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.

Differences in phase synchronization between intermittent photic stimulation and electrical activity of the brain recorded as electroencephalographic (EEG) patterns were studied in two groups of patients with chronically elevated arterial pressure with and without cognitive impairment. It was found that the parameters of phase synchronization calculated using synchrosqueezed wavelet transform of the light stimulus and the EEG pattern can be used as neurophysiological markers of moderate cognitive impairment.

Based on the analysis of joint recurrences, differences in phase synchronization between rhythmic photostimulation and brain responses were revealed in individuals with atrial fibrillation of paroxysmal and persistent types. As a measure of phase synchronization between two signals, the cross-correlation coefficient between the probabilities of recurrences of the corresponding phase trajectories is considered. With a lengthening of the lifetime of atrial fibrillation and an increase in the degree of decline in cognitive functions, the value of this coefficient increases for brain responses to theta-range frequencies.

The differences in phase synchronization between intermittent photostimulation and bioelectrical activity of the brain in the form of electroencephalographic patterns in two groups of people with chronically elevated blood pressure, with and without moderate manifestations of cognitive functions are investigated. It was found that the phase synchronization parameters calculated on the basis of the synchrosqueezed wavelet transform of the electroencephalographic patterns and photostimulus can be markers of mild cognitive impairment.

С помощью методов вейвлетного и мультифрактального анализа показано, что при двигательных нарушениях в структуре паттернов непроизвольных колебаний руки человека, возникающих в процессе выполнения им двигательной задачи, появляются перестройки, сопровождающиеся возникновением долговременных корреляций между последовательными значениями этих колебаний. Эти перестройки являются причиной значительного повышения амплитуды непроизвольных колебаний руки человека с болезнью Паркинсона, по сравнению с такими колебаниями руки здорового человека. Механизм возникновения коррелированной динамики связан с повышением вклада сильных флуктуаций последовательных значений непроизвольных колебаний. Уменьшение амплитуды этих колебаний и энергии их вейвлетного спектра на фоне приема антипаркинсонических препаратов сопровождается уменьшением долговременных корреляций и смещением мультифрактальных характеристик в диапазон, характерный для здорового человека.

Wavelet and multifractal analysis has shown that alterations in oscillatory activity accompanied by long-term correlations between successive values of these oscillations occur as a result of movement disorders in the structure of patterns of involuntary oscillatory hand movements that arise during motor task performance. These alterations cause a significant increase in the variation of the amplitude of involuntary oscillatory hand movements in an individual with Parkinson’s disease compared to healthy individuals. The mechanism of the appearance of correlated dynamics is associated with an increase in the contribution of strong fluctuations of successive values of involuntary oscillations. A decrease in the variation of the amplitude of these oscillations and the energy of their wavelet spectrum in association with antiparkinsonian drugs is accompanied by a decrease in long-term correlations; multifractal characteristics tend to be attributed to the range that is characteristic of healthy individuals.

The mechanisms of transition from the rhythmic to bursting activity upon a change in the external stimulating current are investigated in the membrane model of a nociceptive neuron. It is found that the saddle-node bifurcation of the limit cycle in the structure of the bifurcation diagram of the fast subsystem and the torus bifurcation in the structure of the bifurcation diagram of the total system lead to the emergence of particular solutions of the torus canards type in these transitions. This confirms the assumption that torus canards-type solutions are a common feature in transitions between rhythmic activity and bursts.

A bifurcation analysis of a nociceptive neuron model was performed to study how the firing activity pattern changes when an antinociceptive response to damaging pain stimulation arises in rat dorsal ganglia. Ectopic train activity was found to arise in the model. Suppression of train activity was demonstrated to proceed solely through modification of the activation gating structure of the Na V 1.8 slow sodium channel in response to comenic acid, which exerts an analgesic effect and is an active ingredient of the new nonopioid analgesic Anoceptin.

A model of the membrane of a nociceptive neuron from a rat dorsal ganglion has been used to address the problem of analyzing the regulation of nociceptive signals by 5-hydroxy-γ-pyrone-2-carboxylic acid, which is the active pharmaceutic ingredient of the analgesic Anoceptin. The study has applied bifurcation analysis to report the relationship between the values of model parameters and the type of problem solution before and after the parameters change in response to analgesic modulation.

The patch clamp method is used for studying the characteristics of slow sodium channels responsible for coding of nociceptive signals. Quantitative estimates of rate constants of transitions of “normal” and pharmacologically modified activation gating mechanisms of these channels are obtained. A mathematical model of the type of Hogdkin–Huxley nociceptive neuron membrane is constructed. Cometic acid, which is a drug substance of a new nonopioid analgesic, is used as a pharmacological agent. The application of bifurcation analysis makes it possible to outline the boundaries of the region in which a periodic impulse activity is generated. This boundary separates the set of values of the model parameter for which periodic pulsation is observed from the values for which such pulsations are absent or damped. The results show that the finest effect of modulation of physical characteristic of a part of a protein molecule and its effective charge suppresses the excitability of the nociceptive neuron membrane and, hence, leads to rapid reduction of pain.

Методами нелинейной динамики исследованы мультифрактальные и вейвлетные свойства непроизвольных колебаний (тремора), возникающих при выполнении человеком определенной двигательной задачи (при поддержании изометрического усилия пальцами руки). Показано, что вейвлетная характеристика (максимум глобальной энергии вейвлетного спектра) и мультифрактальные параметры (ширина и асимметрия спектра сингулярности) значимо различаются в норме и при двигательной патологии. Получены соотношения между изменением состояния пациентов с болезнью Паркинсона, связанного с лекарственным облегчением клинических признаков паркинсонического тремора, и изменением значений этих показателей. Предложенный аналитический подход к неинвазивному изучению интегративной активности центральной нервной системы, формируемой как моторный выход в ходе реализации двигательной задачи, позволяет не только количественно оценить степень отклонения двигательной функции от нормы, но и помогает клиницисту определиться с выбором оптимального варианта лечения в каждом конкретном случае.