Desynchronization of Large Scale Delayed Neural Networks

We consider a ring of identical neurons with delayed nearest neighborhood inhibitory interaction. Under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay di erential equation with negative feedback. Despite the fact that the slowly oscillatory periodic solution of the scalar equation is stable, we show that the associated synchronous solution is unstable if the size of the network is large

[[alternative]]Numerical and Mathematical Analysis of Neural Network

計畫編號：NSC93-2119-M032-001研究期間：200409~200507研究經費：270,000[[abstract]]在NN 方面: 對於CNN 動態解，我們將發展好的，可信的數值方法來模擬大型的神經 網路，同時考慮其中所引發的數學問題；動態解中的週期解的存在、唯一、 穩定性，並進一步透過Shadowing Lamma 做嚴格的數學證明週期解存在性， homoclinic point 的存在性，並計算Lyapunov exponent，對temporal chaos 做嚴格的數學證明；另外我們的方程將導入真實情況常常發生的遲滯項 (delay term)[5]，考慮週期解的存在性。 關於Brain Dynamics: 我們要利用Dynamical Systems 及CNN 的技巧來modeling brain structures 和brain activity。Dynamical Systems 的方法包括了資料處理及度量複雜性裡 相關統計量兩部分；其中資料處理包括了delay coordinate 和embedding theory 等方法，複雜性度量裡相關統計量則包含了correlation dimension 、 Lyapunov exponents 、synchronization 及phase synchronization 等。[[sponsorship]]行政院國家科學委員

Traveling Wave Solutions for Planar Lattice Differential Systems with Applications to Neural Networks

AbstractWe obtain some existence results for traveling wave fronts and slowly oscillatory spatially periodic traveling waves of planar lattice differential systems with delay. Our approach is via Schauder's fixed-point theorem for the existence of traveling wave fronts and via S1-degree and equivarant bifurcation theory for the existence of periodic traveling waves. As examples, the obtained abstract results will be applied to a model arising from neural networks and explicit conditions for traveling wave fronts and global continuation of periodic waves will be obtained

Stability of the stationary solutions of neural field equations with propagation delays

In this paper, we consider neural field equations with space-dependent delays. Neural fields are continuous assemblies of mesoscopic models arising when modeling macroscopic parts of the brain. They are modeled by nonlinear integro-differential equations. We rigorously prove, for the first time to our knowledge, sufficient conditions for the stability of their stationary solutions. We use two methods 1) the computation of the eigenvalues of the linear operator defined by the linearized equations and 2) the formulation of the problem as a fixed point problem. The first method involves tools of functional analysis and yields a new estimate of the semigroup of the previous linear operator using the eigenvalues of its infinitesimal generator. It yields a sufficient condition for stability which is independent of the characteristics of the delays. The second method allows us to find new sufficient conditions for the stability of stationary solutions which depend upon the values of the delays. These conditions are very easy to evaluate numerically. We illustrate the conservativeness of the bounds with a comparison with numerical simulation

A simulation-based approach to business model design and organizational Change

While several practice-based approaches of business model design suggest ways to create new business models, there is limited understanding of why and how business models change. This exploratory study employs neural network analysis to simulate business model design and business model change. We conceptualise business model design as a schema of the organisation’s critical resources, transactions, and value proposition. Elements of the schema are connected in a simple neural network. The network evolves based on a constraint satisfaction network until it converges to a stable state of a coherent business model. An in-depth case study of an entrepreneurial venture provides a real-world example to test the analytical framework. Using data from the case study, we run multiple simulations of business model design and business model change. The results suggest that business model change can be understood as a form of constraint satisfaction, linking managerial cognition with business model change. The simulation approach also helps identify possible, but unrealized business models. This novel approach paves the way for new research and practice in business model design and change

New Results for Periodic Solution of High-Order BAM Neural Networks with Continuously Distributed Delays and Impulses

By M-matrix theory, inequality techniques, and Lyapunov functional method, certain sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of periodic solution for a new type of high-order BAM neural networks with continuously distributed delays and impulses. These novel conditions extend and improve some previously known results in the literature. Finally, an illustrative example and its numerical simulation are given to show the feasibility and correctness of the derived criteria

Reduced dynamics for delayed systems with harmonic or stochastic forcing

International audienceThe analysis of nonlinear delay-differential equations (DDEs) subjected to external forcing is difficult due to the infinite dimensionality of the space in which they evolve. To simplify the analysis of such systems, the present work develops a non-homogeneous center manifold (CM) reduction scheme, which allows the derivation of a time-dependent order parameter equation in finite dimension. This differential equation captures the major dynamical features of the delayed system. The forcing is assumed to be small compared to the amplitude of the autonomous system, in order to cause only small variations of the fixed points and of the autonomous CM. The time-dependent CM is shown to satisfy a non-homogeneous partial differential equation. We first briefly review CM theory for DDEs. Then we show, for the general scalar case, how an ansatz that separates the CM into one for the autonomous problem plus an additional time-dependent order-two correction leads to satisfying results. The paper then details the application to a transcritical bifurcation subjected to single or multiple periodic forcings. The validity limits of the reduction scheme are also highlighted. Finally, we characterize the specific case of additive stochastic driving of the transcritical bifurcation, where additive white noise shifts the mode of the probability density function of the state variable to larger amplitudes