The type II phase resetting curve is optimal for stochastic synchrony

The phase-resetting curve (PRC) describes the response of a neural oscillator to small perturbations in membrane potential. Its usefulness for predicting the dynamics of weakly coupled deterministic networks has been well characterized. However, the inputs to real neurons may often be more accurately described as barrages of synaptic noise. Effective connectivity between cells may thus arise in the form of correlations between the noisy input streams. We use constrained optimization and perturbation methods to prove that PRC shape determines susceptibility to synchrony among otherwise uncoupled noise-driven neural oscillators. PRCs can be placed into two general categories: Type I PRCs are non-negative while Type II PRCs have a large negative region. Here we show that oscillators with Type II PRCs receiving common noisy input sychronize more readily than those with Type I PRCs.Comment: 10 pages, 4 figures, submitted to Physical Review

Global synchronization for delayed complex networks with randomly occurring nonlinearities and multiple stochastic disturbances

This is the post print version of the article. The official published version can be obained from the link - Copyright 2009 IOP Publishing LtdThis paper is concerned with the synchronization problem for a new class of continuous time delayed complex networks with stochastic nonlinearities (randomly occurring nonlinearities), interval time-varying delays, unbounded distributed delays as well as multiple stochastic disturbances. The stochastic nonlinearities and multiple stochastic disturbances are investigated here in order to reflect more realistic dynamical behaviors of the complex networks that are affected by the noisy environment. By utilizing a new matrix functional with the idea of partitioning the lower bound h1 of the time-varying delay, we employ the stochastic analysis techniques and the properties of the Kronecker product to establish delay-dependent synchronization criteria that ensure the globally asymptotically mean-square synchronization of the addressed stochastic delayed complex networks. The sufficient conditions obtained are in the form of linear matrix inequalities (LMIs) whose solutions can be readily solved by using the standard numerical software. A numerical example is exploited to show the applicability of the proposed results.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, an International Joint Project sponsored by the Royal Society of the UK, the National 973 Program of China under Grant 2009CB320600, the National Natural Science Foundation of China under Grant 60804028, the Specialized Research Fund for the Doctoral Program of Higher Education for New Teachers under Grant 200802861044, the Teaching and Research Fund for Excellent Young Teachers at Southeast University of China, and the Alexander von Humboldt Foundation of Germany

Stochastic averaging using elliptic functions to study nonlinear stochastic systems

In this paper, a new scheme of stochastic averaging using elliptic functions is presented that approximates nonlinear dynamical systems with strong cubic nonlinearities in the presence of noise by a set of Itô differential equations. This is an extension of some recent results presented in deterministic dynamical systems. The second order nonlinear differential equation that is examined in this work can be expressed as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qeguuDJXwAKbacfiGaf8hEaGNbamaacqGHRaWkcaWGJbadcaaIXaGc% cqWF4baEcqGHRaWkcaWGJbadcaaIZaGccqWF4baEdaahaaWcbeqaai% aaiodaaaGccqGHRaWkcqaH1oqzcaWGMbGaaiikaiab-Hha4jaacYca% cqWFGaaicuWF4baEgaGaaiaacMcacqGHRaWkcqaH1oqzdaahaaWcbe% qaaiaaigdacaGGVaGaaGOmaaaaruWrL9MCNLwyaGGbcOGaa43zaiaa% cIcacqWF4baEcaGGSaGae8hiaaIaf8hEaGNbaiaacaGGSaGae8hiaa% IaeqOVdGNaaeikaiaadshacaqGPaGaaiykaiabg2da9iaaicdaaaa!645D![ddot x + c1x + c3x^3 + varepsilon f(x, dot x) + varepsilon ^{1/2} g(x, dot x, xi {text{(}}t{text{)}}) = 0] where c 1 and c 3 are given constants, ξ( t ) is stationary stochastic process with zero mean and ε≪1 is a small parameter. This method involves the laborious manipulation of Jacobian elliptic functions such as cn, dn and sn rather than the usual trigonometric functions. The use of a symbolic language such as Mathematica reduces the computational effort and allows us to express the results in a convenient form. The resulting equations are Markov approximations of amplitude and phase involving integrals of elliptic functions. Finally, this method was applied to study some standard second order systems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43328/1/11071_2004_Article_BF00120672.pd

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Topics in nonparametric estimation

This book contains papers presented at the Seminar on Mathematical Statistics held at the Institute for Problems of Information Transmission of the Academy of Sciences in the former Soviet Union. Founded in the mid-1960s, this seminar is still active today and attracts most of the researchers in Moscow who are interested in mathematical statistics. The topics covered include density, regression, and image estimation, adaptive estimation, stochastic approximation, median estimation, sequential experimental design, and large deviations for empirical measures. This collection is distinguished by the high scientific level of the papers and their modern approach. This book will be of interest to scientists and engineers who use probability and statistics, to mathematicians and applied statisticians who work in approximation theory, and to computer scientists who work in image analysis