Estimation of Instantaneous Complex Dynamics through Lyapunov Exponents: A Study on Heartbeat Dynamics

Measures of nonlinearity and complexity, and in particular the study of Lyapunov exponents, have been increasingly used to characterize dynamical properties of a wide range of biological nonlinear systems, including cardiovascular control. In this work, we present a novel methodology able to effectively estimate the Lyapunov spectrum of a series of stochastic events in an instantaneous fashion. The paradigm relies on a novel point-process high-order nonlinear model of the event series dynamics. The long-term information is taken into account by expanding the linear, quadratic, and cubic Wiener-Volterra kernels with the orthonormal Laguerre basis functions. Applications to synthetic data such as the Hénon map and Rössler attractor, as well as two experimental heartbeat interval datasets (i.e., healthy subjects undergoing postural changes and patients with severe cardiac heart failure), focus on estimation and tracking of the Instantaneous Dominant Lyapunov Exponent (IDLE). The novel cardiovascular assessment demonstrates that our method is able to effectively and instantaneously track the nonlinear autonomic control dynamics, allowing for complexity variability estimations.     

Correlations in one-dimensional fully developed chaos

Correlations in the baker map and the tent map as examples of one-dimensional, fully developed chaos are considered. It is shown, utilizing symbolic dynamical systems derived from these maps, that the vanishing second-order correlation function is not sufficient to guarantee uncorrelatedness. Importance of the higher-order, especially third-order, correlation functions is emphasized for chaotic systems. In search of the quantities that grasp correlational behaviors as a whole in chaotic systems, it is proposed to use the fixed-separation correlation integral, which is a modified quantity of the usual correlation integral devised to calculate the fractal dimension of strange attractors, for these maps. It is shown that the new quantity contains all the even-number orders of autocorrelation function that are commonly considered.     

Complex Generalized Synchronization and Parameter Identification of Nonidentical Nonlinear Complex Systems

In this paper, generalized synchronization (GS) is extended from real space to complex space, resulting in a new synchronization scheme, complex generalized synchronization (CGS). Based on Lyapunov stability theory, an adaptive controller and parameter update laws are designed to realize CGS and parameter identification of two nonidentical chaotic (hyperchaotic) complex systems with respect to a given complex map vector. This scheme is applied to synchronize a memristor-based hyperchaotic complex Lü system and a memristor-based chaotic complex Lorenz system, a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system, as well as a memristor-based hyperchaotic complex Lü system and a chaotic complex Lü system with fully unknown parameters. The corresponding numerical simulations illustrate the feasibility and effectiveness of the proposed scheme.     

Subharmonics and Chaos in Simple Periodically Forced Biomolecular Models

This article uncovers a remarkable behavior in two biochemical systems that commonly appear as components of signal transduction pathways in systems biology. These systems have globally attracting steady states when unforced, so they might have been considered uninteresting from a dynamical standpoint. However, when subject to a periodic excitation, strange attractors arise via a period-doubling cascade. Quantitative analyses of the corresponding discrete chaotic trajectories are conducted numerically by computing largest Lyapunov exponents, power spectra, and autocorrelation functions. To gain insight into the geometry of the strange attractors, the phase portraits of the corresponding iterated maps are interpreted as scatter plots for which marginal distributions are additionally evaluated. The lack of entrainment to external oscillations, in even the simplest biochemical networks, represents a level of additional complexity in molecular biology, which has previously been insufficiently recognized but is plausibly biologically important.     

Chaotic neural network applied to two-dimensional motion control

Chaotic dynamics generated in a chaotic neural network model are applied to 2-dimensional (2-D) motion control. The change of position of a moving object in each control time step is determined by a motion function which is calculated from the firing activity of the chaotic neural network. Prototype attractors which correspond to simple motions of the object toward four directions in 2-D space are embedded in the neural network model by designing synaptic connection strengths. Chaotic dynamics introduced by changing system parameters sample intermediate points in the high-dimensional state space between the embedded attractors, resulting in motion in various directions. By means of adaptive switching of the system parameters between a chaotic regime and an attractor regime, the object is able to reach a target in a 2-D maze. In computer experiments, the success rate of this method over many trials not only shows better performance than that of stochastic random pattern generators but also shows that chaotic dynamics can be useful for realizing robust, adaptive and complex control function with simple rules.     

