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A simple method for detecting chaos in nature

Daniel Toker, Friedrich T. Sommer, Mark D'Esposito

arXiv:1904.00986v3q-bio.QMnlin.CD

TL;DR

Detecting chaos in biological recordings has been difficult. This paper presents an automated, modular pipeline for classifying noisy time series, and reports high accuracy across diverse systems while finding heart-rate variability to be stochastic in most tested recordings.

  • Problem

    Detecting the presence and degree of chaos in biological recordings has been difficult.

    Source: Discussion · p. 6

  • Method

    The Chaos Decision Tree Algorithm combines processing steps in an automated, modular pipeline for classifying time-series recordings.

    Source: Results · p. 4, Discussion · p. 6

  • Results

    The algorithm performed with high accuracy across diverse systems, and heart-rate variability supported a stochastic interpretation except in two congestive heart failure patients.

    Source: Results · p. 5, Discussion · p. 6

  • Takeaways & Limitations

    The modular pipeline can accommodate edge cases by adding subroutines, and the heart-rate results support the hypothesis that cardiac rhythm variability is stochastic.

    Source: Discussion · p. 6

    The 0–1 test may classify very weakly chaotic systems as periodic when time series are short; longer series are needed for accurate results.

    Source: Discussion · p. 6

Abstract

from arXiv · show

Chaos, or exponential sensitivity to small perturbations, appears everywhere in nature. Moreover, chaos is predicted to play diverse functional roles in living systems. A method for detecting chaos from empirical measurements should therefore be a key component of the biologist's toolkit. But, classic chaos-detection tools are highly sensitive to measurement noise and break down for common edge cases, making it difficult to detect chaos in domains, like biology, where measurements are noisy. However, newer tools promise to overcome these limitations. Here, we combine several such tools into an automated processing pipeline, and show that our pipeline can detect the presence (or absence) of chaos in noisy recordings, even for difficult edge cases. As a first-pass application of our pipeline, we show that heart rate variability is not chaotic as some have proposed, and instead reflects a stochastic process in both health and disease. Our tool is easy-to-use and freely available.

Results

Across ground-truth simulations and held-out recordings, the algorithm classified stochastic, periodic, and chaotic systems with near-perfect accuracy even at high measurement noise, except for the noise-driven sine map. It also tracked changing degrees of chaos and classified most heart-rate recordings as stochastic.

Results

  • The automated pipeline tests for stochasticity, denoises and corrects oversampling, then applies a modified 0–1 chaos test.It defaults to the tools selected for highest classification performance, while allowing users to choose alternatives.Source: Results · p. 4
  • The evaluation used systems with established ground-truth classifications, and held-out datasets were reserved after other datasets guided tool selection.The simulated datasets contained 10,000 time-points; some systems were evaluated using linear combinations of their variables.Source: Results · p. 4
  • Permutation entropy tracked degree of chaos in five systems, but its correlation with largest Lyapunov exponents breaks down in continuous systems without downsampling.The tested systems included the logistic and Hénon maps, Lorenz system, a cortical mean-field model, and an electronic circuit.Source: Results · p. 4
  • Biological simulations were classified as periodic or chaotic with near-perfect accuracy, while non-biological simulations spanning stochastic, periodic, and chaotic classes were classified with high accuracy.The non-biological simulations included strange non-chaotic attractors and linear and nonlinear stochastic processes.Source: Table 1 · p. 4, Table 2 · p. 5
  • Held-out simulated and empirical systems were classified with near-perfect accuracy at high measurement noise, except for the noise-driven sine map.Performance remained high on subsamples of 1,000 and 5,000 points, although some systems did worse with less data.Source: Results · p. 4, Table 3 · p. 5
  • Heart-rate signals from healthy subjects and atrial fibrillation patients, and all but two congestive heart failure patients, were classified as stochastic rather than chaotic.This finding agrees with the view that heart-rate variability reflects a nonlinear stochastic process.Source: Results · p. 5

Discussion

The algorithm detects stochastic, periodic, or chaotic time series and tracks changing chaos levels; it performed with high accuracy on known-ground-truth datasets despite relatively high measurement noise. For heart rate variability, the findings support stochasticity, while short records can obscure very weak chaos.

Discussion

  • The pipeline identifies stochastic, periodic, or chaotic signals and tracks changing chaos levels, using surrogate tests, denoising, downsampling when needed, and a modified 0–1 chaos test.Users can specify alternative subroutines or parameters; otherwise, the automated pipeline uses the choices found most accurate across diverse data.Source: Discussion · p. 5
  • For heart rate variability, the results support the hypothesis that cardiac rhythm variability is stochastic.Source: Discussion · p. 6
  • Short time series can lead the 0–1 test to classify very weakly chaotic systems as periodic; the noise-driven sine map was consistently misclassified as chaotic.The authors note that longer time series are guaranteed to yield accurate results for the very weakly chaotic case.Source: Discussion · p. 6
  • Held-out empirical systems spanning stochastic, periodic, SNA, and chaotic classes had perfect classification accuracy; known-ground-truth datasets overall remained highly accurate under relatively high measurement noise.The heart rate variability datasets were excluded from the known-ground-truth comparisons.Source: Table 4 · p. 6, Discussion · p. 6
  • Some observables represent system dynamics better than others, and the authors recommend using the pipeline alongside computer-model analyses when possible.They state that only noise-free computer models allow absolute certainty about chaos presence, absence, or degree.Source: Discussion · p. 6
  • The modular decision tree can accommodate new subroutines for edge cases, and the authors invite efforts to identify cases where it breaks down.Source: Discussion · p. 6

The Chaos Decision Tree Algorithm.

