Attractors In State Space Research Articles

Signaling networks, network pathology and computational chemotherapy

Cellular signal transduction pathways are highly interconnected and form a complex network where many signaling molecules interact with each other in a complicated manner [1-2]. The complexity of the network structure suggests that the signal transduction network may conduct sophisticated information processing and decision making for cell fate determination rather than simple passing of an outer signal to the interior of the cell [3-4]. Cell fate, the stable cellular phenotype, is determined by the characteristic stable state of the complex network that is defined by a set of state values containing the activity states of all molecules in the network. Thus, systems biological investigation is crucial to understand the cell fate decision mechanism encoded in the nonlinear dynamics of the signal transduction network [5-6]. The recent study by Choi et al [7] shows that state space modeling and attractor landscape analysis of complex molecular networks can reveal a hidden cell fate decision mechanism and identify key network components that together could synergistically force cancer cells to undergo apoptosis. Choi et al investigated the regulatory mechanisms underlying p53 dynamics and its function in modulating cell fate outcome at the system level. In this study, the state transition dynamics of the p53 regulatory network were analyzed using Boolean network modeling and attractor landscape analysis where each network state is represented as a point in the cellular state space and the network state eventually converges to a fixed point (or a set of points) called the attractor state which represents the most stable state with the lowest potential energy in the state space. The analysis results revealed that the most critical network components, which determine both p53 dynamics and the cell fate outcome in response to DNA damage, are feedbacks and interactions between p53, Mdm2, Wip1, Cyclin G and ATM. Disruption of the above critical feedback controls was found to not only result in change of p53 dynamics, i.e. pulsing vs. sustained elevation, but also alteration of cell fate outcome. The attractor landscape analysis was then employed to investigate the DNA damage response of a representative breast cancer cell line, MCF7, and the effect of nutlin-3, a well-known inhibitor of Mdm2, in comparison with normal cells. Limited efficacy of nutlin-3 was indicated by its induction of small basin of attraction to apoptotic attractor in the state-space landscape with p53 dynamics mostly being oscillatory. In contrast, the analysis revealed that treatment of nutlin-3 in combination with inhibition of Wip1 would engender strong synergistic effect in activating p53-mediated apoptosis, as such combined perturbations resulted in a larger basin of attraction to apoptotic attractors with sustained elevation of p53 (Figure ​(Figure1).1). The predicted synergistic effect was validated by single-cell imaging experiment, using a fluorescent p53 reporter line of MCF7. The experimental data suggest that a combinatorial treatment of nutlin-3 and Wip1 inhibitor is a more effective strategy for inducing apoptosis in response to DNA damaging chemotherapeutics. This study demonstrates that system-level analysis of p53 network dynamics and its regulation using attractor landscape can be employed to understand the complex cell fate decision mechanism and identify novel therapeutic strategies for treating cancer. This study also suggests a possible paradigm shift to ‘network pathology’ where we use network information instead of molecular information for personalized diagnosis and treatment of complex disease. The idea of network pathology is that the different mutation profile of each patient can be reflected in the modified network structure for individualized state space analysis. It further suggests the new concept of ‘computational chemotherapy’ that can provide an optimal therapeutic strategy for each individual patient on the basis of network pathology. Figure 1 The state space analysis of the p53 network and the resulting potential energy landscapes for MCF7 cells after different inhibitory treatments that modify the network and induce cell death in the presence of DNA damage

An Evaluation of Damage Features Extraction from Attractors in Structural Health Monitoring

Structural health monitoring provides accurate information about structure’s safety and integrity. The vibration-based structural health monitoring involves extracting a feature which robustly quantifies damage induced change to the structure. Recent work has focused on damage features extracted from the state space attractor of the structural response. Some of these features involve prediction error and local variance ratio. In the present paper, a five degree of freedom spring damper system forced by a Lorenz excitation is used to evaluate these two typical damage features. Their ability of identification damage level and location is characterized and compared.

