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Experimental evidence of anomalous phase synchronization in two diffusively coupled Chua oscillators
(2006)
We study the transition to phase synchronization in two diffusively coupled, nonidentical Chua oscillators. In the experiments, depending on the used parameterization, we observe several distinct routes to phase synchronization, including states of either in-phase, out-of-phase, or antiphase synchronization, which may be intersected by an intermediate desynchronization regime with large fluctuations of the frequency difference. Furthermore, we report the first experimental evidence of an anomalous transition to phase synchronization, which is characterized by an initial enlargement of the natural frequency difference with coupling strength. This results in a maximal frequency disorder at intermediate coupling levels, whereas usual phase synchronization via monotonic decrease in frequency difference sets in only for larger coupling values. All experimental results are supported by numerical simulations of two coupled Chua models
Extreme Rainfall of the South American Monsoon System: A Dataset Comparison Using Complex Networks
(2015)
In this study, the authors compare six different rainfall datasets for South America with a focus on their representation of extreme rainfall during the monsoon season (December February): the gauge-calibrated TRMM 3B42 V7 satellite product; the near-real-time TRMM 3B42 V7 RT, the GPCP 1 degrees daily (1DD) V1.2 satellite gauge combination product, the Interim ECMWF Re-Analysis (ERA-Interim) product; output of a high-spatial-resolution run of the ECHAM6 global circulation model; and output of the regional climate model Eta. For the latter three, this study can be understood as a model evaluation. In addition to statistical values of local rainfall distributions, the authors focus on the spatial characteristics of extreme rainfall covariability. Since traditional approaches based on principal component analysis are not applicable in the context of extreme events, they apply and further develop methods based on complex network theory. This way, the authors uncover substantial differences in extreme rainfall patterns between the different datasets: (i) The three model-derived datasets yield very different results than the satellite gauge combinations regarding the main climatological propagation pathways of extreme events as well as the main convergence zones of the monsoon system. (ii) Large discrepancies are found for the development of mesoscale convective systems in southeastern South America. (iii) Both TRMM datasets and ECHAM6 indicate a linkage of extreme rainfall events between the central Amazon basin and the eastern slopes of the central Andes, but this pattern is not reproduced by the remaining datasets. The authors' study suggests that none of the three model-derived datasets adequately captures extreme rainfall patterns in South America.
Recurrence-plot-based recurrence networks are an approach used to analyze time series using a complex networks theory. In both approaches - recurrence plots and recurrence networks -, a threshold to identify recurrent states is required. The selection of the threshold is important in order to avoid bias of the recurrence network results. In this paper, we propose a novel method to choose a recurrence threshold adaptively. We show a comparison between the constant threshold and adaptive threshold cases to study period-chaos and even period-period transitions in the dynamics of a prototypical model system. This novel method is then used to identify climate transitions from a lake sediment record.
We study prebifurcation fluctuation amplification in nonlinear oscillators subject to bifurcations of spontaneous symmetry breaking which are manifest in the doubling of stable equilibrium states. Our theoretical estimates of both the linear growth and the nonlinear saturation of the fluctuations are in good agreement with our results from numerical simulations. We show that in the saturation mode, the fluctuation variance is proportional to the standard deviation of the external noise, whereas in the linear mode, the fluctuation variance is proportional to the noise variance. It is demonstrated that the phenomenon of prebifurcation noise amplification is more pronounced in the case of a slow transition through the bifurcation point. The amplification of fluctuations in this case makes it easier to form a symmetric probability of the final equilibrium states. In contrast, for a fast transition through the bifurcation point, the effect of amplification is much less pronounced. Under backward and forward passages through the bifurcation point, a loop of noise-dependent hysteresis emerges here. We find that for a fast transition of the nonlinear oscillator through the bifurcation point, the probability symmetry of the final equilibrium states is destroyed
We give evidence of frequency entrainment of dominant peaks in the chaotic spectra of two coupled chaotic nonautonomous oscillators. At variance with the autonomous case, the phenomenon is here characterized by the vanishing of a previously positive Lyapunov exponent in the spectrum, which takes place for a broad range of the coupling strength parameter. Such a state is studied also for the case of chaotic oscillators with ill-defined phases due to the absence of a unique center of rotation. Different phase synchronization indicators are used to circumvent this difficulty
We study frequency selectivity in noise-induced subthreshold signal processing in a system with many noise- supported stochastic attractors which are created due to slow variable diffusion between identical excitable elements. Such a coupling provides coexisting of several average periods distinct from that of an isolated oscillator and several phase relations between elements. We show that the response of the coupled elements under different noise levels can be significantly enhanced or reduced by forcing some elements in resonance with these new frequencies which correspond to appropriate phase relations
We study synchronization transitions in a system of two coupled self-sustained chaotic oscillators. We demonstrate that with the increase of coupling strength the system first undergoes the transition to phase synchronization. With a further increase of coupling, a new synchronous regime is observed, where the states of two oscillators are nearly identical, but one system lags in time to the other. We describe thisregime as a state with correlated amplitudes and a constant phase shift. These transitions are traced in the Lyapunov spectrum.
