Browsing by Author "Shih, Chih-Wen"

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A General Approach to Synchronization of Coupled Cells
(2013)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
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Department of Applied Mathematics
This investigation presents a general framework to establish synchronization of coupled cells and coupled systems. Each individual subsystem is represented by nonlinear differential equations with or without internal or intracellular delay. A general coupling function is employed to depict the communication or interaction between subsystems or cells. Under this framework, the problem of establishing the synchronization for delayed coupled nonlinear systems is transformed to solving a corresponding linear system of algebraic equations. We start by considering a cell-to-cell system under symmetric coupling to present the main idea of the approach. The framework is then extended to the N-cell system under circulant coupling. Delay-dependent, delay-independent, and network-scale-dependent criteria for global synchronization will be established, respectively. The developed scheme can accommodate a wide range of coupled systems. We demonstrate the applications of the present approach to establish synchronization for a gene regulation model, a neuronal model, and some neural networks.
A LATTICE MODEL ON SOMITOGENESIS OF ZEBRAFISH
(2012-11-01)
Liao, Kang-Ling
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
Somitogenesis is the process of the development of somites which are segmental structure in vertebrate embryos. This process depends on the expression of segmentation clock genes. In this investigation, we consider lattice systems which describe the kinetics of the chief segmentation clock genes in zebrafish under negative feedback regulation with delay through interaction with the Delta-Notch signaling among neighboring cells. We first derive the analytical theories for the oscillation-arrested and synchronous oscillation in an autonomous lattice model. Based on the parameter regimes in the theories, we design suitable gradients of degradation rates and delays in a non-autonomous lattice model. Such a lattice system can generate synchronous oscillations, oscillatory traveling waves, oscillation slowing down, oscillation-arrested, and high-low expression levels. We further distinguish between different gradient structures which lead to normal and abnormal segmentations respectively and connect these structures to the dynamical regimes in the cell-cell model.
Absolute stability and synchronization in neural field models with transmission delays
(2016-08-01)
Kao, Chiu-Yen
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Shih, Chih-Wen
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Wu, Chang-Hong
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應用數學系
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Department of Applied Mathematics
Neural fields model macroscopic parts of the cortex which involve several populations of neurons. We consider a class of neural field models which are represented by integro-differential equations with transmission time delays which are space-dependent. The considered domains underlying the systems can be bounded or unbounded. A new approach, called sequential contracting, instead of the conventional Lyapunov functional technique, is employed to investigate the global dynamics of such systems. Sufficient conditions for the absolute stability and synchronization of the systems are established. Several numerical examples are presented to demonstrate the theoretical results. (C) 2016 Elsevier B.V. All rights reserved.
Asymptotic limit in a cell differentiation model with consideration of transcription
(2012-05-15)
Friedman, Avner
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Kao, Chiu-Yen
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet (xi) and GATA-3 (x2). These factors are translated by the mRNA whose levels of expression, y(1) and y(2), depend, respectively, on xi and x2 in a nonlinear nonlocal way. The population density of T cells, phi(t, x(1), x(2), y(1), y(2)), satisfies a hyperbolic conservation law with coefficients depending nonlinearly and nonlocally on (t, x(1), x(2), y(1), y(2)). while the x(i), y(i) satisfy a system of ordinary differential equations. We study the long time behavior of and show, under some conditions on the parameters of the system of differential equations, that the gene expressions in the T-cell population aggregate at one, two or four points, which connect to various cell differentiation scenarios. (C) 2012 Elsevier Inc. All rights reserved.
Asymptotic phases in a cell differentiation model
(2009-08-01)
Friedman, Avner
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Kao, Chiu-Yen
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet (x(1)) and GATA-3 (x(2)). The population density of the T cells, phi(t, x(1), x(2)), satisfies a conservation law partial derivative phi/partial derivative t + (partial derivative/partial derivative x(1))(f(1)phi) + (partial derivative/partial derivative x(2))(f(2)phi) = g phi where f(i) depends on (t, x(1), x(2)) and, in a nonlinear nonlocal way, on phi. It is proved that, as t -> infinity, (t, x(1), x(2)) converges to a linear combination of 1, 2, or 4 Dirac measures. Numerical simulations and their biological implications are discussed. (C) 2009 Elsevier Inc. All rights reserved.
