Serge F. Timashev

Anomalous diffusion as a stochastic component in the dynamics of complex processes

Anomalous diffusion, process in which the mean-squared displacement of system states is a non-linear function of time, is usually identified in real stochastic processes by comparing experimental and theoretical displacements at relatively small time intervals. This paper proposes an interpolation expression for the identification of anomalous diffusion in complex signals for the cases when the dynamics of the system under study reaches a steady state (large time intervals). This interpolation expression uses the chaotic difference moment (transient structural function) of the second order as an average characteristic of displacements. A general procedure for identifying anomalous diffusion and calculating its parameters in real stochastic signals, which includes the removal of the regular (low-frequency) components from the source signal and the fitting of the chaotic part of the experimental difference moment of the second order to the interpolation expression, is presented. The procedure was applied to the analysis of the dynamics of magnetoencephalograms, blinking fluorescence of quantum dots, and X-ray emission from accreting objects. For all three applications, the interpolation was able to adequately describe the chaotic part of the experimental difference moment, which implies that anomalous diffusion manifests itself in these natural signals. The results of this study make it possible to broaden the range of complex natural processes in which anomalous diffusion can be identified. The relation between the interpolation expression and a diffusion model, which is derived in the paper, allows one to simulate the chaotic processes in the open complex systems with anomalous diffusion.

Analysis of cross-correlations in electroencephalogram signals as an approach to proactive diagnosis of schizophrenia

We apply flicker-noise spectroscopy (FNS), a time series analysis method operating on structure functions and power spectrum estimates, to study the clinical electroencephalogram (EEG) signals recorded in children/adolescents (11 to 14 years of age) with diagnosed schizophrenia-spectrum symptoms at the National Center for Psychiatric Health (NCPH) of the Russian Academy of Medical Sciences. The EEG signals for these subjects were compared with the signals for a control sample of chronically depressed children/adolescents. The purpose of the study is to look for diagnostic signs of subjects’ susceptibility to schizophrenia in the FNS parameters for specific electrodes and cross-correlations between the signals simultaneously measured at different points on the scalp. Our analysis of EEG signals from scalp-mounted electrodes at locations F3 and F4, which are symmetrically positioned in the left and right frontal areas of cerebral cortex, respectively, demonstrates an essential role of frequency–phase synchronization, a phenomenon representing specific correlations between the characteristic frequencies and phases of excitations in the brain. We introduce quantitative measures of frequency–phase synchronization and systematize the values of FNS parameters for the EEG data. The comparison of our results with the medical diagnoses for 84 subjects performed at NCPH makes it possible to group the EEG signals into 4 categories corresponding to different risk levels of subjects’ susceptibility to schizophrenia. We suggest that the introduced quantitative characteristics and classification of cross-correlations may be used for the diagnosis of schizophrenia at the early stages of its development.

Regular and stochastic behavior of Parkinsonian pathological tremor signals

Regular and stochastic behavior in the time series of Parkinsonian pathological tremor velocity is studied on the basis of the statistical theory of discrete non-Markov stochastic processes and flicker-noise spectroscopy. We have developed a new method of analyzing and diagnosing Parkinsons disease (PD) by taking into consideration discreteness, fluctuations, long- and short-range correlations, regular and stochastic behavior, Markov and non-Markov effects and dynamic alternation of relaxation modes in the initial time signals. The spectrum of the statistical non-Markovity parameter reflects Markovity and non-Markovity in the initial time series of tremor. The relaxation and kinetic parameters used in the method allow us to estimate the relaxation scales of diverse scenarios of the time signals produced by the patient in various dynamic states. The local time behavior of the initial time correlation function and the first point of the non-Markovity parameter give detailed information about the variation of pathological tremor in the local regions of the time series. The obtained results can be used to find the most effective method of reducing or suppressing pathological tremor in each individual case of a PD patient. Generally, the method allows one to assess the efficacy of the medical treatment for a group of PD patients.

Analysis of biomedical signals by flicker-noise spectroscopy: Identification of photosensitive epilepsy using magnetoencephalograms

The flicker-noise spectroscopy (FNS) approach is used to determine the dynamic characteristics of neuromagnetic responses by analyzing the magnetoencephalographic (MEG) signals recorded as the response of a group of control human subjects and a patient with photosensitive epilepsy (PSE) to equiluminant flickering stimuli of different color combinations. Parameters characterizing the analyzed stochastic biomedical signals for different frequency bands are identified. It is shown that the classification of the parameters of analyzed MEG responses with respect to different frequency bands makes it possible to separate the contribution of the chaotic component from the overall complex dynamics of the signals. It is demonstrated that the chaotic component can be adequately described by the anomalous diffusion approximation in the case of control subjects. On the other hand, the chaotic component for the patient is characterized by a large number of high-frequency resonances. This implies that healthy organisms can suppress the perturbations brought about by the flickering stimuli and reorganize themselves. The organisms affected by photosensitive epilepsy no longer have this ability. This result also gives a way to simulate the separate stages of the brain cortex activity in vivo. The examples illustrating the use of the “FNS device” for identifying even the slightest individual differences in the activity of human brains using their responses to external standard stimuli show a unique possibility to develop the “individual medicine” of the future.