Multiple attractors in stage-structured population models with birth pulses

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. A single species stage-structured model with density-dependent maturation rate and birth pulse is formulated. Using the discrete dynamical system determined by its Poincaré map, we report a detailed study of the various dynamics, including (a) existence and stability of nonnegative equilibria, (b) nonunique dynamics, meaning that several attractors coexist, (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of attractor), (d) supertransients, and (e) chaotic attractors. The occurrence of these complex dynamic behaviour is related to the fact that minor changes in parameter or initial values can strikingly change the dynamic behaviours of system. Further, it is shown that periodic birth pulse, in effect, provides a natural period or cyclicity that allows multiple oscillatory solutions in the continuous dynamical systems.     

Territorial ants depress plant growth through cascading non‐trophic effects in an alpine meadow

Understanding non‐trophic interactions is critical to mechanistically linking community structure and ecosystem functioning. Despite the widespread occurrence of territoriality across animal taxa and ecosystems, the cascading ecological consequences of non‐trophic interactions between territorial animals and intruders have been poorly studied. We experimentally investigated the non‐trophic interaction between territorial ants and members of a dung decomposer community (i.e. predatory arthropods, maggots and coprophagous beetles) in an alpine meadow. We further examined how this non‐trophic interaction cascaded to influence ecosystem properties including dung removal rate, soil nutrient status and aboveground plant biomass surrounding dung pats. Results indicated that territorial interference of ants on key decomposers cascaded to affect plant growth. Specifically, ants significantly decreased the abundance of coprophagous beetles at the time of their peak‐abundance and hence decreased dung removal rates and soil nitrogen concentrations, ultimately decreasing aboveground plant biomass. The strength of this non‐trophic cascading effect was comparable to those reported in studies addressing trophic cascades triggered by predator–prey interactions. Our findings suggest that the non‐trophic interactions and associated cascading effects stemming from territorial behavior should be incorporated into ecological network modeling and research addressing biodiversity–ecosystem functioning relationships.     

Analysis of Chaotic Resonance in Izhikevich Neuron Model

In stochastic resonance (SR), the presence of noise helps a nonlinear system amplify a weak (sub-threshold) signal. Chaotic resonance (CR) is a phenomenon similar to SR but without stochastic noise, which has been observed in neural systems. However, no study to date has investigated and compared the characteristics and performance of the signal responses of a spiking neural system in some chaotic states in CR. In this paper, we focus on the Izhikevich neuron model, which can reproduce major spike patterns that have been experimentally observed. We examine and classify the chaotic characteristics of this model by using Lyapunov exponents with a saltation matrix and Poincaré section methods in order to address the measurement challenge posed by the state-dependent jump in the resetting process. We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state. In order to assess the signal responses of CR in these classified states, we introduced an extended Izhikevich neuron model by considering weak periodic signals, and defined the cycle histogram of neuron spikes as well as the corresponding mutual correlation and information. Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals. Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.     

Construction of a high-density genetic map using specific length amplified fragment markers and identification of a quantitative trait locus for anthracnose resistance in walnut (Juglans regia L.)

Background

Walnut (Juglans regia, 2n = 32, approximately 606 Mb per 1C genome) is an economically important tree crop. Resistance to anthracnose, caused by Colletotrichum gloeosporioides, is a major objective of walnut genetic improvement in China. The recently developed specific length amplified fragment sequencing (SLAF-seq) is an efficient strategy that can obtain large numbers of markers with sufficient sequence information to construct high-density genetic maps and permits detection of quantitative trait loci (QTLs) for molecular breeding.

Results

SLAF-seq generated 161.64 M paired-end reads. 153,820 SLAF markers were obtained, of which 49,174 were polymorphic. 13,635 polymorphic markers were sorted into five segregation types and 2,577 markers of them were used to construct genetic linkage maps: 2,395 of these fell into 16 linkage groups (LGs) for the female map, 448 markers for the male map, and 2,577 markers for the integrated map. Taking into account the size of all LGs, the marker coverage was 2,664.36 cM for the female map, 1,305.58 cM for the male map, and 2,457.82 cM for the integrated map. The average intervals between two adjacent mapped markers were 1.11 cM, 2.91 cM and 0.95 cM for three maps, respectively. ‘SNP_only’ markers accounted for 89.25 % of the markers on the integrated map. Mapping markers contained 5,043 single nucleotide polymorphisms (SNPs) loci, which corresponded to two SNP loci per SLAF marker. According to the integrated map, we used interval mapping (Logarithm of odds, LOD > 3.0) to detect our quantitative trait. One QTL was detected for anthracnose resistance. The interval of this QTL ranged from 165.51 cM to 176.33 cM on LG14, and ten markers in this interval that were above the threshold value were considered to be linked markers to the anthracnose resistance trait. The phenotypic variance explained by each marker ranged from 16.2 to 19.9 %, and their LOD scores varied from 3.22 to 4.04.

