Finite-time Lyapunov exponents and metabolic control coefficients for threshold detection of stimulus–response curves

In biochemical networks transient dynamics plays a fundamental role, since the activation of signalling pathways is determined by thresholds encountered during the transition from an initial state (e.g. an initial concentration of a certain protein) to a steady-state. These thresholds can be defined in terms of the inflection points of the stimulus–response curves associated to the activation processes in the biochemical network. In the present work, we present a rigorous discussion as to the suitability of finite-time Lyapunov exponents and metabolic control coefficients for the detection of inflection points of stimulus–response curves with sigmoidal shape.     

Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type

We discuss how to characterize long-time dynamics of non-smooth dynamical systems, such as integrate-and-fire (I&F) like neuronal network, using Lyapunov exponents and present a stable numerical method for the accurate evaluation of the spectrum of Lyapunov exponents for this large class of dynamics. These dynamics contain (i) jump conditions as in the firing-reset dynamics and (ii) degeneracy such as in the refractory period in which voltage-like variables of the network collapse to a single constant value. Using the networks of linear I&F neurons, exponential I&F neurons, and I&F neurons with adaptive threshold, we illustrate our method and discuss the rich dynamics of these networks.     

Detecting chaotic dynamics of insect populations from long-term survey data

1. Estimates of the Lyapunov exponent, a statistic that measures the sensitive dependence of the dynamic behaviour of a system on its initial conditions, are used to characterize several sets of insect time series.
2. A new method is described to overcome the difficulty of defining the dynamics of an observed, noisy, short ecological time series. This method provides two test statistics for the estimated Lyapunov exponent.
3. This method is applied to forty-six time series comprising six aphid species from five sites and four moth species from six sites. There are few positive Lyapunov exponents and none is sufficiently large to characterize its time series as chaotic.
4. Two methods to estimate the Lyapunov exponent are compared; that based on logarithmically transformed counts yields less variable estimates for highly variable insect data than that based on untransformed counts.     

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.     

Parameter-dependent transitions and the optimal control of dynamical diseases

Many theoretical studies in biological and physical sciences consider the dynamical behavior of ann-dimensional ordinary differential equation that contains a large number of independent parameters. A frequently asked question is, are there permissible parameter sets that result in periodic or chaotic behavior? The large number of distinct parameters often limits the feasibility of trial and error calculations. The large dimension and nonlinearity of the system make application of analytic methods at best difficult and at worst effectively impossible. It is shown here that a computational search for parameter-dependent transitions of attractor topology can be effected by constrained optimization of quantitative measures of dynamical behavior (Hurwitz polynomials, Floquet coefficients, Lyapunov exponents and correlation dimension). As an example, we examine a three-dimensional nonlinear ordinary differential equation containing seven parameters that was constructed by Goldbeter and Segel to model periodic synthesis of cyclic AMP inDictyostelium. A search for bifurcations to periodic solutions is made by minimizing Hurwitz coefficients subject to parameter constraints. By comparing four optimization algorithms, the defects and advantages of the procedure are identified. It is also argued that it may be possible to use this characterization of dynamics to construct optimal responses to dynamical diseases (those disorders that result from parameter-dependent bifurcations in physiological control systems).     

Nonlinear dynamics of the blood flow studied by Lyapunov exponents

In order to gain an insight into the dynamics of the cardiovascular system throughout which the blood circulates, the signals measured from peripheral blood flow in humans were analyzed by calculating the Lyapunov exponents. Over a wide range of algorithm parameters, paired values of both the global and the local Lyapunov exponents were obtained, and at least one exponent equaled zero within the calculation error. This may be an indication of the deterministic nature and finite number of degrees of freedom of the cardiovascular system governing the blood-flow dynamics on a time scale of minutes. A difference was observed in the Lyapunov dimension of controls and athletes.     

An analysis of the reliability phenomenon in the FitzHugh-Nagumo model

The reliability of single neurons on realistic stimuli has been experimentally confirmed in a wide variety of animal preparations. We present a theoretical study of the reliability phenomenon in the FitzHugh-Nagumo model on white Gaussian stimulation. The analysis of the model's dynamics is performed in three regimes—the excitable, bistable, and oscillatory ones. We use tools from the random dynamical systems theory, such as the pullbacks and the estimation of the Lyapunov exponents and rotation number. The results show that for most stimulus intensities, trajectories converge to a single stochastic equilibrium point, and the leading Lyapunov exponent is negative. Consequently, in these regimes the discharge times are reliable in the sense that repeated presentation of the same aperiodic input segment evokes similar firing times after some transient time. Surprisingly, for a certain range of stimulus intensities, unreliable firing is observed due to the onset of stochastic chaos, as indicated by the estimated positive leading Lyapunov exponents. For this range of stimulus intensities, stochastic chaos occurs in the bistable regime and also expands in adjacent parts of the excitable and oscillating regimes. The obtained results are valuable in the explanation of experimental observations concerning the reliability of neurons stimulated with broad-band Gaussian inputs. They reveal two distinct neuronal response types. In the regime where the first Lyapunov has negative values, such inputs eventually lead neurons to reliable firing, and this suggests that any observed variance of firing times in reliability experiments is mainly due to internal noise. In the regime with positive Lyapunov exponents, the source of unreliable firing is stochastic chaos, a novel phenomenon in the reliability literature, whose origin and function need further investigation.     

