The method of identifying the direction of the coupling between oscillators based on the simulation of phase dynamics of weakly coupled and weakly noisy periodic processes is used to determine the direction of the coupling between time series extracted from biological signals. Neuronal activity, arterial pressure and respiratory rate of anesthetized rats were used as initial data. Various options for the direction of the coupling between the analyzed time series were revealed.

This review addresses the application of the methods of nonlinear dynamics to the analysis of dynamic changes in the patterns of the physiological rhythms of the brain when disorders associated with chronic high blood pressure and heart rhythm disturbances such as atrial fibrillation occur with and without moderate cognitive impairment. These methods are shown to be suitable for use in identifying markers for these disorders. These markers are associated with measures of phase synchronization between rhythmic photostimulation and cerebral responses in the form of electroencephalographic patterns.

The work is devoted to the application of various nonlinear dynamics methods to identify interactions in short noisy time series extracted from biological rhythms related to the respiratory, cardiovascular and nervous systems. These interactions are considered as transitions to synchronized states, as the coupling directionality and the delay time in the influence of one system on another. For the analyzed data, the relationship between these rhythms is due to the fact that the neurons of medulla are related to the brain response to changes in blood pressure and respiration during external pain stimulation. Phase synchronization between the variability of blood pressure and respiratory fluctuations in response to these stimuli was established in half of the analyzed time series using methods for assessing phase synchronization based on finding instantaneous phases by the synchrosqueezed wavelet transform and based on calculating the recurrences of phase trajectories. For unsynchronized time series, a predominantly unidirectional influence of respiratory rhythm fluctuations on the blood pressure variability was obtained by the method of phase dynamics modeling and the method of estimating recurrence probabilities. For two third of the data, a unidirectional influence of the blood pressure variability on the neuronal variability was determined, and the remaining data were characterized by a bidirectional relationship between the neuronal and blood pressure variability. Delay time estimation by phase dynamics modeling showed reduced time of influence of the blood pressure variability on the neuronal variability in bidirectional coupling compared to the influence of the neuronal variability on the blood pressure variability.

The purpose of this work is to determine the possibility of detecting changes in the relationships between such physiological rhythms as the activity of neurons in the reticular formation of the medulla oblongata, fluctuations in the blood pressure and respiration in anesthetized rats before and during the development of a pathological state associated with painful colorectal distension. This stretch mimics the pain localized in the lower abdomen in patients with irritable bowel syndrome and it is accompanied by responses of the brain neurons, fluctuations in the blood pressure and respiration. The analysis of changes in the relationships of these rhythms consisted in identifying phase synchronization between the time series of the variability of neuronal activity intervals and the variability of blood pressure intervals at the respiratory rate before and during pain exposure. Methods. To solve this problem, the synchrosqueezed wavelet transform method was applied, which makes it possible to effectively calculate the instantaneous frequencies and phases of non-stationary signals. As indicators of synchronization, we used the values of the index and the duration of phase synchronization as a time interval during which the value of the synchronization index is close to 1. Results. It has been established that the pain effect provides an adjustment of the frequency of the neuronal activity variability and the occurrence of synchronization between this activity and the blood pressure variability at the respiratory rate or causes an adjustment of the frequency of the blood pressure variability and the occurrence of synchronization between the blood pressure variability and the respiratory rhythm. It was found that the pain effect increases the duration of phase synchronization between the variability of the blood pressure and the respiratory rhythm or reduces the duration of phase synchronization between the variability of neuronal activity and the respiratory rhythm. Conclusion. The effect of painful colorectal distension on changes in the parameters of phase synchronization between physiological rhythms in anesthetized rats was studied in detail.

The task is to determine the indices of phase synchronization between biological data extracted from neuronal activity and fluctuations in blood pressure and respiration, based on the calculation of instantaneous frequencies and phases using the synchrosqueezed wavelet analysis. Differences in synchronization indices between the analyzed data in two groups of rats (the control group and the group with experimentally induced colitis) were revealed. Pain exposure was found to be associated with frequency adjustment of the variability of the neuronal activity or the blood pressure, followed by the onset of phase synchronization.

