Long lasting effect of acute restraint stress on behavior and brain anti-oxidative status

Nouhaila Chaoui; Hammou Anarghou; Meriem Laaroussi; Oumaima Essaidi; Mohamed Najimi; Fatiha Chigr; Nouhaila Chaoui; Hammou Anarghou; Meriem Laaroussi; Oumaima Essaidi; Mohamed Najimi; Fatiha Chigr

doi:10.3934/Neuroscience.2022005

AIMS Neuroscience

2022, Volume 9, Issue 1: 57-75. doi: 10.3934/Neuroscience.2022005

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Research article

Long lasting effect of acute restraint stress on behavior and brain anti-oxidative status

Biological Engineering Laboratory, Faculty of Sciences and Techniques, Sultan Moulay Slimane University, Beni Mellal, Morocco

Received: 08 November 2021 Revised: 18 January 2022 Accepted: 18 January 2022 Published: 27 January 2022

Exposure to certain acute stressors results in an immediate behavioral and physiological response to these situations during a significant period of days. The goal of the current study is to evaluate the long-lasting effect of single exposure of restraint stress among mice after 0 h, 24 h, 48 h and 72 h. Five groups of mice are under experiment: a control group and four groups exposed to one session of restraint stress. All these groups have been studied for behavioral tests in order to evaluate their memories. This is done through a Y-labyrinth and an object recognition test, and anxiety by using open field device. In the second part of the study, enzymatic assays (concerning catalase, glutathione s transferase, glutathione peroxidase and superoxide dismutase) are used to evaluate oxidative stress. The enzymatic activity of the antioxidant system is assessed in five brain structures, including the cerebellum, olfactory bulb, spinal bulb, hypothalamus, and hippocampus.

The obtained results show that acute restraint stress leads to a decrease in memory function and to the development of an anxious state; concomitant to an increase of locomotor activity afterword. It causes disturbance of antioxidant balance in the brain by developing a state of oxidative stress. Indeed, restraint stress causes a change in anti-oxidant stress enzymatic activity in the brain, notably in post-stress period. In conclusion, acute restraint stress is responsible for altering cognitive functions, especially memory, and the development of anxious behavior, which could be a result of the generation of oxidative stress; effects that are persistent over an important period after the cessation of stress.

Keywords:

Citation: Nouhaila Chaoui, Hammou Anarghou, Meriem Laaroussi, Oumaima Essaidi, Mohamed Najimi, Fatiha Chigr. Long lasting effect of acute restraint stress on behavior and brain anti-oxidative status[J]. AIMS Neuroscience, 2022, 9(1): 57-75. doi: 10.3934/Neuroscience.2022005

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Abstract

1. Introduction

Let $n$ and $k$ be two positive integers. Denote by $p\left(n, k\right)$ the number of partitions of the positive number $n$ on exactly $k$ parts. Then the partition class $k$ is the sequence $p\left(1, k\right), p\left(2, k\right), \dots, p\left(n, k\right), \dots$ We already know, see ^[1], all these values can be divided into the highest $d_0 = LCM\left(1, 2, \dots, k\right)$ sub sequences, each of which is calculated by the same polynomial.

Choose a sequence of $k$ natural numbers such that: the first member is arbitrary, and the rest form an arithmetic progression with a difference $d = m\cdot d_0, \enspace m \in \mathbb{N}$ , starting from the chosen first member. For example:

$\begin{equation} x_1 = j, \enspace x_2 = j+d, \enspace \dots, \enspace x_k = j+\left(k-1\right)\cdot d, \quad j\in \mathbb{N}. \end{equation}$

(1.1)

The corresponding number of partitions of the class $k$ for the elements of the previous arithmetic progression's values is:

$\begin{equation} p\left(x_1, k\right), \enspace p\left(x_2, k\right), \enspace \dots \enspace, \enspace p\left(x_k, k\right). \end{equation}$

(1.2)

If the values, which are calculated using the same polynomial, multiplied by the corresponding binomial coefficients, form the alternate sum, we notice that the sum always has a value which is independent of $x_1$ , no matter how we form the sequence (1.1).

For the partition function of classes we already know the following results, see ^[1,2] for some details:

ⅰ) The values of the partition function of classes is calculated with one quasi polynomial.

ⅱ) For each class $k$ the quasi polynomial consists of at most $LCM\left(1, 2, \dots, k\right)$ different polynomials, each of them consists of a strictly positive and an alternating part.

ⅲ) All polynomials within one quasi polynomial $p\left(n, k\right)$ are of degree $k-1$ .

ⅳ) All the coefficients with the highest degrees down to $\left[\frac{k}{2}\right]$ are equal for all polynomials (all of strictly positive) and all polynomials differ only in lower coefficients (alternating part).

