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Gonzalez CE, Lainscsek C, Sejnowski TJ, Letellier C. Assessing observability of chaotic systems using Delay Differential Analysis. CHAOS (WOODBURY, N.Y.) 2020; 30:103113. [PMID: 33138467 PMCID: PMC7556884 DOI: 10.1063/5.0015533] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2020] [Accepted: 09/24/2020] [Indexed: 06/11/2023]
Abstract
Observability can determine which recorded variables of a given system are optimal for discriminating its different states. Quantifying observability requires knowledge of the equations governing the dynamics. These equations are often unknown when experimental data are considered. Consequently, we propose an approach for numerically assessing observability using Delay Differential Analysis (DDA). Given a time series, DDA uses a delay differential equation for approximating the measured data. The lower the least squares error between the predicted and recorded data, the higher the observability. We thus rank the variables of several chaotic systems according to their corresponding least square error to assess observability. The performance of our approach is evaluated by comparison with the ranking provided by the symbolic observability coefficients as well as with two other data-based approaches using reservoir computing and singular value decomposition of the reconstructed space. We investigate the robustness of our approach against noise contamination.
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Affiliation(s)
| | | | | | - Christophe Letellier
- CORIA, Rouen Normandie Université, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France
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Cohen MX. Using spatiotemporal source separation to identify prominent features in multichannel data without sinusoidal filters. Eur J Neurosci 2017; 48:2454-2465. [DOI: 10.1111/ejn.13727] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2017] [Revised: 09/06/2017] [Accepted: 09/18/2017] [Indexed: 01/14/2023]
Affiliation(s)
- Michael X Cohen
- Donders Center for Neuroscience; Radboud University and Radboud University Medical Center; Kapittelweg 29 6525 EN Nijmegen The Netherlands
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3
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Matsuda T, Komaki F. Multivariate Time Series Decomposition into Oscillation Components. Neural Comput 2017; 29:2055-2075. [PMID: 28562213 DOI: 10.1162/neco_a_00981] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
Many time series are considered to be a superposition of several oscillation components. We have proposed a method for decomposing univariate time series into oscillation components and estimating their phases (Matsuda & Komaki, 2017 ). In this study, we extend that method to multivariate time series. We assume that several oscillators underlie the given multivariate time series and that each variable corresponds to a superposition of the projections of the oscillators. Thus, the oscillators superpose on each variable with amplitude and phase modulation. Based on this idea, we develop gaussian linear state-space models and use them to decompose the given multivariate time series. The model parameters are estimated from data using the empirical Bayes method, and the number of oscillators is determined using the Akaike information criterion. Therefore, the proposed method extracts underlying oscillators in a data-driven manner and enables investigation of phase dynamics in a given multivariate time series. Numerical results show the effectiveness of the proposed method. From monthly mean north-south sunspot number data, the proposed method reveals an interesting phase relationship.
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Affiliation(s)
- Takeru Matsuda
- Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan
| | - Fumiyasu Komaki
- Department of Mathematical Informatics, Graduate School of Information Science and Technology, Tokyo 113-5656, Japan, and RIKEN Brain Science Institute, Wako 351-0198, Japan
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Das A, Sampson AL, Lainscsek C, Muller L, Lin W, Doyle JC, Cash SS, Halgren E, Sejnowski TJ. Interpretation of the Precision Matrix and Its Application in Estimating Sparse Brain Connectivity during Sleep Spindles from Human Electrocorticography Recordings. Neural Comput 2017; 29:603-642. [PMID: 28095202 PMCID: PMC5424817 DOI: 10.1162/neco_a_00936] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
Abstract
The correlation method from brain imaging has been used to estimate functional connectivity in the human brain. However, brain regions might show very high correlation even when the two regions are not directly connected due to the strong interaction of the two regions with common input from a third region. One previously proposed solution to this problem is to use a sparse regularized inverse covariance matrix or precision matrix (SRPM) assuming that the connectivity structure is sparse. This method yields partial correlations to measure strong direct interactions between pairs of regions while simultaneously removing the influence of the rest of the regions, thus identifying regions that are conditionally independent. To test our methods, we first demonstrated conditions under which the SRPM method could indeed find the true physical connection between a pair of nodes for a spring-mass example and an RC circuit example. The recovery of the connectivity structure using the SRPM method can be explained by energy models using the Boltzmann distribution. We then demonstrated the application of the SRPM method for estimating brain connectivity during stage 2 sleep spindles from human electrocorticography (ECoG) recordings using an [Formula: see text] electrode array. The ECoG recordings that we analyzed were from a 32-year-old male patient with long-standing pharmaco-resistant left temporal lobe complex partial epilepsy. Sleep spindles were automatically detected using delay differential analysis and then analyzed with SRPM and the Louvain method for community detection. We found spatially localized brain networks within and between neighboring cortical areas during spindles, in contrast to the case when sleep spindles were not present.