The review is devoted to the analysis of the relationship between dynamic changes in patterns of electrical activity of the brain during the occurrence of mental disorders in the form of paranoid schizophrenia and depression and in patterns of brain activity in cardiovascular pathology associated with permanent atrial fibrillation, as well as indicators of multifractality of the studied patterns. To assess these indicators of electroencephalographic patterns, we describe a method of multifractal analysis based on the search for maxima of wavelet coefficient modules, and to isolate the fractal component of the signal in the power spectrum we describe a method of autospectral analysis with irregular resampling. It has been shown that the main differences between the multifractal properties of the electrical activity of the brain in health and in pathology are the different widths of the multifractality spectrum and its location, associated with different types of sequential pattern values. In this regard, the multifractality indicators can serve as informative markers of neuronal disorders and can be included in a set of tests for studying various pathologies.

Using a method based on averaging the conditional probabilities of recurrence intervals, a comparative analysis was performed to estimate the direction of coupling between different physiological rhythms (fluctuations in blood pressure, respiration, and neuronal activity in the reticular formation of the medulla oblongata) in two groups of anesthetized rats with or without painful stimulation. Different variants were observed for the directions of coupling between the rhythms. A statistically significant unidirectionality of coupling was detected with most of the data obtained from rats of the control group and rats with experimentally induced colitis. Painful stimulation did not affect the direction of coupling, but slightly reduced the amount of data with unidirectional coupling. Rhythms associated with the cardiovascular system (blood pressure variability) were controlled by rhythms of the respiratory system, and the respiratory rhythm controlled the variability of neuronal activity in the reticular formation of the medulla oblongata.

This review addresses the application of wavelet, multifractal, and recurrence analysis methods to the study of changes occurring in the patterns of electrical activity of the human brain, recorded as electroencephalograms, in maladjustment disorders associated with anxious-phobic disorders, panic attacks, and moderate cognitive impairment. The possibility of using these methods to identify objective indicators of the correction of psychogenic pain in anxious-phobic states and to improve the functional state of the nervous system after stimulatory actions directed to activating the functional connections of the brain in people with panic attacks and moderate cognitive impairment is demonstrated .

The wavelet transform modulus maxima (WTMM) method was used to comparatively analyze multifractality of various components of the electroencephalogram (EEG) in normal conditions and mental disorders, such as schizophrenia and depression. It was shown that, for each mental disorder, the position of the singularity spectrum is stable in most areas of the brain and the main differences between multifractal properties are associated with the α components of the EEG. Anticorrelated dynamics of successive values of the α components was characteristic of a group of depressive individuals, while a combination of anticorrelated and correlated dynamics was observed in a group of schizophrenics. The findings may be of use for clinical diagnosis of neuronal disorders.

We applied methods for determining the recurrence based phase synchronization index and the coupling directionality based on conditional probabilities of recurrence to analyze the direction of coupling and synchronization of time series obtained from experimental data. The activity of brain neurons, fluctuations in arterial blood pressure, and the respiratory rhythm of anesthetized rats were used as experimental data. For the first time, variants of changing the direction of the coupling between the analyzed time series before and during pain exposure were revealed. The relationship of pain exposure with the adjustment of the frequency of the variability of neuron activity of the brain to the frequency of the variability of arterial blood pressure or with the adjustment of the frequency of the variability of blood pressure to the respiratory rhythm with the subsequent occurrence of phase synchronization has been established.

The dynamic changes in the patterns of electroencephalograms of patients with moderate cognitive impairments and the quantitative indicators of these patterns, determined by the methods of wavelet and recurrent analysis before and after therapy using rhythmic influence have been investigated. It has been shown that the calculated indices of wavelet spectra and recurrence plots (values of the coefficient of photic driving, the percentage of recurrent points located on diagonal lines and the cross-correlation coefficient between probabilities of recurrences of the light signal and the brain response) are very sensitive to changes in reactive patterns of electroencephalograms in response to stimulating influences.

This work was aimed at a comparative analysis of the degree of multifractality of electroencephalographic time series obtained from a group of healthy subjects and from patients with mental disorders. We analyzed long-term records of patients with paranoid schizophrenia and patients with depression. To evaluate the properties of multifractal scaling of various electroencephalographic time series, the method of maximum modulus of the wavelet transform and multifractal analysis of fluctuations without a trend were used. The stability of the width and position of the singularity spectrum for each of the test groups was revealed, and a relationship was established between the correlation and anticorrelation dynamics of successive values of the electroencephalographic time series and the type of mental disorders. It was shown that the main differences between the multifractal properties of brain activity in normal and pathological conditions lie in the different width of the multifractality spectrum and its location associated with the correlated or anticorrelated dynamics of the values of successive time series. It was found that the schizophrenia group is characterized by a greater degree of multifractality compared to the depression group. Thus, the degree of multifractality can be included in a set of tests for differential diagnosis and research of mental disorders.

