Petri net representation of multi-valued logical regulatory graphs

Relying on a convenient logical representation of regulatory networks, we propose a generic method to qualitatively model regulatory interactions in the standard elementary and coloured Petri net frameworks. Logical functions governing the behaviours of the components of logical regulatory graphs are efficiently represented by Multivalued Decision Diagrams, which are also at the basis of the translation of logical models in terms of Petri nets. We further delineate a simple strategy to sort trajectories through the introduction of priority classes (in the logical framework) or priority functions (in the Petri net framework). We also focus on qualitative behaviours such as multistationarity or sustained oscillations, identified as specific structures in state transition graphs (for logical models) or in marking graphs (in Petri nets). Regulatory circuits are known to be at the origin of such properties. In this respect, we present a method that allows to determine the functionality contexts of regulatory circuits, i.e. constraints on external regulator states enabling the corresponding dynamical properties. Finally, this approach is illustrated through an application to the modelling of a regulatory network controlling T lymphocyte activation and differentiation.     

基于双格的多值模型的精化关系与对称化简

多值模型是传统布尔模型的扩展。与布尔模型相比,多值模型更适合对包含不确定和不一致信息的软件系统进行建模。为了解决模型检测时的状态爆炸问题,研究了对基于双格的多值模型的对称化简方法。提出了一种新的多值模型的精化关系,证明其保持对[μ]演算公式的模型检测结果的正确性。定义多值模型的对称化简商结构,证明商结构与原模型之间存在互为精化的关系,因此对[μ]演算公式的模型检测在二者上可以得到相同的结果。     

多值逻辑网络的输出跟踪牵制控制

本文利用代数状态空间方法,研究了多值逻辑控制网络的输出跟踪牵制控制.首先利用矩阵的半张量积给出了带牵制控制的多值逻辑控制网络的代数表示.其次基于该代数表示,定义了一组合适的能达集,并建立了多值逻辑控制网络输出跟踪牵制控制器的设计方法.再次,利用多值逻辑哑算子的性质,给出了多值逻辑控制网络分布式输出跟踪控制问题可解的充要条件.最后将所得的理论结果应用于网络演化博弈的演化行为分析.     

一种实现三值逻辑电路的DNA计算模型

近年来,随着生物计算和量子计算研究的深入,多值逻辑电路的各种实现成为一个热门的研究方向.发夹结构是DNA分子一种特殊杂交方式的产物,具有结果稳定、特异性强的优点.本文首次提出了一种利用DNA分子来实现多值逻辑电路的方法,用DNA分子的多发夹结构来表示三值逻辑的值,并给出"与"运算和"或"运算的计算模型,该模型适合应用于大规模的多值逻辑电路.     

WANG氏RC互连线模型平衡截断化简的分解算法

平衡截断（Balanced Truncation）是一种有效的模型简化（Model Reduction）方法,它的优点是简化模型有一个误差上限,使简化模型的性能得到保证.该文介绍一种平衡实现的分解算法,并应用Matlab对一4阶的Wang氏RC互连线模型的平衡实现及其模型截断简化进行编程仿真.单位阶跃响应和波特图显示原始模型与平衡模型曲线重合.给出了1阶、2阶、3阶平衡截断简化模型的阶跃响应和波特图,并对简化模型的理论误差上限与模型的实际最大误差进行了对比.方法可用于对IC互连线等模型的简化.     

A reduction method and qualitative analysis of dynamic systems: II

In the first part of this series, the method of reduction was proposed as a logical engine for hypotheses formulation as the development of the well-known algorithms of the comparison method. In this part, its application for the qualitative analysis of different properties of dynamical systems given in form of motion systems, differential equations, and automata models with various depths of delay is demonstrated.     

Model order reduction of MIMO bilinear systems by multi-order Arnoldi method

In this paper, we present a time domain model order reduction method for multi-input multi-output (MIMO) bilinear systems by general orthogonal polynomials. The proposed method is based on a multi-order Arnoldi algorithm applied to construct the projection matrix. The resulting reduced model can match a desired number of expansion coefficient terms of the original system. The approximate error estimate of the reduced model is given. And we also briefly discuss the stability preservation of the reduced model in some cases. Additionally, in combination with Krylov subspace methods, we propose a two-sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously. The effectiveness of the proposed methods is demonstrated by two numerical examples.     

