Possible mechanism of bursting suppression in nociceptive neurons.

The use of the mathematical model of rat nociceptive neuron membrane allowed us to predict a new mechanism of suppression of ectopic bursting discharges, which arise in neurons of dorsal root ganglia and are one of the causes of neuropathic pain. The treatment with comenic acid leads to switching off the ectopic bursting discharges due to a decrease in the effective charge transferring via the activation gating structure of the slow sodium channels (Na V1.8a). Comenic acid is a drug substance of a new non-opioid analgesic [1] Thus, this analgesic not only reduces the frequency of rhythmic discharges of nociceptive neuron membrane [2] but also it suppresses its ectopic bursting discharges.

[Features of Parkinsonian and essential tremor of the human hand].

Using spectral, wavelet, multifractal, and recurrence analyses we examined the features of involuntary shaking (tremor) arising during the performance of a given motor task. The task was to maintain the efforts of fingers under isometric conditions by a healthy subject, a patient with primary bilateral parkinsonism, and a patient with a syndrome of the essential tremor. The physiological tremor was characterized by the minimal amplitude, the broad power spectrum, the minimal energy of the wavelet spectrum, the maximal degree of multifractality, the minimal degree of determinism, and the maximal entropy of recurrence time density. In the case of the essential tremor we observed the significant enhancement of the wavelet spectrum energy and the decrease of the oscillation complexity. It was evident as the appearance of clear peaks in the power spectra, the fall in the multifractality degree, the emergence of a quasiperiodic structure in the recurrence diagrams, the grow of determinism and the decrease of the entropy of recurrence time density. All these trends were enhanced for the parkinsonian tremor. The suggested characteristics enable us to estimate the degree of deviation of the motor function from the healthy one.

[Nonlinear dynamics of involuntary shaking of the human hand under motor dysfunction].

Using nonlinear dynamic methods we examined wavelet and multifractal features of involuntary shaking (tremor) arising during the performance the motor task (sustaining effort of fingers under isometric conditions). The wavelet score (the maximum of the global wavelet spectrum) and multifractal parameters (the width and asymmetry of the singularity spectrum) significantly differ in tremor of healthy subjects and patients with akinetic-rigid form of Parkinson's disease. The relations between the change of the patient state connected with the drug relief of parkinsonian symptoms and the variations of the parameter values have been obtained. The suggested analytic approach for noninvasive study of integrative activity of the central nervous system, formed as the motor exit during realization of the motor task, enables not only to estimate quantitatively the degree of deviation of the motor function from the healthy one, but it can help to a clinician to choose the optimal treatment in every particular case.

[The Analysis of Changes in EEG Patterns in Response to Transmittent Photic Stimulation in Different Disorders of Central Nervous System].

In this study, for the first time, the photic driving reaction in patients with different disorders of the functional state of central nervous system caused by cerebrovascular conditions of different severity was examined by wavelet analysis of electroencephalographic (EEG) fragments. The background fragments of EEG in the group of patients with vegetative-vascular dystonia, are characterized by lower values of the wavelet spectrum energy in the alpha range than the fragments in the control group. In patients from the groups of vertebrobasilar insufficiency and atherosclerotic damage of cerebral blood vessels, the values of energy in delta range is even lower. The wavelet analysis of reactive patterns has demonstrated the different photic driving of beta, theta and alpha ranges in the patients of various groups. The study demonstrates the possibility of quantitative estimation of the human brain lability of light stimulus perception by the wavelet analysis. The results can be used for the adequate choice of treatment for a patient with cerebrovascular disorder.

Analysis of differences between physiological and pathological tremor of human fingers.

The study tried to reveal the differences between physiological and pathological tremor appearing during maintenance of isometric force in healthy individual and parkinsonian patients with tremor. Six-level wavelet expansion was used presenting the tremor as a sum of smoothed (slow) component and high-frequency detailed components of 6 expansion levels. The differences in the high-frequency components were assessed by calculating the energetic parameters in the power spectrum of these components. These parameters make it possible not only to reliably distinguish the energetic spectra of the detail components of the physiological and pathological tremor, but also to correlate changes in patient state caused by drug treatment of parkinsonian trembling with the decrease in these parameters.

