Quantification of evolution from order to randomness in practical time series analysis

Influence of QRS complex detection errors on entropy algorithms. Application to heart rate variability discrimination

Signal entropy measures such as approximate entropy (ApEn) and sample entropy (SampEn) are widely used in heart rate variability (HRV) analysis and biomedical research. In this article, we analyze the influence of QRS detection errors on HRV results based on signal entropy measures. Specifically, we study the influence that QRS detection errors have on the discrimination power of ApEn and SampEn using the cardiac arrhythmia suppression trial (CAST) database. The experiments assessed the discrimination capability of ApEn and SampEn under different levels of QRS detection errors. The results demonstrate that these measures are sensitive to the presence of ectopic peaks: from a successful classification rate of 100%, down to a 75% when spikes are present. The discriminating capability of the metrics degraded as the number of misdetections increased. For an error rate of 2% the segmentation failed in a 12.5% of the experiments, whereas for a 5% rate, it failed in a 25%.

Approximate entropy of network parameters

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We first define a purely structural entropy obtained by computing the approximate entropy of the so-called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erdös-Rényi networks. By using classical results of Pincus, we show that our entropy measure often converges with network size to a certain binary Shannon entropy. As a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs allow us to naturally associate with a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.

Eukaryotic evolutionary transitions are associated with extreme codon bias in functionally-related proteins

Codon bias in the genome of an organism influences its phenome by changing the speed and efficiency of mRNA translation and hence protein abundance. We hypothesized that differences in codon bias, either between-species differences in orthologous genes, or within-species differences between genes, may play an evolutionary role. To explore this hypothesis, we compared the genome-wide codon bias in six species that occupy vital positions in the Eukaryotic Tree of Life. We acquired the entire protein coding sequences for these organisms, computed the codon bias for all genes in each organism and explored the output for relationships between codon bias and protein function, both within- and between-lineages. We discovered five notable coordinated patterns, with extreme codon bias most pronounced in traits considered highly characteristic of a given lineage. Firstly, the Homo sapiens genome had stronger codon bias for DNA-binding transcription factors than the Saccharomyces cerevisiae genome, whereas the opposite was true for ribosomal proteins – perhaps underscoring transcriptional regulation in the origin of complexity. Secondly, both mammalian species examined possessed extreme codon bias in genes relating to hair – a tissue unique to mammals. Thirdly, Arabidopsis thaliana showed extreme codon bias in genes implicated in cell wall formation and chloroplast function – which are unique to plants. Fourthly, Gallus gallus possessed strong codon bias in a subset of genes encoding mitochondrial proteins – perhaps reflecting the enhanced bioenergetic efficiency in birds that co-evolved with flight. And lastly, the G. gallus genome had extreme codon bias for the Ciliary Neurotrophic Factor – which may help to explain their spontaneous recovery from deafness. We propose that extreme codon bias in groups of genes that encode functionally related proteins has a pathway-level energetic explanation.

Functional isolation within the cerebral cortex in the vegetative state: a nonlinear method to predict clinical outcomes

BACKGROUND

Establishing prognosis in patients in a persistent vegetative state (VS) is still challenging. Neural networks underlying consciousness may be regarded as complex systems whose outputs show a degree of unpredictability experimentally quantifiable by means of nonlinear parameters such as approximate entropy (ApEn).

OBJECTIVE

The authors propose that the VS might be the result of derangement of the above neural networks, with an ensuing decrease in complexity and mutual interconnectivity: this might lead to a functional isolation within the cerebral cortex and to a reduction in the chaotic behavior of its outputs, with monotony taking the place of unpredictability. To test this hypothesis, the authors investigated whether nonlinear dynamics methods applied to electroencephalography (EEG) recordings may be able to predict outcomes.

METHODS

A total of 38 vegetative patients and 40 matched healthy controls were investigated. At admission, all patients were assessed by means of the Extended Glasgow Outcomes Coma Scale (E-GOS) and the Coma Recovery Scale-Revised (CRS-R). At the same time an EEG recording was performed and used for time series analysis and ApEn computation. Patients were clinically reassessed at 6 months from the first evaluation.

RESULTS

Mean ApEn values (0.73, standard deviation [SD] = 0.12 vs 0.97, SD = 0.02; P < .001) were lower in patients than in controls. Patients with the lowest ApEn values either died (n = 14) or remained in a VS (n = 12), whereas patients with the highest ApEn values became minimally conscious (n = 5) or showed partial (n = 4) or full recovery (n = 3).

