邊界條件影響下之細胞類神經網路花樣複雜度

Abstract

在本篇論文中，我們討論由(1.4－1.6)的三個templates所產生的mosaic patterns的邊界影響。也就是說，對(1.1a)考慮在有限長方格子點上的穩定點的邊界影響，如：Neumann, periodic 和 Dirichlet 邊界條件。將這些滿足邊界條件的穩定點個數表示成 $\Gamma^N_{\bf k}$, $\Gamma^P_{\bf k}$ and 和$\Gamma^D_{\bf k}$ 。在這裡，我們所討論的就是，是否spatial entropy $h=\lim\limits_{{\bf k} \to \infty} \frac{1}{k_1k_2} \ln \Gamma^B_{\bf k}:=h_B $ ，B=N, P and D。 由我們的結果可以看出，當m,n大的時候不會有邊界影響，而當剛從沒有chaos到有chaos的幾區和中間的幾區，我們則無法判別有無影響。

In this thesis, we study how the complexity of the mosaic patterns obtained by three templates given in (1.4---1.6) is affected by the boundary effects. Specifically, consider the steady state of (1.1a) to be defined in the finite rectangular lattice, say $T_{(k_1,k_2)}:=T_{\bf k}$ of size $k_1 \times k_2$, with various boundary conditions such as Neumann, periodic and Dirichlet boundary conditions. Let the corresponding numbers of such mosaic solutions be denoted by $\Gamma^N_{\bf k}$, $\Gamma^P_{\bf k}$ and $\Gamma^D_{\bf k}$. Here the superindex N denotes the Neumann boundary conditions are imposed. $\Gamma^P_{\bf k}$ and $\Gamma^D_{\bf k}$ are defined similarly. The question addressed here is whether the spatial entropy $h$ of mosaic patterns generated by (1.1a) is equal to $h=\lim\limits_{{\bf k} \to \infty} \frac{1}{k_1k_2} \ln \Gamma^B_{\bf k}:=h_B $ where $B$ = $N$, $P$ and $D$. Our results seem to indicate that at two extreme cases of chaos, i.e., the phase of ``the end of chaos", there has no boundary effects as far as spatial entropy is concerned. However, during the phase of ``beginning of chaos" and the phase of ``transition chaos", boundary effects are much more complex and difficult to determine.