The joint recurrence analysis is used to reveal differences in the phase synchronization between intermittent photic stimulation and brain responses of patients with cardiac fibrillation of paroxysmal and persistent types. As a measure of phase synchronization between two signals, we consider the mutual correlation factor between recurrence probabilities of corresponding phase trajectories. The value of this coefficient increases upon an increase in the cardiac fibrillation time and in the extent of decline of cognitive functions for brain responses to theta-range frequencies.

The aim of this work is to search for markers of moderate cognitive impairment in electroencephalographic patterns based on the study of the degree of synchronization between the stimulus and the response of the brain in various forms of vascular genesis and the presence or absence of cognitive disorders. To search for such markers, the methods of nonlinear dynamics associated with the synchrosqueezed wavelet transform and the analysis of joint signal recurrences were used. It was shown that the parameters of phase synchronization differ significantly in individuals with cardiac arrhythmias and moderate cognitive disorders.

The task is to identify differences in phase synchronization between the intermittent photic stimulation and a response of the brain of patients with arterial hypertension and manifestations of moderate cognitive impairment and without such manifestations. To solve the task we used the recurrence based phase synchronization index, the ratio of instantaneous frequencies, the duration and the phase synchronization index obtained after the synchrosqueezed wavelet transform of the EEG patterns and the light time series. We have shown that moderate cognitive impairment correlates with a greater degree of phase synchronization. This is manifested in an increase in the index and duration of phase synchronization between intermittent photostimulation and bioelectric activity of the brain, as well as with a shift in this activity to a lower frequency range compared to the excitation frequency. The results can be used for clinical assessment of the degree of moderate cognitive impairment of vascular origin.

Using the method of recurrent analysis of electroencephalograms of the brain with subsequent one-way analysis of variance, the effectiveness of the therapy method used in the clinic, associated with the formation of stable functional connections of the brain in patients with disorders of the functional state of the central nervous system corresponding to the initial stage of vascular pathology, was assessed. Decrease in the percentage of recurrent points located on the diagonal lines of recurrent diagrams and the coefficient of mutual correlation between the probabilities of recurrences of the light signal and the response of the brain to the theta and beta frequencies, as well as an increase in these indicators for the alpha-range frequencies after therapeutic effects, confirm the effectiveness of the considered therapy.

The dynamics of the photic driving reaction was studied in subjects with panic attacks, using wavelet and recurrence analyses of electroencephalographic (EEG) patterns. Quantitative measures of the driving reaction to the given frequency were estimated from the increase in the energy of the wavelet spectrum during photostimulation, the coefficient of photic driving, and parameters of a joint recurrence plot of the light stimulus and EEG pattern (the rate and time of recurrence of the EEG trajectory into the vicinity of a previous point). The parameters were determined before and after non-drug therapy, which included activation and training of artificial stable functional connections (ASFCs) in the brain. The formation of ASFCs was found to decrease both asymmetry in occipital lobe responses to the photostimulus and the degree of neuroticism in subjects with panic attacks. The changes were reflected in decreasing values of the photic driving coefficient and recurrence rate and increasing values of the recurrence time. The improvement of the subject’s psychophysiological state after a course of ASFC activation was confirmed by a positive dynamics of psychophysiological testing data.

The sixth chapter examines the mechanisms of changes in the dynamic complexity of hand tremor with an increase in the degree of motor disorders. For this, multifractal and wavelet properties of involuntary oscillations that arise during the performance of certain motor tasks by a healthy person and a person with varying degrees of motor pathology are investigated. It was shown that clinical manifestations of pathological tremor correlate, firstly, with a significant increase in the global energy of the wavelet spectrum and, secondly, with a decrease in the width of the singularity spectrum. At the same time, an improvement in the functional state (a decrease in the amplitude of a pathological tremor) corresponds to a decrease in long-term correlations and an increase in the randomness of up-down patterns, which corresponds to a partial shift of the singularity spectrum into the range of values typical for a tremor of the hand of a healthy person. The mechanism for reducing the dynamic complexity of patterns of involuntary oscillations of the human hand with an increase in the degree of motor disorders is based on a decrease in the contribution of strong fluctuations, accompanied by the appearance of long-term correlations of sequential values of tremor. The considered features in changes in tremor patterns make it possible to quantify the degree of movement disorders and can be used to facilitate the differential diagnosis of the parkinsonian and essential tremor in complex clinical cases.