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Generalized synchronization in the presence of dynamical noise and its detection via recurrent neural networks

José M. Amigó, Roberto Dale, Juan C. King, Klaus Lehnertz

TL;DR

This work extends generalized synchronization theory to drivers perturbed by dynamical noise by modeling noise as finitely parameterized stochasticity and framing it as stochastic forcing. It introduces higher-period cross and synchronization maps and proves their correspondence under noise, using state-space reconstructions via Takens-type embeddings. Detection is achieved with recurrent neural networks (LSTM), validated on two coupled Hénon maps and on real intracranial EEG data to reveal dominant directionality and focal driving. The approach demonstrates robustness of synchronization detection to noise and nonstationarity, and provides a practical toolkit for analyzing coupled nonlinear systems in physics and neuroscience. Overall, the study advances both the theory of noisy synchronization and its application to complex, real-world time series.

Abstract

Given two unidirectionally coupled nonlinear systems, we speak of generalized synchronization when the responder \textquotedblleft follows\textquotedblright\ the driver. Mathematically, this situation is implemented by a map from the driver state space to the responder state space termed the synchronization map. In nonlinear times series analysis, the framework of the present work, the existence of the synchronization map amounts to the invertibility of the so-called cross map, which is a continuous map that exists in the reconstructed state spaces for typical time-delay embeddings. The cross map plays a central role in some techniques to detect functional dependencies between time series. In this paper, we study the changes in the \textquotedblleft noiseless scenario\textquotedblright\ just described when noise is present in the driver, a more realistic situation that we call the \textquotedblleft noisy scenario\textquotedblright . Noise will be modeled using a family of driving dynamics indexed by a finite number of parameters, which is sufficiently general for practical purposes. In this approach, it turns out that the cross and synchronization maps can be extended to the noisy scenario as families of maps that depend on the noise parameters, and only for \textquotedblleft generic\textquotedblright\ driver states in the case of the cross map. To reveal generalized synchronization in both the noiseless and noisy scenarios, we check the existence of synchronization maps of higher periods (introduced in this paper) using recurrent neural networks and predictability. The results obtained with synthetic and real world data demonstrate the capability of our method.

Generalized synchronization in the presence of dynamical noise and its detection via recurrent neural networks

TL;DR

This work extends generalized synchronization theory to drivers perturbed by dynamical noise by modeling noise as finitely parameterized stochasticity and framing it as stochastic forcing. It introduces higher-period cross and synchronization maps and proves their correspondence under noise, using state-space reconstructions via Takens-type embeddings. Detection is achieved with recurrent neural networks (LSTM), validated on two coupled Hénon maps and on real intracranial EEG data to reveal dominant directionality and focal driving. The approach demonstrates robustness of synchronization detection to noise and nonstationarity, and provides a practical toolkit for analyzing coupled nonlinear systems in physics and neuroscience. Overall, the study advances both the theory of noisy synchronization and its application to complex, real-world time series.

Abstract

Given two unidirectionally coupled nonlinear systems, we speak of generalized synchronization when the responder \textquotedblleft follows\textquotedblright\ the driver. Mathematically, this situation is implemented by a map from the driver state space to the responder state space termed the synchronization map. In nonlinear times series analysis, the framework of the present work, the existence of the synchronization map amounts to the invertibility of the so-called cross map, which is a continuous map that exists in the reconstructed state spaces for typical time-delay embeddings. The cross map plays a central role in some techniques to detect functional dependencies between time series. In this paper, we study the changes in the \textquotedblleft noiseless scenario\textquotedblright\ just described when noise is present in the driver, a more realistic situation that we call the \textquotedblleft noisy scenario\textquotedblright . Noise will be modeled using a family of driving dynamics indexed by a finite number of parameters, which is sufficiently general for practical purposes. In this approach, it turns out that the cross and synchronization maps can be extended to the noisy scenario as families of maps that depend on the noise parameters, and only for \textquotedblleft generic\textquotedblright\ driver states in the case of the cross map. To reveal generalized synchronization in both the noiseless and noisy scenarios, we check the existence of synchronization maps of higher periods (introduced in this paper) using recurrent neural networks and predictability. The results obtained with synthetic and real world data demonstrate the capability of our method.

Paper Structure

This paper contains 20 sections, 6 theorems, 80 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The noiseless scenario
  3. Embedding theorems
  4. The cross map
  5. Generalized synchronization
  6. Relationship between generalized synchronization and the cross map
  7. Dynamical noise as stochastic forcing
  8. Coupled dynamics and noise
  9. The noisy scenario
  10. State reconstruction
  11. Cross map
  12. Synchronization map
  13. Numerical simulations
  14. Models
  15. Results
  16. ...and 5 more sections

Key Result

Theorem 2

[Takens Theorem Takens1981] If $d\geq 2\dim _{X}+1$, then the map $E_{f,\varphi _{X}}:\mathcal{M}_{X}\rightarrow \mathbb{R}^{d}$ defined as is an embedding for generic $f$ and $\varphi _{X}$.

Figures (4)

  • Figure 1: Numerical results for the model Hénon 0.3-0.3, i.e., $b_{1}=0.3$ in (\ref{['HenonMapX']}) and $b_{2}=0.3$ in (\ref{['HenonMap2']}). (a) MSE vs the coupling strength $C$ for a noiseless driver (noise amplitude $A=0$) and a noisy driver ($A=0.005$, $0.013$) obtained with an LSTM net. (b) $\Delta r$ vs $C$ for a noiseless driver ($A=0$) and a noisy driver ($A=0.005$, $0.013$) obtained via $6$-nearest-neighbor cross prediction. See text for more detail.
  • Figure 2: Numerical results for the model Hénon 0.3-0.1, i.e., $b_{1}=0.3$ in (\ref{['HenonMapX']}) and $b_{2}=0.1$ in (\ref{['HenonMap2']}). The information displayed in the panels (a) and (b) is the same as in Figure \ref{['figure1']}.
  • Figure 3: Numerical results for the model Hénon 0.1-0.3, i.e., $b_{1}=0.1$ in (\ref{['HenonMapX']}) and $b_{2}=0.3$ in (\ref{['HenonMap2']}). The information displayed in the panels (a) and (b) is the same as in Figure \ref{['figure1']}.
  • Figure 4: Top to bottom: plots of $\Delta \mathrm{MSE}(n)$, the directionality indicator (\ref{['direct index']}), obtained using the segments $S_{X,n}$ and $S_{Y,n}$, $1\leq n\leq 1435$, given in equations (\ref{['S_X,n']})-(\ref{['S_X,1435']}), for Cases 1 to 5. The clinical seizure occurs in the segment $n=1433$, too close to the right margin to be marked. See Section \ref{['sec7.1']} for detail.

Theorems & Definitions (18)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Proposition 6
  • Remark 7
  • Definition 8
  • Remark 9
  • Definition 10
  • ...and 8 more