CONCLUSIONS

These findings suggest that dynamic correlates of neural residual complexity might help in predicting outcomes in vegetative patients.

Clock gene expression during chronic inflammation induced by infection with Trypanosoma brucei brucei in rats

African sleeping sickness is characterized by alterations in rhythmic functions. It is not known if the disease affects the expression of clock genes, which are the molecular basis for rhythm generation. We used a chronic rat model of experimental sleeping sickness, caused by the extracellular parasite Trypanosoma brucei brucei (Tb brucei), to study the effects on clock gene expression. In tissue explants of pituitary glands from Period1-luciferase (Per1-luc) transgenic rats infected with Tb brucei, the period of Per1-luc expression was significantly shorter. In explants containing the suprachiasmatic nuclei (SCN), the Per1-luc rhythms were flat in 21% of the tissues. We also examined the relative expression of Per1, Clock, and Bmal1 mRNA in the SCN, pineal gland, and spleen from control and infected rats using qPCR. Both Clock and Bmal1 mRNA expression was reduced in the pineal gland and spleen following Tb brucei infection. Infected rats were periodic both in core body temperature and in locomotor activity; however, early after infection, we observed a significant decline in the amplitude of the locomotor activity rhythm. In addition, both activity and body temperature rhythms exhibited decreased regularity and "robustness." In conclusion, although experimental trypanosome infection has previously been shown to cause functional disturbances in SCN neurons, only 21% of the SCN explants had disturbed Per1-luc rhythms. However, our data show that the infection overall alters molecular clock function in peripheral clocks including the pituitary gland, pineal gland, and spleen.

Increased hypothalamic-pituitary-adrenal axis activity in Huntington's disease

CONTEXT

Huntington's disease (HD) is a fatal hereditary neurodegenerative disorder characterized by motor, cognitive, and behavioral disturbances. Hypothalamic-pituitary-adrenal (HPA) axis dysfunction could contribute to a number of HD signs and symptoms; however, no data are available on cortisol diurnal variations and secretory dynamics in HD patients.

OBJECTIVE

The aim of the study was to perform a detailed analysis of HPA axis function in HD patients in relation to clinical signs and symptoms.

DESIGN, SETTING, AND PARTICIPANTS

Twenty-four-hour cortisol secretion was studied in eight early-stage, medication-free HD patients and eight age-, sex-, and body mass index-matched controls in a clinical research laboratory. Cortisol levels were measured every 10 min.

MAIN OUTCOME MEASURES

Multiparameter autodeconvolution and cosinor regression were applied to quantify basal, pulsatile, and total cortisol secretion rates as well as diurnal variations in cortisol levels.

RESULTS

Total cortisol secretion rate and the amplitude of the diurnal cortisol profile were both significantly higher in HD patients compared with controls (3490 +/- 320 vs. 2500 +/- 220 nmol/liter/24 h, P = 0.023; and 111 +/- 14 vs. 64 +/- 8 nmol/liter, P = 0.012, respectively). Cortisol concentrations in patients were particularly increased in the morning and early afternoon period. In HD patients, mean 24-h cortisol levels significantly correlated with total motor score, total functional capacity, as well as body mass index.

CONCLUSIONS

HPA axis hyperactivity is an early feature of HD and is likely to result from a disturbed central glucocorticoid feedback due to hypothalamic pathology. HPA axis dysfunction may contribute to some signs and symptoms in HD patients.

Diabetes: Models, Signals, and Control

The control of diabetes is an interdisciplinary endeavor, which includes a significant biomedical engineering component, with traditions of success beginning in the early 1960s. It began with modeling of the insulin-glucose system, and progressed to large-scale in silico experiments, and automated closed-loop control (artificial pancreas). Here, we follow these engineering efforts through the last, almost 50 years. We begin with the now classic minimal modeling approach and discuss a number of subsequent models, which have recently resulted in the first in silico simulation model accepted as substitute to animal trials in the quest for optimal diabetes control. We then review metabolic monitoring, with a particular emphasis on the new continuous glucose sensors, on the analyses of their time-series signals, and on the opportunities that they present for automation of diabetes control. Finally, we review control strategies that have been successfully employed in vivo or in silico, presenting a promise for the development of a future artificial pancreas and, in particular, discuss a modular architecture for building closed-loop control systems, including insulin delivery and patient safety supervision layers. We conclude with a brief discussion of the unique interactions between human physiology, behavioral events, engineering modeling and control relevant to diabetes.

