To study the dynamical mechanism which generates Parkinsonian resting tremor, we apply coupling directionality analysis to local field potentials (LFP) and accelerometer signals recorded in an ensemble of 48 tremor epochs in four Parkinsonian patients with depth electrodes implanted in the ventro-intermediate nucleus of the thalamus (VIM) or the subthalmic nucleus (STN). Apart from the traditional linear Granger causality method we use two nonlinear techniques: phase dynamics modelling and nonlinear Granger causality. We detect a bidirectional coupling between the subcortical (VIM or STN) oscillation and the tremor, in the theta range (around 5 Hz) as well as broadband (>2 Hz). In particular, we show that the theta band LFP oscillations definitely play an efferent role in tremor generation, while beta band LFP oscillations might additionally contribute. The brain-->tremor driving is a complex, nonlinear mechanism, which is reliably detected with the two nonlinear techniques only. In contrast, the tremor-->brain driving is detected with any of the techniques including the linear one, though the latter is less sensitive. The phase dynamics modelling (applied to theta band oscillations) consistently reveals a long delay in the order of 1-2 mean tremor periods for the brain-->tremor driving and a small delay, compatible with the neural transmission time, for the proprioceptive feedback. Granger causality estimation (applied to broadband signals) does not provide reliable estimates of the delay times, but is even more sensitive to detect the brain-->tremor influence than the phase dynamics modelling.

We investigate synchronization between the low-frequency oscillations of heart rate and blood pressure having in humans a basic frequency close to 0.1 Hz. A method is proposed for quantitative estimation of synchronization between these oscillating processes based on calculation of relative time of phase synchronization of oscillations. It is shown that healthy subjects exhibit on average substantially longer epochs of internal synchronization between the low-frequency oscillations in heart rate and blood pressure than patients after acute myocardial infarction.

We propose a method that allows one to estimate the parameters of model scalar time-delay differential equations from time series. The method is based on a statistical analysis of time intervals between extrema in the time series. We verify our method by using it for the reconstruction of time-delay differential equations from their chaotic solutions and for modeling experimental systems with delay-induced dynamics from their chaotic time series.

A mathematical model is proposed for the autonomic control of cardiovascular system, which takes into account two separated self-exciting sympathetic control loops of heart rate and peripheral vascular tone. The control loops are represented by self-exciting time-delay systems and their tone depends on activity of the aortic, carotid, and lower-body baroreceptors. The model is used to study the dynamics of the adaptive processes that manifest in a healthy cardiovascular system during the passive head-up tilt test. Computer simulation provides continuous observation of the dynamics of the indexes and variables that cannot be measured in the direct experiment, including the noradrenaline concentration in vessel wall and heart muscle, tone of the sympathetic and parasympathetic control, peripheral vascular resistance, and blood pressure. In the supine and upright positions, we estimated the spectral characteristics of the model variables, especially in the low-frequency band, and the original index of total percent of phase synchronization between the low-frequency oscillations in heart rate and blood pressure signals. The model demonstrates good quantitative agreement with the dynamics of the experimentally observed indexes of cardiovascular system that were averaged for 50 healthy subjects.