In this paper we present an approach to recover the dynamics from recurrences of a system and then generate (multivariate) twin surrogate (TS) trajectories. In contrast to other approaches, such as the linear-like surrogates, this technique produces surrogates which correspond to an independent copy of the underlying system, i. e. they induce a trajectory of the underlying system visiting the attractor in a different way. We show that these surrogates are well suited to test for complex synchronization, which makes it possible to systematically assess the reliability of synchronization analyses. We then apply the TS to study binocular fixational movements and find strong indications that the fixational movements of the left and right eye are phase synchronized. This result indicates that there might be one centre only in the brain that produces the fixational movements in both eyes or a close link between two centres.
In this paper, we present an approach to recover the dynamics from recurrences of a system and then generate (multivariate) twin surrogate (TS) trajectories. In contrast to other approaches, such as the linear-like surrogates, this technique produces surrogates which correspond to an independent copy of the underlying system, i.e. they induce a trajectory of the underlying system visiting the attractor in a different way. We show that these surrogates are well suited to test for complex synchronization, which makes it possible to systematically assess the reliability of synchronization analyses. We then apply the TS to study binocular fixational movements and find strong indications that the fixational movements of the left and right eye are phase synchronized. This result indicates that there might be only one centre in the brain that produces the fixational movements in both eyes or a close link between the two centres.
We report the identification of global phase synchronization (GPS) in a linear array of unidirectionally coupled Mackey-Glass time-delay systems exhibiting highly non-phase-coherent chaotic attractors with complex topological structure. In particular, we show that the dynamical organization of all the coupled time-delay systems in the array to form GPS is achieved by sequential synchronization as a function of the coupling strength. Further, the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster. We have confirmed these results by estimating instantaneous phases including phase difference, average phase, average frequency, frequency ratio, and their differences from suitably transformed phase coherent attractors after using a nonlinear transformation of the original non-phase-coherent attractors. The results are further corroborated using two other independent approaches based on recurrence analysis and the concept of localized sets from the original non-phase-coherent attractors directly without explicitly introducing the measure of phase.
Sensory information entering the nervous system follows independent paths of processing such that specific features are individually detected. However, sensory perception, awareness, and cognition emerge from the combination of information. Here we have analyzed the corticocortical network of the cat, looking for the anatomical substrate which permits the simultaneous segregation and integration of information in the brain. We find that cortical communications are mainly governed by three topological factors of the underlying network: (i) a large density of connections, (ii) segregation of cortical areas into clusters, and (iii) the presence of highly connected hubs aiding the multisensory processing and integration. Statistical analysis of the shortest paths reveals that, while information is highly accessible to all cortical areas, the complexity of cortical information processing may arise from the rich and intricate alternative paths in which areas can influence each other.
How do diverse dynamical patterns arise from the topology of complex networks? We study synchronization dynamics in the cortical brain network of the cat, which displays a hierarchically clustered organization, by modeling each node (cortical area) with a subnetwork of interacting excitable neurons. We find that in the biologically plausible regime the dynamics exhibits a hierarchical modular organization, in particular, revealing functional clusters coinciding with the anatomical communities at different scales. Our results provide insights into the relationship between network topology and functional organization of complex brain networks.
