神經網路上的同步方程之多個穩定狀況的分析

Abstract

在這篇論文中，我們利用幾何奇異擾動理論研究來自Hindmarsh-Rose 網路之同步化方程的多重穩定狀態。 我們的主要結果如下: 首先，我們給出同步化Hindmarsh-Rose方程多重穩定狀態的解釋，例如我們能下結論說在爆裂(bursting)解與具有canard現象的週期解能共存。 其次，我們可以充分了解從初始狀態至穩定狀態的過程。 最後，我們可識別出穩定狀態的吸引範圍。 這些都說明了用幾何奇異擾動理論理解實際生物系統的全域動態性質是相當有用的。

In this thesis, geometric singular perturbation theory is applied to investigate multistate stability of synchronous equations derived from Hindmarsh-Rose Networks. Our main results contain the following. First, explanation of multistability of the synchronous Hindmarsh-Rose equation can be given. For instance, we are able to conclude among other things that a bursting solution and a periodic solution with canard explosion can coexistence. The transition from initial states toward stable states can be fully predicted. Finally, the attraction region with respect to each stable state can be identified. This illustrates the power of using singular perturbation theory to understand the global dynamical properties of realistic biological systems.