過去的文獻中我們發現不同的神經模型皆展現了鎖模的現象，這暗示了神經 元中隱含了相角動力學的機制，然而現今的機器學習領域中卻忽略了此項重要 的特性。本論文將以保留了離散化相角動力學特性的Kinouchi-Copelli 模型為切 入點研究其在正方形晶格網絡中的動力學行為，並發現它展現了神經編碼中重 要的線性-非線性-卜瓦松(LNP)模型的特性。另一方面，再經過引入抑制型突觸 概念的延拓後，Kinouchi-Copelli 神經元網絡重現了在視網膜神經節細胞中墨西 哥帽型的接受域，這些結果都說明了引入相角動力學機制的神經元網絡，在研 究神經資訊處理的前潛力。

In previous literature, it found that different neuron models demonstrate the mode-locking phenomenon, this indicates there exists the phase dynamics behind neural activities. However, modern machine learning models only consider the rate model and drop this important property. In this thesis, we start from the Kinouchi-Copelli neuron located on the square lattice network and study its dynamic behaviors, which we observe that it presents the property of the LinearNonlinear-Poisson(LNP) model in neural encoding. On the other hand, after including the concept of inhibitory synapse, the Kinouchi-Copelli neuronal network produces the Mexican-hat shape receptive field which can be found in retina ganglion cells. Those results demonstrate the phase dynamics neuronal networks have the potential to study the information processing in neuroscience.

Abstract (Chinese) ----------------------------------- i
Abstract --------------------------------------------- ii
Acknowledgements (Chinese) --------------------------- iii
Contents --------------------------------------------- iv
List of Figures -------------------------------------- vi
1 Introduction --------------------------------------- 1
2 Phase dynamics in neural activities ---------------- 4
2.1 Mode locking ------------------------------------- 4
2.2 Phase dynamics behind neurons -------------------- 7
2.3 U(1) neuron -------------------------------------- 9
3 Kinouchi-Copelli Neuron ---------------------------- 12
3.1 Model introduction ------------------------------- 12
3.2 Model property ----------------------------------- 14
4 Kinouchi-Copelli Neuronal Network ------------------ 19
4.1 Phase transition in KCNN ------------------------- 21
4.2 Finite-size scaling ------------------------------ 23
5 Poor man’s scaling in KCNN ------------------------- 27
5.1 Mode-counting method ----------------------------- 27
5.2 Poor man’s scaling ------------------------------- 29
6 Discussions and Future Works ----------------------- 32
6.1 Inhibitory synapse for Kinouchi-Copelli neuron --- 32
6.2 Viewing angle and local network structure -------- 34
Bibliography ----------------------------------------- 37

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