O. E. Dick

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 plane sections of the boundary have been constructed in the three-parameter spaces (I, gNa,gK), (I, Gm, Gh), (I, Gm, Gn).

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.

Fractal characteristics of the functional state of the brain in patients with anxiuos phobic disorders

The task of estimating the functional state of the CNS with the pain syndrome in patients with anxious phobic disorders is examined. For solving the task, the methods of spectral and multifractal analyses of EEG segments are applied during the perception of psychogenic pain and its removal by the psychorelaxation technique. It has been demonstrated that contrary to power spectra, singularity spectra allow distinguishing EEGs quantitatively in the examined functional states of the brain. Pain suppression in patients with anxious phobic disorders during psychorelaxation is accompanied by changing the width of the singularity spectrum and approximation of the multifractal parameter to the value corresponding to healthy subjects.

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.

Energy and fractal characteristics of physiological and pathological tremors of the human hand

Tremors (involuntary shaking) were compared in healthy subjects and patients with the tremulous form of Parkinsonism sustaining isometric effort of the hand. For this purpose, fractal analysis and multilevel wavelet decomposition of tremor were used, and the energy characteristics determined on the basis of the spectral density of the energy of detailed components obtained at different levels of decomposition. The calculated fractal and energy characteristics of the tremor of healthy subjects were significantly lower than those of patients. If the patients took antiparkinsonian drugs at their usual doses, the characteristics shifted towards normal values. The decrease in the fractal dimension indicates an increase in the strength of the correlation in the dynamics of involuntary shaking. Thus, the characteristics studied allow not only physiological and pathological tremors to be discriminated, but also the strategy of selection of optimal drugs for relieving Parkinsonian tremor to be developed.

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

Abstract. 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.

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.

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) that occur 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 essential tremor syndrome. The physiological tremor was characterized by the lowest amplitude, a broad power spectrum, the lowest energy of the wavelet spectrum, the highest degree of multifractality, the lowest degree of determinism, and the highest entropy of the recurrence time density. In the case of the essential tremor we observed a significant enhancement of the wavelet spectrum energy and a decrease of the oscillation complexity. This was evident via the occurrence of clear peaks in the power spectra, a decrease in the degree of multifractality, the emergence of a quasi-periodic structure in the recurrence diagrams, an increase in determinism and a decrease of the entropy of recurrence time density. All these trends were increased for the parkinsonian tremor data. These characteristics enable us to quantitatively estimate the degree of deviation of motor function from the healthy case.

Nonlinear dynamics of involuntary shaking of the human hand under motor dysfunction

Wavelet and multifractal features of involuntary shaking (tremor) arising during the performance of the motor task (under sustaining isometric effort of fingers of the human hand) have been examined by nonlinear dynamic methods. The wavelet score (the maximum of the global energy of a wavelet spectrum) and multifractal parameters (the width and asymmetry of a singularity spectrum) significantly differ in tremor of healthy subjects and patients with motor dysfunction. The relations between the change of the state of the patients with Parkinson’s disease 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 us 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.

Analysis of changes in EEG patterns in response to rhythmic photic stimulation under various disruptions of the functional state of the central nervous system

Using the wavelet transform of the electroencephalographic (EEG) fragments we have examined for the first time the photic driving reaction for patients with disruptions of the functional state of the central nervous system arising by cerebrovascular disturbances of various intensity degrees. Background fragments of EEG of patients in the vegetovascular dystonia group are characterized by the smaller values of the wavelet spectrum energy in the α range than the fragments in the control group. Patients in the groups of vertebrobasilar insufficiency and arteriosclerotic damage of cerebral blood vessels have much less energy values in δ range. The wavelet analysis of reactive patterns has demonstrated the different photic driving for β, θ and α ranges for the patients of various groups. The study demonstrates the possibility of the wavelet analysis to estimate quantitatively the human brain lability to percept light stimuli. The results can be applied for appropriate choice of treatment of a patient having cerebrovascular disturbances.