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Intrinsic time-scale decomposition (ITD) is a new nonlinear method of time-frequency representation which can decipher the minute changes in the nonlinear EEG signals. In this work, we have automatically classified normal, interictal and ictal EEG signals using the features derived from the ITD representation. The energy, fractal dimension and sample entropy features computed on ITD representation coupled with decision tree classifier has yielded an average classification accuracy of 95.67%, sensitivity and specificity of 99% and 99.5%, respectively using 10-fold cross validation scheme. With application of the nonlinear ITD representation, along with conceptual advancement and improvement of the accuracy, the developed system is clinically ready for mass screening in resource constrained and emerging economy scenarios.

In the present study, both single channel electroencephalography (EEG) complexity and two channel interhemispheric dependency measurements have newly been examined for classification of patients with obsessive–compulsive disorder (OCD) and controls by using support vector machine classifiers. Three embedding entropy measurements (approximate entropy, sample entropy, permutation entropy (PermEn)) are used to estimate single channel EEG complexity for 19-channel eyes closed cortical measurements. Mean coherence and mutual information are examined to measure the level of interhemispheric dependency in frequency and statistical domain, respectively for eight distinct electrode pairs placed on the scalp with respect to the international 10–20 electrode placement system. All methods are applied to short EEG segments of 2 s. The classification performance is measured 20 times with different 2-fold cross-validation data for both single channel complexity features (19 features) and interhemispheric dependency features (eight features). The highest classification accuracy of 85 ±5.2% is provided by PermEn at prefrontal regions of the brain. Even if the classification success do not provided by other methods as high as PermEn, the clear differences between patients and controls at prefrontal regions can also be obtained by using other methods except coherence. In conclusion, OCD, defined as illness of orbitofronto-striatal structures [Beucke et al., JAMA Psychiatry70 (2013) 619–629; Cavedini et al., Psychiatry Res.78 (1998) 21–28; Menzies et al., Neurosci. Biobehav. Rev.32(3) (2008) 525–549], is caused by functional abnormalities in the pre-frontal regions. Particularly, patients are characterized by lower EEG complexity at both pre-frontal regions and right fronto-temporal locations. Our results are compatible with imaging studies that define OCD as a sub group of anxiety disorders exhibited a decreased complexity (such as anorexia nervosa [Toth et al., Int. J. Psychophysiol.51(3) (2004) 253–260] and panic disorder [Bob et al., Physiol. Res.55 (2006) S113–S119]).

There has been considerable interest in quantifying the complexity of different time series, such as physiologic time series, traffic time series. However, these traditional approaches fail to account for the multiple time scales inherent in time series, which have yielded contradictory findings when applied to real-world datasets. Then multi-scale entropy analysis (MSE) is introduced to solve this problem which has been widely used for physiologic time series. In this paper, we first apply the MSE method to different correlated series and obtain an interesting relationship between complexity and Hurst exponent. A modified MSE method called multiscale permutation entropy analysis (MSPE) is then introduced, which replaces the sample entropy (SampEn) with permutation entropy (PE) when measuring entropy for coarse-grained series. We employ the traditional MSE method and MSPE method to investigate complexities of different traffic series, and obtain that the complexity of weekend traffic time series differs from that of the workday time series, which helps to classify the series when making predictions.

In this paper, we consider the chaotic phenomenon and Kolomogorov complexity in computing the environmental interface temperature. First, the environmental interface is defined in the context of the complex system, in particular for autonomous dynamical systems. Then we consider the following issues in modeling procedure: (i) how to replace given differential equations by appropriate difference equations in modeling of phenomena in the environmental world? (ii) whether a mathematically correct solution to the corresponding differential equation or system of equations is always physically possible and (iii) phenomenon of chaos in autonomous dynamical systems in environmental problems, in particular in solving the energy balance equation to calculate environmental interface temperature. The difference form of this equation for computing the environmental interface temperature is discussed and analyzed depending on parameters of equation, using the Lyapunov exponent and sample entropy. Finally, the Kolmogorov complexity of time series obtained from this difference equation is analyzed.