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Affiliation(s)
- Anup Das
- Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093, U.S.A., and Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - Aaron L Sampson
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - Claudia Lainscsek
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - Lyle Muller
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - Wutu Lin
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - John C Doyle
- Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, U.S.A.
| | - Sydney S Cash
- Cortical Physiology Laboratory, Massachusetts General Hospital, Boston, MA 02114, U.S.A.
| | - Eric Halgren
- Department of Radiology, University of California, San Diego, La Jolla, CA 92093, U.S.A.
| | - Terrence J Sejnowski
- Division of Biological Sciences and Institute of Neural Computation, University of California, San Diego, La Jolla, CA 92093, U.S.A., and Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
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Das A, Sampson AL, Lainscsek C, Muller L, Lin W, Doyle JC, Cash SS, Halgren E, Sejnowski TJ. Interpretation of the Precision Matrix and Its Application in Estimating Sparse Brain Connectivity during Sleep Spindles from Human Electrocorticography Recordings. Neural Comput 2017; 29:603-642. [PMID: 28095202 DOI: 10.1162/neco{\_}a{\_}00936] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The correlation method from brain imaging has been used to estimate functional connectivity in the human brain. However, brain regions might show very high correlation even when the two regions are not directly connected due to the strong interaction of the two regions with common input from a third region. One previously proposed solution to this problem is to use a sparse regularized inverse covariance matrix or precision matrix (SRPM) assuming that the connectivity structure is sparse. This method yields partial correlations to measure strong direct interactions between pairs of regions while simultaneously removing the influence of the rest of the regions, thus identifying regions that are conditionally independent. To test our methods, we first demonstrated conditions under which the SRPM method could indeed find the true physical connection between a pair of nodes for a spring-mass example and an RC circuit example. The recovery of the connectivity structure using the SRPM method can be explained by energy models using the Boltzmann distribution. We then demonstrated the application of the SRPM method for estimating brain connectivity during stage 2 sleep spindles from human electrocorticography (ECoG) recordings using an [Formula: see text] electrode array. The ECoG recordings that we analyzed were from a 32-year-old male patient with long-standing pharmaco-resistant left temporal lobe complex partial epilepsy. Sleep spindles were automatically detected using delay differential analysis and then analyzed with SRPM and the Louvain method for community detection. We found spatially localized brain networks within and between neighboring cortical areas during spindles, in contrast to the case when sleep spindles were not present.
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Affiliation(s)
- Anup Das
- Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093, U.S.A., and Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - Aaron L Sampson
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - Claudia Lainscsek
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - Lyle Muller
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - Wutu Lin
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
| | - John C Doyle
- Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, U.S.A.
| | - Sydney S Cash
- Cortical Physiology Laboratory, Massachusetts General Hospital, Boston, MA 02114, U.S.A.
| | - Eric Halgren
- Department of Radiology, University of California, San Diego, La Jolla, CA 92093, U.S.A.
| | - Terrence J Sejnowski
- Division of Biological Sciences and Institute of Neural Computation, University of California, San Diego, La Jolla, CA 92093, U.S.A., and Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A.