ⅴ) The form of any polynomial $p\left(n, k\right)$ is:

$\begin{equation} p\left(n, k\right) = a_1n^{k-1}+a_2n^{k-2}+\dots+a_k, \end{equation}$

(1.3)

where the coefficients $a_1, a_2, \dots, a_k$ are calculated in the general form.

Let us forget for a moment that the coefficients $a_1, a_2, \dots$ are known in general form. Knowing that all values for partitions class of the sequence (1.1) are obtained by one polynomial $p\left(n, k\right)$ , it is possible to determine all unknown coefficients in a completely different way from that given in papers ^[1,2]. To determine $k$ unknowns, a $k$ equation is required. For this purpose, it is sufficient to know all the values of the sequence (1.2). To this end, we must form the system (1.4) and solve it. (For $k = 10$ , see ^[3]).

$\begin{equation} \begin{split} & a_1\cdot x_1^{k-1}+a_2\cdot x_1^{k-2}+\dots+a_k = p\left(x_1, k\right)\\ & a_1\cdot x_2^{k-1}+a_2\cdot x_2^{k-2}+\dots+a_k = p\left(x_2, k\right)\\ & \dots \quad \dots \quad \dots\\ & a_1\cdot x_k^{k-1}+a_2\cdot x_k^{k-2}+\dots+a_k = p\left(x_k, k\right) \end{split} \end{equation}$

(1.4)

The system (1.4) can be solved by Cramer's Rule. For further analysis, we need to find the following determinants. We will start with the known Vandermonde determinant, see ^[4].

$\begin{equation} \Delta_m = \begin{vmatrix} x_{1}^{m-1} & x_{1}^{m-2} & \dots & 1 \\ x_{2}^{m-1} & x_{2}^{m-2} & \dots & 1 \\ \dots & \dots & \dots \\ x_m^{m-1} & x_m^{m-2} & \dots & 1 \\ \end{vmatrix} = \prod\limits_{1 \le i \lt j \le m}\left(x_i-x_j\right), \enspace m \gt 1. \end{equation}$

(1.5)

When we remove the first column and an arbitrary row from the previous determinant we obtain the Vandermonde determinant of one order less. The following results are known, see ^[4] and are needed for further exposure. If we remove the second column and an arbitrary $a$ -th row from the determinant (1.5) we get

$\begin{equation} \begin{vmatrix} x_1^{m-1} & x_1^{m-3} & \dots & x_1 & 1 \\ \dots & \dots & \dots & \dots & \dots \\ x_{a-1}^{m-1} & x_{a-1}^{m-3} & \dots & x_{a-1} & 1 \\ x_{a+1}^{m-1} & x_{a+1}^{m-3} & \dots & x_{a+1} & 1 \\ \dots & \dots & \dots & \dots & \dots \\ x_m^{m-1} & x_m^{m-3} & \dots & x_m & 1 \\ \end{vmatrix} = \left(\sum\limits_{0 \le i\le m}^{i\ne a}x_i\right)\cdot \prod\limits_{0 \le i \lt j \le m}^{i, j\ne a}\left(x_i-x_j\right) \end{equation}$

(1.6)

If we remove the third column and an arbitrary $a$ -th row from the determinant (1.5) we get

$\begin{equation} \begin{vmatrix} x_1^{m-1} & x_1^{m-2} & x_1^{m-4} &\dots & x_1 & 1 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ x_{a-1}^{m-1} & x_{a-1}^{m-2} & x_{a-1}^{m-4} \dots & x_{a-1} & 1 \\ x_{a+1}^{m-1} & x_{a+1}^{m-2} & x_{a+1}^{m-4} \dots & x_{a+1} & 1 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ x_m^{m-1} & x_m^{m-2} & x_m^{m-4} & \dots & x_m & 1 \\ \end{vmatrix} = \sum\limits_{1 \le i \lt j \le m}^{i, j \ne a}x_i \cdot x_j \prod\limits_{0 \le i \lt j\le m}^{i, j \ne a}\left(x_i-x_j\right) \end{equation}$

(1.7)

Generaly, if we remove the $b$ -th column and an arbitrary $a$ -th row from the determinant (1.5) we get