The Chaos Decision Tree Algorithm checks whether recordings are predominantly stochastic before testing deterministic data for chaos, with denoising and downsampling when needed. Its biological simulations include systems configured for periodic or chaotic behavior.

The Chaos Decision Tree Algorithm.

  • The modified 0–1 test drives a two-dimensional system from the input series and classifies chaos from the median K statistic across 100 values of c.K approaches 1 for chaotic systems and 0 for periodic systems; the pipeline sets σ to 0.5 by default and selects a cutoff if none is supplied.Source: The Chaos Decision Tree Algorithm · pp. 6–7
  • Before applying the 0–1 test, the algorithm screens for stochasticity because stochastic signals can be misclassified as chaotic.It tests permutation entropy against amplitude-adjusted Fourier and cyclic phase permutation surrogate distributions.Source: The Chaos Decision Tree Algorithm · p. 7
  • Schreiber denoising is the default for signals that pass the stochasticity test, because it outperforms moving-average and wavelet alternatives in recovering deterministic components.It also improves the modified 0–1 test, although it runs more slowly.Source: The Chaos Decision Tree Algorithm · p. 7
  • For oversampled data, the pipeline iteratively downsamples by two when η > 10, stopping at η ≤ 10 or fewer than 100 points.Downsampling after denoising was more accurate than no downsampling or the tested local-extrema approach for oversampled signals with measurement noise.Source: The Chaos Decision Tree Algorithm · p. 7
  • After denoising and any needed downsampling, permutation entropy serves as a proxy for degree of chaos and tracks it in the tested systems.The paper reports that Table 5 evaluates this tracking against largest Lyapunov exponents.Source: Table 5 · p. 6, The Chaos Decision Tree Algorithm · p. 7

Biological simulations.

  • Simulated biological signals were tested with white measurement noise up to 40% of the original data’s standard deviation.For each simulated system and noise level, the authors created 100 datasets of 10,000 time points.Source: Biological simulations · p. 8
  • The cortical model’s coupling settings produced periodic or chaotic dynamics, but analysis used only one firing-rate variable from one of its 120 neural populations.The model contains 14 first-order differential equations per population.Source: Biological simulations · p. 8
  • Biological simulations included neuron, white-blood-cell, and NF-κB models configured for periodic or chaotic dynamics.The neuron’s membrane potential displayed chaotic spikes at the stated parameter values; Mackey–Glass and NF-κB model settings provided periodic and chaotic cases.Source: Biological simulations · p. 8

Non-biological simulations.

The non-biological simulation suite covers diverse dynamical systems and difficult classification cases, including hyperchaos, strange non-chaotic dynamics, and stochastic processes that can challenge existing tests.

Non-biological simulations.

  • The suite includes periodic, chaotic, period-doubled, and strange non-chaotic regimes in logistic, Lorenz, cubic, and other map or oscillator systems.The generalized Hénon map is included as a hyperchaotic case, which can be mistaken for stochasticity by determinism tests.Source: Non-biological simulations · pp. 8–9
  • Two Freitas systems are included because noise titration incorrectly classifies them as chaotic despite their non-chaotic, noise-driven or correlated-noise character.The set also includes a bounded random walk with local nonstationarities and a cyclostationary process.Source: Non-biological simulations · p. 9
  • The cyclostationary process tests a known weakness of surrogate tests that use only Fourier-based surrogates, which assume stationarity.Its cyclically varying probability distributions make it non-stationary.Source: Non-biological simulations · p. 9
  • The benchmarks include ARMA(1) processes with c=0 and ϕ=0.99, non-stationary random walks, and a chaotic Rössler system configured with a=0.2, b=0.2, and c=5.7.When ϕ is close to but less than 1 and θ ≠ 0, stationary ARMA processes are often misclassified as non-stationary by stationarity tests.Source: Non-biological simulations · p. 9

Empirical data.

The empirical dataset spans real recordings previously classified as stochastic, chaotic, or strange non-chaotic, alongside heart-rate recordings from healthy subjects and patients.

Empirical data.