Direct Lyapunov exponent analysis enables parametric study of transient signalling governing cell behaviour

Computational models aid in the quantitative understanding of cell signalling networks. One important goal is to ascertain how multiple network components work together to govern cellular responses, that is, to determine cell 'signal-response' relationships. Several methods exist to study steady-state signals in the context of differential equation-based models. However, many biological networks influence cell behaviour through time-varying signals operating during a transient activated state that ultimately returns to a basal steady-state. A computational approach adapted from dynamical systems analysis to discern how diverse transient signals relate to alternative cell fates is described. Direct finite-time Lyapunov exponents (DLEs) are employed to identify phase-space domains of high sensitivity to initial conditions. These domains delineate regions exhibiting qualitatively different transient activities that would be indistinguishable using steady-state analysis but which correspond to different outcomes. These methods are applied to a physicochemical model of molecular interactions among caspase-3, caspase-8 and X-linked inhibitor of apoptosis--proteins whose transient activation determines cell death against survival fates. DLE analysis enabled identification of a separatrix that quantitatively characterises network behaviour by defining initial conditions leading to apoptotic cell death. It is anticipated that DLE analysis will facilitate theoretical investigation of phenotypic outcomes in larger models of signalling networks.     

On the dynamics of a simple biochemical control circuit

The quantitative dynamics of a biochemical control circuit that regulates enzyme or protein synthesis by end-product feedback is analyzed. We first study a simplified repressible system, which is known to exhibit either a steady state or an oscillatory solution. By showing the analogy of thisn-dimensional system with a time-delay equation for a single variable the mechanism of the self-sustained oscillations becomes transparent. In a more sophisticated system we will find as well either steady state or oscillatory solutions. We determine the role of the parameters with respect to stability and frequency. The most general case will be treated by means of the concept of Lyapunov exponents.     

Predictability of human EEG: a dynamical approach

The electroencephalogram recordings from human scalp are analysed in the framework of recent methods of nonlinear dynamics. Three stages of brain activity are considered: the alpha waves (eyes closed), the deep sleep (stage four) and the Creutzfeld-Jakob coma. Two dynamical parameters of the attractors are evaluated. These are the Lyapunov exponents, which measure the divergence or convergence of trajectories in phase space and the Kolmogorov or metric entropy, whose inverse gives the mean predicting time of a given EEG signal. In all the stages considered, the results reveal the presence of at least two positive Lyapunov exponents, which are the footprints of chaos. This number increases to three positive exponents in the case of alpha waves, indicating that although for very short episodes the alpha waves seem extremely coherent, the variability of the brain increases markedly over larger periods of activity. The degree of entropy/chaos increases from coma to deep sleep and then to alpha waves. The large predicting time observed for deep sleep suggests that these waves are related to a slow rate of information processing. The predicting time of the alpha waves is much smaller, indicating a rapid loss of information. Finally, with the help of the Lyapunov exponents, the attractor's dimensions are evaluated using two different conjectures and compared to values obtained previously by the Grassberger-Procaccia algorithm.     

Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction

Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.     

Numerical simulation of collapsible-tube flows with sinusoidal forced oscillations

Collapsible-tube flow with self-excited oscillations has been extensively investigated. Though physiologically relevant, forced oscillation coupled with self-excited oscillation has received little attention in this context. Based on an ODE model of collapsible-tube flow, the present study applies modern dynamics methods to investigate numerically the responses of forced oscillation to a limit-cycle oscillation which has topological characteristics discovered in previous unforced experiments. A devil's staircase and period-doubling cascades are presented with forcing frequency and amplitude as control parameters. In both cases, details are provided in a bifurcation diagram. Poincaré sections, a frequency spectrum and the largest Lyapunov exponents verify the existence of chaos in some circumstances. The thin fractal structure found in the strange attractors is believed to be a result of high damping and low stiffness in such systems.     

Quasicycles revisited: apparent sensitivity to initial conditions

Environmental noise is known to sustain cycles by perturbing a deterministic approach to equilibrium that is itself oscillatory. Quasicycles produced in this way display a regular period but varied amplitude. They were proposed by Nisbet and Gurney (Nature 263 (1976) 319) as one possible explanation for population fluctuations in nature. Here, we revisit quasicyclic dynamics from the perspective of nonlinear time series analysis. Time series are generated with a predator-prey model whose prey's growth rate is driven by environmental noise. A method for the analysis of short and noisy data provides evidence for sensitivity to initial conditions, with a global Lyapunov exponent often close to zero characteristic of populations 'at the edge of chaos'. Results with methods restricted to long time series are consistent with a finite-dimensional attractor on which dynamics are sensitive to initial conditions. These results are compared with those previously obtained for quasicycles in an individual-based model with heterogeneous spatial distributions. Patterns of sensitivity to initial conditions are shown to differentiate phase-forgetting from phase-remembering quasicycles involving a periodic driver. The previously reported mode at zero of Lyapunov exponents in field and laboratory populations may reflect, in part, quasicyclic dynamics.     