Coexistence and extinction for two competing species in patchy environments
(2019-01-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
A system of two competing species u and v that diffuse over a two-patch environment is investigated. When u-species has smaller birth rate in the first patch and larger birth rate in the second patch than v-species, and the average birth rate for u-species is larger than or equal to v-species, it was shown in a previous publication that two species coexist in a slow diffusion environment, whereas u-species drives v-species into extinction in a fast diffusion environment. In this paper, we analyze global dynamics and bifurcations for the same model with identical order of birth rates, but with opposite order of average birth rates, i.e., the average birth rate of u-species is less than that of v-species. We observe richer dynamics with two scenarios, depending on the relative difference between the variation in the birth rates of v-species on two patches and the variation in the average birth rates of two species. When the variation in average birth rates is relatively large, there is no stability switch for the semitrivial equilibria. On the other hand, such a stability switch takes place when the variation in average birth rates is relatively mild. In both cases, v-species, with larger average birth rate, prevails in a fast diffusion environment, whereas in a slow diffusion environment, the two species can coexist or u-species that has the greatest birth rate among both species and patches will persist and drive v-species to extinction.
Complete Stability in Multistable Delayed Neural Networks
(2009-03-01)
Cheng, Chang-Yuan
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
We investigate the complete stability for multistable delayed neural networks. A new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property. An iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity. The existence of 3(n) equilibria and 2(n) positively invariant sets for the n-neuron system remains valid under the new formulation. The theory is demonstrated by a numerical illustration.
Convergent dynamics for multistable delayed neural networks
(2008-10-01)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
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Department of Applied Mathematics
This investigation aims at developing a methodology to establish convergence of dynamics for delayed neural network systems with multiple stable equilibria. The present approach is general and can be applied to several network models. We take the Hopfield-type neural networks with both instantaneous and delayed feedbacks to illustrate the idea. We shall construct the complete dynamical scenario which comprises exactly 2(n) stable equilibria and exactly (3(n)-2(n)) unstable equilibria for the n-neuron network. In addition, it is shown that every solution of the system converges to one of the equilibria as time tends to infinity. The approach is based on employing the geometrical structure of the network system. Positively invariant sets and componentwise dynamical properties are derived under the geometrical configuration. An iteration scheme is subsequently designed to confirm the convergence of dynamics for the system. Two examples with numerical simulations are arranged to illustrate the present theory.
FROM APPROXIMATE SYNCHRONIZATION TO IDENTICAL SYNCHRONIZATION IN COUPLED SYSTEMS
(2020-09-01)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
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Department of Applied Mathematics
We establish a framework to investigate approximate synchronization of coupled systems under general coupling schemes. The units comprising the coupled systems may be nonidentical and the coupling functions are nonlinear with delays. Both delay-dependent and delay-independent criteria for approximate synchronization are derived, based on an approach termed sequential contracting. It is explored and elucidated that the synchronization error, the distance between the asymptotic state and the synchronous set, decreases with decreasing difference between subsystems, difference between the row sums of connection matrix, and difference of coupling time delays between different units. This error vanishes when these factors decay to zero, and approximate synchronization becomes identical synchronization for the coupled system comprising identical subsystems and connection matrix with identical row sums, and with identical coupling delays. The application of the present theory to nonlinearly coupled heterogeneous FitzHugh-Nagumo neurons is illustrated. We extend the analysis to study approximate synchronization and asymptotic synchronization for coupled Lorenz systems and show that for some coupling schemes, the synchronization error decreases as the coupling strength increases, whereas in another case, the error remains at a substantial level for large coupling strength.
Global consensus for discrete-time competitive systems
(2009-07-15)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
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Department of Applied Mathematics
Grossberg established a remarkable convergence theorem for a class of competitive systems without knowing and using Lyapunov function for the systems. We present the parallel investigations for the discrete-time version of the Grossberg's model. Through developing an extended component-competing analysis for the coupled system, without knowing a Lyapunov function and applying the LaSalle's invariance principle, the global pattern formation or the so-called global consensus for the system can be achieved. A numerical simulation is performed to illustrate the present theory. (C) 2008 Elsevier Ltd. All rights reserved.
