transition in the population and chlorophyll dynamics of phytoplankton. We offer a possible explanation for these observations by deriving scaling laws in the vicinity of the critical point of a new universality class of non-local cell growth and division models. This `criticality hypothesis' can be tested through new scaling predictions derived for our model class, for the response of chlorophyll distributions to perturbations. The derived scaling laws may also be generalized to other cellular traits and environmental drivers relevant to phytoplankton ecology., Comment: Companion paper in BioRxiv https://doi.org/10.1101/679209

European Biophysics Journal, 35 (6), ISSN:0175-7571, ISSN:1432-1017

Im Rahmen dieser Arbeit wird ein Konzept zur Ordnungsreduktion von höherdimensionalen nichtlinearen Oszillatormodellen vorgestellt. Hierbei werden zwei wesentliche Ziele verfolgt. Zum einen wird eine höherdimensionale Modellierung der Oszillatorschaltung verwendet. Hierdurch lassen sich die Einflüsse parasitärer Effekte sowie struktureller Erweiterungen auf das dynamische Verhalten des Systems berücksichtigen. Zum anderen wird durch eine anschließende Ordnungsreduktion über die Methode der Zentrumsmannigfaltigkeit eine zweidimensionale Systembeschreibung erzeugt, deren wesentliche Dynamik derjenigen des höherdimensionalen Systems entspricht. Durch diese, in der Ordnung reduzierte, nichtlineare und parameterabhängige Systembeschreibung wird die Anwendbarkeit nichtlinearer Analysemethoden ermöglicht bzw. vereinfacht. Mit der Anwendung der Andronov-Hopf-Bifurkationsanalyse auf das reduzierte System lässt sich eine Stabilitätsuntersuchung durchführen sowie die Amplitude und Frequenz aller Zustandsgrößen approximieren. Das vorgestellte Konzept wird anhand des Beispielsystems eines LC-Tank-VCOs durchgeführt. In this paper, an order reduction technique for higher-dimensional nonlinear oscillator models, based on a center manifold approach, is presented. By modeling the oscillator circuit in the higher-dimensional state space, influences of parasitic elements and of structural extensions of the oscillator architecture on the dynamical system behavior can be examined. Using the proposed order reduction technique, a generalized second order model will be derived, which includes selected design parameters of the higher order model. By using an Andronov-Hopf bifurcation analysis, the reduced system can be studied with respect to stability as well as the amplitude and frequency of the individual state variables. The concept is applied to the design of LC-tank VCOs.

‘Big’ high-dimensional data are commonly analyzed in low-dimensions, after performing a dimensionality reduction step that inherently distorts the data structure. For a similar analysis, clustering methods are also often used. These methods introduce a bias as well, either by starting from the assumption of a particular, often geometric, property of the clusters, or by using iterative schemes to enhance cluster contours, with consequences that are hard to control. The goal of data analysis should, however, be to encode and detect structural data features at all scales and densities simultaneously, without assuming a parametric form of data point distances, or modifying them. Here, we propose a novel approach that directly encodes data point neighborhood similarities as a sparse graph. Our non-iterative framework permits a transparent interpretation of data, without altering the original data dimension and metric. Several natural and synthetic data applications demonstrate the efficacy of our novel method.

A dominant view holds that the outer and middle ear are the determining factors for the frequency dependence of mammalian hearing sensitivity, but this view has been challenged. In the ensuing debate, there has been a missing element regarding in what sense and to what degree the biophysics of the inner ear might contribute to this frequency dependence. Here, we show that a simple model of the inner ear based on fundamental physical principles, reproduces, alone, the experimentally observed frequency dependence of the hearing threshold. This provides direct cochlea modeling support of the possibility that the inner ear could have a substantial role in determining the frequency dependence of mammalian hearing., Scientific Reports, 7, ISSN:2045-2322