In this paper, we have used the Kolmogorov complexity and sample entropy measures to estimate the complexity of the UV-B radiation time series in the Vojvodina region (Serbia) for the period 1990–2007. We have defined the Kolmogorov complexity spectrum and have introduced the Kolmogorov complexity spectrum highest value (KCH). We have established the UV-B radiation time series on the basis of their daily sum (dose) for seven representative places in this region using: (i) measured data, (ii) data calculated via a derived empirical formula and (iii) data obtained by a parametric UV radiation model. We have calculated the Kolmogorov complexity (KC) based on the Lempel–Ziv algorithm (LZA), KCH and sample entropy (SE) values for each time series. We have divided the period 1990–2007 into two subintervals: (i) 1990–1998 and (ii) 1999–2007 and calculated the KC, KCH and SE values for the various time series in these subintervals. It is found that during the period 1999–2007, there is a decrease in the KC, KCH and SE, compared to the period 1990–1998. This complexity loss may be attributed to (i) the increased human intervention in the post civil war period causing increase of the air pollution and (ii) the increased cloudiness due to climate changes.

Detecting epileptic seizure is a very time consuming and costly task if a support vector machine (SVM) hardware processor is used. In this paper, an automated seizure detection scheme is developed by combining discrete wavelet transform (DWT), sample entropy (SampEn) and a novel classification algorithm based on each wavelet coefficient and voting strategy. In order to save circuit area, a Daubechies order 4 (db4) filter of lattice structure is introduced in DWT, only half elements of the symmetric distance matrix in the SampEn are stored and module reusing strategy is used. To speed up the detection, intermediate results are reused by reasonably organizing the SampEn calculation procedures. The seizure detection scheme is implemented in a field-programmable gate array (FPGA) and its classification performance is tested with publicly available epilepsy dataset.

The Multiscale Entropy (MSE) is an effective measure to quantify the dynamical complexity of complex systems, which has many successful applications in physiological and physical fields. It uses different scales to mean-coarse-grain the original series, and then calculates the sample entropy for each coarse-grained series. Inspired by the MSE, we in this paper propose the Multi-Moment Multiscale Local Sample Entropy (MMMLSE), which considers both mean-coarse-grained and standard-deviation-coarse-grained characteristics of the original series for each scale, to quantify the dynamical complexity of complex systems. We use simulated data ($1/f$ noise, white noise and logistic map) to test the performance of our proposed method, with results showing that the MMMLSE can accurately and effectively characterize these complex systems. The ability to preserve nonlinear dynamics of the proposed method is also proved by surrogate data and nonlinearity test experiment. Furthermore, we apply the MMMLSE to analyze physiological signals, and the MMMLSE reveals that the ill individuals have lower dynamical complexity at larger scales than the healthy ones, and the elder individuals have lower dynamical complexity at larger scales than the younger ones, which are consistent with the reality.

Coronavirus disease (COVID-19) is a pandemic disease that has affected almost all around the world. The most crucial step in the treatment of patients with COVID-19 is to investigate about the coronavirus itself. In this research, for the first time, we analyze the complex structure of the coronavirus genome and compare it with the other two dangerous viruses, namely, dengue and HIV. For this purpose, we employ fractal theory, sample entropy, and approximate entropy to analyze the genome walk of coronavirus, dengue virus, and HIV. Based on the obtained results, the genome walk of coronavirus has greater complexity than the other two deadly viruses. The result of statistical analysis also showed the significant difference between the complexity of genome walks in case of all complexity measures. The result of this analysis opens new doors to scientists to consider the complexity of a virus genome as an index to investigate its danger for human life.

Severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) is the most dangerous type of coronavirus and has infected over 25.3 million people around the world (including causing 848,000 deaths). In this study, we investigated the similarity between the genome walks of coronaviruses in various animals and those of human SARS-CoV-2. Based on the results, although bats show a similar pattern of coronavirus genome walks to that of SARS-CoV-2 in humans, decoding the complex structure of coronavirus genome walks using sample entropy and fractal theory showed that the complexity of the pangolin coronavirus genome walk has a 94% match with the complexity of the SARS-CoV-2 genome walk in humans. This is the first reported study that found a similarity between the hidden characteristics of pangolin coronavirus and human SARS-CoV-2 using complexity-based analysis. The results of this study have great importance for the analysis of the origin and transfer of the virus.

