Dynamical Behavior of Nonautonomous Stochastic Reaction-Diffusion Neural Network Models

This brief investigates nonautonomous stochastic reaction-diffusion neural-network models with S-type distributed delays. First, the existence and uniqueness of mild solution are studied under the Lipschitz condition without the linear growth condition. Due to the existence of a nonautonomous reaction-diffusion term and the infinite dimensional Wiener process, the criteria for the well-posedness of the models are established based on the evolution system theory. Then, the S-type distributed delay, which is an infinite delay, is handled by the truncation method, and sufficient conditions for the global exponential stability are obtained by constructing a simple Lyapunov-Krasovskii functional candidate. Finally, neural-network examples and an illustrative example are given to show the applications of the obtained results.</p

Asymptotic stability for neural networks with mixed time-delays: The discrete-time case

This is the post print version of the article. The official published version can be obtained from the link - Copyright 2009 Elsevier LtdThis paper is concerned with the stability analysis problem for a new class of discrete-time recurrent neural networks with mixed time-delays. The mixed time-delays that consist of both the discrete and distributed time-delays are addressed, for the first time, when analyzing the asymptotic stability for discrete-time neural networks. The activation functions are not required to be differentiable or strictly monotonic. The existence of the equilibrium point is first proved under mild conditions. By constructing a new Lyapnuov–Krasovskii functional, a linear matrix inequality (LMI) approach is developed to establish sufficient conditions for the discrete-time neural networks to be globally asymptotically stable. As an extension, we further consider the stability analysis problem for the same class of neural networks but with state-dependent stochastic disturbances. All the conditions obtained are expressed in terms of LMIs whose feasibility can be easily checked by using the numerically efficient Matlab LMI Toolbox. A simulation example is presented to show the usefulness of the derived LMI-based stability condition.This work was supported in part by the Biotechnology and Biological Sciences Research Council (BBSRC) of the UK under Grants BB/C506264/1 and 100/EGM17735, the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grants GR/S27658/01 and EP/C524586/1, an International Joint Project sponsored by the Royal Society of the UK, the Natural Science Foundation of Jiangsu Province of China under Grant BK2007075, the National Natural Science Foundation of China under Grant 60774073, and the Alexander von Humboldt Foundation of Germany

Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

Stochastic Representations of Ion Channel Kinetics and Exact Stochastic Simulation of Neuronal Dynamics

In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation with a commonly used numerical approximation strategy. The first representation we present is a random time change representation, popularized by Thomas Kurtz, with the second being analogous to a "Gillespie" representation. Exact stochastic algorithms are provided for the different representations, which are preferable to either (a) fixed time step or (b) piecewise constant propensity algorithms, which still appear in the literature. As examples, we provide versions of the exact algorithms for the Morris-Lecar conductance based model, and detail the error induced, both in a weak and a strong sense, by the use of approximate algorithms on this model. We include ready-to-use implementations of the random time change algorithm in both XPP and Matlab. Finally, through the consideration of parametric sensitivity analysis, we show how the representations presented here are useful in the development of further computational methods. The general representations and simulation strategies provided here are known in other parts of the sciences, but less so in the present setting.Comment: 39 pages, 6 figures, appendix with XPP and Matlab cod

Mini-Workshop: Dynamics of Stochastic Systems and their Approximation

The aim of this workshop was to bring together specialists in the area of stochastic dynamical systems and stochastic numerical analysis to exchange their ideas about the state of the art of approximations of stochastic dynamics. Here approximations are considered in the analytical sense in terms of deriving reduced dynamical systems, which are less complex, as well as in the numerical sense via appropriate simulation methods. The main theme is concerned with the efficient treatment of stochastic dynamical systems via both approaches assuming that ideas and methods from one ansatz may prove beneficial for the other. A particular goal was to systematically identify open problems and challenges in this area

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...