Conclusions

High-density genetic maps for walnut containing 16 LGs were constructed using the SLAF-seq method with an F1 population. One QTL for walnut anthracnose resistance was identified based on the map. The results will aid molecular marker-assisted breeding and walnut resistance genes identification.

Electronic supplementary material

The online version of this article (doi:10.1186/s12864-015-1822-8) contains supplementary material, which is available to authorized users.     

Pattern Selection in Network of Coupled Multi-Scroll Attractors

Multi-scroll chaotic attractor makes the oscillator become more complex in dynamic behaviors. The collective behaviors of coupled oscillators with multi-scroll attractors are investigated in the regular network in two-dimensional array, which the local kinetics is described by an improved Chua circuit. A feasible scheme of negative feedback with diversity is imposed on the network to stabilize the spatial patterns. Firstly, the Chua circuit is improved by replacing the nonlinear term with Sine function to generate infinite aquariums so that multi-scroll chaotic attractors could be generated under appropriate parameters, which could be detected by calculating the Lyapunov exponent in the parameter region. Furthermore, negative feedback with different gains (D₁, D₂) is imposed on the local square center area A₂ and outer area A₁ of the network, it is found that spiral wave, target wave could be developed in the network under appropriate feedback gain with diversity and size of controlled area. Particularly, homogeneous state could be reached after synchronization by selecting appropriate feedback gain and controlled size in the network. Finally, the distribution for statistical factors of synchronization is calculated in the two-parameter space to understand the transition of pattern region. It is found that developed spiral waves, target waves often are associated with smaller factor of synchronization. These results show that emergence of sustained spiral wave and continuous target wave could be effective for further suppression of spatiotemporal chaos in network by generating stable pacemaker completely.     

Pollen coupling of forest trees: forming synchronized and periodic reproduction out of chaos

Many of the tree species in mature forests show masting; their reproductive activity has a large variance between years and is often synchronized between different individuals. In this paper, we analyse a globally coupled map model in which trees accumulate photosynthate every year, produce flowers when the energy reserve level exceeds a threshold, and set seeds and fruits at a rate limited by pollen availability. Without pollen limitation, the trees in the forest show independent chaotic fluctuation. Coupling of trees via pollen exchange results in reproduction being synchronized partially or completely over the forest. The whole forest shows diverse dynamical behaviors determined by the values of two essential parameters; the depletion coefficient k and the coupling strength beta. We find perfectly synchronized periodic reproduction, synchronized reproduction with a chaotic time series, clustering phenomena, and chaotic reproduction of trees without synchronization over individuals. There are many parameter windows in which synchronized reproduction of trees shows a stable periodic fluctuation. For perfectly synchronized forests, we can calculate all the Lyapunov exponents analytically. They show that synchronized reproduction of all the trees in the forest can only occur when trees flower at low (but positive) levels in a significant fraction of years, resulting in small fruit sets due to outcrossed pollen limitation. This is consistent with the observation that the distinction between mast years and non-mast years is often not clear cut.     

Kinetics of formation of hypoxanthine containing base pairs by HIV-RT: RNA template effects on the base substitution frequencies

Hypoxanthine (H), the deamination product of adenine, has been implicated in the high frequency of A to G transitions observed in retroviral and other RNA genomes. Although H·C base pairs are thermodynamically more stable than other H·N pairs, polymerase selection may be determined in part by kinetic factors. Therefore, the hypoxanthine induced substitution pattern resulting from replication by viral polymerases may be more complex than that predicted from thermodynamics. We have examined the steady-state kinetics of formation of base pairs opposite template H in RNA by HIV-RT, and for the incorporation of dITP during first- and second-strand synthesis. Hypoxanthine in an RNA template enhances the k_2app for pairing with standard dNTPs by factors of 10–1000 relative to adenine at the same sequence position. The order of base pairing preferences for H in RNA was observed to be H·C >> H·T > H·A > H·G. Steady-state kinetics of insertion for all possible mispairs formed with dITP were examined on RNA and DNA templates of identical sequence. Insertion of dITP opposite all bases occurs 2–20 times more frequently on RNA templates. This bias for higher insertion frequencies on RNA relative to DNA templates is also observed for formation of mispairs at template A. This kinetic advantage afforded by RNA templates for mismatches and pairing involving H suggests a higher induction of mutations at adenines during first-strand synthesis by HIV-RT.     

