Range Entropy: A Bridge between Signal Complexity and Self-Similarity

Diverging functional connectivity timescales: Capturing distinct aspects of cognitive performance in early psychosis

BACKGROUND

Psychosis spectrum disorders (PSDs) are marked by cognitive impairments, the neurobiological correlates of which remain poorly understood. Here, we investigate the entropy of time-varying functional connectivity (TVFC) patterns from resting-state functional magnetic resonance imaging (rs-fMRI) as potential biomarker for cognitive performance in PSDs. By combining our results with multimodal reference data, we hope to generate new insights into the mechanisms underlying cognitive dysfunction in PSDs. We hypothesized that low-entropy TVFC patterns (LEN) would be more behaviorally informative than high-entropy TVFC patterns (HEN), especially for tasks that require extensive integration across diverse cognitive subdomains.

METHODS

rs-fMRI and behavioral data from 97 patients in the early phases of psychosis and 53 controls were analyzed. Positron emission tomography (PET) and magnetoencephalography (MEG) data were taken from a public repository (Hansen et al., 2022). Multivariate analyses were conducted to examine relationships between TVFC patterns at multiple spatial scales and cognitive performance in patients.

RESULTS

Compared to HEN, LEN explained significantly more cognitive variance on average in PSD patients, driven by superior encoding of information on psychometrically more integrated tasks. HEN better captured information in specific subdomains of executive functioning. Nodal HEN-LEN transitions were spatially aligned with neurobiological gradients reflecting monoaminergic transporter densities and MEG beta-power. Exploratory analyses revealed a close statistical relationship between LEN and positive symptom severity in patients.

CONCLUSION

Our entropy-based analysis of TVFC patterns dissociates distinct aspects of cognition in PSDs. By linking topographies of neurotransmission and oscillatory dynamics with cognitive performance, it enhances our understanding of the mechanisms underlying cognitive deficits in PSDs.

The brain's "dark energy" puzzle: How strongly is glucose metabolism linked to resting-state brain activity?

Brain glucose metabolism, which can be investigated at the macroscale level with [18F]FDG PET, displays significant regional variability for reasons that remain unclear. Some of the functional drivers behind this heterogeneity may be captured by resting-state functional magnetic resonance imaging (rs-fMRI). However, the full extent to which an fMRI-based description of the brain's spontaneous activity can describe local metabolism is unknown. Here, using two multimodal datasets of healthy participants, we built a multivariable multilevel model of functional-metabolic associations, assessing multiple functional features, describing the 1) rs-fMRI signal, 2) hemodynamic response, 3) static and 4) time-varying functional connectivity, as predictors of the human brain's metabolic architecture. The full model was trained on one dataset and tested on the other to assess its reproducibility. We found that functional-metabolic spatial coupling is nonlinear and heterogeneous across the brain, and that local measures of rs-fMRI activity and synchrony are more tightly coupled to local metabolism. In the testing dataset, the degree of functional-metabolic spatial coupling was also related to peripheral metabolism. Overall, although a significant proportion of regional metabolic variability can be described by measures of spontaneous activity, additional efforts are needed to explain the remaining variance in the brain's 'dark energy'.

Individual characteristics outperform resting-state fMRI for the prediction of behavioral phenotypes

In this study, we aimed to compare imaging-based features of brain function, measured by resting-state fMRI (rsfMRI), with individual characteristics such as age, gender, and total intracranial volume to predict behavioral measures. We developed a machine learning framework based on rsfMRI features in a dataset of 20,000 healthy individuals from the UK Biobank, focusing on temporal complexity and functional connectivity measures. Our analysis across four behavioral phenotypes revealed that both temporal complexity and functional connectivity measures provide comparable predictive performance. However, individual characteristics consistently outperformed rsfMRI features in predictive accuracy, particularly in analyses involving smaller sample sizes. Integrating rsfMRI features with demographic data sometimes enhanced predictive outcomes. The efficacy of different predictive modeling techniques and the choice of brain parcellation atlas were also examined, showing no significant influence on the results. To summarize, while individual characteristics are superior to rsfMRI in predicting behavioral phenotypes, rsfMRI still conveys additional predictive value in the context of machine learning, such as investigating the role of specific brain regions in behavioral phenotypes.

