High-dimensional Attractor Research Articles

Featurizing Koopman mode decomposition for robust forecasting.

This article introduces an advanced Koopman mode decomposition (KMD) technique-coined Featurized Koopman Mode Decomposition (FKMD)-that uses delay embedding and a learned Mahalanobis distance to enhance analysis and prediction of high-dimensional dynamical systems. The delay embedding expands the observation space to better capture underlying manifold structures, while the Mahalanobis distance adjusts observations based on the system's dynamics. This aids in featurizing KMD in cases where good features are not a priori known. We show that FKMD improves predictions for a high-dimensional linear oscillator, a high-dimensional Lorenz attractor that is partially observed, and a cell signaling problem from cancer research.

A New Method for Determining the Embedding Dimension of Financial Time Series Based on Manhattan Distance and Recurrence Quantification Analysis.

Identification of embedding dimension is helpful to the reconstruction of phase space. However, it is difficult to calculate the proper embedding dimension for the financial time series of dynamics. By this Letter, we suggest a new method based on Manhattan distance and recurrence quantification analysis for determining the embedding dimension. By the advantages of the above two tools, the new method can calculate the proper embedding dimension with the feature of stability, accuracy and rigor. Besides, it also has a good performance on the chaotic time series which has a high-dimensional attractors.

A new approach to measure the fractal dimension of a trajectory in the high-dimensional phase space

In this paper, we introduce a new approach, which measures the fractal dimension (FD) of a trajectory in the multi-dimensional phase space based on the self-similarity of the sub-trajectories. Actually, we first compute the length of the sub-trajectories extracted from zooming out the trajectory in the phase space and then estimate the average length of the sub-trajectories in these zooms. Finally, we also calculate the fractal dimension of the trajectory based on the exponent of the power-law between the average length and the zoom-out size. For validating this approach, we also use the Weierstrass cosine function, which can generate fractured (fractal) trajectories with different dimensions. A set of the EEG segments recorded under the eyes-open and eyes-closed resting conditions is also employed to validate this new method by the data of a natural system. Generally, the outcomes of this method represent that it can well follow variations create in the dimension of a fractal trajectory. Therefore, since this new dimension can be estimated in every high-dimensional phase space, it is a good choice for investigating the dimension and the behavior of the high-dimensional strange attractors.

Analysing nystagmus waveforms: a computational framework

We present a new computational approach to analyse nystagmus waveforms. Our framework is designed to fully characterise the state of the nystagmus, aid clinical diagnosis and to quantify the dynamical changes in the oscillations over time. Both linear and nonlinear analyses of time series were used to determine the regularity and complexity of a specific homogenous phenotype of nystagmus. Two-dimensional binocular eye movement recordings were carried out on 5 adult subjects who exhibited a unilateral, uniplanar, vertical nystagmus secondary to a monocular late-onset severe visual loss in the oscillating eye (the Heimann-Bielschowsky Phenomenon). The non-affected eye held a central gaze in both horizontal and vertical planes (± 10 min. of arc). All affected eyes exhibited vertical oscillations, with mean amplitudes and frequencies ranging from 2.0°–4.0° to 0.25–1.5 Hz, respectively. Unstable periodic orbit analysis revealed only 1 subject exhibited a periodic oscillation. The remaining subjects were found to display quasiperiodic (n = 1) and nonperiodic (n = 3) oscillations. Phase space reconstruction allowed attractor identification and the computation of a time series complexity measure—the permutation entropy. The entropy measure was found to be able to distinguish between a periodic oscillation associated with a limit cycle attractor, a quasiperiodic oscillation associated with a torus attractor and nonperiodic oscillations associated with higher-dimensional attractors. Importantly, the permutation entropy was able to rank the oscillations, thereby providing an objective index of nystagmus complexity (range 0.15–0.21) that could not be obtained via unstable periodic orbit analysis or attractor identification alone. These results suggest that our framework provides a comprehensive methodology for characterising nystagmus, aiding differential diagnosis and also permitting investigation of the waveforms over time, thereby facilitating the quantification of future therapeutic managements. In addition, permutation entropy could provide an additional tool for future oculomotor modelling.

Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable

We construct a data-driven dynamical system model for a macroscopic variable the Reynolds number of a high-dimensionally chaotic fluid flow by training its scalar time-series data. We use a machine-learning approach, the reservoir computing for the construction of the model, and do not use the knowledge of a physical process of fluid dynamics in its procedure. It is confirmed that an inferred time-series obtained from the model approximates the actual one in each of various time-intervals, and that some characteristics of the chaotic invariant set mimic the actual ones. We investigate the appropriate choice of the delay-coordinate, especially the delay-time and the dimension, which enables us to construct a model having a relatively high-dimensional attractor easily.