A neural model of predictive recognition in form pathway of visual cortex

We present a functional model of form pathway in visual cortex based on predictive coding scheme, in which the prediction is compared with feedforward signals filtered by two kinds of spatial resolution maps, broad and fine resolution map. We propose here the functional role of the prediction and of the two kinds of resolution maps in perception of object form in visual system. The prediction is represented based on memory of dynamical attractors in temporal cortex, categorized by an elemental figure in posterior temporal cortex. The prediction is generated by the feedforward signals of main neurons in broad resolution maps of V(1) and V(4), and then is compared with the feedforward signals of main neurons in fine resolution map of V(1) and V(4).     

Glioma: Application of Whole-Tumor Texture Analysis of Diffusion-Weighted Imaging for the Evaluation of Tumor Heterogeneity

Background and Purpose

To apply a texture analysis of apparent diffusion coefficient (ADC) maps to evaluate glioma heterogeneity, which was correlated with tumor grade.

Materials and Methods

Forty patients with glioma (WHO grade II (n = 8), grade III (n = 10) and grade IV (n = 22)) underwent diffusion-weighted imaging (DWI), and the corresponding ADC maps were obtained. Regions of interest containing the lesions were drawn on every section of the ADC map containing the tumor, and volume-based data of the entire tumor were constructed. Texture and first order features including entropy, skewness and kurtosis were derived from the ADC map using in-house software. A histogram analysis of the ADC map was also performed. The texture and histogram parameters were compared between low-grade and high-grade gliomas using an unpaired student’s t-test. Additionally, a one-way analysis of variance analysis with a post-hoc test was performed to compare the parameters of each grade.

Results

Entropy was observed to be significantly higher in high-grade gliomas than low-grade tumors (6.861±0.539 vs. 6.261±0.412, P  = 0.006). The fifth percentiles of the ADC cumulative histogram also showed a significant difference between high and low grade gliomas (836±235 vs. 1030±185, P = 0.037). Only entropy proved to be significantly different between grades III and IV (6.295±0.4963 vs. 7.119±0.3165, P<0.001). The diagnostic accuracy of ADC entropy was significantly higher than that of the fifth percentile of the ADC histogram (P = 0.0034) in distinguishing high- from low-grade glioma.

Conclusion

A texture analysis of the ADC map based on the entire tumor volume can be useful for evaluating glioma grade, which provides tumor heterogeneity.     

Analysis of chaotic oscillations induced in two coupled Wilson–Cowan models

Although it is known that two coupled Wilson–Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson–Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson–Cowan oscillators.     

Red spectra from white and blue noise

The value of maps of the interval in modelling population dynamics has recently been called into question because temporal variations from such maps have blue or white power spectra, whereas many observations of real populations show time-series with red spectra. One way to deal with this discrepancy is to introduce chaotic or stochastic fluctuations in the parameters of the map. This leads to on–off intermittency and can markedly redden the spectrum produced by a model that does not by itself have a red spectrum. The parameter fluctuations need not themselves have a red spectrum in order to achieve this effect. Because the power spectrum is not invariant under a change of variable, another way to redden the spectrum is by a suitable transformation of the variables used. The question this poses is whether spectra are the best means of characterizing a fluctuating variable.     