We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function. (C) 2006 American Institute of Physics
Higher resilience to climatic disturbances in tropical vegetation exposed to more variable rainfall
(2019)
With ongoing global warming, the amount and frequency of precipitation in the tropics is projected to change substantially. While it has been shown that tropical forests and savannahs are sustained within the same intermediate mean annual precipitation range, the mechanisms that lead to the resilience of these ecosystems are still not fully understood. In particular, the long-term impact of rainfall variability on resilience is as yet unclear. Here we present observational evidence that both tropical forest and savannah exposed to a higher rainfall variability-in particular on interannual scales-during their long-term past are overall more resilient against climatic disturbances. Based on precipitation and tree cover data in the Brazilian Amazon basin, we constructed potential landscapes that enable us to systematically measure the resilience of the different ecosystems. Additionally, we infer that shifts from forest to savannah due to decreasing precipitation in the future are more likely to occur in regions with a precursory lower rainfall variability. Long-term rainfall variability thus needs to be taken into account in resilience analyses and projections of vegetation response to climate change.
We investigate a network of influences connected to global mean temperature. Considering various climatic factors known to influence global mean temperature, we evaluate not only the impacts of these factors on temperature but also the directed dependencies among the factors themselves. Based on an existing recurrence-based connectivity measure, we propose a new and more general measure that quantifies the level of dependence between two time series based on joint recurrences at a chosen time delay. The measures estimated in the analysis are tested for statistical significance using twin surrogates. We find, in accordance with earlier studies, the major drivers for global mean temperature to be greenhouse gases, ENSO, volcanic activity, and solar irradiance. We further uncover a feedback between temperature and ENSO. Our results demonstrate the need to involve multiple, delayed interactions within the drivers of temperature in order to develop a more thorough picture of global temperature variations.
Recurrence plots have recently been recognized as a powerful tool for the analysis of data. Not only the visualization of structures of the time series but also the possibility to estimate invariants from them and the possibility to analyze non-stationary data sets are remarkable. However, the question of how much information is encoded in such a two-dimensional and binary representation has not been discussed so far. In this Letter we show that-under some conditions-it is possible to reconstruct an attractor from the recurrence plot, at least topologically. This means that all relevant dynamical information is contained in the plot. (C) 2004 Elsevier B.V. All rights reserved
Human comment is studied using data from 'tianya' which is one of the most popular on-line social systems in China. We found that the time interval between two consecutive comments on the same topic, called inter-event time, follows a power-law distribution. This result shows that there is no characteristic decay time on a topic. It allows for very long periods without comments that separate bursts of intensive comments. Furthermore, the frequency of a different ID commenting on a topic also follows a power-law distribution. It indicates that there are some "hubs" in the topic who lead the direction of the public opinion. Based on the personal comments habit, a model is introduced to explain these phenomena. The numerical simulations of the model fit well with the empirical results. Our findings are helpful for discovering regular patterns of human behavior in on-line society and the evolution of the public opinion on the virtual as well as real society.
The method of twin surrogates has been introduced to test for phase synchronization of complex systems in the case of passive experiments. In this paper we derive new analytical expressions for the number of twins depending on the size of the neighborhood, as well as on the length of the trajectory. This allows us to determine the optimal parameters for the generation of twin surrogates. Furthermore, we determine the quality of the twin surrogates with respect to several linear and nonlinear statistics depending on the parameters of the method. In the second part of the paper we perform a hypothesis test for phase synchronization in the case of experimental data from fixational eye movements. These miniature eye movements have been shown to play a central role in neural information processing underlying the perception of static visual scenes. The high number of data sets (21 subjects and 30 trials per person) allows us to compare the generated twin surrogates with the "natural" surrogates that correspond to the different trials. We show that the generated twin surrogates reproduce very well all linear and nonlinear characteristics of the underlying experimental system. The synchronization analysis of fixational eye movements by means of twin surrogates reveals that the synchronization between the left and right eye is significant, indicating that either the centers in the brain stem generating fixational eye movements are closely linked, or, alternatively that there is only one center controlling both eyes.