The recent explosion of high dimensional, high resolution ‘big-data’ from automated bioinformatics measurement techniques demands new methods for unsupervised data processing. An essential analysis step is the identification of groups of similar data, or ‘clusters’, in noisy high-dimensional data spaces, as this permits to perform some analysis steps at the group level. Popular clustering algorithms introduce an undesired cluster shape bias, require prior knowledge of the number of clusters, and are unable to properly deal with noise. Manual data gating, often used to assist these methods, is based on low-dimensional projection techniques, which is prone to obscure the underlying data structure. While Hebbian Learning Clustering successfully overcomes all of these limitations (by using only local similarities to infer global structure), previous implementations were unsuited to deal with big data sets. Here, we present a novel implementation based on realistic neuronal dynamics that removes also this obstacle. By a performance that scales favourably compared to all standard clustering algorithms, unbiased large data analysis becomes feasible on standard desktop hardware.

Neurons communicate using patterns of impulses (action potentials); practically all these patterns can, upon suitably chosen parameters, be reproduced by Rulkov’s phenomenological, low-dimensional, map-based neuron models. Here, using phase response curves, we show that Rulkov map neurons also respond to transient pulse stimulation in a way that is compatible with the biological examples. This is important because Rulkov maps are computationally very inexpensive, allowing to perform large-scale simulations of the brain.

We record spontaneous activity from a developing culture of dissociated rat hippocampal neurons in vitro using a multi electrode array. To statistically characterize activity, we look at the time intervals between recorded spikes, which, unlike neuronal avalanche sizes, do not require the selection of a time bin. The distribution of inter event intervals in our data approximate power laws at all recorded stages of development, with exponents that can be used to characterize the development of the culture. Synchronized bursting emerges as the culture matures, and these bursts show activity that decays approximately exponentially. From this, we propose a model for neuronal activity within bursts based on the consumption of a shared resource. Our model produces power law distributed avalanches in simulations, and is analytically demonstrated to produce power law distributed inter event intervals with an exponent close to that observed in our data. This indicates that power law distributions in neuronal avalanche size and other observables, can be also an artefact of exponentially decaying activity within synchronized bursts.

Frequency changes in the partial tones of a sound can affect the way we perceive it, a phenomenon generally understood to be cortical in origin. A mesoscopic model now attributes perceived pitch to a physical mechanism linked to the presence of the cochlear fluid.

By adding some kinds of frustrations to coupled oscillatory networks, interesting synchronization phenomena can be obtained. In this study, we propose amplitude analysis by using whole power consumption in coupled oscillatory networks with frustrations.

Scientific investigations in medicine and beyond increasingly require observations to be described by more features than can be simultaneously visualized. Simply reducing the dimensionality by projections destroys essential relationships in the data. Similarly, traditional clustering algorithms introduce data bias that prevents detection of natural structures expected from generic nonlinear processes. We examine how these problems can best be addressed, where in particular we focus on two recent clustering approaches, Phenograph and Hebbian learning clustering, applied to synthetic and natural data examples. Our results reveal that already for very basic questions, minimizing clustering bias is essential, but that results can benefit further from biased post-processing. This article is part of the themed issue ‘Mathematical methods in medicine: neuroscience, cardiology and pathology’.

The cochlea provides a biological information-processing paradigm that we only begin to under- stand in its full complexity. Our work reveals an interacting network of strongly nonlinear dynami- cal nodes, on which even simple sound input triggers subnetworks of activated elements that follow power-law size statistics ('avalanches'). From dynamical systems theory, power-law size distribu- tions relate to a fundamental ground-state of biological information processing. Learning destroys these power laws. These results strongly modify the models of mammalian sound processing and provide a novel methodological perspective for understanding how the brain processes information., Videos are not included, please ask authors

Astounding properties of biological sensors can often be mapped onto a dynamical system below the occurrence of a bifurcation. For mammalian hearing, a Hopf bifurcation description has been shown to work across a whole range of scales, from individual hair bundles to whole regions of the cochlea. We reveal here the origin of this scale invariance, from a general level, applicable to all dynamics in the vicinity of a Hopf bifurcation (embracing, e.g., neuronal Hodgkin-Huxley equations). When subject to natural "signal coupling," ensembles of Hopf systems below the bifurcation threshold exhibit a collective Hopf bifurcation. This collective Hopf bifurcation occurs at parameter values substantially below where the average of the individual systems would bifurcate, with a frequency profile that is sharpened if compared to the individual systems.

