Method for detecting quantitative differences in the parameters of involuntary effort fluctuations in healthy volunteers and patients with Parkinson's syndrome.

The aim of the study was to detect quantitative differences in the parameters of involuntary effort fluctuations in healthy volunteers and patients with Parkinson's syndrome. Using a mathematical model, the process of isometric effort maintenance was separated into rapid (involuntary) and slow (voluntary) constituents. Significant differences were detected for the attenuation coefficient of fluctuations in the involuntary constituent, but not for the frequency.

Role of signal-dependent noise during maintenance of isometric force.

The assumption that signal-dependent noise during isometric force production controls the stabilization of voluntary isometric force is considered. To verify the assumption the trajectory of isometric force is decomposed into voluntary and involuntary components and the mathematical model describing the relationship between them is developed. It is shown that the integral of an involuntary component (signal-dependent noise) plays the role of the controlling parameter realizing the stabilization of a voluntary component.

Method for studying the processes maintaining equilibrium at rest standing.

According to current concepts, two processes are responsible for the maintenance of body equilibrium: shifts of the reference point caused by changes in the posture and stabilization of oscillations of the center of mass relative to this point. Exact changes in these processes occurring at various forms of locomotor disorders remain unclear. We proposed an original mathematical method allowing separate evaluation of these two processes responsible for the maintenance of body equilibrium.

How is bursting mode maintained under the action of exo- and endogenous factors?

The investigation is concerned with slow oscillations of the intracellular calcium concentration, fast oscillations of the membrane potential, and their interactions during bursting generation. The question associated with functioning of the bursting mode is considered: how is the condition of simultaneous existence of the slow and fast membrane potential oscillations (necessary for the bursting mode) realized under exo- and endogenous actions that result in changes in the neuron membrane parameters? To answer the question the regions of existence of fast, slow and bursting oscillations have been constructed on different parameter planes. The study has revealed that in spite of the fact the regions of the existence of the slow and fast oscillations are sensitive to parameter variations, the condition of the bursting mode existence remains due to invariance of the spatial vicinity of these regions.

Method for constructing the boundary of the bursting oscillations region in the neuron model.

We examine the problem of constructing the boundary of bursting oscillations on a parameter plane for the system of equations describing the electrical behaviour of the membrane neuron arising from the interaction of fast oscillations of the cytoplasma membrane potential and slow oscillations of the intracellular calcium concentration. As the boundary point on the parameter plane we consider the values at which the limit cycle of the slow subsystem is tangent to the Hopf bifurcation curve of the fast subsystem. The method suggested for determining the boundary is based on the dissection of the system variables into slow and fast.