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Kim K, Lim SH, Lee J, Kang WS, Moon C, Choi JW. Joint Maximum Likelihood Time Delay Estimation of Unknown Event-Related Potential Signals for EEG Sensor Signal Quality Enhancement. SENSORS 2016; 16:s16060891. [PMID: 27322267 PMCID: PMC4934317 DOI: 10.3390/s16060891] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/01/2016] [Revised: 06/03/2016] [Accepted: 06/09/2016] [Indexed: 12/02/2022]
Abstract
Electroencephalograms (EEGs) measure a brain signal that contains abundant information about the human brain function and health. For this reason, recent clinical brain research and brain computer interface (BCI) studies use EEG signals in many applications. Due to the significant noise in EEG traces, signal processing to enhance the signal to noise power ratio (SNR) is necessary for EEG analysis, especially for non-invasive EEG. A typical method to improve the SNR is averaging many trials of event related potential (ERP) signal that represents a brain’s response to a particular stimulus or a task. The averaging, however, is very sensitive to variable delays. In this study, we propose two time delay estimation (TDE) schemes based on a joint maximum likelihood (ML) criterion to compensate the uncertain delays which may be different in each trial. We evaluate the performance for different types of signals such as random, deterministic, and real EEG signals. The results show that the proposed schemes provide better performance than other conventional schemes employing averaged signal as a reference, e.g., up to 4 dB gain at the expected delay error of 10°.
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Affiliation(s)
- Kyungsoo Kim
- Department of Information & Communication Engineering, Daegu Gyeongbuk Institute of Science and Technology, Daegu 771-813, Korea.
| | - Sung-Ho Lim
- Department of Information & Communication Engineering, Daegu Gyeongbuk Institute of Science and Technology, Daegu 771-813, Korea.
| | - Jaeseok Lee
- Department of Information & Communication Engineering, Daegu Gyeongbuk Institute of Science and Technology, Daegu 771-813, Korea.
| | - Won-Seok Kang
- Wellness Convergence Research Center, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 771-813, Korea.
| | - Cheil Moon
- Department of Brain & Cognitive Sciences, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 771-813, Korea.
| | - Ji-Woong Choi
- Department of Information & Communication Engineering, Daegu Gyeongbuk Institute of Science and Technology, Daegu 771-813, Korea.
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Abstract
Nonlinear dynamical system analysis based on embedding theory has been used for modeling and prediction, but it also has applications to signal detection and classification of time series. An embedding creates a multidimensional geometrical object from a single time series. Traditionally either delay or derivative embeddings have been used. The delay embedding is composed of delayed versions of the signal, and the derivative embedding is composed of successive derivatives of the signal. The delay embedding has been extended to nonuniform embeddings to take multiple timescales into account. Both embeddings provide information on the underlying dynamical system without having direct access to all the system variables. Delay differential analysis is based on functional embeddings, a combination of the derivative embedding with nonuniform delay embeddings. Small delay differential equation (DDE) models that best represent relevant dynamic features of time series data are selected from a pool of candidate models for detection or classification. We show that the properties of DDEs support spectral analysis in the time domain where nonlinear correlation functions are used to detect frequencies, frequency and phase couplings, and bispectra. These can be efficiently computed with short time windows and are robust to noise. For frequency analysis, this framework is a multivariate extension of discrete Fourier transform (DFT), and for higher-order spectra, it is a linear and multivariate alternative to multidimensional fast Fourier transform of multidimensional correlations. This method can be applied to short or sparse time series and can be extended to cross-trial and cross-channel spectra if multiple short data segments of the same experiment are available. Together, this time-domain toolbox provides higher temporal resolution, increased frequency and phase coupling information, and it allows an easy and straightforward implementation of higher-order spectra across time compared with frequency-based methods such as the DFT and cross-spectral analysis.
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Affiliation(s)
- Claudia Lainscsek
- Howard Hughes Medical Institute, Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A. and Institute for Neural Computation, University of California San Diego, La Jolla, CA 92093, U.S.A.
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