Co-Algebraic Models for Quantitative Spatial Logics

We introduce a class of coalgebraic models and a family of modal logics that support the specification of spatial properties of distributed applications. The evaluation of a formula yields a value in a suitable multi-valued algebraic structure, giving a measure of the satisfaction of a requirement, induced by the decomposition of a system into subsystems, meant as available resources. As semantic domain we consider certain algebraic structures, called c-semirings, that allow us to generalize boolean logics to the multi-valued case, while keeping a number of the axioms of boolean algebras. Under suitable conditions on the structure of c-semirings, we show that, even if our logical formalisms are equipped with spatial operators, the interpretation of formulas fully characterizes bisimilarity.     

Model reduction by the low-frequency approximation balancing methodfor unstable systems

The unstable system model reduction technique is investigated by the low-frequency approximation balancing method. The reduced models retain the DC gain of the original model which offer a good approximation at low frequencies. Various relevant properties of the reduced model are available. In particular, the reduced models ensure the minimality and the number of the unstable poles as the original model. An upper bound for model reduction error is also provided in the frequency domain     

Stable Model Order Reduction Method for Fractional-Order Systems Based on Unsymmetric Lanczos Algorithm

This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.      

Parameter independent model order reduction

Several recently developed model order reduction methods for fast simulation of large-scale dynamical systems with two or more parameters are reviewed. Besides, an alternative approach for linear parameter system model reduction as well as a more efficient method for nonlinear parameter system model reduction are proposed in this paper. Comparison between different methods from theoretical elegancy to complexity of implementation are given. By these methods, a large dimensional system with parameters can be reduced to a smaller dimensional parameter system that can approximate the original large sized system to a certain degree for all the parameters.     

Learning as a nonlinear line of attraction in a recurrent neural network

A method to embed N dimensional, multi-valued patterns into an auto-associative memory represented as a nonlinear line of attraction in a fully connected recurrent neural network is presented in this paper. The curvature of the nonlinear attractor is defined by the Kth degree polynomial line which best fits the training data in N dimensional state space. The width of the nonlinear line is then characterized by the statistical characteristics of the training patterns. Stability of the recurrent network is verified by analyzing the trajectory of the points in the state space during convergence. The performance of the network is benchmarked through the reconstruction of original gray-scale images from their corrupted versions. It is observed that the proposed method can quickly and successfully reconstruct each image with an average convergence rate of 3.10 iterations.     

Simplification of a complex signal transduction model using invariants and flow equivalent servers

In this paper we consider the modeling of a portion of the signal transduction pathway involved in the angiogenic process. The detailed model of this process is affected by a high level of complexity due to the functional properties that are represented and the size of its state space. To overcome these problems, we suggest approaches to simplify the detailed representation that result in models with a lower computational and structural complexity, while still capturing the overall behavior of the detailed one.The simplification process must take into account both the structural aspects and the quantitative behavior of the original model. To control the simplification from a structural point of view, we propose a set of reduction steps that maintain the invariants of the original model. To ensure the correspondence between the simplified and the original models from a quantitative point of view we use the flow equivalent method that provides a way of obtaining the parameters of the simplified model on the basis of those of the original one.To support the proposed methodology we show that a good agreement exists among the temporal evolutions of the relevant biological products in the simplified and detailed model evaluated with a large set of input parameters.     

Algebraic method of simplifying Boolean networks using semi‐tensor product of Matrices

The huge state space of large Boolean networks makes analysis and synthesis difficult. This paper, using a new matrix analysis tool called semi‐tensor product of matrices, to explain a simplification method of Boolean networks in a mathematical manner. The idea consists of two steps. First, remove the nodes whose logical dynamics are independent of themselves directly; second, use the logical functions (LFs) of the removed nodes to substitute for their corresponding variables in the LFs of other nodes; such nodes evolve directly with both themselves and the removed nodes. We discover that the simplified and original Boolean networks share some important topological structures such as attractor cycles, steady states and paths. An algebraic algorithm is provided to find all of the cycles and steady states of simplified Boolean networks. Finally we apply the results to the metastatic melanoma network to check the effect of the simplification method.     

