目前，機器學習僅處理實數，其學習機制是透過以線性代數為基礎的理論建構
而成。然而真實神經元的行為相當複雜，Hodgkin–Huxley 神經元模型需要用四維的非線性動力系統才能較完整描述神經元在固定刺激底下的行為。它藉由不同種類的離子通道的動力演化捕捉到神經元放電的機制。本論文著重於真實神經元的最簡模型化。從三種典型的神經元模型出發，我們發現每個模型所描述的神經的行為模式都存在著相變。透過分岔理論的分析，我們提出了神經元的最簡化模型是以複數型態存在的，並命名為U(1) 神經元模型。藉由極限圈與定點的數學詮釋，此模型得以直觀的描述神經元放電與靜止等行為，並提供行為之間的相變機制。接下來，我們發現U(1) 神經元模型可以很好地抓住Hodgkin–Huxley 模型中的電壓變數。最後，在考慮神經元之間的連結後，我們提出由此神經元模型造出的神經網絡，並在簡單條件下模擬此網絡的行為表現。U(1) 神經元模型涵蓋兩種神經元型態，統一了典型神經模型的精華。此外，其複數性質，更開啟了機器學習領域的新可能性。

At present, the machine learning processes only real numbers. Its mechanism for learning is constructed through the theory based on linear algebra. However, the behavior of real neurons is quite complicated. Hodgkin–Huxley model, which describes the neurons more comprehensively under a fixed stimulus, is a four-dimensional nonlinear dynamic system. It captures the dynamics of the ion channels, which allow the ions to pass by, to generate the electric signals. In this paper, we propose a minimum model for real neurons. Starting from three typical neuron models, we find that the behavioral patterns of the neuron appearing in each model have phase transitions. Through the analysis of bifurcation theory, we propose that the minimum model of neurons should be in complex form, and we name this model as the U(1) neuron model. With the mathematical interpretation of limit cycles and fixed points, this model can intuitively describe the behavior of neurons, such as firing and resting state, and the mechanism for their phase transition. Next, we show that the U(1) neuron model can capture the action potentials well by making a comparison to the Hodgkin–Huxley model. At last, we consider the connections
between neurons to create a neuronal network consisting of the U(1) neurons and give a demonstration of its performance under simple conditions. The U(1) neuron model unifies the essence of typical neuron models. Besides, its complex-number nature opens up new possibilities in the field of machine learning.

Abstract
Contents
1 Introduction -------------------------------------------- 1
2 Neuron Models ------------------------------------------- 3
2.1 Hodgkin–Huxley model ---------------------------------- 5
2.2 FitzHugh-Nagumo model --------------------------------- 11
2.3 Ermentrout-Kopell model ------------------------------- 15
3 Dynamics Phase Transition ------------------------------- 17
3.1 Saddle-node on invariant circle ----------------------- 18
3.2 Hopf bifurcation -------------------------------------- 21
4 U(1) Neuron Model --------------------------------------- 23
4.1 Capture the Essential Dynamics variables -------------- 24
4.2 Comparison with the Hodgkin–Huxley model -------------- 28
5 U(1) Neuronal Network ----------------------------------- 38
6 Simulate U(1) Neuronal Network -------------------------- 42
6.1 Homogeneous Fully-Connected UNN ----------------------- 44
6.2 Inhomogeneous Fully-Connected UNN --------------------- 49
7 Discussion and Future Work ------------------------------ 54
Appendix -------------------------------------------------- 56
A. Stability for fixed point in FitzHugh-Nagumo model ----- 57
B. Decreasing Lyapunov Potential -------------------------- 60
References ------------------------------------------------ 62

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