How Complex is COVID-19: Spatiotemporal Variation of Chaotic Nature of the Virus in Indian States

Like other pandemics, COVID‑19 also created a huge socioeconomic imbalance and distress in people. Often, every pandemic is characterized as chaotic and complex. Hence, the nature of the virus spread and deaths should be analyzed to prepare for the next similar pandemic. In this analysis, the popular and well‑known time series in chaos theory is implemented, and the results are deduced for the states of India. The phase space reconstruction algorithm is implemented, and false nearest neighbor (FNN) method is applied to determine the dimensionality, and also Lyapunov exponent of the time series is estimated. The chaotic nature of COVID‑19 cases showed a less severe and low complexity, with the FNN dimension range of 3–5, whereas the COVID‑19 deaths showed moderate complexity with FNN dimensions 2–7. Policymakers should take action on medical availability in rural states and control people’s movement in highly populated areas.

Non-linear Analysis of Single Electroencephalography (EEG) for Sleep-Related Healthcare Applications / 대한의료정보학회지

OBJECTIVES: Soft-computing techniques are commonly used to detect medical phenomena and to help with clinical diagnoses and treatment. The purpose of this paper is to analyze the single electroencephalography (EEG) signal with the chaotic methods in order to identify the sleep stages. METHODS: Data acquisition (polysomnography) was performed on four healthy young adults (all males with a mean age of 27.5 years). The evaluated algorithm was designed with a correlation dimension and Lyapunov's exponent using a single EEG signal that detects differences in chaotic characteristics. RESULTS: The change of the correlation dimension and the largest Lyapunov exponent over the whole night sleep EEG was performed. The results show that the correlation dimension and largest Lyapunov exponent decreased from light sleep to deep sleep and they increased during the rapid eye movement stage. CONCLUSIONS: These results suggest that chaotic analysis may be a useful adjunct to linear (spectral) analysis for identifying sleep stages. The single EEG based nonlinear analysis is suitable for u-healthcare applications for monitoring sleep.

Function Analysis of the Largest Lyapunov Exponent of Masseter Muscle / 医疗卫生装备

Objective To apply chaos characteristics to prediction of the unilateral mastication.Methods The paper calculated the Largest Lyapunov Exponent of the masseter muscle of some males and females with a method from small data sets,which then was processed by reusable two-factor analysis of variance.Results The results shows that the signal of the masseter muscle has chaos characteristics,the male's Largest Lyapunov Exponent is higher that the female's,and the chaos degree of the masseter muscle on both sides is consistent nearly.Conclusion The Largest Lyapunov Exponent can be used to characterize the signal of the masseter muscle.Comparative Analysis of the Largest Lyapunov Exponent on both sides can be used as reference when to predict and diagnose the unilateral mastication.

The Changes of EEG Nonlinear Parameter in Sleep and Wakefulness States

BACKGROUND: Up to now, sleep stages have traditionally been determined by the visual inspection of individual EEG waves. However, the exact physiological meaning of the sleep waves is not known. The purpose of this study was to try and find out the physiological parameters of the EEG of the sleep and wakefulness states by calculating one of the non-linear chaos parameter, the largest Lyapunov exponent (LLE), of EEG time series. METHODS: The digital EEG of the wakefulness with eye opening (WEO), wakefulness with eye closure (WEC), stage1 (S1), stage2 (S2), stage3 or 4 (S34) were recorded at centroparietal region (C4-P4 bipolar derivation) in 10 normal subjects. Lyapunov exponents of 50 EEG time series in different states were compared. RESULTS: LLE's of WEO, WEC, S1, S2, S34 showed an increas-ing tendency as states switched from wakefulness to sleep. LLE of sleep was larger than that of awake state. CONCLUSIONS: The EEG of the sleep state appeared to be more chaotic than that of the awake state. This nonlinear chaos parameter can be used as a physiological parameter of normal sleep and awake states.

effects of various degrees of coronary stenosis on Lyapunov exponents of ECG and R-R interval time series / 中国医学物理学杂志

Purpose:It has been shown that the heart is a chaotic oscillator. So it is appropriiate to use the Lyapunov exponent, an important parameter to identify the nature of non-linear dynamical systems, for identifying the state of human heart. Methods:Preliminary results are obtained in this paper using Wolf's algorithm for 8 normal and 107 abnormal ECG recordings. Results:Significant differences are found between the Lyapunov exponents of normal ECG and ECG with obvious coronary stenosis (OCS), but there is no significant difference between the Lyapunov exponents of normal ECG and ECG with mild coronary stenosis (MCS);Significant differences are also found between the Lyapunov exponents of R-R interval series of normal ECG、ECG with MCS and ECG with OCS. Conclusions:It is apparent that the R-R interval series can give us more messages about human heart, and the Lyapunov exponents of ECG and R-R interval series are the appropriate parameters for the identification of the physiological states of human heart. It is possible to use Lyapunov exponent for early diagnosis of Coronary Heart Disease.