The analysis of palaeoclimate time series is usually affected by severe methodological problems, resulting primarily from non-equidistant sampling and uncertain age models. As an alternative to existing methods of time series analysis, in this paper we argue that the statistical properties of recurrence networks - a recently developed approach - are promising candidates for characterising the system's nonlinear dynamics and quantifying structural changes in its reconstructed phase space as time evolves. In a first order approximation, the results of recurrence network analysis are invariant to changes in the age model and are not directly affected by non-equidistant sampling of the data. Specifically, we investigate the behaviour of recurrence network measures for both paradigmatic model systems with non-stationary parameters and four marine records of long-term palaeoclimate variations. We show that the obtained results are qualitatively robust under changes of the relevant parameters of our method, including detrending, size of the running window used for analysis, and embedding delay. We demonstrate that recurrence network analysis is able to detect relevant regime shifts in synthetic data as well as in problematic geoscientific time series. This suggests its application as a general exploratory tool of time series analysis complementing existing methods.
The effect of additive noise on transitions in nonlinear systems far from equilibrium is studied. It is shown that additive noise in itself can induce a hidden phase transition, which is similar to the transition induced by multiplicative noise in a nonlinear oscillator [P. Landa and A. Zaikin, Phys. Rev. E 54, 3535 (1996)]. Investigation of different nonlinear models that demonstrate phase transitions induced by multiplicative noise shows that the influence of additive noise upon such phase transitions can be crucial: additive noise can either blur such a transition or stabilize noise-induced oscillations.
A new globally uniform Lagrangian transport scheme for large ensembles of passive tracer particles is presented and applied to wind data from a coupled atmosphere-ocean climate model that includes interactive dynamical feedback with stratospheric chemistry. This feedback from the chemistry is found to enhance large-scale meridional air mass exchange in the northern winter stratosphere as well as intrusion of stratospheric air into the troposphere, where both effects are due to a weakened polar vortex.
We show many versatile phase synchronous configurations that emerge in an array of coupled chaotic elements due to the presence of a periodic stimulus. Then, we explain the relevance of these configurations to the understanding of how information about such a. stimulus is transmitted from one side to the other in this array. The stimulus actively creates the ways to be transmitted, by making the chaotic elements to phase synchronize
Identifying causal links (couplings) is a fundamental problem that facilitates the understanding of emerging structures in complex networks. We propose and analyze inner composition alignment-a novel, permutation-based asymmetric association measure to detect regulatory links from very short time series, currently applied to gene expression. The measure can be used to infer the direction of couplings, detect indirect (superfluous) links, and account for autoregulation. Applications to the gene regulatory network of E. coli are presented.
Correlations, as observed between the concentrations of metabolites in a biological sample, may be used to gain additional information about the physiological state of a given tissue. in this mini-review, we discuss the integration of these observed correlations into metabolomic networks and their relationships with the underlying biochemical pathways
We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincare map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents. (C) 2004 Elsevier Ltd. All rights reserved
Objectives: Scoring sleep visually based on polysomnography is an important but time-consuming element of sleep medicine. Where-as computer software assists human experts in the assignment of sleep stages to polysomnogram epochs, their performance is usually insufficient. This study evaluates the possibility to fully automatize sleep staging considering the reliability of the sleep stages available from human expert sleep scorers. Methods: We obtain features from EEG, ECG and respiratory signals of polysomnograms from ten healthy subjects. Using the sleep stages provided by three human experts, we evaluate the performance of linear discriminant analysis on the entire polysomnogram and:only on epochs where the three experts agree in their-sleep stage scoring. Results: We show that in polysomnogram intervals, to which all three scorers assign the same sleep stage, our algorithm achieves 90% accuracy. This high rate of agreement with the human experts is accomplished with only a small set of three frequency features from the EEG. We increase-the performance to 93% by including ECG and respiration features. In contrast, on intervals of ambiguous sleep stage, the sleep stage classification obtained from our algorithm, agrees with the human consensus scorer in approximately 61%. Conclusions: These findings suggest that machine classification is highly consistent with human sleep staging and that error in the algorithm's assignments is rather a problem of lack of well-defined criteria for human experts to judge certain polysomnogram epochs than an insufficiency of computational procedures
The incidence of cardiovascular diseases increases with the growth of the human population and an aging society, leading to very high expenses in the public health system. Therefore, it is challenging to develop sophisticated methods in order to improve medical diagnostics. The question whether the normal heart rate is chaotic or not is an attempt to elucidate the underlying mechanisms of cardiovascular dynamics and therefore a highly controversial topical challenge. In this contribution we demonstrate that linear and nonlinear parameters allow us to separate completely the data sets of the three groups provided for this controversial topic in nonlinear dynamics. The question whether these time series are chaotic or not cannot be answered satisfactorily without investigating the underlying mechanisms leading to them. We give an example of the dominant influence of respiration on heart beat dynamics, which shows that observed fluctuations can be mostly explained by respiratory modulations of heart rate and blood pressure (coefficient of determination: 96%). Therefore, we recommend reformulating the following initial question: "Is the normal heart rate chaotic?" We rather ask the following: " Is the normal heart rate 'chaotic' due to respiration?"