We propose that a handle could be put on big data by looking at the systems that actually generate the data, rather than the data itself, realizing that there may be only few generic processes involved in this, each one imprinting its very specific structures in the space of systems, the traces of which translate into feature space. From this, we propose a practical computational clustering approach, optimized for coping with such data, inspired by how the human cortex is known to approach the problem.

Parameter inference is a fundamental problem in data-driven modeling. Given observed data that is believed to be a realization of some parameterized model, the aim is to find parameter values that are able to explain the observed data. In many situations, the dominant sources of uncertainty must be included into the model, for making reliable predictions. This naturally leads to stochastic models. Stochastic models render parameter inference much harder, as the aim then is to find a distribution of likely parameter values. In Bayesian statistics, which is a consistent framework for data-driven learning, this so-called posterior distribution can be used to make probabilistic predictions. We propose a novel, exact and very efficient approach for generating posterior parameter distributions, for stochastic differential equation models calibrated to measured time-series. The algorithm is inspired by re-interpreting the posterior distribution as a statistical mechanics partition function of an object akin to a polymer, where the measurements are mapped on heavier beads compared to those of the simulated data. To arrive at distribution samples, we employ a Hamiltonian Monte Carlo approach combined with a multiple time-scale integration. A separation of time scales naturally arises if either the number of measurement points or the number of simulation points becomes large. Furthermore, at least for 1D problems, we can decouple the harmonic modes between measurement points and solve the fastest part of their dynamics analytically. Our approach is applicable to a wide range of inference problems and is highly parallelizable., 15 pages, 8 figures

The Hopf Cochlea is hard- and software implemented model of the mammalian hearing organ, constructed from a series of feedforward coupled nonlinear Hopf system amplifier sections. The nonlinearity of the Hopf amplifiers removes a superposition principle among the sections. Instead, the sections couple extensively, e.g. by generating combination tones that have been shown to be essential for pitch perception. Here, we demonstrate exemplarily how such a complex network can be tuned for the dedicated amplification of a desired complex sound. Surprisingly, despite the complicated network structure of the cochlea, it is exactly the network's nonlinearity that renders the network simpler tunable than if the cochlea obeyed the superposition principle.

The separation of mixed auditory signals into their sources is an eminent neuroscience and engineering challenge. We reveal the principles underlying a deterministic, neural network–like solution to this problem. This approach is orthogonal to ICA/PCA that views the signal constituents as independent realizations of random processes. We demonstrate exemplarily that in the absence of salient frequency modulations, the decomposition of speech signals into local cosine packets allows for a sparse, noise-robust speaker separation. As the main result, we present analytical limitations inherent in the approach, where we propose strategies of how to deal with this situation. Our results offer new perspectives toward efficient noise cleaning and auditory signal separation and provide a new perspective of how the brain might achieve these tasks.

Synchronization phenomena in coupled oscillatory systems are very important models to describe various higher-dimensional nonlinear phenomena in the field of natural science. In this paper, phase synchronization in a ring of van der Pol oscillators coupled by time-varying resistors is studied. The coexistence of in-phase and anti/N-phase states and various interesting phase synchronization patterns are observed when the parameters are changed. Further, the influence of duty cycle of time-varying resistors for the observed phase synchronization patterns is investigated.

Durability describes the ability of a device to operate properly in imperfect conditions. We have recently proposed a novel neural network structure called an “Affordable Neural Network” (AfNN), in which affordable neurons of the hidden layer are considered as the elements responsible for the robustness property as is observed in human brain function. Whereas earlier we have shown that AfNNs can still generalize and learn, here we show that these networks are robust against damages occurring after the learning process has terminated. The results support the view that AfNNs embody the important feature of durability. In our contribution, we investigate the durability of the AfNN when some of the neurons in the hidden layer are damaged after the learning process.