Diverging functional connectivity timescales: Capturing distinct aspects of cognitive performance in early psychosis

Background

Psychosis spectrum disorders (PSDs) are marked by cognitive impairments, the neurobiological correlates of which remain poorly understood. Here, we investigate the entropy of time-varying functional connectivity (TVFC) patterns from resting-state fMRI (rfMRI) as potential biomarker for cognitive performance in PSDs. By combining our results with multimodal reference data, we hope to generate new insights into the mechanisms underlying cognitive dysfunction in PSDs. We hypothesized that low-entropy TVFC patterns (LEN) would be more behaviorally informative than high-entropy TVFC patterns (HEN), especially for tasks that require extensive integration across diverse cognitive subdomains.

Methods

rfMRI and behavioral data from 97 patients in the early phases of psychosis and 53 controls were analyzed. Positron-Emission Tomography (PET) and magnetoencephalography (MEG) data were taken from a public repository (Hansen et al., 2022). Multivariate analyses were conducted to examine relationships between TVFC patterns at multiple spatial scales and cognitive performance in patients.

Results

Compared to HEN, LEN explained significantly more cognitive variance on average in PSD patients, driven by superior encoding of information on psychometrically more integrated tasks. HEN better captured information in specific subdomains of executive functioning. Nodal HEN-LEN transitions were spatially aligned with neurobiological gradients reflecting monoaminergic transporter densities and MEG beta power. Exploratory analyses revealed a close statistical relationship between LEN and positive PSD symptoms.

Conclusion

Our entropy-based analysis of TVFC patterns dissociates distinct aspects of cognition in PSDs. By linking topographies of neurotransmission and oscillatory dynamics with cognitive performance, it enhances our understanding of the mechanisms underlying cognitive deficits in PSDs.

CRediT Authorship Contribution Statement

Fabian Hirsch: Conceptualization, Methodology, Software, Formal analysis, Writing - Original Draft, Writing - Review & Editing, Visualization; Ângelo Bumanglag: Methodology, Software, Formal analysis, Writing - Review & Editing; Yifei Zhang: Methodology, Software, Formal analysis, Writing - Review & Editing; Afra Wohlschlaeger: Methodology, Writing - Review & Editing, Supervision, Project administration.

Benefits of Zero-Phase or Linear Phase Filters to Design Multiscale Entropy: Theory and Application

In various applications, multiscale entropy (MSE) is often used as a feature to characterize the complexity of the signals in order to classify them. It consists of estimating the sample entropies (SEs) of the signal under study and its coarse-grained (CG) versions, where the CG process amounts to (1) filtering the signal with an average filter whose order is the scale and (2) decimating the filter output by a factor equal to the scale. In this paper, we propose to derive a new variant of the MSE. Its novelty stands in the way to get the sequences at different scales by avoiding distortions during the decimation step. To this end, a linear-phase or null-phase low-pass filter whose cutoff frequency is well suited to the scale is used. Interpretations on how the MSE behaves and illustrations with a sum of sinusoids, as well as white and pink noises, are given. Then, an application to detect attentional tunneling is presented. It shows the benefit of the new approach in terms of p value when one aims at differentiating the set of MSEs obtained in the attentional tunneling state from the set of MSEs obtained in the nominal state. It should be noted that CG versions can be replaced not only for the MSE but also for other variants.