Nonlinear Analysis of EEG in Various Mental States of Normal Person

BACKGROUND: In order to compare the differences between various mental activities and to determine the significance of chaotic patterns, we performed a fractal dimension and Lyapunov exponent analysis of digital EEG data which are known to be non linear biological signals. METHODS: During the EEG recordings, different kinds of tasks were performed, including eye closing, eye opening, calculation, listening to music, and remembering a picture. Performance signals were recorded for each test as a digital EEG for more than one minute. We used our own software, CHASIM, to calculate the fractal dimension and Lyapunov exponent. The statistical analysis performed was an ANOVA using SAS 6.12 for Windows (9.0). RESULTS: The fractal dimensions during the "listening to music and noise", and "moving both toes" tasks were increased. During the "eye opening", "calculation", and "moving both fingers" tasks, the fractal dimensions were decreased. CONCLUSIONS: Auditory stimulation generated a higher correlation dimension than visual stimulation. The interpretation of these results is still not totally clear, but we hope to find future applications for non-linear chaotic analysis for the functional evaluation of the central nervous system.

A Preliminary Study for the Estimation of the First Positive Lyapunov Exponent of the EEG in Experimental Status Epilepticus Rats / 대한간질학회지

BACKGROUND AND OBJECTIVES: The hypothesis that brain is a nonlinear dynamic system exhibiting deterministic chaos has offered new methods to the investigation of information processing in the brain by analysis and classification of EEG signals. We used the first positiveLyapunov exponent (L1) which is one of indicator of nonlinear dynamic to evaluate the brain function in chemical seizure models METHODS: Lithium-Pilocarpine induced seizure model and kainic acid induced seizure model are used. From serial EEG according to seizure stages. 32.768 sec of data (16.384 data point) were recorded and digirized by a 12-bit analog-digital converter in an IBM PC. The data from serial EEG according to seizure stageswere analyzed for determining the L1. We used the time delays calculated by the method of mutual information to reconstruct the attactor. Time delays of 46-58 msec and enbedding dimensions of 13-19 were used for chemical seizure model. The L1 were calculated for 4 channels. RESULTS: The averaged valued of L1 were serially decreased in both lithium-pilocarpin model and kainic acid model according to increasing seizure stages. CONCLUSION: Our results reveal the decrease of the chaotic activity according to increasing seizure stage. It is suggested that the brain has decreased information procedding and a less flexible neural network during seizure. However further evaluation is required because the significance of these changes are not confirmed.

A preliminary Study for Nonlinear Dynamic Analysis of EEG in Patients with Dementia of Alzheimer

The changes of electroencephalogram(EEG) in patients with dementia of Alzheimer's type are most commonly studied by analyzing power or magnitude in traditionally defined frequency bands. However because of the absence of an identified metric which quantifies the complex amount of information, there are many limitations in using such a linear method. According to the chaos theory, irregular signals of EEG can be also resulted from low dimensional deterministic chaos. Chaotic nonlinear dynamics in the EEG can be studied by calculating the largest Lyapunov exponent(L1). The authors have analyzed EEG epochs from three patients with dementia of Alzheimer's type and three matched control subjects. The largest L1 is calculated from EEG epochs consisting of 16.384 data points per channel in 15 channels. The results showed that patients with dementia of Alzheimer's type had significantly lower L1 than non-demented controls on 8 channels. Topographic analysis showed that the L1 were significantly lower in patients with Alzheimer's disease on all the frontal, temporal, central, and occipital head regions. These results show that brains of patients with dementia of Alzheimer's type have a decreased chaotic quality of electrophysiological behavior. We conclude that the nonlinear analysis such as calculating the L1 can be a promising tool for detecting relative changes in the complexity of brain dynamics.

Nonlinear Dynamic Analysis Using Lyapunov Exponent of EEG in Patients with Schizophrenia / 대한정신약물학회지

OBJECT: It seemed worthwhile to estimate nonlinear measures of the electroencephalogram (EEG) in schizophrenic patients, because nonlinear measures might serve as indicators of the specific brain function in schizophrenia. METHOD: Previous studies which estimated the chaoticity in the brain of schizophrenia with nonlinear methods recorded the EEGs at limited electrodes, so we tried to record EEGs from 16 channels for nonlinear analysis in 19 patients with Schizophrenia and 8 healthy control subjects. We employed a new method to calculate the nonlinear invariant measures. For limited noisy data, this algorithm was strikingly faster and more accurate than previous ones. RESULTS: Our results showed that the schizophrenic patients had lower values of the largest positive Lyapunov exponent at the left inferior frontal and anterior temporal head regions compared with normal controls. CONCLUSIONS: These results suggest that the nonlinear analysis of the EEGs such as the estimation of the largest positive Lyapunov exponent seems to be a useful tool in analyzing EEG data to explore the neurodynamics of the brain of schizophrenic patients.