The 300 year record of the yearly sunspot numbers and numerically generated trajectory of the solar inertial motion (SIM) were subjects of a synchronization analysis. Phase synchronization of the sunspot cycle and a fast component of the SIM have been found and confirmed with statistical significance in three epochs (1727-1757, 1802-1832 and 1863-1922) of the entire 1700-1997 record. This result can be considered as a quantitative support for the hypothesis that there is a weak interaction of gravity and solar activity.
In this work, we reanalyze the heart rate variability (HRV) data from the 2002 Computers in Cardiology (CiC) Challenge using the concept of large-scale dimension densities and additionally apply this technique to data of healthy persons and of patients with cardiac diseases. The large-scale dimension density (LASDID) is estimated from the time series using a normalized Grassberger-Procaccia algorithm, which leads to a suitable correction of systematic errors produced by boundary effects in the rather large scales of a system. This way, it is possible to analyze rather short, nonstationary, and unfiltered data, such as HRV. Moreover, this method allows us to analyze short parts of the data and to look for differences between day and night. The circadian changes in the dimension density enable us to distinguish almost completely between real data and computer-generated data from the CiC 2002 challenge using only one parameter. In the second part we analyzed the data of 15 patients with atrial fibrillation (AF), 15 patients with congestive heart failure (CHF), 15 elderly healthy subjects (EH), as well as 18 young and healthy persons (YH). With our method we are able to separate completely the AF (rho(mu)(ls)=0.97 +/- 0.02) group from the others and, especially during daytime, the CHF patients show significant differences from the young and elderly healthy volunteers (CHF, 0.65 +/- 0.13; EH, 0.54 +/- 0.05; YH, 0.57 +/- 0.05; p < 0.05 for both comparisons). Moreover, for the CHF patients we find no circadian changes in rho(mu)(ls) (day, 0.65 +/- 0.13; night, 0.66 +/- 0.12; n.s.) in contrast to healthy controls (day, 0.54 +/- 0.05; night, 0.61 +/- 0.05; p=0.002). Correlation analysis showed no statistical significant relation between standard HRV and circadian LASDID, demonstrating a possibly independent application of our method for clinical risk stratification
During the last glacial period, climate records from the North Atlantic region exhibit a pronounced spectral component corresponding to a period of about 1470 years, which has attracted much attention. This spectral peak is closely related to the recurrence pattern of Dansgaard-Oeschger (DO) events. In previous studies a red noise random process, more precisely a first-order autoregressive (AR1) process, was used to evaluate the statistical significance of this peak, with a reported significance of more than 99%. Here we use a simple mechanistic two-state model of DO events, which itself was derived from a much more sophisticated ocean-atmosphere model of intermediate complexity, to numerically evaluate the spectral properties of random (i.e., solely noise-driven) events. This way we find that the power spectral density of random DO events differs fundamentally from a simple red noise random process. These results question the applicability of linear spectral analysis for estimating the statistical significance of highly non-linear processes such as DO events. More precisely, to enhance our scientific understanding about the trigger of DO events, we must not consider simple "straw men" as, for example, the AR1 random process, but rather test against realistic alternative descriptions.