Predicting regional carbon price in China based on multi-factor HKELM by combining secondary decomposition and ensemble learning

Accurately predicting carbon price is crucial for risk avoidance in the carbon financial market. In light of the complex characteristics of the regional carbon price in China, this paper proposes a model to forecast carbon price based on the multi-factor hybrid kernel-based extreme learning machine (HKELM) by combining secondary decomposition and ensemble learning. Variational mode decomposition (VMD) is first used to decompose the carbon price into several modes, and range entropy is then used to reconstruct these modes. The multi-factor HKELM optimized by the sparrow search algorithm is used to forecast the reconstructed subsequences, where the main external factors innovatively selected by maximum information coefficient and historical time-series data on carbon prices are both considered as input variables to the forecasting model. Following this, the improved complete ensemble-based empirical mode decomposition with adaptive noise and range entropy are respectively used to decompose and reconstruct the residual term generated by VMD. Finally, the nonlinear ensemble learning method is introduced to determine the predictions of residual term and final carbon price. In the empirical analysis of Guangzhou market, the root mean square error(RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE) of the model are 0.1716, 0.1218 and 0.0026, respectively. The proposed model outperforms other comparative models in predicting accuracy. The work here extends the research on forecasting theory and methods of predicting the carbon price.

Forecasting of Beijing PM2.5 with a hybrid ARIMA model based on integrated AIC and improved GS fixed-order methods and seasonal decomposition

Accurate particulate matter 2.5 (PM_2.5) prediction plays a crucial role in the accurate management of air pollution and prevention of respiratory diseases. However, PM_2.5, as a nonlinear time series with great volatility, is difficult to achieve accurate prediction. In this paper, a hybrid autoregressive integrated moving average (ARIMA) model is proposed based on the Augmented Dickey-Fuller test (ADF root test) of annual PM_2.5 data, thus demonstrating the necessity of first-order difference. The new method of using integrated akaike information criterion (AIC) and improved grid search (GS) methods is proposed to avoid the bias caused by using AIC alone to determine the order because the data are not exactly normally distributed. The comprehensive evaluation coefficient (CEC) is used to select the optimal parameter structure of the prediction model by considering multiple evaluation perspectives. The entropy value of the decomposed series is obtained by using range entropy A (RangeEn_A), and the series is reconstructed according to the entropy value, and finally the reconstructed series is predicted. We used Beijing PM_2.5 data for validation and the results showed that the new hybrid ARIMA model improved values of RMSE 99.23%, MAE 99.20%, R² 118.61%, TIC 99.28%, NMAE 98.71%, NMSE 99.97%, OPC 43.13%, MOPC 98.43% and CEC 99.25% compared with the traditional ARIMA model. The results show that the method does greatly improve the prediction performance and provides a convincing tool for policy formulation and governance.

The complexity of spontaneous brain activity changes in schizophrenia, bipolar disorder, and ADHD was examined using different variations of entropy

Adult attention deficit/hyperactivity disorder (ADHD), schizophrenia (SCHZ), and bipolar disorder (BP) have common symptoms and differences, and the underlying neural mechanisms are still unclear. This article will thoroughly discuss the differences between ADHD, BP, and SCHZ (31 healthy control and 31 ADHD; 34 healthy control and 34 BP; 42 healthy control and 42 SCHZ) relative to healthy subjects in combination with three atlases (et al., the Brainnetome atlas, the Dosenbach atlas, the Power atlas) and seven entropies (et al., approximate entropy (ApEn), sample entropy (SaEn), permutation entropy (PeEn), fuzzy entropy (FuEn), differential entropy (DiffEn), range entropy (RaEn), and dispersion entropy (DispEn)), as well as the prominent significant brain regions, in the hope of giving information that is more suitable for analyzing different diseases' entropy. First, the reliability (et al., intraclass correlation coefficient [ICC]) of seven kinds of entropy is calculated and analyzed by using the MSC dataset (10 subjects and 100 sessions in total) and simulation data; then, seven types of entropy and multiscale entropy expanded based on seven kinds of entropy are used to explore the differences and brain regions of ADHD, BP, and SCHZ relative to healthy subjects; and finally, by verifying the classification performance of the seven information entropies on ADHD, BP, and SCHZ, the effectiveness of the seven entropy methods is evaluated through these three methods. The core brain regions that affect the classification are given, and DiffEn performed best on ADHD, SaEn for BP, and RaEn for SCHZ.