The Effects of Graded Levels of Calorie Restriction XV: Phase Space Attractors Reveal Distinct Behavioral Phenotypes.

Calorie restriction (CR) has a positive impact on health and life span. Previous work, however, does not reveal the whole underlying mechanism of behavioral phenotypes under CR. We propose a new approach based on phase space reconstruction (PSR) to analyze the behavioral responses of mice to graded CR. This involved reconstructing high-dimensional attractors which topologically represent the intrinsic dynamics of mice based on low-dimensional time series of movement counts observed during the 90-day time course of restriction. PSR together with correlation dimensions (CD), Kolmogorov entropy (KE), and multifractal spectra builds a map from internal attractors to the phenotype of mice and reveals the mice with increasing CR levels undergo significant changes from a normal to a new state. Features of the attractors (CD and KE) were significantly associated with gene expression profiles in the hypothalamus of the same individuals.

Low-dimensional paradigms for high-dimensional hetero-chaos.

The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time-while typical trajectories wander throughout the attractor. Furthermore, if arbitrarily close to each point of the attractor there are points on periodic orbits that have different unstable dimensions, then we say such an attractor is "hetero-chaotic" (i.e., it has heterogeneous chaos). This is hard to picture but we believe that most physical systems possessing a high-dimensional attractor are of this type. We have created simplified models with that behavior to give insight into real high-dimensional phenomena.

Crisis of the chaotic attractor of a climate model: a transfer operator approach

The destruction of a chaotic attractor leading to rough changes in the dynamics of a dynamical system is studied. Local bifurcations are known to be characterised by a single or a pair of characteristic exponents crossing the imaginary axis. As a result, the approach of such bifurcations in the presence of noise can be inferred from the slowing down of the decay of correlations (Held and Kleinen 2004 Geophys. Res. Lett. 31 1–4). On the other hand, little is known about global bifurcations involving high-dimensional attractors with several positive Lyapunov exponents.It is known that the global stability of chaotic attractors may be characterised by the spectral properties of the Koopman (Mauroy and Mezić 2016 IEEE Trans. Autom. Control 61 3356–69) or the transfer operators governing the evolution of statistical ensembles. Accordingly, it has recently been shown (Tantet 2017 J. Stat. Phys. 1–33) that a boundary crisis in the Lorenz flow coincides with the approach to the unit circle of the eigenvalues of these operators associated with motions about the attractor, the stable resonances. A second class of resonances, the unstable resonances, are responsible for the decay of correlations and mixing on the attractor. In the deterministic case, these cannot be expected to be affected by general boundary crises.Here, however, we give an example of a chaotic system in which slowing down of the decay of correlations of some observables does occur at the approach of a boundary crisis. The system considered is a high-dimensional, chaotic climate model of physical relevance. Moreover, coarse-grained approximations of the transfer operators on a reduced space, constructed from a long time series of the system, give evidence that this behaviour is due to the approach of unstable resonances to the unit circle. That the unstable resonances are affected by the crisis can be physically understood from the fact that the process responsible for the instability, the ice-albedo feedback, is also active on the attractor. Finally, we discuss implications regarding response theory and the design of early-warning signals.

Higher-dimensional attractors with absolutely continuous invariant probability

Consider a dynamical system given by , where E is a linear expanding map of , C is a linear contracting map of and f is in . We provide sufficient conditions for E that imply the existence of an open set of pairs for which the corresponding dynamic T admits a unique absolutely continuous invariant probability.A geometrical characteristic of transversality between self-intersections of images of is present in the dynamic of the maps in . In addition, we give a condition between E and C under which it is possible to perturb f to obtain a pair in .

Cell Fate Decision as High-Dimensional Critical State Transition.

Cell fate choice and commitment of multipotent progenitor cells to a differentiated lineage requires broad changes of their gene expression profile. But how progenitor cells overcome the stability of their gene expression configuration (attractor) to exit the attractor in one direction remains elusive. Here we show that commitment of blood progenitor cells to the erythroid or myeloid lineage is preceded by the destabilization of their high-dimensional attractor state, such that differentiating cells undergo a critical state transition. Single-cell resolution analysis of gene expression in populations of differentiating cells affords a new quantitative index for predicting critical transitions in a high-dimensional state space based on decrease of correlation between cells and concomitant increase of correlation between genes as cells approach a tipping point. The detection of “rebellious cells” that enter the fate opposite to the one intended corroborates the model of preceding destabilization of a progenitor attractor. Thus, early warning signals associated with critical transitions can be detected in statistical ensembles of high-dimensional systems, offering a formal theory-based approach for analyzing single-cell molecular profiles that goes beyond current computational pattern recognition, does not require knowledge of specific pathways, and could be used to predict impending major shifts in development and disease.

