In the modern industrialized countries every year several hundred thousands of people die due to sudden cardiac death. The individual risk for this sudden cardiac death cannot be defined precisely by common available, noninvasive diagnostic tools like Holter monitoring, highly amplified ECG and traditional linear analysis of heart rate variability (HRV). Therefore, we apply some rather unconventional methods of nonlinear dynamics to analyze the HRV. Especially, some complexity measures that are based on symbolic dynamics as well as a new measure, the renormalized entropy, detect some abnormalities in the HRV of several patients who have been classified in the low risk group by traditional methods. A combination of these complexity measures with the parameters in the frequency domain seems to be a promising way to get a more precise definition of the individual risk. These findings have to be validated by a representative number of patients. (c) 1995 American Institute of Physics.

The methods of NLD describe complex rhythm fluctuations and separate structures of non-linear behavior in the heart rate time series more successfully than classical methods of time and frequency domains. This leads to an improved discrimination between a normal (healthy persons) and an abnormal (high risk patients) type of heart beat generation. Some patients with an unknown risk exhibit similar patterns to high risk patients and this suggests a hidden high risk. The methods of symbolic dynamics and renormalized entropy were particularly useful measures for classifying the dynamics of HRV.

A general mechanism of coherence resonance that occurs in noisy dynamical systems close to the onset of bifurcation is demonstrated through examples of period-doubling and torus-birth bifurcations. Near the bifurcation of a periodic orbit, noise produces the characteristic peaks of ''noisy precursors'' in the power spectrum. The signal-to-noise ratio evaluated at these peaks is maximal for a certain optimal noise intensity in a manner that resembles a stochastic resonance. ͓S1063-651X͑97͒06307-1͔ PACS number͑s͒: 05.40.ϩj, 05.20.Ϫy Nonlinear systems perturbed by noise have the potential to display a wide range of complex responses including, somewhat paradoxically, an enhancement of net order and coherence as noise levels increase. A distinguished example of this phenomenon is stochastic resonance ͑SR͒ ͓1͔ which has attracted considerable attention over the last decade ͑see for references the reviews ͓2͔͒. Conventional SR occurs in noisy dynamical systems when perturbed by a weak external periodic signal. For such systems, significant amplification of the weak periodic signal may occur solely by increasing the level of the noise intensity. The signal-to-noise ratio ͑SNR͒, and other appropriate measures of signal coherence, pass through a maximum at an optimal noise strength when the noise-controlled time scale of the system matches the period of the external signal.A similar effect of noise-induced coherence may also be observed in systems which lack an external signal, but whose intrinsic dynamics are controlled by noise intensity. In earlier studies ͓3,4͔ the noise-induced enhancement of coherence in underdamped nonlinear oscillators has been found. The noise-induced peak at zero frequency appeared in the vicinity of a pitchfork bifurcation ͓3͔, whereas the decrease of the width of a fluctuating peak in the power spectrum is shown for an underdamped oscillator, whose eigenfrequency possesses an extreme in energy, in ͓4͔. Recently, a noiseinduced coherent motion has been observed for autonomous systems in ͓5͔, where the effect of noise on a nonuniform limit cycle has been studied, and in ͓6͔, where a coherence resonance in a noise-driven excitable system has been reported. This group of phenomena can be called coherence resonance or ''internal'' SR, which underlines the fact that one can observe SR-like phenomena without an external periodic signal.In the present paper we study the response of nonlinear dynamical systems to noise excitation near the onset of dynamical instabilities of periodic orbits. Our starting point is the key paper of Wiesenfeld ͓7͔, which carefully elaborates the way in which noise controls the qualitative structure of the power spectrum. In brief, Wiesenfeld demonstrates that the power spectrum of a system observed after a bifurcation point can, nevertheless, be visible even before the bifurcation actually occurs if there is noise present. We thus observe a noisy precursor of the bifurcation.To follow this line of thought further, let us suppose that noise induces a peak of height H at ...

