论文标题
二维空间结构域中延迟神经场模型的动力学
Dynamics of delayed neural field models in two-dimensional spatial domains
论文作者
论文摘要
在适当的功能分析环境中，延迟的神经场模型可以看作是动态系统。在二维矩形空间域，对于特殊的连接性和延迟功能，我们描述了线性化方程的光谱特性。我们将延迟微分方程（DDE）的特征积分方程转换为具有边界条件的线性部分微分方程（PDE）。我们证明，DDE的特征值和特征向量与获得该边界价值问题（BVP）的非平凡解相等。当连接性内核由单个指数组成时，我们构建了该BVP的解决方案的基础，该解决方案以$ l^2 $形成完整的集合。这给出了光谱的完整表征，并用于构建解决方案问题的解决方案。作为应用程序，我们举例说明了HOPF分叉并计算第一个Lyapunov系数。
Delayed neural field models can be viewed as a dynamical system in an appropriate functional analytic setting. On two dimensional rectangular space domains, and for a special class of connectivity and delay functions, we describe the spectral properties of the linearized equation. We transform the characteristic integral equation for the delay differential equation (DDE) into a linear partial differential equation (PDE) with boundary conditions. We demonstrate that finding eigenvalues and eigenvectors of the DDE is equivalent with obtaining nontrivial solutions of this boundary value problem (BVP). When the connectivity kernel consists of a single exponential, we construct a basis of the solutions of this BVP that forms a complete set in $L^2$. This gives a complete characterization of the spectrum and is used to construct a solution to the resolvent problem. As an application we give an example of a Hopf bifurcation and compute the first Lyapunov coefficient.
