細胞類神經網路：與空間有關的模版、模型以及混沌

Abstract

在本篇論文中，我們考慮的是一個放在R2空間的整數格子點Z2上且含有偏差項z的細胞類神經網路(CNN)。細胞們彼此之間的關係則是些微的與空間相關，這些關係則是在整數格子點上佈滿蜂巢狀的型式所構成。我們使用兩個變數a以及ε來描述細胞間彼此的直接作用力。跟空間無關的細胞類神經網路一樣，對於固定的ε≠0，我們也可以將整個變數空間(z,a;ε)做分割成許多[m,n]ε區。這亦解釋了細胞類神經網路中的一所謂「學習間題上而在這些Mosaic模型的複雜度方面，只要m,n的最小值都不小於2的時候，便會發生空間上的混沌。 在時間上的混沌，我們是針對一維、三細胞且偏差值為0的細胞類神經網路。若是跟空間無關且考慮邊界條件，如Dirchlet，Neumann，Periodic邊界條件的話，我們無法找到Shil'nikov定理中所述的混沌；另外。我們亦有找一組跟空間有關的系統，存在有Shil'nikov定理中所敘述的混沌。

We consider a Cellular Neural Network (CNN) with a bias term z in the integer lattice Z2 on the plane R2 in this thesis. The coupling between cells is "weakly" spatially-dependent. Such coupling is motivated by filling the plane with honeycomb type of lattice. We use two parameters, a and ε to describe the weights between such interacting cells. Like spatially independent template, for fixed ε≠0, we can still partition the parameter space (z,a;ε) into [m,n]ε regions. This, in turn, addresses the so-called "Learning Problem" in CNNs. In the case of complexity of the mosaic pattern, we find that when minimum of m,n is no less than 2, the [m,n] mosaic patterns have spatial chaos. In the temporal chaos part, we mainly consider a Cellular Neural Network with 1-dimension, 3-cell, and z=0. In the spatially independent case with boundary various conditions, such as Direhiet, Neumann, and Periodic boundary condition, we show that the system have no chaotic behavior in the Shil'nikov sense. In addition, we give one example of spatially dependent system that has chaotic behavior in the Shil'nikov sense.