線性耦合系統的耗散性與同步化

Abstract

一個關於耦合系統的基本想法是由個別子系統的性質去發展出耦合系統的性質，而在動態系統中的一個基本性質為解的有界性，又因為semi-passiveity的這套理論可以得到此結果，因此在過去的幾年中，這套理論持續發展著。這套理論的目標是利用個別的子系統滿足類似的耗散條件而得到耦合系統的解會是有界的，並且應用在擴散耦合的系統，且隨後可以得到此耦合系統會是同步化，另外這套理論可以應用在幾個具代表性的神經網路的模型上：Hodgkin-Huxley、Morris-Lecar、FitzHugh-Nagumo、Hindmarsh-Rose，然而在文獻中semi-passivity的定義並不足以得到定理之結果，因此，我們修改定義並修正理論之證明，此外，我們推廣這套理論由文獻中的擴散耦合至線性耦合，其中耦合矩陣的每列總和為非負值，最後利用修正後的理論來操作推廣後的四個神經網路系統，並給予一些數值的模擬。

A basic idea to study coupled systems is to develop the properties for coupled systems from the properties for individual subsystems. Among these basic dynamical properties is the boundedness of solutions. A theory termed semi-passivity to conclude such a property has been developed in the past few years. Basically, its goal is to conclude that all solutions of coupled systems are bounded in forward time, if the isolated subsystems satisfy some dissipative-like condition. The theory has been applied in diffusively coupled systems and subsequently synchronization for the system was asserted. The synchronization theory is then implemented in some representative neural models: Hodgkin-Huxley, Morris-Lecar, FitzHugh-Nagumo and Hindmarsh-Rose. However, the definition for semi-passivity in the literature is not sufficient for the theory reported. In this thesis, we modify the definition of semi-passivity and then straighten the justification of the dissipative theory for linearly coupled systems. In addition, we extend the theory to linearly coupled systems whose row sums of coupling matrix are nonnegative from the requirement of diffusive coupling in the literature. We demonstrate the straightened theory and extended framework for Hodgkin-Huxley, Morris-Lecar, FitzHugh-Nagumo and Hindmarsh-Rose neuronal models, and provide some numerical simulations.