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. The stabilization is carried out both in the absence and in the presence of visual feedback. Changes of experimental conditions are accompanied by essential changes in the amplitude of the involuntary component oscillations.

Changes in sensitivities of amplitude and frequency curves of the Hodgkin-Huxley system produced by variations of maximal conductances.

The question of sensitivity of amplitude and frequency curves of the Hodgkin-Huxley system to variations of maximal values of sodium (gNa) and potassium (gK) conductances is considered. Changes in the sensitivity produced by simultaneous changes in gNa and gK values are examined. It has been shown that simultaneous increase of the values of gNa and gK is accompanied by decrease of the sensitivity and, therefore, by improvement of quality of nerve fiber functioning.

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.

Functional states of an excitable membrane and their dependence on its parameter values.

The property of an excitable membrane of a nerve cell to change the type of electrical activity has been examined with the change of the value of applied current (I). The dependence of this property on the values of the membrane parameters is determined. Two different functional states depending on the values of the membrane parameters are considered. For one of the states a change in the value of I is accompanied by a change in the type of activity (damped periodic oscillations jump to undamped periodic oscillations or vice versa). For the other state the type of activity remains phasic (damped periodic oscillations) for each value of I. For the mathematical model of a membrane we have considered the problem of obtaining the boundary, partitioning the parameter space into the regions to which these functional states correspond. We suggest a mathematical set of this problem and give its algorithm. These boundaries have been constructed for two different variable parameters of the model. A good agreement between the boundaries and the experimental values of sodium and potassium conductances for different excitable membranes has been obtained.

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. Furthermore, it has been demonstrated that there exists the possibility to enhance the maximal conductances of the calcium-dependent currents simultaneously such that even strong variations of their values do not induce changes of the oscillations mode (i.e. it remains bursting).

On the relationship between the number of negative slope regions in the voltage-current curve of the Hodgkin-Huxley model and its parameter values.

We consider the problem of the existence of a negative slope region (NSR) in the voltage-current curve of the neuronal membrane and the relationship between this phenomenon and the membrane parameters. For the Hodgkin-Huxley model it is proposed to determine the dependence of the number of NSR on the values of the maximal sodium (gNa) and potassium (gk) conductances. The method is suggested for constructing the boundaries on the (gNa, gk) plane, in passing through which the number of NSR changes to 1. Using the method we partition the (gNa, gk) plane into the regions corresponding to the curves with the different number of NSR. This number can be changed from 0 to 2 in changing the values of gNa and gk over the physiologically possible range.

Partition of the Hodgkin-Huxley type model parameter space into the regions of qualitatively different solutions.

We have examined the problem of obtaining relationships between the type of stable solutions of the Hodgkin-Huxley type system, the values of its parameters and a constant applied current (I). As variable parameters of the system the maximal Na+(-gNa), K+(-gK) conductances and shifts (Gm, Gh, Gn) of the voltage-dependences have been chosen. To solve this problem it is sufficient to find points belonging to the boundary, partitioning the parameter space of the system into the regions of the qualitatively different types of stable solutions (steady states and stable periodic oscillations). Almost all over the physiological range of I, a type of stable solution is determined by the type of steady state (stable or unstable). Using this fact, the approximate solution of this problem could be obtained by analyzing the spectrum of eigenvalues of the Jacobian matrix for the linearized system. The families of the plan sections of the boundary have been constructed in the three-parameter spaces (I, -gNa, -gK), (I, Gm, Gh), (I, Gm, Gn).

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. The strong point of the method is that it requires the integration of the slow subsystem only. An example of the application of the method for the stomatogastric neuron model [Guckenheimer J, Gueron S, Harris-Warrick RM (1993) Philos Trans R Soc Lond B 341: 345-359] is given.