Temporal complexity of EEG encodes human alertness

OBJECTIVE

Automatic human alertness monitoring has recently become an important research topic with important applications in many areas such as the detection of drivers' fatigue, monitoring of monotonous tasks that require a high level of alertness such as traffic control and nuclear power plant monitoring, and sleep staging. In this study, we propose that balanced dynamics of Electroencephalography (EEG) (so called EEG temporal complexity) is a potentially useful feature for identifying human alertness states. Recently, a new signal entropy measure, called Range Entropy (RangeEn), was proposed to overcome some limitations of two of the most widely used entropy measures, namely Approximate Entropy (ApEn) and Sample Entropy (SampEn), and showed its relevance for the study of time domain EEG complexity. In this paper, we investigated whether the RangeEn holds discriminating information associated with human alertness states, namely Awake, Drowsy, and Sleep and compare its performance against those of SampEn and ApEn.

APPROACH

We used EEG data from 60 healthy subjects of both sexes and different ages acquired during whole night sleeps. Using a 30-second sliding window, we computed the three entropy measures of EEG and performed statistical analyses to evaluate the ability of these entropy measures to discriminate among the different human alertness states.

MAIN RESULTS

Although the three entropy measures contained useful information about human alertness, RangeEn showed a higher discriminative capability compared to ApEn and SampEn especially when using EEG within the Beta frequency band.

SIGNIFICANCE

Our findings highlight the EEG temporal complexity evolution through the human alertness states. This relationship can potentially be exploited for the development of automatic human alertness monitoring systems and diagnostic tools for different neurological and sleep disorders, including insomnia.

On the Spatial Distribution of Temporal Complexity in Resting State and Task Functional MRI

Measuring the temporal complexity of functional MRI (fMRI) time series is one approach to assess how brain activity changes over time. In fact, hemodynamic response of the brain is known to exhibit critical behaviour at the edge between order and disorder. In this study, we aimed to revisit the spatial distribution of temporal complexity in resting state and task fMRI of 100 unrelated subjects from the Human Connectome Project (HCP). First, we compared two common choices of complexity measures, i.e., Hurst exponent and multiscale entropy, and observed a high spatial similarity between them. Second, we considered four tasks in the HCP dataset (Language, Motor, Social, and Working Memory) and found high task-specific complexity, even when the task design was regressed out. For the significance thresholding of brain complexity maps, we used a statistical framework based on graph signal processing that incorporates the structural connectome to develop the null distributions of fMRI complexity. The results suggest that the frontoparietal, dorsal attention, visual, and default mode networks represent stronger complex behaviour than the rest of the brain, irrespective of the task engagement. In sum, the findings support the hypothesis of fMRI temporal complexity as a marker of cognition.

Forecasting Regional Carbon Prices in China Based on Secondary Decomposition and a Hybrid Kernel-Based Extreme Learning Machine

Accurately forecasting carbon prices is key to managing associated risks in the financial market for carbon. To this end, the traditional strategy does not adequately decompose carbon prices, and the kernel extreme learning machine (KELM) with a single kernel function struggles to adapt to the nonlinearity, nonstationarity, and multiple frequencies of regional carbon prices in China. This study constructs a model, called the VMD-ICEEMDAN-RE-SSA-HKELM model, to forecast regional carbon prices in China based on the idea of ‘decomposition–reconstruction–integration’. The VMD is first used to decompose carbon prices and the ICEEMDAN is then used to decompose the residual term that contains complex information. To reduce the systematic error caused by increases in the mode components of carbon price, range entropy (RE) is used to reconstruct the results of its secondary decomposition. Following this, HKELM is optimized by the sparrow search algorithm and used to forecast each subseries of carbon prices. Finally, predictions of the price of carbon are obtained by linearly superimposing the results of the forecasts of each of its subseries. The results of experiments show that the secondary decomposition strategy proposed in this paper is superior to the traditional decomposition strategy, and the proposed model for forecasting carbon prices has significant advantages over a considered reference group of models.