Novel method for measuring the complexity of schizophrenic EEG based on symbolic entropy analysis

Symbolic dynamics is a useful tool in several fields of complexity analysis in nonlinear science. In order to investigate complexities of the human brain electrical activities under different brain functional states, a novel method in terms of symbolic entropy is defined and proposed in this paper. The novel algorithm based on symbolic dynamics is developed for quantitatively measuring the complexity of the EEG data. Simulated signals derived from chaotic systems and several real EEG data under normal and pathological brain functional states are examined and compared. The experimental results show that the proposed method can distinguish not only the complexities of simulated signals but also the complexities of two groups of EEG data under different brain functional states. It is superior to the traditional entropy methods. Moreover, the algorithm can be easily completed and fast computed.

Nonlinear analysis of discharge patterns in monkey basal ganglia

Spontaneous discharge of basal ganglia neurons is often analyzed with time- or frequency-domain methods. However, it has been shown that sequences of inter-spike interval series are not fully described by such linear procedures. We therefore carried out a characterization of the nonlinear features of spontaneous discharge of neurons in the primate basal ganglia. We studied the spontaneous activity of neurons in the subthalamic nucleus (22 cells), as well as neurons in the external and internal pallidal segments (53 and 39 cells, respectively), recorded with standard extracellular recording methods in two awake Rhesus monkeys. As a measure of the statistical irregularity of neuronal discharge, we compared the approximate entropy of inter-spike interval sequences with that of shuffled representations of the same data. In all three basal ganglia structures, approximately 95% of the original data showed lower approximate entropy values than the shuffled data, suggesting a temporal organization in the original sequence. Fano factor analysis confirmed the presence of a temporal organization of inter-spike interval sequences, and indicated the presence of self-similarity in the great majority of them. In addition, Hurst exponent analysis showed that the inter-spike interval series are persistent. Hurst exponents often differ between short and long scaling ranges. Subsequent principal component analyses allowed us to identify three distinct patterns of the temporal evolution of inter-spike interval sequences in the phase space. These types were found in varying distributions in all three nuclei. Our analyses demonstrate that the discharge of most neurons in the basal ganglia of awake monkeys has nonlinear features that may be important for information coding in the basal ganglia.

Intra-limb segmental influences on random-like movements in humans

The experiment investigated the influence of segmental coupling on the ability to produce random-like movements in individual limb segments. Adult participants were instructed to move randomly (2 min trials) in the sagittal plane their index finger, hand, and lower arm as "frozen" effector units or where the individual links within the upper limb complex were free to move independently. The findings showed that the distal finger movements were more random-like when the proximal joints were also free to vary, but the reverse directional segmental effects were not present. Analysis of the movement frequency structure of the coordination between limb segments showed that patterns of modal frequencies were preserved even though the participants were trying to produce with equal probability a wide range of frequencies. These findings provide further evidence that: (1) the boundary conditions on the degrees of freedom of the neural output of an effector are relatively restrictive; (2) inter-limb reactive forces can enhance the limits on the dynamical degrees of freedom; and (3) the intrinsic dynamics influence movement output even when the task goal is a random output.

Acipimox enhances spontaneous growth hormone secretion in obese women

We hypothesized that a high circulating free fatty acid (FFA) concentration is involved in the pathogenesis of hyposomatotropism associated with obesity. To evaluate this hypothesis, 10 healthy premenopausal women (body mass index 33.8 +/- 1.0 kg/m(2)) were studied in the follicular phase of their menstrual cycle at two occasions with a time interval of at least 8 wk, where body weight remained stable. Subjects were randomly assigned to treatment with either acipimox (an inhibitor of lipolysis, 250 mg orally 4 times daily) or placebo in a double-blind crossover design, starting 1 day before admission until the end of the blood sampling period. Blood samples were taken during 24 h with a sampling interval of 10 min for assessment of growth hormone (GH) concentrations, and GH secretion was estimated by deconvolution analysis. Identical methodology was used to study GH secretion in a historical control group of age-matched normal weight women. GH secretion was clearly blunted in obese women (total daily release 66 +/- 10 vs. lean controls: 201 +/- 23 mU x l(Vd)(-1) x 24 h(-1), P = 0.005, where l(Vd) is lite of distribution volume). Acipimox considerably enhanced total (113 +/- 50 vs. 66 +/- 10 mU x l(Vd)(-1) x 24 h(-1), P = 0.02) and pulsatile GH secretion (109 +/- 49 vs. 62 +/- 30 mU x l(Vd)(-1) x 24 h(-1), P = 0.02), but GH output remained lower compared with lean controls. Further analysis did not show any relationship between the effects of acipimox on GH secretion and regional body fat distribution. In conclusion, acipimox unleashes spontaneous GH secretion in obese women. It specifically enhances GH secretory burst mass. This might mean that lowering of systemic FFA concentrations by acipimox modulates neuroendocrine mechanisms that orchestrate the activity of the somatotropic ensemble.