Attractor analysis of asynchronous Boolean models of signal transduction networks

Prior work on the dynamics of Boolean networks, including analysis of the state space attractors and the basin of attraction of each attractor, has mainly focused on synchronous update of the nodes’ states. Although the simplicity of synchronous updating makes it very attractive, it fails to take into account the variety of time scales associated with different types of biological processes. Several different asynchronous update methods have been proposed to overcome this limitation, but there have not been any systematic comparisons of the dynamic behaviors displayed by the same system under different update methods. Here we fill this gap by combining theoretical analysis such as solution of scalar equations and Markov chain techniques, as well as numerical simulations to carry out a thorough comparative study on the dynamic behavior of a previously proposed Boolean model of a signal transduction network in plants. Prior evidence suggests that this network admits oscillations, but it is not known whether these oscillations are sustained. We perform an attractor analysis of this system using synchronous and three different asynchronous updating schemes both in the case of the unperturbed (wild-type) and perturbed (node-disrupted) systems. This analysis reveals that while the wild-type system possesses an update-independent fixed point, any oscillations eventually disappear unless strict constraints regarding the timing of certain processes and the initial state of the system are satisfied. Interestingly, in the case of disruption of a particular node all models lead to an extended attractor. Overall, our work provides a roadmap on how Boolean network modeling can be used as a predictive tool to uncover the dynamic patterns of a biological system under various internal and environmental perturbations.

Dynamical role of the degree of intraspecific cooperation: A simple model for prebiotic replicators and ecosystems

We present a simple mean field model to analyze the dynamics of competition between two populations of replicators in terms of the degree of intraspecific cooperation (i.e., autocatalysis) in one of these populations. The first population can only replicate with Malthusian kinetics while the second one can reproduce with Malthusian or autocatalytic replication or with a combination of both reproducing strategies. The model consists of two coupled, nonlinear, autonomous ordinary differential equations. We investigate analytically and numerically the phase plane dynamics and the bifurcation scenarios of this ecologically coupled system, focusing on the outcome of competition for several degrees of intraspecific cooperation, σ , in the second population of replicators. We demonstrate that the dynamics of both populations can not be governed by a limit cycle, and also that once cooperation is considered, the topology of phase space does not allow for coexistence. Even for low values of the degree of intraspecific cooperation, for large enough autocatalytic replication rates, the second population of replicators is able to outcompete the first one, having a wide basin of attraction in state space. We characterize the same power law dependence between the outcompetition extinction times, τ , and the degree of intraspecific cooperation for both populations, given by τ ∼ c i σ − 1 . Our results suggest that, under some kinetic conditions, the appearance of autocatalysis might be favorable in a population of replicators growing with Malthusian kinetics competing with another population also reproducing exponentially.

Metodología para evaluar el difeomorfismo de un atractor caótico usando el filtro de Kalman en señales fisiológicas

Para la caracterización de señales fisiológicas, las cuales contienen estructuras fuertemente no lineales, es común usar procedimientos derivados de técnicas fractales que hacen parte del análisis de complejidad. En este trabajo se propone una función de evaluación basada en el filtro de Kalman para predecir puntos de un atractor reconstruido en el espacio de estados y medir la capacidad de esos puntos para reconstruir la señal en el tiempo, y así, evaluar la calidad de ese atractor a partir de una señal unidimensional. Se propone el uso de medidas estadísticas como Kullback–Leibler, Kolmogorov–Smirnov y Hellinger para determinar a diferencia entre la estructura estadística embebida en los puntos que fueron resultado de la predicción usando el atractor reconstruido y los puntos correspondientes de la señal original.Los resultados fueron obtenidos sobre la reconstrucción de atractores provenientes de señales ECG usando la base de datos del MIT-BIH y señales EEG obtenidas en la Clínica para Epilepsia de la Universidad de Bonn. Así, fue posible evaluar la capacidad depredicción correspondiente a los atractores reconstruidos a partir de los registros, con lo que se puede concluir que un atractor con alta capacidad de predicción en el tiempo implica buenas propiedades de embebimiento en el espacio de estados.