生态阈值确定方法综述

生态系统在环境条件变化时表现出的剧变或阈值现象是当前生态学研究的热点,但是生态阈值定量检测的困难阻碍了这一主题的研究与应用。本文从典型案例入手,通过分析潜在生态阈值的S型曲线式、补给-压力式和跃迁式驱动-响应机理,归纳了局部加权回归散点平滑法、分段回归、高斯模型、拐点分析软件、稳态转换检测软件、指示种阈值分析和系统动力学仿真模型7种生态阈值确定方法,并评述了其优缺点和适用性,以期为生态阈值的定量分析研究提供方法借鉴。     

Stability and motor adaptation in human arm movements

In control, stability captures the reproducibility of motions and the robustness to environmental and internal perturbations. This paper examines how stability can be evaluated in human movements, and possible mechanisms by which humans ensure stability. First, a measure of stability is introduced, which is simple to apply to human movements and corresponds to Lyapunov exponents. Its application to real data shows that it is able to distinguish effectively between stable and unstable dynamics. A computational model is then used to investigate stability in human arm movements, which takes into account motor output variability and computes the force to perform a task according to an inverse dynamics model. Simulation results suggest that even a large time delay does not affect movement stability as long as the reflex feedback is small relative to muscle elasticity. Simulations are also used to demonstrate that existing learning schemes, using a monotonic antisymmetric update law, cannot compensate for unstable dynamics. An impedance compensation algorithm is introduced to learn unstable dynamics, which produces similar adaptation responses to those found in experiments.     

Complexity analysis of electrocardiographic signals

Two types of electrocardiographic data series were investigated using appropriate tests based on a selection of semi-quantitative analysis algorithms. Distribution histograms, power spectra, auto-correlation functions, state-space portraits, Lyapunov exponents and wavelet transformations were applied to electrocardiograms of normal and stressed subjects. Statistical analysis using the Student's t-test revealed significant and non-significant alterations in stress-loaded cases compared to normal ones. Higher levels of adrenaline may account for a more complex dynamics (deterministic chaos) revealed in the stressed subjects.     

Bifurcations and chaos in a time-discrete integral model of population dynamics.

The properties of a time-discrete integral model of population dynamics are studied. The model equations define a dynamic system in the cone of nonnegative functions in L2. The presence of dissipator caused by spatial interaction is detected. Several bifurcations of codimensions 1 and 2 are investigated. The model may go over to chaotic behavior through a sequence of period-doubling bifurcations. The Lyapunov exponents and dimension of an attractor are found. The higher dimension of an attractor is shown to be a common feature of chaotic behavior.     

Design of ant-inspired stochastic control policies for collective transport by robotic swarms

In this paper, we present an approach to designing decentralized robot control policies that mimic certain microscopic and macroscopic behaviors of ants performing collective transport tasks. In prior work, we used a stochastic hybrid system model to characterize the observed team dynamics of ant group retrieval of a rigid load. We have also used macroscopic population dynamic models to design enzyme-inspired stochastic control policies that allocate a robotic swarm around multiple boundaries in a way that is robust to environmental variations. Here, we build on this prior work to synthesize stochastic robot attachment–detachment policies for tasks in which a robotic swarm must achieve non-uniform spatial distributions around multiple loads and transport them at a constant velocity. Three methods are presented for designing robot control policies that replicate the steady-state distributions, transient dynamics, and fluxes between states that we have observed in ant populations during group retrieval. The equilibrium population matching method can be used to achieve a desired transport team composition as quickly as possible; the transient matching method can control the transient population dynamics of the team while driving it to the desired composition; and the rate matching method regulates the rates at which robots join and leave a load during transport. We validate our model predictions in an agent-based simulation, verify that each controller design method produces successful transport of a load at a regulated velocity, and compare the advantages and disadvantages of each method.     

Phase resetting curves and oscillatory stability in interneurons of rat somatosensory cortex

Synchronous oscillations in neural activity are found over wide areas of the cortex. Specific populations of interneurons are believed to play a significant role in generating these synchronized oscillations through mutual synaptic and gap-junctional interactions. Little is known, though, about the mechanism of how oscillations are maintained stably by particular types of interneurons and by their local networks. To obtain more insight into this, we measured membrane-potential responses to small current-pulse perturbations during regular firing, to construct phase resetting curves (PRCs) for three types of interneurons: nonpyramidal regular-spiking (NPRS), low-threshold spiking (LTS), and fast-spiking (FS) cells. Within each cell type, both monophasic and biphasic PRCs were observed, but the proportions and sensitivities to perturbation amplitude were clearly correlated to cell type. We then analyzed the experimentally measured PRCs to predict oscillation stability, or firing reliability, of cells for a complex stochastic input, as occurs in vivo. To do this, we used a method from random dynamical system theory to estimate Lyapunov exponents of the simplified phase model on the circle. The results indicated that LTS and NPRS cells have greater oscillatory stability (are more reliably entrained) in small noisy inputs than FS cells, which is consistent with their distinct types of threshold dynamics.