## 数学代写|数学分析代写Mathematical Analysis代考|MA4551

2022年7月13日

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• Statistical Computing 统计计算
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• Foundations of Data Science 数据科学基础
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## 数学代写|数学分析代写Mathematical Analysis代考|Analysis and Illustration of the Model Dynamics

In [1] we assumed $g$ in the form of Hill function with Hill coefficient equal to 2 , that is $g(\zeta)=\frac{\zeta^{2}}{1+\zeta^{2}}$. Moreover, without additional input we considered the model being fully symmetric, meaning that all coefficients are the same with reference values $\alpha_{i}=3, a_{i}=0.4, b_{1}=1, c=1$. The main aim of the analysis presented in [1] was to check if the model is able to correctly predict to which neuronal population an additional input was applied.

Looking at the model dynamics for the basic symmetric case it is easy to see that the phase space portrait is symmetric, as exchanging the variables $x_{1} \leftrightarrow x_{2}$ does not change the model. Moreover, the straight line $x_{2}=x_{1}$ is a specific trajectory on which the following equation is satisfied:
$$\dot{\zeta}=\alpha\left(a-\zeta+\zeta g\left(\zeta^{2}\right)\right)=3\left(0.4-\frac{\zeta}{1+\zeta^{2}}\right) .$$
Due to the properties of the function $\frac{\zeta}{1+\zeta^{2}}$ (which is unimodal with one maximum equal to $0.5$ at $\zeta=1$ we see that there are two steady states of Eq. (6), $0<\zeta_{1}<1<$ $\zeta_{2}$ and $\zeta_{1}$ is stable while $\zeta_{2}$ is unstable. Within the straight line $x_{2}=x_{1}$ all solutions with initial value below $\zeta_{2}$ tends to $\zeta_{1}$, while those above $\zeta_{2}$ tends to $\infty$ as $t \rightarrow \infty$. These corresponds to the dynamics in the whole phase space where $\left(\zeta_{1}, \zeta_{1}\right)$ is a stable node while $\left(\zeta_{2}, \zeta_{2}\right)$ is a saddle. The stable curve of a saddle divides the phase space into two regions: below this curve solutions are attracted by the first steady state, while above this curve solutions tend to $\infty$. It should be noticed that we are interested in the model dynamics for the values of variables near the first steady state, as large values are not biologically plausible. Hence, in Fig. 7 we present only a part of the phase space corresponding to such values.

In Fig. 7a we see our reference symmetric case with all solutions attracted by the steady state with coordinates near $0.4$. Giving additional input to the first variable results in moving this state to the right (cf. Fig. 7b), while when the additional input is given to the second variable, then the steady state moves up. This shows that the system correctly recognizes to which population this additional input is applied.
In [1] we presented an extended analysis of System (5) also for parameter values different than reference values, especially focusing on the role of parameters $a$ and $c$ (denoted by $I$ and $\epsilon$ in the original paper) for the number of steady states. It occurs that there can be from one to three steady states, similarly to other models considered in this chapter. The stability of these states is also similar to the cases studied above in previous sections.

## 数学代写|数学分析代写Mathematical Analysis代考|Universal Generating Function

UGF is a form of ordinary moment generating function which represents the probability mass function of discrete random variables. Let a discrete random variable $X$ has $M$ possible values $\left(x_{1}, x_{2}, \ldots, x_{M}\right)$ and $\left(p_{1}, p_{2}, \ldots, p_{M}\right)$ be the corresponding probabilities. Then UGF of discrete random variable $X=\left(x_{1}, x_{2}, \ldots, x_{M}\right)$ is répreesenentêd by a poolynônial
$$U(z)=\sum_{j=1}^{M} p_{j} z^{\left(X=x_{j}\right)}, j=1,2,3, \ldots, M$$
Consider a system having $X_{i}(i=1,2, \ldots, n)$ components and every component has $v\left(v=1,2, \ldots, M_{j}\right)$ possible state denoted by $x_{j v}$. Let probability distribution of each variable $x_{j v}$ (jth variable at $v$ th state) is represented by $p_{j v}$. Then the UGF of random variable $X_{j}$ is defined as
$$u_{X j}(z)=\sum_{v=1}^{M_{j}} p_{j v} z^{x_{j v}}$$
Combination of $m$ universal generating functions of $m$ variables is defined by composition operator $\otimes_{f}$, where the properties of the composition operator $\otimes_{f}$ totally depend on properties of the function $f\left(X_{1}, X_{2}, \ldots, X_{m}\right)$. Therefore, the composition is given by
\begin{aligned} U(z) &=\otimes_{f}\left(u_{X_{1}}(z), u_{X_{2}}(z), \ldots, u_{X_{m}}(z)\right) \ &=\otimes_{f}\left(\sum_{v_{1}=1}^{M_{1}} \sum_{v_{2}=1}^{M_{2}} \sum_{v_{3}=1}^{M_{3}} \cdots \sum_{v_{m}=1}^{M_{m}}\left(\prod_{i=1}^{m} p_{i v_{i}} z^{f\left(x_{i_{1}}, \ldots, x_{i_{m}}\right)}\right)\right. \end{aligned}

## 数学代写|数学分析代写Mathematical Analysis代考|Analysis and Illustration of the Model Dynamics

$$\dot{\zeta}=\alpha\left(a-\zeta+\zeta g\left(\zeta^{2}\right)\right)=3\left(0.4-\frac{\zeta}{1+\zeta^{2}}\right) \text {. }$$

## 数学代写|数学分析代写Mathematical Analysis代考|Universal Generating Function

UGF 是普通矩生成函数的一种形式，它表示离散随机变量的概率质量函数。设一个离散随

$$U(z)=\sum_{j=1}^{M} p_{j} z^{\left(X=x_{j}\right)}, j=1,2,3, \ldots, M$$

$$u_{X j}(z)=\sum_{v=1}^{M_{j}} p_{j v} z^{x_{j v}}$$

$$U(z)=\otimes_{f}\left(u_{X_{1}}(z), u_{X_{2}}(z), \ldots, u_{X_{m}}(z)\right) \quad=\otimes_{f}\left(\sum_{v_{1}=1}^{M_{1}} \sum_{v_{2}=1}^{M_{2}} \sum_{v_{3}=1}^{M_{3}} \cdots \sum_{v_{m}=1}^{M_{m}}\left(\prod_{i=1}^{m} p_{i v_{i}} z^{f\left(x_{i_{1}}, \ldots, x_{i m}\right.}\right)\right)$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。