A new framework for consensus for discrete-time directed networks of multi-agents with distributed delays

Copyright @ 2012 Taylor & FrancisIn this article, the distributed consensus problem is considered for discrete-time delayed networks of dynamic agents with fixed topologies, where the networks under investigation are directed and the time-delays involved are distributed time delays including a single or multiple time delay(s) as special cases. By using the invariance principle of delay difference systems, a new unified framework is established to deal with the consensus for the discrete-time delayed multi-agent system. It is shown that the addressed discrete-time network with arbitrary distributed time delays reaches consensus provided that it is strongly connected. A numerical example is presented to illustrate the proposed methods.This work was supported in part by City University of Hong Kong under Grant 7008114, the Royal Society of the UK, the National Natural Science Foundation of China under Grants 60774073 and 61074129, and the Natural Science Foundation of Jiangsu Province of China under Grant BK2010313

New Stability Criterion for Takagi-Sugeno Fuzzy Cohen-Grossberg Neural Networks with Probabilistic Time-Varying Delays

A new global asymptotic stability criterion of Takagi-Sugeno fuzzy Cohen-Grossberg neural networks with probabilistic time-varying delays was derived, in which the diffusion item can play its role. Owing to deleting the boundedness conditions on amplification functions, the main result is a novelty to some extent. Besides, there is another novelty in methods, for Lyapunov-Krasovskii functional is the positive definite form of p powers, which is different from those of existing literature. Moreover, a numerical example illustrates the effectiveness of the proposed methods

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

Local/global analysis of the stationary solutions of some neural field equations

Neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integro-differential equations. The solutions of these equations represent the state of activity of these populations when submitted to inputs from neighbouring brain areas. Understanding the properties of these solutions is essential in advancing our understanding of the brain. In this paper we study the dependency of the stationary solutions of the neural fields equations with respect to the stiffness of the nonlinearity and the contrast of the external inputs. This is done by using degree theory and bifurcation theory in the context of functional, in particular infinite dimensional, spaces. The joint use of these two theories allows us to make new detailed predictions about the global and local behaviours of the solutions. We also provide a generic finite dimensional approximation of these equations which allows us to study in great details two models. The first model is a neural mass model of a cortical hypercolumn of orientation sensitive neurons, the ring model. The second model is a general neural field model where the spatial connectivity isdescribed by heterogeneous Gaussian-like functions.Comment: 38 pages, 9 figure

Dynamical systems applied to consciousness and brain rhythms in a neural network

This thesis applies the great advances of modern dynamical systems theory (DST) to consciousness. Consciousness, or subjective experience, is faced here in two different ways: from the global dynamics of the human brain and from the integrated information theory (IIT), one of the currently most prestigious theories on consciousness. Before that, a study of a numerical simulation of a network of individual neurons justifies the use of the Lotka-Volterra model for neurons assemblies in both applications. All these proposals are developed following this scheme: • First, summarizing the structure, methods and goal of the thesis. • Second, introducing a general background in neuroscience and the global dynamics of the human brain to better understand those applications. • Third, conducting a study of a numerically simulated network of neurons. This network, which displays brain rhythms, can be employed, among other objectives, to justify the use of the Lotka-Volterra model for applications. • Fourth, summarizing concepts from the mathematical DST such as the global attractor and its informational structure, in addition to its particularization to a Lotka-Volterra system. • Fifth, introducing the new mathematical concepts of model transform and instantaneous parameters that allow the application of simple mathematical models such as Lotka-Volterra to complex empirical systems as the human brain. • Sixth, using the model transform, and specifically the Lotka-Volterra transform, to calculate global attractors and informational structures in global dynamics of the human brain. • Seventh, knowing the probably most prestigious theory on consciousness, the IIT developed by G. Tononi. • Eighth, using informational structures to develop a continuous version of IIT. And ninth, establishing some final conclusions and commenting on new open questions from this work. These nine points of this scheme correspond to the nine chapters of this thesis