The use of regularity as estimated by approximate entropy to distinguish saltatory growth

A nonlinear dynamics metric, approximate entropy (ApEn), is investigated as a diagnostic method for distinguishing between mathematical models, and the underlying mechanistic hypotheses that purport to describe the same time series experimental observations. ApEn measures the occurrence of pattern regularity within a time series, and is used here to investigate growth patterns in daily length growth. The notion investigated is that ApEn distributions for competing time series patterns expressed as mathematical formulations can be modelled by Monte Carlo and bootstrap methods and compared to the ApEn values for an original experimental data series. If the ApEn values for the different models do not overlap, then it is expected that ApEn can be utilized to distinguish these models and hypotheses, and to provide statistical assessment for the underlying biological patterns in experimental data. The conclusion is that the ApEn metric is successful as a time series diagnostic tool. It is a model-independent statistic that clearly differentiates saltatory growth from slowly varying continuous models of growth and serves to further document the saltatory nature of growth. This is a unique application of approximate entropy, illustrating the broad applicability of ApEn to biological time series, with the specific example of discriminating a saltatory growth process in longitudinal growth data. Future investigations of regularity in longitudinal time series in human biology with ApEn statistics are suggested.

On learning to move randomly

In 2 experiments, the authors examined whether and to what degree young adults can learn to produce random planar motion of the index finger or fingers. Three different types of information feedback were provided to the participants (N = 8 in each experiment) over up to 5 days of practice across the 2 experiments. The results from both experiments revealed that the participants produced a relatively low level of movement randomness in finger motion and that they did not learn through practice to enhance the stochastic properties of their movement under any feedback conditions. The findings provide further evidence that there are relatively tight constraints on the number of dimensions that are regulating single-limb planar motion and that those constraints are not susceptible to change through typical learning protocols.

Entropies of short binary sequences in heart period dynamics

Dynamic aspects of R-R intervals have often been analyzed by means of linear and nonlinear measures. The goal of this study was to analyze binary sequences, in which only the dynamic information is retained, by means of two different aspects of regularity. R-R interval sequences derived from 24-h electrocardiogram (ECG) recordings of 118 healthy subjects were converted to symbolic binary sequences that coded the beat-to-beat increase or decrease in the R-R interval. Shannon entropy was used to quantify the occurrence of short binary patterns (length N = 5) in binary sequences derived from 10-min intervals. The regularity of the short binary patterns was analyzed on the basis of approximate entropy (ApEn). ApEn had a linear dependence on mean R-R interval length, with increasing irregularity occurring at longer R-R interval length. Shannon entropy of the same sequences showed that the increase in irregularity is accompanied by a decrease in occurrence of some patterns. Taken together, these data indicate that irregular binary patterns are more probable when the mean R-R interval increases. The use of surrogate data confirmed a nonlinear component in the binary sequence. Analysis of two consecutive 24-h ECG recordings for each subject demonstrated good intraindividual reproducibility of the results. In conclusion, quantification of binary sequences derived from ECG recordings reveals properties that cannot be found using the full information of R-R interval sequences.