Additive noise destroys the random attractor close to bifurcation

We provide an example for stabilization by noise. Due to the presence of higher order differential operators our approach does not rely on monotonicity arguments, i.e. the preserved order of solutions. Moreover, as the noise is highly degenerate mixing properties of the system might not be available.In our examples already a scalar additive noise destroys the complexity of a high-dimensional deterministic attractor of a PDE on an unbounded domain. The main result shows that by adding a certain amount of noise all trajectories converge to a single stationary solution. Close to bifurcation there is a lower bound on the amount of noise necessary for this stabilization, which depends on the distance to bifurcation, and the presence of small (but not arbitrarily small) noise already suffices.We focus on stochastic PDEs posed on unbounded domains without any decay condition at infinity. This setting allows for spatially constant or periodic solutions of arbitrary period. But we need to work in weighted spaces and establish the existence of random attractors in that setting first.

Non-Darwinian dynamics in therapy-induced cancer drug resistance.

Development of drug resistance, the prime cause of failure in cancer therapy, is commonly explained by the selection of resistant mutant cancer cells. However, dynamic non-genetic heterogeneity of clonal cell populations continuously produces meta-stable phenotypic variants (persisters), some of which represent stem-like states that confer resistance. Even without genetic mutations, Darwinian selection can expand these resistant variants, which would explain the invariably rapid emergence of stem-like resistant cells. Here, using quantitative measurements and modeling we show that appearance of multi-drug resistance in HL60 leukemic cells following treatment with vincristine is not explained by Darwinian selection but by Lamarckian induction. Single-cell longitudinal monitoring confirms the induction of multi-drug resistance in individual cells. Associated transcriptome changes indicate a lasting stress-response consistent with a drug-induced switch between high-dimensional cancer attractors. Resistance-induction correlates with Wnt-pathway up-regulation and is suppressed by β-catenin knock-down, revealing a new opportunity for early therapeutic intervention against resistance development.

Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

Why do we need a new paradigm in radiobiology?

Over the past 20 or so years experimental evidence, which questions the fundamentals of some 50 years standing, of both biology and radiobiology has accrued. In order to accommodate this new evidence within a framework that encompasses existing knowledge, attention has to be paid to the organisational or epigenetic, features of the cell. In recent years the high dimensional dynamic attractor has emerged as a potential organisational or regulatory agent that represents phenotype. It is argued here that its limited robustness provides a modus operandi to account for stress induced genomic instability.When radiation deposition events act to overcome the robustness of a normal or “home” attractor in the cell of an established species and cause a transition to a variant attractor or phenotype, the consequences are unrelated to any specific molecular damage to the genomic DNA. Rather they correspond to the loss of evolutionarily acquired stability (genotypic replicative integrity) and robustness. Such processes are termed type B events and give rise to a separate category of effects and risk to those associated with the conventional effects of radiation, type A effects. How type B risks might be assessed is discussed.

Using cell fate attractors to uncover transcriptional regulation of HL60 neutrophil differentiation

BackgroundThe process of cellular differentiation is governed by complex dynamical biomolecular networks consisting of a multitude of genes and their products acting in concert to determine a particular cell fate. Thus, a systems level view is necessary for understanding how a cell coordinates this process and for developing effective therapeutic strategies to treat diseases, such as cancer, in which differentiation plays a significant role. Theoretical considerations and recent experimental evidence support the view that cell fates are high dimensional attractor states of the underlying molecular networks. The temporal behavior of the network states progressing toward different cell fate attractors has the potential to elucidate the underlying molecular mechanisms governing differentiation.ResultsUsing the HL60 multipotent promyelocytic leukemia cell line, we performed experiments that ultimately led to two different cell fate attractors by two treatments of varying dosage and duration of the differentiation agent all-trans-retinoic acid (ATRA). The dosage and duration combinations of the two treatments were chosen by means of flow cytometric measurements of CD11b, a well-known early differentiation marker, such that they generated two intermediate populations that were poised at the apparently same stage of differentiation. However, the population of one treatment proceeded toward the terminally differentiated neutrophil attractor while that of the other treatment reverted back toward the undifferentiated promyelocytic attractor. We monitored the gene expression changes in the two populations after their respective treatments over a period of five days and identified a set of genes that diverged in their expression, a subset of which promotes neutrophil differentiation while the other represses cell cycle progression. By employing promoter based transcription factor binding site analysis, we found enrichment in the set of divergent genes, of transcription factors functionally linked to tumor progression, cell cycle, and development.ConclusionSince many of the transcription factors identified by this approach are also known to be implicated in hematopoietic differentiation and leukemia, this study points to the utility of incorporating a dynamical systems level view into a computational analysis framework for elucidating transcriptional mechanisms regulating differentiation.