Multistability equivalence between gene regulatory networks of different dimensionality with application to a differentiation network

This paper formulates and addresses the problem of equivalence in terms of multistability properties between nonlinear models of gene regulatory systems of different dimensionality. Given a nonlinear dynamical model of a gene regulatory network and the structure of another higher‐dimensional gene regulatory network, the aim is to find a dynamical model for the latter that has the same equilibria and stability properties as the former. We propose construction rules for the dynamics of a high‐dimensional system, given the low‐dimensional system and the high‐dimensional network structure. These construction rules yield a multistability‐equivalent system, as we prove in this work. We demonstrate the value of our method by applying it to an example of a multistable gene regulatory network involved in mesenchymal stem cell differentiation. Here, differentiation is described by a core motif of three genetic regulators, but the detailed network contains at least nine genes. The proposed construction method allows to transfer the multistability based differentation mechanism of the core motif to the more detailed gene regulatory network. Copyright © 2016 John Wiley & Sons, Ltd.     

Poisson‐Based Weight Reduction of Animated Meshes

While animation using barycentric coordinates or other automatic weight assignment methods has become a popular method for shape deformation, the global nature of the weights limits their use for real‐time applications. We present a method that reduces the number of control points influencing a vertex to a user‐specified number such that the deformations created by the reduced weight set resemble that of the original deformation. To do so we show how to set up a Poisson minimization problem to solve for a reduced weight set and illustrate its advantages over other weight reduction methods. Not only does weight reduction lower the amount of storage space necessary to deform these models but also allows GPU acceleration of the resulting deformations. Our experiments show that we can achieve a factor of 100 increase in speed over CPU deformations using the full weight set, which makes real‐time deformations of large models possible.     

Fractal dimensions of random attractors for stochastic Benjamin–Bona–Mahony equation on unbounded domains

This paper considers random attractor and its fractal dimension for Benjamin–Bona–Mahony equation driven by additive white noise on unbounded domains . Firstly, we investigate the existence of random attractor for the random dynamical system defined on an unbounded domain. Secondly, we present criterion for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable Banach space. Finally, we apply expectations of some random variables and these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic Benjamin–Bona–Mahony equation driven by additive white noise.     

An approach for decomposition and reduction of dynamical models of large-scale power systems

This paper describes an approach for decomposition and reduction of dynamical models of largo scale power systems. In this approach the system is decomposed into two subsystems, the first one is described by a linear model while the second is described by a non-linear model. This decomposition is based on a derived criteria for linearization by which we can know the two subsystems a priori. Further, the linear subsystem model is reduced by aggregation and hence the order of the system is reduced. The results of the validity of this approach as applied to two different largo power systems are indicated.     

Neural learning of chaotic dynamics

In recent years, considerable progress has been made in modeling chaotic time series with neural networks. Most of the work concentrates on the development of architectures and learning paradigms that minimize the prediction error. A more detailed analysis of modeling chaotic systems involves the calculation of the dynamical invariants which characterize a chaotic attractor. The features of the chaotic attractor are captured during learning only if the neural network learns the dynamical invariants. The two most important of these are the largest Lyapunov exponent which contains information on how far in the future predictions are possible, and the Correlation or Fractal Dimension which indicates how complex the dynamical system is. An additional useful quantity is the power spectrum of a time series which characterizes the dynamics of the system as well, and this in a more thorough form than the prediction error does. In this paper, we introduce recurrent networks that are able to learn chaotic maps, and investigate whether the neural models also capture the dynamical invariants of chaotic time series. We show that the dynamical invariants can be learned already by feedforward neural networks, but that recurrent learning improves the dynamical modeling of the time series. We discover a novel type of overtraining which corresponds to the forgetting of the largest Lyapunov exponent during learning and call this phenomenondynamical overtraining. Furthermore, we introduce a penalty term that involves a dynamical invariant of the network and avoids dynamical overtraining. As examples we use the Hénon map, the logistic map and a real world chaotic series that corresponds to the concentration of one of the chemicals as a function of time in experiments on the Belousov-Zhabotinskii reaction in a well-stirred flow reactor.