Sample Entropy as a Tool to Assess Lumbo-Pelvic Movements in a Clinical Test for Low-Back-Pain Patients

Low back pain (LBP) obviously reduces the quality of life but is also the world’s leading cause of years lived with disability. Alterations in motor response and changes in movement patterns are expected in LBP patients when compared to healthy people. Such changes in dynamics may be assessed by the nonlinear analysis of kinematical time series recorded from one patient’s motion. Since sample entropy (SampEn) has emerged as a relevant index measuring the complexity of a given time series, we propose the development of a clinical test based on SampEn of a time series recorded by a wearable inertial measurement unit for repeated bending and returns (b and r) of the trunk. Twenty-three healthy participants were asked to perform, in random order, 50 repetitions of this movement by touching a stool and another 50 repetitions by touching a box on the floor. The angular amplitude of the b and r movement and the sample entropy of the three components of the angular velocity and acceleration were computed. We showed that the repetitive b and r “touch the stool” test could indeed be the basis of a clinical test for the evaluation of low-back-pain patients, with an optimal duration of 70 s, acceptable in daily clinical practice.

Combining Measures of Signal Complexity and Machine Learning for Time Series Analyis: A Review

Measures of signal complexity, such as the Hurst exponent, the fractal dimension, and the Spectrum of Lyapunov exponents, are used in time series analysis to give estimates on persistency, anti-persistency, fluctuations and predictability of the data under study. They have proven beneficial when doing time series prediction using machine and deep learning and tell what features may be relevant for predicting time-series and establishing complexity features. Further, the performance of machine learning approaches can be improved, taking into account the complexity of the data under study, e.g., adapting the employed algorithm to the inherent long-term memory of the data. In this article, we provide a review of complexity and entropy measures in combination with machine learning approaches. We give a comprehensive review of relevant publications, suggesting the use of fractal or complexity-measure concepts to improve existing machine or deep learning approaches. Additionally, we evaluate applications of these concepts and examine if they can be helpful in predicting and analyzing time series using machine and deep learning. Finally, we give a list of a total of six ways to combine machine learning and measures of signal complexity as found in the literature.

Determination of Parameters for an Entropy-Based Atrial Fibrillation Detector

Entropy algorithm is an important nonlinear method for cardiovascular disease detection due to its power in analyzing short-term time series. In previous a study, we proposed a new entropy-based atrial fibrillation (AF) detector, i.e., Entropy_AF, which showed a high classification accuracy in identifying AF and non-AF rhythms. As a variation of entropy measures, Entropy_AF has two parameters that need to be initialized before the calculation: (1) tolerance threshold r and (2) similarity weight n. In this study, a comprehensive analysis for the two parameters determination was presented, aiming to achieve a high detection accuracy for AF events. Data were from the MIT-BIH AF database. RR interval recordings were segmented using a 30-beat time window. The parameters r and n were initialized from a relatively small value, then gradually increased, and finally the best parameter combination was determined using grid searching. AUC (area under curve) values from the receiver operator characteristic curve (ROC) were compared under different parameter combinations of parameters r and n, and the results demonstrated that the selection of these two parameters plays an important role in AF/non-AF classification. Small values of parameters r and n can lead to a better detection accuracy than other selections. The best AUC value for AF detection was 98.15%, and the corresponding parameter combinations for Entropy_AF were as follows: r = 0.01, n = 0.0625, 0.125, 0.25, or 0.5; r = 0.05 and n = 0.0625, 0.125, or 0.25; and r = 0.10 and n = 0.0625 or 0.125.