  • The neuron integrated-circuit recordings include 10 datasets each from chaotic, strange non-chaotic, and periodic states previously classified by response consistency to external input.The circuit’s state varies with parameter b.Source: Empirical data · p. 10
  • The stellar-flux recording is from KIC 5520878, identified as a strange non-chaotic system, and the analysis uses a detrended interval selected between large pixel-related shifts.The selected interval covers time points 11,620–14,003.Source: Empirical data · p. 10
  • The empirical set includes North Atlantic Oscillation data previously identified as stochastic and Parkinson’s and essential tremor recordings previously characterized as nonlinear stochastic oscillators.The NAO series covers monthly means from January 1950 through December 2018.Source: Empirical data · p. 10
  • Heart-rate recordings cover healthy subjects, congestive heart failure patients, and patients with atrial fibrillation; the analyzed segments span the first 2.78 hours.Healthy and congestive-heart-failure recordings are in sinus rhythm, whereas the atrial-fibrillation recordings are not.Source: Empirical data · p. 10

Parameters and largest Lyapunov exponents for data in Table 5.

To assess tracking of changing chaos levels, the authors generated parameterized signals and compared permutation entropy with largest Lyapunov exponents, using analytic calculations where available and an approximation for the cortical model.

Parameters and largest Lyapunov exponents for data in Table 5.

  • The logistic map, Hénon map, and Lorenz system were varied across parameter ranges to generate signals with different degrees of chaos and calculate largest Lyapunov exponents.The logistic and Hénon analyses generated 501 and 401 signals, respectively; Lorenz signals were generated over a range in which the system remained chaotic.Source: Parameters and largest Lyapunov… · pp. 10–11
  • For the cortical model, 585 configurations varied postsynaptic inhibitory response and inhibitory diffusion, with 25,000 simulated time points per configuration.Because no analytical Lyapunov-exponent tool was available, estimates came from divergence between two noise-free simulations with slightly different initial conditions.Source: Parameters and largest Lyapunov… · p. 11
  • Cortical-model configurations with estimated largest Lyapunov exponents below −5 or above 5 were excluded as likely poor estimates.The estimation procedure automated the choice of fitting interval by locating the abrupt mean change associated with saturation.Source: Parameters and largest Lyapunov… · p. 11
  • For the neuron integrated circuit, ground-truth largest Lyapunov exponents for its periodic, strange non-chaotic, and chaotic states came from the circuit’s known equations.The equations describe a physical implementation of a two-dimensional system.Source: Parameters and largest Lyapunov… · p. 11

Statistics and reproducibility.

The paper reports classification accuracy across datasets, with statistical testing limited to specified correlation and pipeline tests. It also describes dataset reuse and code and data resources for reproducibility.

Statistics and reproducibility

  • Tables 1–2 datasets optimized the algorithm, which was then retested on datasets in Tables 3–4.No data were excluded, and no sample-size calculation was performed.Source: Statistics and reproducibility · p. 11
  • Spearman-correlation p-values in Table 5 and Supplementary Table 14 were Bonferroni-corrected because correlations used the same ground-truth largest Lyapunov exponents.Elsewhere, results report fractions correctly classified, with exact sample sizes in all tables.Source: Statistics and reproducibility · p. 11
  • When stationarity testing is selected, stochasticity testing proceeds only after the signal passes a two-tailed stationarity test at α = 0.05.For determinism, permutation entropy outside the distribution from 1000 surrogates yields classification as predominantly deterministic; this corresponds to a two-tailed test at α = 9.99e−4.Source: Statistics and reproducibility · p. 11
  • The 0–1 test for chaos provides a K-statistic without an associated p-value.No statistical analyses beyond the specified tests and classification accuracy were reported.Source: Statistics and reproducibility · p. 11
  • Code and links for generating simulated datasets are provided, alongside links to most empirical datasets analyzed.The passage notes that links to most empirical datasets are in the Methods.Source: Statistics and reproducibility · p. 11

Reporting summary.

Further research-design information is available in the linked Nature Research Reporting Summary.

Reporting summary

Data availability

The paper provides simulation scripts and makes nearly all analyzed empirical datasets freely available online.

Data availability

  • Matlab scripts for simulating the mean-field cortical model and NF-κB transcription are linked in the Methods; scripts for other simulated systems accompany the algorithm code.The accompanying scripts are available at the stated Figshare DOI.Source: Data availability · p. 11
  • All empirical datasets except the integrated circuit recordings are freely available online, with data-source URLs provided in the Methods.Source: Data availability · p. 11

Code availability

The Chaos Decision Tree Algorithm code is available through Figshare.

Code availability

Additional information

Supplementary information is available online; the paper also provides correspondence, reprint, publisher-neutrality, and open-access licensing information.

Additional information

  • Supplementary information is available at the paper’s DOI.Correspondence and requests for materials should be addressed to D.T.; reprint information is available from Nature.Source: Additional information · p. 13
  • The publisher states that it remains neutral regarding jurisdictional claims in published maps and institutional affiliations.Source: Additional information · p. 13
  • The article is licensed under Creative Commons Attribution 4.0, subject to its attribution and third-party-material terms.The license permits sharing and adaptation with appropriate credit, a license link, and an indication of changes; some third-party material may require separate permission.Source: Additional information · p. 13
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