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The anisotropic arrangement of trabeculae in the proximal femur of humans and primates is seen as striking evidence for the functional adaptation of trabecular bone architecture. Quantitative evidence to demonstrate this adaptation for trabecular bone is still scarce, because experimental design of controlled load change is difficult. In this work, we use the natural variation of loading caused by a different main locomotor behavior of primates. Using high-resolution computed tomography and advanced image analysis techniques, we analyze the heterogeneity of the architecture in four proximal femora of four primate species. Although the small sample number does not allow an interspecies comparison, the very differently loaded bones are well suited to search for common structural features as a result of adaptation. A cubic volume of interest of size (5 mm) 3 was moved through the proximal femur and a morphometric analysis including local anisotropy was performed on 209 positions on average. The correlation of bone volume fraction (BV/TV) with trabecular number (Tb.N) and trabecular thickness (Tb.Th) leads to the suggestion of two different mechanisms of trabecular bone adaptation. Higher values of BV/TV in highly loaded regions of the proximal femur are due to a thickening of the trabeculae, whereas Tb.N does not change. In less loaded regions, however, lower values of BV/TV are found, caused by a reduction of the number of the trabeculae, whereas Tb.Th remains constant. This reduction in Tb.N goes along with an increase in the degree of anisotropy, indicating an adaptive selection of trabeculae. Anat Rec, 294:55-67, 2011. V V C 2010 Wiley-Liss, Inc.

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In this study we generalize symbolic dynamics to analyze two-dimensional objects and utilize measures of complexity to quantify the structure of symbol-encoded images. This technique is applied to study quantitatively the structure of human cancellous bone by analyzing computed tomography images. First, the preprocessed images are transformed into symbols, applying a mixture of static and dynamic encoding. Next, the spatial distribution of cancellous bone is evaluated using measures of complexity. New parameters are introduced to quantify the cancellous bone architecture as a whole. The results exhibit that the complexity of the structure declines more rapidly than density during the loss of bone in osteoporosis, strongly suggesting an exponential relationship between bone mass and architecture. It is found that normal bone has complex ordered structure, while the structure during the initial stage of bone loss is characterized by lower complexity and a significantly higher level of disorder, which is maximal there. A strong grade of the bone loss leads again to ordered structure, however its complexity is minimal. In addition, this method is significantly sensitive to changes in structure of natural composite materials. ͓S1063-651X͑98͒08911-9͔

A nondestructive and noninvasive method for numeric characterization (quantification) of the structural composition of human bone tissue has been developed and tested. In order to quantify and to compare the structural composition of bones from 2D computed tomography (CT) images acquired at different skeletal locations, a series of robust, versatile, and adjustable image segmentation and structure assessment algorithms were developed. The segmentation technique facilitates separation from cortical bone and standardizes the region of interest. The segmented images were symbol-encoded and different aspects of the bone structural composition were quantified using six different measures of complexity. These structural examinations were performed on CT images of bone specimens obtained at the distal radius, humeral mid-diaphysis, vertebral body, femoral head, femoral neck, proximal tibia, and calcaneus. In addition, the ability of the noninvasive and nondestructive measures of complexity to quantify trabecular bone structure was verified by comparing them to conventional static histomorphometry performed on human fourth lumbar vertebral bodies. Strong correlations were established between the measures of complexity and the histomorphometric parameters except for measures expressing trabecular thickness. Furthermore, the ability of the measures of complexity to predict vertebral bone strength was investigated by comparing the outcome of the complexity analysis of the CT images with the results of a biomechanical compression test of the third lumbar vertebral bodies from the same population as used for histomorphometry. A multiple regression analysis using the proposed measures including structure complexity index, structure disorder index, trabecular network index, index of a global ensemble, maximal L-block, and entropy of x-ray attenuation distribution revealed an excellent relationship (r=0.959, r2=0.92) between the measures of complexity and compressive bone strength. In conclusion, the image segmentation techniques and the assessment of bone architecture by measures of complexity have been successfully applied to analyze high-resolution peripheral quantitative computed tomography (pQCT) and CT images obtained from the distal radius, humeral mid-diaphysis, third and fourth lumbar vertebral bodies, proximal femur, proximal tibia, and calcaneus. The proposed approach is of broad interest as it can be applied for the quantification of structures and textures originating from different imaging modalities in other fields of science.