Optimality and stability in a class of bang–bang controlled biochemical reaction systems

This paper deals with the optimization of biochemical reaction systems of rank one. Two optimization problems are solved: the problem of optimal operation for maximum productivity in steady state and the problem of the start-up to the optimal steady state. Application of Pontryagin's maximum principle shows that the controller is of the bang–bang type, with no singular intervals. The determination of the optimal switching surface involves the solution of a two point boundary value problem. Solving such a difficult problem is avoided by choosing candidate switching surfaces on a heuristic basis. This study shows that switching on the stability boundary of the nominal operating point corresponding to the maximum dilution rate is the best choice. Here the value of the cost index is minimum amongst the various switching surfaces considered and the stability boundary satisfies the conditions imposed on a candidate switching surface for proper operation. Simple, robust algorithms are formulated for accurately estimating the system's stability boundary. The obtained results display the influence of feedback control on the stability of the set point. The bang–bang controller substantially increases the set point's region of attraction in state space as compared to the uncontrolled bioreactor.

Boolean Network Model Predicts Cell Cycle Sequence of Fission Yeast

A Boolean network model of the cell-cycle regulatory network of fission yeast (Schizosaccharomyces Pombe) is constructed solely on the basis of the known biochemical interaction topology. Simulating the model in the computer faithfully reproduces the known activity sequence of regulatory proteins along the cell cycle of the living cell. Contrary to existing differential equation models, no parameters enter the model except the structure of the regulatory circuitry. The dynamical properties of the model indicate that the biological dynamical sequence is robustly implemented in the regulatory network, with the biological stationary state G1 corresponding to the dominant attractor in state space, and with the biological regulatory sequence being a strongly attractive trajectory. Comparing the fission yeast cell-cycle model to a similar model of the corresponding network in S. cerevisiae, a remarkable difference in circuitry, as well as dynamics is observed. While the latter operates in a strongly damped mode, driven by external excitation, the S. pombe network represents an auto-excited system with external damping.

Experimental Enhanced Nonlinear Dynamics and Identification of Attractor Morphing Modes for Damage Detection

This paper demonstrates two novel methods for identifying small parametric variations in an experimental system based on the analysis of sensitivity vector fields (SVFs) and probability density functions (PDFs). The experimental system includes a smart sensing beam excited by a nonlinear feedback excitation through two lead zirconate titanate patches symmetrically bonded on both sides at the root of the beam. The nonlinear feedback excitation requires the measurement of the dynamics (e.g., velocity of one point at the tip of the beam) and a nonlinear feedback loop, and is designed such that the beam vibrates in a chaotic regime. Changes in the state space attractor of the dynamics due to small parametric variations can be captured by SVFs, which, in turn, are collected by applying point cloud averaging to points distributed in the attractors for nominal and changed parameters. Also, the PDFs characterize statistically the distribution of points in the attractors. The differences between the PDFs of the attractors for different changed parameters and the base line attractor can provide different attractor morphing modes for identifying variations in distinct parameters. Experimental results based on the proposed approaches show that very small amounts of added mass at different locations along the beam can be accurately identified.2

Numerically evaluated functional equivalence between chaotic dynamics in neural networks and cellular automata under totalistic rules

Chaotic dynamics in a recurrent neural network model and in two-dimensional cellular automata, where both have finite but large degrees of freedom, are investigated from the viewpoint of harnessing chaos and are applied to motion control to indicate that both have potential capabilities for complex function control by simple rule(s). An important point is that chaotic dynamics generated in these two systems give us autonomous complex pattern dynamics itinerating through intermediate state points between embedded patterns (attractors) in high-dimensional state space. An application of these chaotic dynamics to complex controlling is proposed based on an idea that with the use of simple adaptive switching between a weakly chaotic regime and a strongly chaotic regime, complex problems can be solved. As an actual example, a two-dimensional maze, where it should be noted that the spatial structure of the maze is one of typical ill-posed problems, is solved with the use of chaos in both systems. Our computer simulations show that the success rate over 300 trials is much better, at least, than that of a random number generator. Our functional simulations indicate that both systems are almost equivalent from the viewpoint of functional aspects based on our idea, harnessing of chaos.