Phase-and-frequency-locking phenomena among coupled biological oscillators are a topic of current interest, in particular to neuroscience. In the case of mono-directionally pulse-coupled oscillators, phase-locking is well understood, where the phenomenon is globally described by Arnold tongues. Here, we develop the tools that allow corresponding investigations to be made for more general pulse-coupled networks. For two bi-directionally coupled oscillators, we prove the existence of three-dimensional Arnold tongues that mediate from the mono- to the bi-directional coupling topology. Under this transformation, the coupling strength at which the onset of chaos is observed is invariant. The developed framework also allows us to compare information transfer in feedforward versus recurrent networks. We find that distinct laws govern the propagation of phase-locked spike-time information, indicating a qualitative difference between classical artificial vs. biological computation.

An important step towards understanding the working principles of the mammalian hearing sensor is the concept of an active cochlear amplifier. Theoretical arguments and physiological measurements suggest that the active cochlear amplifiers originate from systems close to a Hopf bifurcation. Efforts to model the mammalian hearing sensor on these grounds have, however, either had problems in reproducing sufficiently close essential aspects of the biological example (Magnasco, M.O. Phys. Rev. Lett. 90, 058101 (2003); Duke, T. & Jülicher, F. Phys. Rev. Lett. 90, 158101 (2003)), or required complicated spatially coupled differential equation systems that are unfeasible for transient signals (Kern, A. & Stoop, R. Phys. Rev. Lett. 91, 128101 (2003)). Here, we demonstrate a simple system of electronically coupled Hopf amplifiers that not only leads to the desired biological response behavior, but also has real-time capacity. The obtained electronic Hopf cochlea shares all salient signal processing features exhibited by the mammalian cochlea and thus provides a simple and efficient design of an artificial mammalian hearing sensor., Sensors, 7 (12), ISSN:1424-8220

Biological neocortical neurons are arranged in a columnar clustered architecture. Using a mathematical model in which the clustering properties can be monitored by means of a connectivity probability function, we investigate the information propagation in the associated networks, by means of simulations and a semi-analytical approach. Our analysis demonstrates that for systems with n-nearest neighbor coupling, the information propagation increases linearly in the neighbor order n. For fractal coupling, shown to give rise to small-world network characteristics, in contrast, an enhanced dependence is found, that, in our model of the neocortex, quickly saturates at a high level, indicating the superiority of this network type.

Activity pattern modalities of neuronal ensembles are determined by node properties as well as network structure. For many purposes, it is of interest to be able to relate activity patterns to either node properties or to network properties (or to a combination of both). When in physiological neural networks we observe bursting on a coarse-grained time and space scale, a proper decision on whether bursts are the consequence of individual neurons with an inherent bursting property or whether we are dealing with a genuine network effect has generally not been possible because of the noise in these systems. Here, by linking different orders of time and space scales, we provide a simple coarse-grained criterion for deciding this question., New Journal of Physics, 17, ISSN:1367-2630

The study of complex networks has pursued an understanding of macroscopic behavior by focusing on power-laws in microscopic observables. Here, we uncover two universal fundamental physical principles that are at the basis of complex networks generation. These principles together predict the generic emergence of deviations from ideal power laws, which were previously discussed away by reference to the thermodynamic limit. Our approach proposes a paradigm shift in the physics of complex networks, toward the use of power-law deviations to infer meso-scale structure from macroscopic observations., 14 pages, 7 figures

Bursting neurons fire rapid sequences of action potential spikes followed by a quiescent period. The basic dynamical mechanism of bursting is the slow currents that modulate a fast spiking activity caused by rapid ionic currents. Minimal models of bursting neurons must include both effects. We considered one of these models and its relation with a generalized Kuramoto model, thanks to the definition of a geometrical phase for bursting and a corresponding frequency. We considered neuronal networks with different connection topologies and investigated the transition from a non-synchronized to a partially phase-synchronized state as the coupling strength is varied. The numerically determined critical coupling strength value for this transition to occur is compared with theoretical results valid for the generalized Kuramoto model., Comment: 31 pages, 5 figures