Identification of earthquake precursors in the hydrogeochemical and geoacoustic data for the Kamchatka peninsula by flicker-noise spectroscopy

Abstract. A phenomenological systems approach for identifying potential precursors in multiple signals of different types for the same local seismically active region is proposed based on the assumption that a large earthquake may be preceded by a system reconfiguration (preparation) on different time and space scales. A nonstationarity factor introduced within the framework of flicker-noise spectroscopy, a statistical physics approach to the analysis of time series, is used as the dimensionless criterion for detecting qualitative (precursory) changes within relatively short time intervals in arbitrary signals. Nonstationarity factors for chlorine-ion concentration variations in the underground water of two boreholes on the Kamchatka peninsula and geacoustic emissions in a deep borehole within the same seismic zone are studied together in the time frame around a large earthquake on 8 October 2001. It is shown that nonstationarity factor spikes (potential precursors) take place in the interval from 70 to 50 days before the earthquake for the hydrogeochemical data and at 29 and 6 days in advance for the geoacoustic data.

Fundamentals of fluctuation metrology

The possibility in principle of extending metrological concepts to the characteristics of complex objects, the primary information about the state or structure of which is presented in the form of complex chaotic dependences and cannot be expressed using standard metrological images such as directly measured time and length and other dimensional values, is shown. To correctly characterize the dynamic state of such complex objects, including states of objects during nonstationary evolution, or the special features of structures formed under the conditions of external actions of various intensities, it is necessary, first, to introduce autocorrelation dependences averaged over time or spatial intervals on the basis of measured dynamic variables and, next, to use these dependences to find sets of information parameters, which can be presented as metrological characteristics of the dynamic state under study or spatial image to be analyzed. The phenomenological basis of the corresponding analysis is provided by flicker-noise spectroscopy with its possibilities of developing procedures and algorithms that can be used to obtain metrological characteristics over various frequency (time and spatial) ranges of the signals analyzed. This is the basis on which unity of metrological characteristic measurements with a determined uncertainty (error) in measurements can be achieved, standards and reference samples of fluctuation metrology can be created, and methods for the transfer of standard parameters from standards to reference samples and then to working measurement instruments can be developed. This opens up the possibility for solving many practical problems of microelectronics, energetics, nanoindustry, chemical technology, which include standardization of the state of complex systems and articles of various functional purposes, and the quality of products created.

Frequency and phase synchronization in neuromagnetic cortical responses to flickering-color stimuli

In our earlier study dealing with the analysis of neuromagnetic responses (magnetoencephalograms—MEG) to flickering-color stimuli for a group of control human subjects (9 volunteers) and a patient with photosensitive epilepsy (a 12-year old girl), it was shown that Flicker-Noise Spectroscopy (FNS) was able to identify specific differences in the responses of each organism. The high specificity of individual MEG responses manifested itself in the values of FNS parameters for both chaotic and resonant components of the original signal. The present study applies the FNS cross-correlation function to the analysis of correlations between the MEG responses simultaneously measured at spatially separated points of the human cortex processing the red-blue flickering color stimulus. It is shown that the cross-correlations for control (healthy) subjects are characterized by frequency and phase synchronization at different points of the cortex, with the dynamics of neuromagnetic responses being determined by the low-frequency processes that correspond to normal physiological rhythms. But for the patient, the frequency and phase synchronization breaks down, which is associated with the suppression of cortical regulatory functions when the flickering-color stimulus is applied, and higher frequencies start playing the dominating role. This suggests that the disruption of correlations in the MEG responses is the indicator of pathological changes leading to photosensitive epilepsy, which can be used for developing a method of diagnosing the disease based on the analysis with the FNS cross-correlation function.

Stochastic Variability in X-Ray Emission from the Black Hole Binary GRS 1915+105

We examine stochastic variability in the dynamics of X-ray emission from the black hole system GRS 1915+105, a strongly variable microquasar commonly used for studying relativistic jets and the physics of black hole accretion. The analysis of sample observations for 13 different states in both soft (low) and hard (high) energy bands is performed by flicker-noise spectroscopy (FNS), a phenomenological time series analysis method operating on structure functions and power spectrum estimates. We find the values of FNS parameters, including the Hurst exponent, flicker-noise parameter, and characteristic timescales, for each observation based on multiple 2500 s continuous data segments. We identify four modes of stochastic variability driven by dissipative processes that may be related to viscosity fluctuations in the accretion disk around the black hole: random (RN), power-law (1F), one-scale (1S), and two-scale (2S). The variability modes are generally the same in soft and hard energy bands of the same observation. We discuss the potential for future FNS studies of accreting black holes.

Dynamics of stainless steel turning: Analysis by flicker-noise spectroscopy

We use flicker-noise spectroscopy (FNS), a phenomenological method for the analysis of time and spatial series operating on structure functions and power spectrum estimates, to identify and study harmful chatter vibrations in a regenerative turning process. The 3D cutting force components experimentally measured during stainless steel turning are analyzed, and the parameters of their stochastic dynamics are estimated. Our analysis shows that the system initially exhibiting regular vibrations associated with spindle rotation becomes unstable to high-frequency noisy oscillations (chatter) at larger cutting depths. We suggest that the chatter may be attributed to frictional stick-and-slip interactions between the contact surfaces of cutting tool and workpiece. We compare our findings with previously reported results obtained by statistical, recurrence, multifractal, and wavelet methods. We discuss the potential of FNS in monitoring the turning process in manufacturing practice.

Evolution of the Dynamic Dissipative Systems as a Temporal “Colour” Fractal

New possibilities in elaborating the evolution of dynamical dissipative system evolution are associated with the ideas of nonlinear system dynamics and deterministic chaos1, 2, 3, 4. In this paper a new phenomenological method for analysis of non-linear dissipative system dynamics, which may be called flicker-noise spectroscopy5, 6, 7, 8, is presented. An algorithm is developed which allows us to obtain as many phenomenological parameters — “passport data” — as necessary for the description and characterization of the dynamic system’s state and the changes of its state during evolution. We demonstrate the application of this approach for analysis of the time series by describing various phenomena. Spatial space structures may be examined in a similar manner.