Fetal heart rate variability and complexity in the course of pregnancy

Aim of this study was the examination of fetal heart rate variability and complexity measures during pregnancy using fetal magnetocardiography. We registered 80 fetal magnetocardiograms in 19 healthy fetuses between the 16th and 41st week of gestation. On the basis of beat to beat intervals, mean RR interval (mRR), its standard deviation (SD), root mean square of successive differences (RMSSD), as well as complexity variables such as dimension (ApD1), entropy (ApEn), Lyapunov exponent (ApML) and trajectory divergence rate (p) were calculated for each recording. Dependency of these variables on gestational age was evaluated with correlation analysis. All variables changed consistently over time. RMSSD showed the strongest dependency on gestational age, followed closely by ApEn, SD and p. ApD1 and mRR showed only weak dependency. We conclude that magnetocardiography is well suited to register fetal cardiac activity with sufficient accuracy to permit detailed analysis of various heart rate variables during the second and third trimester of pregnancy. The observed increases in heart rate variability and complexity of fetuses most likely reflect differing but overlapping aspects of fetal development. They may be linked to the maturation of the autonomic nervous system and could aid in the timely identification of pathological conditions.

Use of approximate entropy measurements to classify ventricular tachycardia and fibrillation

Implantable cardiac devices which treat arrhythmias require automated detection to decide when to deliver therapy and, in some devices like the implantable cardioverter-defibrillator (ICD), to determine which therapy to deliver. This type of detection often requires the separation of fibrillatory chaotic rhythms from coherent tachycardias. In the ICD, multiple rate zones can be programmed for detection of ventricular tachycardia (VT) and ventricular fibrillation (VF) and for delivery of therapy: antitachycardia pacing and cardioversion for VT, and defibrillation for VF. Previous research determined that current technology is unable to uniquely classify VF. Analysis of typical settings of the ICD revealed that 40% to 80% of VTs were misclassified as VF (depending on device/setting). Improved detection of VF must be sought in order to capitalize on the cost effectiveness of low-energy VT therapies, potentially saving up to 20% of the life of the battery. In this study, a statistical measure of variability, called approximate entropy (ApEn), has been applied to separate fibrillatory and nonfibrillatory rhythms. Although standard deviation is often used as a measure of variability, it fails to capture the level of regularity or complexity. ApEn improves upon standard deviation by quantifying differences between random and regular signals. ApEn was tested on a small patient set containing VF, VT, and sinus rhythm (SR). Intracardiac recordings (bipolar, ventricular) were selected from the Ann Arbor Electrogram Library (Ann Arbor, MI) where filtering (1-500 Hz), amplification (approximately 1000), and digitization (1000 Hz) are tightly controlled. Results demonstrated that ApEn has the ability to quantify subtle differences between VF and other rhythms.

[Fetal magnetocardiography]

Fetal magnetocardiography is a new, alternative method for prenatal surveillance. The fetal magnetocardiogram (FMCG) registers the magnetic field produced by conduction currents in the fetal heart. Compared to the fetal electrocardiogram, the propagation of magnetic fields is relatively undisturbed by surrounding tissue. The FMCG thus has the advantage of a higher signal-to-noise ratio and can be acquired earlier pregnancy. Also, the high temporal resolution of the signal permits a significantly more precise determination of fetal heart rate parameters than fetal ultrasound. FMCG registration using a biomagnetometer is noninvasive and can be performed as of the second trimeter. It can be used to examine signal morphology, cardiac time intervals, heart rate variability as well as cardiac magnetic fields. To date, arrhythmic activity has been observed in the form of supraventricular and ventricular ectopies as well as atrial flutter, atrio-ventricular block, atrial tachycardia and Torsades de Pointes tachycardia. We also report here on the presence of short episodes of bradycardia in the second trimester of normal pregnancy. Measurement of the magnetic field strength at various locations above the abdomen has allowed the reconstruction of the fetal cardiac magnetic field and the determination of its relation to the position of the fetus. Signal averaging has permitted the precise examination of signal amplitude and cardiac time intervals and has shown that they increase in the course of pregnancy. Heart rate variability could be quantified in the time and frequency domain as well as using parameters of nonlinear dynamics. The results demonstrated an increase of variability and complexity over gestational age. Furthermore spectral analysis of fetal heart arte data could be associated with sympathetic and parasympathetic activity as well as, with respiration. Although the studies presenting these results have involved only limited numbers of observations, they demonstrate the potential of the method in the examination of the fetal conductive system, arrhythmias, congential defects, growth, development of the autonomic system, acidosis and distress. Furthermore, first results in pathological cases indicate that it may become a valuable tool in prenatal diagnostics. Improvements in instrumentation as well as prospective multicenter studies with larger numbers of appropriate subjects are required to determine whether magnetocardiography will establish itself as a new tool in clinical fetal, surveillance.