Epigenetic Regulation of the Mammalian Cell

BackgroundUnderstanding how mammalian cells are regulated epigenetically to express phenotype is a priority. The cellular phenotypic transition, induced by ionising radiation, from a normal cell to the genomic instability phenotype, where the ability to replicate the genotype accurately is compromised, illustrates important features of epigenetic regulation. Based on this phenomenon and earlier work we propose a model to describe the mammalian cell as a self assembled open system operating in an environment that includes its genotype, neighbouring cells and beyond. Phenotype is represented by high dimensional attractors, evolutionarily conditioned for stability and robustness and contingent on rules of engagement between gene products encoded in the genetic network.Methodology/FindingsWe describe how this system functions and note the indeterminacy and fluidity of its internal workings which place it in the logical reasoning framework of predicative logic. We find that the hypothesis is supported by evidence from cell and molecular biology.ConclusionsEpigenetic regulation and memory are fundamentally physical, as opposed to chemical, processes and the transition to genomic instability is an important feature of mammalian cells with probable fundamental relevance to speciation and carcinogenesis. A source of evolutionarily selectable variation, in terms of the rules of engagement between gene products, is seen as more likely to have greater prominence than genetic variation in an evolutionary context. As this epigenetic variation is based on attractor states phenotypic changes are not gradual; a phenotypic transition can involve the changed contribution of several gene products in a single step.

Normal Lyapunov exponents and asymptotically stable attractors

Suppose f is a C 1+α map and leaves a lower-dimensional compact attractor A. In this article, we show that if for every f-ergodic probability measure supported on A, the normal Lyapunov exponents are negative, then this attractor could be a high-dimensional attractor. Moreover, we prove that the supremum of the normal Lyapunov exponents on the set of all ergodic measures can be achieved.

Cell Fates as High-Dimensional Attractor States of a Complex Gene Regulatory Network

Cells in multicellular organisms switch between distinct cell fates, such as proliferation or differentiation into specialized cell types. Genome-wide gene regulatory networks govern this behavior. Theoretical studies of complex networks suggest that they can exhibit ordered (stable) dynamics, raising the possibility that cell fates may represent high-dimensional attractor states. We used gene expression profiling to show that trajectories of neutrophil differentiation converge to a common state from different directions of a 2773-dimensional gene expression state space, providing the first experimental evidence for a high-dimensional stable attractor that represents a distinct cellular phenotype.

Reconstruction of systems with delayed feedback: II. Application

We apply a recently proposed method for the analysis of time series from systems with delayed feedback to experimental data generated by a laser. The method allows estimating the delay time with an error of the order of the sampling interval, while an approach based on the peaks of either the autocorrelation function, or the time delayed mutual information would yield systematically larger values. We reconstruct rather accurately the equations of motion and, in turn, estimate the Lyapunov spectrum even for high dimensional attractors. By comparing models constructed for different “embedding dimensions” with the original data, we are able to find the minimal faithful model. For short delays, the results of our procedure have been cross-checked using a conventional Takens time-delay embedding. For large delays, the standard analysis is inapplicable since the dynamics becomes hyperchaotic. In such a regime we provide the first experimental evidence that the Lyapunov spectrum, rescaled according to the delay time, is independent of the delay time itself. This is in full analogy with the independence of the system size found in spatially extended systems.

Estimating the dimension of high-dimensional attractors: A comparison between two algorithms

We compare two algorithms for the numerical estimation of the correlation dimension from a finite set of vectors: the “classical” algorithm of Grassberger and Procaccia (GPA) and the recently proposed algorithm of Judd (JA). Data set size requirements and their relations to systematic and statistical errors of the estimates are investigated. It is demonstrated that correlation dimensions of the order of 6 can correctly be resolved on the basis of the about 100000 data points in the case of a continuous trajectory on a strange attractor; the minimum data set size is, however, noticeably dependent on the geometrical structure of the system from which the vectors were sampled.