Evaluation of the correlation of the activities of various organs is an important area of research in physiology. In this paper, we evaluated the correlation among the brain and facial muscles’ reactions to various auditory stimuli. We played three different music (relaxing, pop, and rock music) to 13 subjects and accordingly analyzed the changes in complexities of EEG and EMG signals by calculating their fractal exponent and sample entropy. Based on the results, EEG and EMG signals experienced more significant changes by presenting relaxing, pop, and rock music, respectively. A strong correlation was observed among the alterations of the complexities of EMG and EEG signals, which indicates the coupling of the activities of facial muscles and brain. This method could be further applied to investigate the coupling of the activities of the brain and other organs of the human body.

The coronavirus has influenced the lives of many people since its identification in 1960. In general, there are seven types of coronavirus. Although some types of this virus, including 229E, NL63, OC43, and HKU1, cause mild to moderate illness, SARS-CoV, MERS-CoV, and SARS-CoV-2 have shown to have severer effects on the human body. Specifically, the recent known type of coronavirus, SARS-CoV-2, has affected the lives of many people around the world since late 2019 with the disease named COVID-19. In this paper, for the first time, we investigated the variations among the complex structures of coronaviruses. We employed the fractal dimension, approximate entropy, and sample entropy as the measures of complexity. Based on the obtained results, SARS-CoV-2 has a significantly different complex structure than SARS-CoV and MERS-CoV. To study the high mutation rate of SARS-CoV-2, we also analyzed the long-term memory of genome walks for different coronaviruses using the Hurst exponent. The results demonstrated that the SARS-CoV-2 shows the lowest memory in its genome walk, explaining the errors in copying the sequences along the genome that results in the virus mutation.

Evaluation of the correlation among the activities of various organs is an important research area in physiology. In this paper, we analyzed the correlation between the brain and skin reactions in response to various auditory stimuli. We played three different music (relaxing, pop, and rock music) to eleven subjects (4 M and 7 F, 18–22 years old) and accordingly analyzed the changes in the complexity of Electroencephalogram (EEG) and Galvanic Skin Response (GSR) signals by calculating their fractal exponent and sample entropy. A strong correlation was observed among the alterations of the complexity of GSR and EEG signals in the case of fractal dimension ($r=0.9971$) and also sample entropy ($r=0.8120$), which indicates the correlation between the activities of skin and brain. This analysis method could be further applied to investigate the correlation among the activities of the brain and other organs of the human body.

It is known that heart activity changes during aging. In this paper, we evaluated alterations of heart activity from the complexity point of view. We analyzed the variations of heart rate of patients with congestive heart failure that are categorized into four different age groups, namely 30–39, 50–59, 60–69, and 70–79 years old. For this purpose, we employed three complexity measures that include fractal dimension, sample entropy, and approximate entropy. The results showed that the trend of increment of subjects’ age is reflected in the trend of increment of the complexity of heart rate variability (HRV) since the values of fractal dimension, approximate entropy, and sample entropy increase as subjects get older. The analysis of the complexity of other physiological signals can be further considered to investigate the variations of activity of other organs due to aging.

Since skin activity, like other organs, is controlled by the brain, we decoded the correlation among the brain and skin responses in auditory stimulation by complexity-based analysis of EEG and GSR signals. Three pieces of music were selected according to the difference in the fractal exponent and sample entropy of embedded noises in them. We calculated the fractal dimension and sample entropy of EEG and GSR signals for 11 subjects in rest and response to these music pieces. The correlation coefficients of 0.9525 and 0.9822 in the case of fractal dimension and sample entropy demonstrated a strong correlation between the complexities of the GSR and EEG signals. Therefore, we can state that the skin and brain responses are coupled. This method can be applied to evaluate the relationship between the human brain and other organs.

Analysis of the variations of heart activity during different human activities is an important area of research in sport sciences. Therefore, in this paper, we evaluated the variations of heart activity for 23 subjects while sitting, hand biking, walking, and running. Since the obtained R-R time series (as the indicator of heart rate variability (HRV)) has a complex structure that contains information, we employed fractal dimension, sample entropy, and Shannon entropy for our analysis. According to the results, doing a harder activity causes a more significant alteration in the complexity and information content of HRV. The results of statistical analyses also verified the obtained results. Similar investigations can be conducted in case of other activities to evaluate the variations in heart activity in different conditions.