Parameter Reconstruction Based on Sensitivity Vector Fields

A novel approach to determine very accurately multiple parameter variations by exploiting the geometric shape of dynamic attractors in state space is presented. The approach is based on the analysis of sensitivity vector fields. These sensitivity vector fields describe changes in the state space attractor of the dynamics and system behavior when parameter variations occur. Distributed throughout the attractor in state space, these fields form a collection of snapshots for known parameter changes. Proper orthogonal decomposition of the parameter space is then employed to distinguish multiple simultaneous parametric variations. The parametric changes are reconstructed by analyzing the deformation of attractors which are characterized by means of the sensitivity vector fields. A set of basis functions in the vector space formed by the sensitivity fields is obtained and is used to successfully identify test cases involving multiple simultaneous parametric variations. The method presented is shown to be robust over a wide range of parameter variations and to perform well in the presence of noise. One of the main applications of the proposed technique is detecting multiple simultaneous damage in vibration-based structural health monitoring.

Behavioral Dynamicsof Human Locomotion

Gibson (1979) argued that “control lies not in the brain, but in the animalenvironment system.” To make good on this claim, we must show how adaptive behavior emerges from the interactions of an agent with a structured environment guided by occurrent information. Here we attempt to model the behavioral dynamics of human walking and show how locomotor paths emerge “online” from simple laws for steering and obstacle avoidance. Our approach is inspired by Schoner, Dose, and Engels’s (1995) control system for mobile robots. By behavioral dynamics, we mean a description of the time evolution of observed behavior. Assume that goal-directed behavior can be described by a few behavioral variables, which define a state space for the system. Goals correspond to attractors in state space to which trajectories converge, whereas states to be avoided correspond to repellors from which trajectories diverge. The problem is to formalize a system of differential equations, or dynamical system, whose solutions capture the observed behavior. We take the current heading direction φ and turning rate as behavioral variables, assuming travel at a constant speed v (see Figure 1). From the agent’s current (x, z) position, a goal lies in the direction ψg at a distance dg; an obstacle may also lie in direction ψo at a distance do. The simplest description of steering toward a goal is for the agent to bring the heading error between the current heading direction and the goal direction to zero (φ – ψg = 0), which defines an attractor in state space.

ヒューマノイドロボットの状態空間におけるアトラクタ設計と自律運動の生成・制御(創発ロボテックス)

ヒューマノイドロボットの状態空間におけるアトラクタ設計と自律運動の生成・制御(創発ロボテックス)

Markov chain Monte Carlo estimation of nonlinear dynamics from time series

Much of nonlinear time series analysis is concerned with inferring unmeasured quantities — e.g., system parameters, the shape of attractors in state space — from a noisy measured time series. From a Bayesian perspective, the time series is a vector sample picked at random from a probability density. The density reflects the system dynamics and our subjective uncertainty about system parameters, the measurement function, dynamical noise and measurement noise. The conditional probability density of the system parameters given the measured data is the basis of a Bayesian estimate of the system parameters. Using illustrative chaotic systems with large-amplitude dynamical and measurement noise, we show here that it is feasible to use the Markov chain Monte Carlo (MCMC) technique to generate the Bayesian conditional probabilities. The resulting parameter estimates are markedly superior to those based on conventional least-squares methods: the MCMC-based estimates are unbiased and allow estimates of dynamical parameters on unmeasured components of the state vector. In addition, the MCMC method enables de-noised attractors to be reconstructed, not just in an embedding based on lags of measured variables but in the state space that includes unmeasured components of the dynamics’ state vector. The general purpose MCMC technique effectively combines techniques of nonlinear noise reduction and nonlinear parameter estimation.

Forecasting high waters at Venice Lagoon using chaotic time series analysis and nonlinear neural networks

Time series analysis using nonlinear dynamics systems theory and multilayer neural networks models have been applied to the time sequence of water level data recorded every hour at ‘Punta della Salute’ from Venice Lagoon during the years 1980–1994. The first method is based on the reconstruction of the state space attractor using time delay embedding vectors and on the characterisation of invariant properties which define its dynamics. The results suggest the existence of a low dimensional chaotic attractor with a Lyapunov dimension, DL, of around 6.6 and a predictability between 8 and 13 hours ahead. Furthermore, once the attractor has been reconstructed it is possible to make predictions by mapping local-neighbourhood to local-neighbourhood in the reconstructed phase space. To compare the prediction results with another nonlinear method, two nonlinear autoregressive models (NAR) based on multilayer feedforward neural networks have been developed. From the study, it can be observed that nonlinear forecasting produces adequate results for the ‘normal’ dynamic behaviour of the water level of Venice Lagoon, outperforming linear algorithms, however, both methods fail to forecast the ‘high water’ phenomenon more than 2–3 hours ahead.