Temporal complexity of fMRI is reproducible and correlates with higher order cognition

It has been hypothesized that resting state networks (RSNs), extracted from resting state functional magnetic resonance imaging (rsfMRI), likely display unique temporal complexity fingerprints, quantified by their multiscale entropy patterns (McDonough and Nashiro, 2014). This is a hypothesis with a potential capacity for developing digital biomarkers of normal brain function, as well as pathological brain dysfunction. Nevertheless, a limitation of McDonough and Nashiro (2014) was that rsfMRI data from only 20 healthy individuals was used for the analysis. To validate this hypothesis in a larger cohort, we used rsfMRI datasets of 987 healthy young adults from the Human Connectome Project (HCP), aged 22-35, each with four 14.4-min rsfMRI recordings and parcellated into 379 brain regions. We quantified multiscale entropy of rsfMRI time series averaged at different cortical and sub-cortical regions. We performed effect-size analysis on the data in 8 RSNs. Given that the morphology of multiscale entropy is affected by the choice of its tolerance parameter (r) and embedding dimension (m), we repeated the analyses at multiple values of r and m including the values used in McDonough and Nashiro (2014). Our results reinforced high temporal complexity in the default mode and frontoparietal networks. Lowest temporal complexity was observed in the subcortical areas and limbic system. We investigated the effect of temporal resolution (determined by the repetition time T_R) after downsampling of rsfMRI time series at two rates. At a low temporal resolution, we observed increased entropy and variance across datasets. Test-retest analysis showed that findings were likely reproducible across individuals over four rsfMRI runs, especially when the tolerance parameter r is equal to 0.5. The results confirmed that the relationship between functional brain connectivity strengths and rsfMRI temporal complexity changes over time scales. Finally, a non-random correlation was observed between temporal complexity of RSNs and fluid intelligence suggesting that complex dynamics of the human brain is an important attribute of high-level brain function.

The thermodynamic brain and the evolution of intellect: the role of mental energy

The living state is low entropy, highly complex organization, yet it is part of the energy cycle of the environment. Due to the recurring presence of the resting state, stimulus and its response form a thermodynamic cycle of perception that can be modeled by the Carnot engine. The endothermic reversed Carnot engine relies on energy from the environment to increase entropy (i.e., the synaptic complexity of the resting state). High entropy relies on mental energy, which represents intrinsic motivation and focuses on the future. It increases freedom of action. The Carnot engine can model exothermic, negative emotional states, which direct the focus on the past. The organism dumps entropy and energy to its environment, in the form of aggravation, anxiety, criticism, and physical violence. The loss of mental energy curtails freedom of action, forming apathy, depression, mental diseases, and immune problems. Our improving intuition about the brain's intelligent computations will allow the development of new treatments for mental disease and novel find applications in robotics and artificial intelligence.

Localization of the epileptogenic zone based on ictal stereo-electroencephalogram: Brain network and single-channel signal feature analysis

Accurate localization of the epileptogenic zone (EZ) is crucial for refractory focal epilepsy patients to achieve freedom from seizures following epilepsy surgery. In this study, ictal stereo-electroencephalography data from 35 patients with refractory focal epilepsy were analyzed. Effective networks based on partial directed coherence were analyzed, and a gray level co-occurrence matrix was applied to extract the time-varying features of the in-degree. These features, combined with the single-channel signal time-frequency features, including approximate entropy and line length, were used to localize the EZ based on a cluster algorithm. For all seizure-free patients (n = 23), the proposed method was effective in identifying the clinical-EZ-contacts and clinical-EZ-blocks, with an F1-score of 62.47 % and 72.18 %, respectively. The sensitivity was 96.00 % for the clinical-EZ-block identification, which provided the information for the decision-making of clinicians, prompting clinicians to focus on the identified EZ-blocks and their nearby contacts. The agreement between the EZ identified by the proposed method and the clinical-EZ was worse for non-seizure-free patients (n = 12) than for seizure-free patients. Furthermore, our method provided better results than using only brain network or single-channel signal features. This suggests that combining these complementary features can facilitate more accurate localization of the EZ.