Interpretation of the Precision Matrix and Its Application in Estimating Sparse Brain Connectivity during Sleep Spindles from Human Electrocorticography Recordings

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 8 x 8 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

Interpretation of the Precision Matrix and Its Application in Estimating Sparse Brain Connectivity during Sleep Spindles from Human Electrocorticography Recordings

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 8 x 8 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

Objective assessment of bradykinesia in Parkinson’s disease using evolutionary algorithms: clinical validation

Background: There is an urgent need for developing objective, effective and convenient measurements to help clinicians accurately identify bradykinesia. The purpose of this study is to evaluate the accuracy of an objective approach assessing bradykinesia in finger tapping (FT) that uses evolutionary algorithms (EAs) and explore whether it can be used to identify early stage Parkinson’s disease (PD). Methods: One hundred and seven PD, 41 essential tremor (ET) patients and 49 normal controls (NC) were recruited. Participants performed a standard FT task with two electromagnetic tracking sensors attached to the thumb and index finger. Readings from the sensors were transmitted to a tablet computer and subsequently analyzed by using EAs. The output from the device (referred to as "PD-Monitor") scaled from − 1 to +1 (where higher scores indicate greater severity of bradykinesia). Meanwhile, the bradykinesia was rated clinically using the Movement Disorder Society- Sponsored Revision of the Unified Parkinson’s Disease Rating Scale (MDS-UPDRS) FT item. Results: With an increasing MDS-UPDRS FT score, the PD-Monitor score from the same hand side increased correspondingly. PD-Monitor score correlated well with MDS-UPDRS FT score (right side: r = 0.819, P = 0.000; left side: r = 0.783, P = 0.000). Moreover, PD-Monitor scores in 97 PD patients with MDS-UPDRS FT bradykinesia and each PD subgroup (FT bradykinesia scored from 1 to 3) were all higher than that in NC. Receiver operating characteristic (ROC) curves revealed that PD-Monitor FT scores could detect different severity of bradykinesia with high accuracy (≥89.7%) in the right dominant hand. Furthermore, PD-Monitor scores could discriminate early stage PD from NC, with area under the ROC curve greater than or equal to 0.899. Additionally, ET without bradykinesia could be differentiated from PD by PD-Monitor scores. A positive correlation of PD-Monitor scores with modified Hoehn and Yahr stage was found in the left hand sides. Conclusions: Our study demonstrated that a simple to use device employing classifiers derived from EAs could not only be used to accurately measure different severity of bradykinesia in PD, but also had the potential to differentiate early stage PD from normality

Delay Differential Analysis of Time Series

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

Equivariance identification using delay differential equations

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