Smoothing data with local instabilities for the identification of chaotic systems

Chaotic systems are characterized by the local divergence of nearby orbits in state-space. This provokes a sensitive dependence on initial conditions and, in turn, drastically limits the accuracy of long-term predictions. This has important implications in the filtering of data generated by such systems. In the present paper, the use of global smoothers for chaotic data is investigated. The ultimate objective is to be able to identify dynamically valid models from smoothed data when the identification from the original noisy data has completely failed. The objective is to produce identified models that faithfully reproduce the dynamical invariants of the original system, such as the geometry of the attractors in state-space, the largest Lyapunov exponent, fractal dimensions and Poincare sections. Numerical examples are included, which illustrate the main points of the paper.

Circle in the round: State space attractors for evolved sighted robots

This paper presents an analysis of an artificially evolved dynamical network-based control system for a simulated autonomous mobile robot engaged in simple visually guided tasks.

Local analysis of time series from the oscillatory electrocatalytic reduction of hydrogen peroxide

Experimental time series from an electrochemical reaction are analyzed using a local method which was developed by Broomhead and King. We discuss the use of the method on experimental data which exhibit both noise and drift. The experiments were done on the reduction of hydrogen peroxide on a platinum rotating disk electrode in acidic solution. The potential was measured as a function of time under galvanostatic conditions and state space attractors were constructed from the time series. The local analysis was used to estimate the eigenvalues of a fixed point of the attractor. We show how the system characteristics change as a slow temporal variation of the oscillations occurs.

HOW MANY DELAY COORDINATES DO YOU NEED?

Theorems on the use of delay coordinates for analyzing experimental data are discussed. To reconstruct a one-to-one correspondence with the state-space attractor, m delay coordinates are sufficient, where m>2D0 (here D0 denotes the box-counting dimension). For calculating the correlation dimension D2, m>D2 delays are sufficient. These results remain true under finite impulse response (FIR) filters.

Dimensional analysis of no-task human EEG using the Grassberger-Procaccia method.

A method developed by Grassberger and Procaccia allows estimation of the dimensional complexity of the state-space attractor of a time series. Saturation of dimensional-complexity estimates with increasing values of embedding dimension is considered a strong indication that the time series is governed by deterministic chaos. The present investigation employed the Grassberger-Procaccia method to estimate EEG dimensional complexity in a multi-subject, factorial experiment. Twelve subjects were tested under two no-task conditions (eyes closed and open), with the block of two conditions being repeated four times. EEG was recorded from the nineteen 10-20 loci. Dimensional complexity declined across Blocks 1-3 and then leveled off, and was higher in the eyes-open than in the eyes-closed condition. Condition also interacted with locus in that the increase in dimensional complexity associated with opening the eyes was greater over occipital loci. Comparison with the results of Fourier analysis indicated that a similar but not identical pattern of effects was obtained for alpha (8-12 Hz) power. Further, across the entire data set, if alpha power exceeded a value of about 70 microV2, dimensional complexity was uniformly low, a finding in concert with previous results indicating that the eyes-closed, occipital alpha rhythm possibly represents deterministic chaos of relatively low dimension.

Chaos as a Limit in a Boundary Value Problem

A piecewise-linear. 3-variable autonomous O.D.E. of C0 type, known to describe constantshape travelling waves in one-dimensional reaction-diffusion media of Rinzel-Keller type, is numerically shown to possess a chaotic attractor in state space. An analytical method proving the possibility of chaos is outlined and a set of parameters yielding Shil'nikov chaos indicated. A symbolic dynamics technique can be used to show how the limiting chaos dominates the behavior even of the finite boundary value problem.