GLOBAL DYNAMICS FOR TWO-SPECIES COMPETITION IN PATCHY ENVIRONMENT
(2014-08-01)
Lin, Kuang-Hui
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Lou, Yuan
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Shih, Chih-Wen
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Tsai, Tze-Hung
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應用數學系
An ODE system modeling the competition between two species in a two-patch environment is studied. Both species move between the patches with the same dispersal rate. It is shown that the species with larger birth rates in both patches drives the other species to extinction, regardless of the dispersal rate. The more interesting case is when both species have the same average birth rate but each species has larger birth rate in one patch. It has previously been conjectured by Gourley and Kuang that the species that can concentrate its birth in a single patch wins if the diffusion rate is large enough, and two species will coexist if the diffusion rate is small. We solve these two conjectures by applying the monotone dynamics theory, incorporated with a complete characterization of the positive equilibrium and a thorough analysis on the stability of the semi-trivial equilibria with respect to the dispersal rate. Our result on the winning strategy for sufficiently large dispersal rate might explain the group breeding behavior that is observed in some animals under certain ecological conditions.
GLOBAL SYNCHRONIZATION AND ASYMPTOTIC PHASES FOR A RING OF IDENTICAL CELLS WITH DELAYED COUPLING
(2011)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
;
Department of Applied Mathematics
We consider a neural network which consists of a ring of identical neurons coupled with their nearest neighbors and is subject to self-feedback delay and transmission delay. We present an iteration scheme to analyze the synchronization and asymptotic phases for the system. Delay-independent, delay-dependent, and scale-dependent criteria are formulated for the global synchronization and global convergence. Under this setting, the possible asymptotic dynamics include convergence to single equilibrium, multiple equilibria, and synchronous oscillation. The study aims at elucidating the effects from the scale of network, self-decay, self-feedback strength, coupling strength, and delay magnitudes upon synchrony, convergent dynamics, and oscillation of the network. The disparity between the contents of synchrony induced by distinct factors is investigated. Two different types of multistable dynamics are distinguished. Moreover, oscillation and desynchronization induced by delays are addressed. We answer two conjectures in the literature.
Integrating conjoint analysis with quality function deployment to carry out customer-driven concept development for ultrabooks
(2013-11-01)
Wang, Chih-Hsuan
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Shih, Chih-Wen
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工業工程與管理學系
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Department of Industrial Engineering and Management
A hybrid framework integrating conjoint analysis (CA) with quality function deployment (QFD) is presented to incorporate customer preferences into the process of product development. In particular, the proposed framework constitutes two sequential phases, namely, concept generation based on CA and prototype evaluation based on QFD. In addition, product features are characterized by customer requirements (CRs) and functional attributes (FAs). By means of DEMATEL (Decision Making and Trial Laboratory), the impacts of FAs on CRs are systematically identified to visualize their causalities. Instead of utilizing FAs, the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is employed to assess potential prototypes in terms of CRs. (C) 2013 Elsevier B.V. All rights reserved.
Multiple almost periodic solutions in nonautonomous delayed neural networks
(2007-12-01)
Lin, Kuang-Hui
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
A general methodology that involves geometric configuration of the network structure for studying multistability and multiperiodicity is developed. We consider a general class of nonautonomous neural networks with delays and various activation functions. A geometrical formulation that leads to a decomposition of the phase space into invariant regions is employed. We further derive criteria under which the n-neuron network admits 2(n) exponentially stable sets. In addition, we establish the existence of 2(n) exponentially stable almost periodic solutions for the system, when the connection strengths, time lags, and external bias are almost periodic functions of time, through applying the contraction mapping principle. Finally, three numerical simulations are presented to illustrate our theory.
Multistability and convergence in delayed neural networks
(2007-01-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
We present the existence of 2(n) stable stationary solutions for a general n-dimensional delayed neural networks with several classes of activation functions. The theory is obtained through formulating parameter conditions motivated by a geometrical observation. Positively invariant regions for the flows generated by the system and basins of attraction for these stationary solutions are established. The theory is also extended to the existence of 2(n) limit cycles for the n-dimensional delayed neural networks with time-periodic inputs. It is further confirmed that quasiconvergence is generic for the networks through justifying the strongly order preserving property as the self-feedback time lags are small for the neurons with negative self-connection weights. Our theory on existence of multiple equilibria is then incorporated into this quasiconvergence for the network. Four numerical simulations are presented to illustrate our theory. (c) 2006 Elsevier B.V. All rights reserved.
Multistability for Delayed Neural Networks via Sequential Contracting
(2015-12-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
In this paper, we explore a variety of new multistability scenarios in the general delayed neural network system. Geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics. Criteria derived from different geometric configurations lead to disparate numbers of equilibria. A new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system. The formulation accommodates both smooth sigmoidal and piecewise-linear activation functions. Several numerical examples illustrate the present analytic theory.