Recurrence plots exhibit line structures which represent typical behaviour of the investigated system. The local slope of these line structures is connected with a specific transformation of the time scales of different segments of the phase-space trajectory. This provides us a better understanding of the structures occurring in recurrence plots. The relationship between the time-scales and line structures are of practical importance in cross recurrence plots. Using this relationship within cross recurrence plots, the time-scales of differently sampled or time- transformed measurements can be adjusted. An application to geophysical measurements illustrates the capability of this method for the adjustment of time-scales in different measurements. (C) 2005 Elsevier B.V. All rights reserved
We analyze the variability in the x-ray lightcurves of the black hole candidate Cygnus X-1 by linear and nonlinear time series analysis methods. While a linear model describes the overall second order properties of the observed data well, surrogate data analysis reveals a significant deviation from linearity. We discuss the relation between shot noise models usually applied to analyze these data and linear stochastic autoregressive models. We debate statistical and interpretational issues of surrogate data testing for the present context. Finally, we suggest a combination of tools from linear and nonlinear time series analysis methods as a procedure to test the predictions of astrophysical models on observed data.
Locking-based frequency measurement and synchronization of chaotic oscillators with complex dynamics
(2002)
Aims. Sunspot distribution in the northern and southern solar hemispheres exibit striking synchronous behaviour on the scale of a Schwabe cycle. However, sometimes the bilateral symmetry of the Butterfly diagram relative to the solar equatorial plane breaks down. The investigation of this phenomenon is important to explaining the almost-periodic behaviour of solar cycles. Methods. We use cross-recurrence plots for the study of the time-varying phase asymmetry of the northern and southern hemisphere and compare our results with the latitudinal distribution of the sunspots. Results. We observe a long-term persistence of phase leading in one of the hemispheres, which lasts almost 4 solar cycles and probably corresponds to the Gleissberg cycle. Long-term variations in the hemispheric-leading do not demonstrate clear periodicity but are strongly anti-correlated with the long-term variations in the magnetic equator.
We present conditions for the local and global synchronizations in coupled-map networks using the matrix measure approach. In contrast to many existing synchronization conditions, the proposed synchronization criteria do not depend on the solution of the synchronous state and give less limitation on the network connections. Numerical simulations of the coupled quadratic maps demonstrate the potentials of our main results.
Phase synchronization is an important phenomenon that occurs in a wide variety of complex oscillatory processes. Measuring phase synchronization can therefore help to gain fundamental insight into nature. In this Letter we point out that synchronization analysis techniques can detect spurious synchronization, if they are fed with a superposition of signals such as in electroencephalography or magnetoencephalography data. We show how techniques from blind source separation can help to nevertheless measure the true synchronization and avoid such pitfalls
We study several algorithms to simulate bone mass loss in two-dimensional and three-dimensional computed tomography bone images. The aim is to extrapolate and predict the bone loss, to provide test objects for newly developed structural measures, and to understand the physical mechanisms behind the bone alteration. Our bone model approach differs from those already reported in the literature by two features. First, we work with original bone images, obtained by computed tomography (CT); second, we use structural measures of complexity to evaluate bone resorption and to compare it with the data provided by CT. This gives us the possibility to test algorithms of bone resorption by comparing their results with experimentally found dependencies of structural measures of complexity, as well as to show efficiency of the complexity measures in the analysis of bone models. For two-dimensional images we suggest two algorithms, a threshold algorithm and a virtual slicing algorithm. The threshold algorithm simulates bone resorption on a boundary between bone and marrow, representing an activity of osteoclasts. The virtual slicing algorithm uses a distribution of the bone material between several virtually created slices to achieve statistically correct results, when the bone-marrow transition is not clearly defined. These algorithms have been tested for original CT 10 mm thick vertebral slices and for simulated 10 mm thick slices constructed from ten I mm thick slices. For three-dimensional data, we suggest a variation of the threshold algorithm and apply it to bone images. The results of modeling have been compared with CT images using structural measures of complexity in two- and three-dimensions. This comparison has confirmed credibility of a virtual slicing modeling algorithm for two-dimensional data and a threshold algorithm for three-dimensional data