Attention deficit hyperactivity disorder (ADHD) is a mental health disorder that is very common among children and may last into their adulthood. It is known that ADHD affects the attention of patients due to problems with short-term memory. Therefore, analysis of attention and memory of these patients should come into consideration. In this study, the complexity and memory of Electroencephalogram (EEG) signals are analyzed to investigate the reduction in attention and memory of patients with ADHD compared to normal subjects while playing a serious game. To achieve this, the fractal dimension and sample entropy of EEG signals are analyzed to evaluate the alterations in the complexity of EEG signals. Moreover, the Hurst exponent of EEG signals for ADHD and non-ADHD subjects is calculated to discuss the memory of EEG signals. The results showed a smaller fractal dimension and sample entropy of EEG signals for patients with ADHD that reflects their lower attention. Besides, the Hurst exponent of EEG signals for these patients reflects their lower memory than normal subjects. Therefore, it can be concluded that the reductions of attention and memory in ADHD subjects are mapped on the reduction of complexity and memory of their EEG signals.

The evaluation of the correlation between the activations of various organs has great importance. This work investigated the synchronization of the brain and heart responses to different auditory stimuli using complexity-based analysis. We selected three pieces of music based on the difference in the complexity of embedded noise (including white noise, brown noise, and pink noise) in them. We played these pieces of music for 11 subjects (7 M and 4 F) and computed the fractal dimension and sample entropy of EEG signals and R–R time series [as heart rate variability (HRV)]. We found strong correlations ($r=0.9999$ in the case of fractal dimension and $r=0.7862$ in the case of sample entropy) among the complexities of EEG signals and HRV. This finding demonstrates the synchronization of the brain and heart responses and auditory stimuli from the complexity perspective.

An important research area in physiological and sport sciences is the analysis of the variations of the muscle reaction due to changes in walking speed. In this paper, we investigated the effect of walking speed variations on leg muscle reaction by the analysis of Electromyogram (EMG) signals at different walking inclines. For this purpose, we benefited from fractal theory and sample entropy to analyze how the complexity of EMG signals changes at different walking speeds. According to the results, although fractal theory could not show a clear trend between the variations of the complexity of EMG signals and the variations of the walking speed, however, based on the results, increasing the speed of walking in the case of different inclines is mapped on to the decrement of the sample entropy of EMG signals. Therefore, sample entropy could decode the effect of walking speed on the reaction of leg muscle. This analysis method could be applied to analyze the variations of other physiological signals of humans durin walking.

This paper analyzed the coupling among the reactions of eyes and brain in response to visual stimuli. Since eye movements and electroencephalography (EEG) signals as the features of eye and brain activities have complex patterns, we utilized fractal theory and sample entropy to decode the correlation between them. In the experiment, subjects looked at a dot that moved on different random paths (dynamic visual stimuli) on the screen of a computer in front of them while we recorded their EEG signals and eye movements simultaneously. The results indicated that the changes in the complexity of eye movements and EEG signals are coupled ($r=-0.8043$ in case of fractal dimension and $r=-0.9259$ in case of sample entropy), which reflects the coupling between the brain and eye activities. This analysis could be extended to evaluate the correlation between the activities of other organs versus the brain.

Decoding of the coupling among the brain and heart activations is an important research area in network physiology. We studied the coupling of brain and heart activations for 48 subjects who performed the NASA Revised Multi-Attribute Task Battery II under three different activity level conditions. During the experiment, the physical activity of subjects was manipulated by changing the speed of a stationary bike (including no movement, 50rpm, and 70rpm) or a treadmill (including no movement, 3km/h, and 5km/h), while their physiological signals were recorded. We analyzed the complex structure of electroencephalogram (EEG) and R-R signals using fractal theory and sample entropy. The results demonstrated that the alterations of the complex structures of EEG and R-R signals are strongly correlated, which indicates the coupling between brain and heart activations. This method of analysis can be applied to evaluate the coupling between different organs.