Snapback repellers and homoclinic orbits for multi-dimensional maps
(2012-02-01)
Liao, Kang-Ling
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
Marotto extended Li-Yorke's theorem on chaos from one-dimension to multi-dimension through introducing the notion of snapback repeller in 1978. Due to a technical flaw, he redefined snapback repeller in 2005 to validate this theorem. This presentation provides two methodologies to facilitate the application of Marotto's theorem. The first one is to estimate the radius of repelling neighborhood for a repelling fixed point. This estimation is of essential and practical significance as combined with numerical computations of snapback points. Secondly, we propose a sequential graphic-iteration scheme to construct homoclinic orbit for a repeller. This construction allows us to track the homoclinic orbit. Applications of the present methodologies with numerical computation to a chaotic neural network and a predator-prey model are demonstrated. (C) 2011 Elsevier Inc. All rights reserved.
Synchronized oscillations in a mathematical model of segmentation in zebrafish
(2012-04-01)
Liao, Kang-Ling
;
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
;
Department of Applied Mathematics
Somitogenesis is a process for the development of somites which are transient, segmental structures that lie along the anterior-posterior axis of vertebrate embryos. The pattern of somites is governed by the segmentation clock and its timing is controlled by the clock genes which undergo synchronous oscillation over adjacent cells in the posterior presomitic mesoderm (PSM). In this paper, we analyze a mathematical model which depicts the kinetics of the zebrafish segmentation clock genes subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta-Notch signalling. Our goal is to elucidate how synchronous oscillations are generated for the cells in the posterior PSM, and how oscillations are arrested for the cells in the anterior PSM. For this system of delayed equations, an iteration technique is employed to derive the global convergence to the synchronous equilibrium, which corresponds to the oscillation-arrested. By applying the delay Hopf bifurcation theory and the center manifold theorem, we derive the criteria for the existence of stable synchronous oscillations for the cells at the tail bud of the PSM. Our analysis provides the basic parameter ranges and delay magnitudes for stable synchronous, asynchronous oscillation and oscillation-arrested. We exhibit how synchronous oscillations are affected by the degradation rates and delays. Extended from the analytic theory, further numerical findings linked to the segmentation process are presented.
The kinetics in mathematical models on segmentation clock genes in zebrafish
(2018-01-01)
Chen, Kuan-Wei
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Liao, Kang-Ling
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
Somitogenesis is the process for the development of somites in vertebrate embryos. This process is timely regulated by synchronous oscillatory expression of the segmentation clock genes. Mathematical models expressed by delay equations or ODEs have been proposed to depict the kinetics of these genes in interacting cells. Through mathematical analysis, we investigate the parameter regimes for synchronous oscillations and oscillation-arrested in an ODE model and a model with transcriptional and translational delays, both with Michaelis-Menten type degradations. Comparisons between these regimes for the two models are made. The delay model has larger capacity to accommodate synchronous oscillations. Based on the analysis and numerical computations extended from the analysis, we explore how the periods and amplitudes of the oscillations vary with the degradation rates, synthesis rates, and coupling strength. For typical parameter values, the period and amplitude increase as some synthesis rate or the coupling strength increases in the ODE model. Such variational properties of oscillations depend also on the magnitudes of time delays in delay model. We also illustrate the difference between the dynamics in systems modeled with linear degradation and the ones in systems with Michaelis-Menten type reactions for the degradation. The chief concerns are the connections between the dynamics in these models and the mechanism for the segmentation clocks, and the pertinence of mathematical modeling on somitogenesis in zebrafish.
Transiently chaotic neural networks with piecewise linear output functions
(2009-01-30)
Chen, Shyan-Shiou
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
Admitting both transient chaotic phase and convergent phase, the transiently chaotic neural network (TCNN) provides superior performance than the classical networks in solving combinatorial optimization problems. We derive concrete parameter conditions for these two essential dynamic phases of the TCNN with piecewise linear output function. The confirmation for chaotic dynamics of the system results front a successful application of the Marotto theorem which was recently clarified. Numerical simulation on applying the TCNN with piecewise linear output function is carried out to find the optimal solution of it travelling salesman problem. It is demonstrated that the performance is even better than the previous TCNN model with logistic output function. (C) 2007 Elsevier Ltd. All rights reserved.