数学代写|数学分析代写Mathematical Analysis代考|MATH 307

2022年7月13日

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数学代写|数学分析代写Mathematical Analysis代考|Influence Function Depending on One Variable Only

In this section we would like to discuss System (1) of the following specific form:
\begin{aligned} &\dot{x_{1}}=f_{1}\left(x_{1}\right)+c_{1} g_{1}\left(x_{2}\right), \ &\dot{x_{2}}=f_{2}\left(x_{2}\right)+c_{2} g_{2}\left(x_{1}\right) \end{aligned}

which is related to the modeling of dyadic interactions. Modeling dyadic interactions using ODEs started with a short note by Strogatz [8] who noticed that fluctuations of emotions in the relationship of Romeo and Juliet could be described using a simple linear mathematical oscillator. However, it is obvious that humans’ emotions are non-linear, and this was a reason to propose a model of the form (4) by Rinaldi and his coauthors; cf. [6] for details and the history of this model. Similar approach but in a little different context was used by Liebovitch et al. [2]. They used the notion of actors instead of partners to mark that they do not describe love relationships. Here we focus on the interpretation presented in our paper [5], which is similar to those by Liebovitch et al.

In System (4), the inner dynamics is described by linear functions $f_{i}(\zeta)=a_{i}-$ $b_{i} \zeta$. In our interpretation (cf. [5]) coefficients $a_{i}$ of the inner dynamics correspond to the natural optimism/pessimism of the $i$ th actor. More precisely, when $a_{i}>0$, then the $i$ th actor is an optimist with the natural level of optimism equal to $\frac{a_{i}}{b_{i}}$ which is called uninfluenced steady state, while for $a_{i}<0$ this steady state reflects the natural level of pessimism for the $i$ th actor. The coefficient $b_{i}>0$ is called a forgetting coefficient in this context.

Both functions $g_{i}$ are the same, that is $g_{1}(\zeta)=g_{2}(\zeta)=g(\zeta)$ and $g$ have the following properties:
(a) $g(0)=0$;
(b) $g^{\prime}(0)=1$;
(c) $g$ is increasing;
(d) $\zeta g^{\prime \prime}(\zeta)<0$ for $\zeta \neq 0$.

数学代写|数学分析代写Mathematical Analysis代考|Influence Function of the Form xig

In this section we consider the model proposed in the context of perceptual decision making in [1]. In that paper we focused on modeling the most basic perceptual decision-making in neuronal networks in which the network needs to disambiguate between two sensory stimuli. We described changes in firing rates $x_{1}, x_{2}$ of two neuronal populations. In the simplest version without time delay this model reads
\begin{aligned} &\dot{x}{1}=\alpha{1}\left(a_{1}-b_{1} x_{1}+c x_{2} g\left(x_{1} x_{2}\right)\right), \ &\dot{x}{2}=\alpha{2}\left(a_{2}-b_{2} x_{2}+c x_{1} g\left(x_{1} x_{2}\right)\right), \end{aligned}
where $\frac{1}{\alpha_{i}}$ are time scales for these neuronal populations, the $i$ th population receives an external input $a_{i}$, our populations are self-inhibited with inhibition coefficients $b_{i}$, $c$ is a coefficient describing the maximal capacity of a synapse, while $g$ characterizes interactions between neuronal populations related to so-called synaptic plasticity for which we assumed a sigmoid shape. In this model we are interested in positive values of the variables $x_{i}$ and this is the reason we define the function $g$ only for non-negative values. However, in general the model could be extended for all real values of $x_{i}$ and then the function $g$ should be considered as a function on $\mathbb{R}$.

We assume that $g: \mathbb{R}{+} \rightarrow \mathbb{R}{+}$of class $\mathbf{C}^{1}$ satisfies the following assumptions:
$-g(0)=0$
$-g^{\prime}(\zeta)>0$ for $\zeta>0$

• there exists $\zeta_{1}>0$ such that in $\left(0, \zeta_{1}\right)$ the function $g$ is convex and for $\zeta>\zeta_{1}$ it is concave;
$-g$ is bounded.

数学代写|数学分析代写Mathematical Analysis代考|Influence Function Depending on One Variable Only

$$\dot{x_{1}}=f_{1}\left(x_{1}\right)+c_{1} g_{1}\left(x_{2}\right), \quad \dot{x_{2}}=f_{2}\left(x_{2}\right)+c_{2} g_{2}\left(x_{1}\right)$$

$a_{i}<0$ 这种稳定状态反映了自然的悲观情绪水平 $i$ 演员。系数 $b_{i}>0$ 在这种情况下称为遗 忘系数。

(a) $g(0)=0$
$\left(\right.$ 二) $g^{\prime}(0)=1$
(C) $g$ 在垾加;
(d) $\zeta g^{\prime \prime}(\zeta)<0$ 为了 $\zeta \neq 0$.

数学代写|数学分析代写Mathematical Analysis代考|Influence Function of the Form xig

$$\dot{x} 1=\alpha 1\left(a_{1}-b_{1} x_{1}+c x_{2} g\left(x_{1} x_{2}\right)\right), \quad \dot{x} 2=\alpha 2\left(a_{2}-b_{2} x_{2}+c x_{1} g\left(x_{1} x_{2}\right)\right),$$

$-g(0)=0$
$-g^{\prime}(\zeta)>0$ 为了 $\zeta>0$

• 那里存在 $\zeta_{1}>0$ 这样在 $\left(0, \zeta_{1}\right)$ 功能 $g$ 是凸的并且对于 $\zeta>\zeta_{1}$ 它是凹的：
$-g$ 是有界的。

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