Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument

We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.Comment: 24 pages, 1 figur

Integration of continuous-time dynamics in a spiking neural network simulator

Contemporary modeling approaches to the dynamics of neural networks consider two main classes of models: biologically grounded spiking neurons and functionally inspired rate-based units. The unified simulation framework presented here supports the combination of the two for multi-scale modeling approaches, the quantitative validation of mean-field approaches by spiking network simulations, and an increase in reliability by usage of the same simulation code and the same network model specifications for both model classes. While most efficient spiking simulations rely on the communication of discrete events, rate models require time-continuous interactions between neurons. Exploiting the conceptual similarity to the inclusion of gap junctions in spiking network simulations, we arrive at a reference implementation of instantaneous and delayed interactions between rate-based models in a spiking network simulator. The separation of rate dynamics from the general connection and communication infrastructure ensures flexibility of the framework. We further demonstrate the broad applicability of the framework by considering various examples from the literature ranging from random networks to neural field models. The study provides the prerequisite for interactions between rate-based and spiking models in a joint simulation

Estimation of Solutions of Differential Systems with Delayed Argument of Neutral Type

Tato disertační práce pojednává o řešení diferenciálních rovnic a systémů diferenciálních rovnic. Hlavní pozornost je věnována asymptotickým vlastnostem rovnic se zpožděním a systémů rovnic se zpožděním. V první kapitole jsou uvedeny fyzikální a technické příklady popsané pomocí diferenciálních rovnic se zpožděním a jejich systémů. Je uvedena klasifikace rovnic se zpožděním a jsou zformulovány základní pojmy stability s důrazem na druhou metodu Ljapunova. Ve druhé kapitole jsou studovány odhady řešení rovnic neutrálního typu. Třetí kapitola se zabývá systémy diferenciálních rovnic neutrálního typu. Jsou odvozeny asymptotické odhady pro řešení i pro derivace řešení. V závěru kapitoly jsou uvedeny příklady a srovnání výsledků s pracemi jiných autorů. Výpočty byly prováděny pomocí programu MATLAB. Poslední, čtvrtá kapitola, se zabývá asymptotickými vlastnostmi systémů se speciálním typem nelinearity, tzv. sektorové nelinearity. Jsou odvozeny vlastnosti řešení a derivace řešení. Základní metodou pro důkazy je v celé práci druhá Ljapunovova metoda a použití funkcionálů Ljapunova-Krasovského.This dissertation discusses the solutions to the differential equation and to systems of differential equations. The main attention is paid to study of asymptotical properties of equations with delay and systems of equations with delay. In the first chapter are given physical and technical examples described by differential equations with delay and their systems. The classification of equations with delay is given and basic notions of theory of stability are formulated (mainly with the emphasis on the Lyapunov second method). In the second chapter estimates of solutions of equations of neutral type are studied. The third chapter deals with systems of differential equations of neutral type. Asymptotic estimates for solutions and their derivatives are proved. At the end of the chapter examples and comparisons of our results and of other authors are given. The calculation were performed with the MATLAB software. Last, the fourth chapter deals with asymptotical properties of systems having a special type of nonlinearities, so called ``sector nonlinearities''. Properties and estimations of solutions and derivatives are derived. The basic tools used in the dissertation are the Lyapunov second method and functionals of Lyapunov-Krasovskii type.

Distributed state estimation in sensor networks with randomly occurring nonlinearities subject to time delays

This is the post-print version of the Article. The official published version can be accessed from the links below - Copyright @ 2012 ACM.This article is concerned with a new distributed state estimation problem for a class of dynamical systems in sensor networks. The target plant is described by a set of differential equations disturbed by a Brownian motion and randomly occurring nonlinearities (RONs) subject to time delays. The RONs are investigated here to reflect network-induced randomly occurring regulation of the delayed states on the current ones. Through available measurement output transmitted from the sensors, a distributed state estimator is designed to estimate the states of the target system, where each sensor can communicate with the neighboring sensors according to the given topology by means of a directed graph. The state estimation is carried out in a distributed way and is therefore applicable to online application. By resorting to the Lyapunov functional combined with stochastic analysis techniques, several delay-dependent criteria are established that not only ensure the estimation error to be globally asymptotically stable in the mean square, but also guarantee the existence of the desired estimator gains that can then be explicitly expressed when certain matrix inequalities are solved. A numerical example is given to verify the designed distributed state estimators.This work was supported in part by the National Natural Science Foundation of China under Grants 61028008, 60804028 and 61174136, the Qing Lan Project of Jiangsu Province of China, the Project sponsored by SRF for ROCS of SEM of China, the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany

Sufficient Conditions for Polynomial Asymptotic Behaviour of the Stochastic Pantograph Equation

This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in $p$-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in $p$-th mean and in the almost sure sense. When polynomial bounds cannot be achieved, we show for a different set of parameters that exponential growth bounds of solutions in $p$-th mean and an almost sure sense can be obtained. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations.Comment: 29 pages, to appear Electronic Journal of Qualitative Theory of Differential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ. (July 1--4, 2015, Szeged, Hungary