Georges Teissier (1900–1972) and the Modern Synthesis in France

This Perspectives is devoted to the ideas of the French zoologist Georges Teissier about the mechanisms of evolution and the relations between micro- and macroevolution. Working in an almost universally neo-Lamarckian context in France, Teissier was one of the very few Darwinians there at the time of the evolutionary synthesis. The general atmosphere of French zoology during the 1920s and the 1930s will first be recalled, to understand the specific conditions in which Teissier became a zoologist. After a brief overview of his joint work with Philippe L’Héritier on the experimental genetics of Drosophila, this article describes the ways Teissier, during the 1950s, conceptualized the mechanisms that could allow for macroevolutionary transitions.IT is usually acknowledged that France did not significantly participate in the elaboration of 20th century evolutionary theory, often designated The Modern Synthesis. In their classical book on the history of the synthesis, Ernst Mayr and William B. Provine devoted a whole—nonetheless small—chapter to this specific issue (Mayr and Provine 1998, pp. 309–328). Mayr clearly stated that “France is the only major scientific nation that did not contribute significantly to the evolutionary synthesis” (Mayr 1998, p. 309). In the absence of a French architect of the synthesis, Mayr and Provine asked Ernest Boesiger, a Swiss population geneticist and a former student of Georges Teissier, to tell the story of what had happened in French biology at the time of the evolutionary synthesis. Boesiger, who died in 1975, wrote a paper in 1974 that provided the firm basis of the chapter. In very strong terms, he depicted French biology as “a kind of living fossil in the rejection of modern evolutionary theories” (Boesiger 1998, p. 309). He insisted on the fact that, even in 1974, most French biologists and philosophers were still reluctant to accept Darwinism. As regards the period of the 1930s, Boesiger was able to think of only two exceptions: Georges Teissier and Philippe L’Héritier. He then referred to their joint research in population genetics, which was based on the new technique of the population cages with the species Drosophila melanogaster, and listed their contributions to this new discipline.If Teissier and L’Héritier’s works on Drosophila are nowadays more widely recognized than in 1974, due in particular to the efforts of Jean Gayon and Michel Veuille (Gayon and Veuille 2001), this recognition could have as an unintended consequence the reduction of both Teissier and L’Héritier to being simply the inventors of a useful technique, namely the population cages (see especially how Mayr presented their work in his other classical book, Mayr 1982, p. 574), or as the founders of a French school of population geneticists (Gayon and Veuille 2001). The aim of this article is to reevaluate the way Georges Teissier (1900–1972) conceived Darwinian natural selection not only as an important mechanism for evolution at the population level but more fundamentally as a general key for the unification of biology, exactly as Julian Huxley or Ernst Mayr did during the same period (1930–1970). However, starting in the early 1950s, Teissier went on to conceive a very specific understanding of the evolutionary synthesis.In this article, I will first describe the general atmosphere of evolutionary issues in French biology at the time when Teissier started working as a zoologist, to understand against what he developed his joint research program with L’Héritier and afterward his general conceptions about evolution. During the 1930s and the 1940s, only a very few scientists in France could be seen as Darwinians. In addition to Teissier and L’Héritier, one may also consider Marcel Prenant, Boris Ephrussi, and the mathematician Gustave Malécot. Building on Jean Gayon and Michel Veuille’s work, I will then give a quick overview of L’Héritier and Teissier’s most important achievements in the field of population genetics. In the third part, I will discuss the discovery made by Teissier and L’Héritier of a case of cytoplasmic inheritance in Drosophila. This unexpected finding led them into the field of non-Mendelian heredity. I will then develop in detail the way Teissier finally went on to conceive the relation between microevolution and macroevolution, in light of the general context of French biology and of the development of the field of cytoplasmic inheritance.     

Arthrite rhumatoide traitée par noréthynodrel associée à mestranol: Aspects cliniques et tests de laboratoire

A cause de l''effet favorable de la grossesse sur l''activité de l''arthrite rhumatoïde, on s''est demandé si une pseudo-grossesse, produite par un progestatif de synthèse n''entraînerait pas une rémission du moins partielle de cette maladie.Noréthynodrel associée à mestranol (Enovid), 30 mg./jour, a été administrée à 44 femmes pendant quatre à 24 mois. A cause d''effets secondaires indésirables, 11 patientes furent soustraites de l''investigation. Les résultats s''appuient sur 33 cas. Une rémission apparente complète s''est manifestée chez sept patientes aux stades précoces de la maladie; chez 15, une amélioration objective des signes inflammatoires a été observée; chez quatre, une amélioration subjective seule a été notée; chez sept, il n''y a eu aucune amélioration. Dix-sept femmes sur 36 ont présenté une altération d''un ou plusieurs tests hépatiques. Trois présentèrent un ictère cholestatique. Les 17-OH plasmatiques se sont élevés à trois ou quatre fois la normale.De cette étude il ressort que noréthynodrel associée à mestranol peut produire une atténuation des signes inflammatoires de l''arthrite rhumatoïde. L''effet est palliatif, mitigé et non curatif et ne résulte pas nécessairement de l''état de pseudo-grossesse en soi.     

Robustness and information propagation in attractors of random boolean networks

Attractors represent the long-term behaviors of Random Boolean Networks. We study how the amount of information propagated between the nodes when on an attractor, as quantified by the average pairwise mutual information ([Formula: see text]), relates to the robustness of the attractor to perturbations ([Formula: see text]). We find that the dynamical regime of the network affects the relationship between [Formula: see text] and [Formula: see text]. In the ordered and chaotic regimes, [Formula: see text] is anti-correlated with [Formula: see text], implying that attractors that are highly robust to perturbations have necessarily limited information propagation. Between order and chaos (for so-called "critical" networks) these quantities are uncorrelated. Finite size effects cause this behavior to be visible for a range of networks, from having a sensitivity of 1 to the point where [Formula: see text] is maximized. In this region, the two quantities are weakly correlated and attractors can be almost arbitrarily robust to perturbations without restricting the propagation of information in the network.