$\begin{multline*} \Delta_{m}^{\left(a, b\right)} = \begin{vmatrix} x_1^{m-1} & x_1^{m-2} & \dots & x_1^{b+1} & x_1^{b-1} & \dots& x_1 & 1 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots\\ x_{a-1}^{m-1} & x_{a-1}^{m-2} & \dots & x_{a-1}^{b+1} & x_{a-1}^{b-1} & \dots & x_{a-1} & 1 \\ x_{a+1}^{m-1} & x_{a+1}^{m-2} & \dots & x_{a+1}^{b+1} & x_{a+1}^{b-1} & \dots & x_{a+1} & 1 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots\\ x_m^{m-1} & x_m^{m-2} & \dots & x_m^{b+1} & x_m^{b-1} & \dots & x_m & 1 \\ \end{vmatrix} = \left(\sum\limits_{1 \le t_1 \lt t_2 \dots \lt t_{b-1} \le m}^{t_1, \dots, t_{b-1} \ne a}x_{t_1}x_{t_2} \dots x_{t_{b-1}}\right) \prod\limits_{0\le i \lt j\le m}^{i, j\ne a}\left(x_i-x_j\right). \end{multline*}$

The label $\Delta_{m}^{\left(a, b\right)}$ means that from $\Delta_m$ remove the $a$ -th row and $b$ -th column from the set of variables $x_a$ .

2. Invariants of the partitions classes

2.1. The first partition invariant of classes

Theorem 1. Let $m, j$ and $k$ be three positive integers and

$I_1\left(k, j, d\right) = \sum\limits_{i = 0}^{k-1}\left(-1\right)^i \binom{k-1}{i}p\left(j+i\cdot d, k\right),$

where $d = m\cdot LCM\left(1, 2, 3, \dots, k\right)$ . Then $I_1\left(k, j, d\right) = \left(-1\right)^{k-1}\frac{d^{k-1}}{k!}$ and is independent of $j$ . ( $I_1\left(k, j, d\right)$ is the first partition invariant which exists in all classes.)

Proof. Among the values of the class $k$ we choose the ones corresponding to the sequence (1.1), and they are given with the sequence (1.2). According to ^[2], all the elements in (1.2) can be calculated using the same polynomial $p\left(n, k\right)$ with degree $k-1$ . Elements of the following sequence:

$q, q+d, \dots, q+\left(k-1\right)\cdot d, \quad q \ne j,$

are calculated with not necessarily the same polynomial as the previous one. Let the polynomial $p\left(n, k\right)$ have the form as in (1.3). To determine the coefficients $a_1, a_2, \dots, a_k$ it suffices to know the $k$ values: $p\left(x_1, k\right), p\left(x_2, k\right), \dots, p\left(x_k, k\right)$ where $x_1 = j, \enspace x_2 = j+d, \dots, \enspace x_k = j+\left(k-1\right)d\enspace$ are different numbers. Since $\Delta_k \ne 0$ , system (1.4) always has a unique solution, because all the elements of the set $\left\{x_1, x_2, \dots, x_k\right\}$ are different from one another. According to Cramer's Rule, to determine the coefficient of the highest degree of the polynomial (1.3), which calculates the value of the number of partitions of class $k$ , we have the following formula:

$\begin{equation} a_1 = \frac{p\left(x_1, k\right)\Delta_{k}^{\left(1, 1\right)}-p\left(x_2, k\right)\Delta_{k}^{\left(2, 1\right)}+\dots+\left(-1\right)^{k-1}p\left(x_k, k\right)\Delta_{k}^{\left(k, 1\right)}} {\Delta_k} \end{equation}$

(2.1)

Determinants $\Delta_{k}^{\left(a, 1\right)}, \left(1 \le a \le k\right)$ are also Vandermonde and their values are equal to $\Delta_{k-1}$ . Let $\left\{x_i\right\}_{1\le i \le k}$ satisfy (1.1) then for $1\le a \le k$ it holds that

$\begin{multline*} \Delta_{k}^{\left(a, 1\right)} = \Delta_{k-1} = \frac{\Delta_k}{\prod\limits_{i = 1}^{a-1}\left(x_i-x_a\right)\prod\limits_{i = a+1}^{k}\left(x_a-x_i\right)}\\ = \frac{\Delta_k}{\left(-1\right)^{a-1}\left(a-1\right)!d^{a-1}\cdot \left(-1\right)^{k-a}\left(k-a\right)!d^{k-a}} \\ = \frac{\left(-1\right)^{k-1}\Delta_k}{\left(a-1\right)!\left(k-a\right)!d^{k-1}}. \end{multline*}$

Replacing in (2.1), after shortening with $\Delta_k$ we have

$\begin{multline*} \quad \quad \quad \quad \quad \quad \quad a_1 = \left(-1\right)^{k-1}\left( \frac{p\left(x_1, k\right)}{0!\left(k-1\right)!d^{k-1}}-\frac{p\left(x_2, k\right)}{1!\left(k-2\right)!d^{k-1}}+\dots \right. \left. +\left(-1\right)^{k-1}\frac{p\left(x_k, k\right)}{\left(k-1\right)!0!d^{k-1}}\right). \end{multline*}$

The coefficient $a_1$ is already defined in ^[2] where it is shown that $a_1 = \frac{1}{k!\left(k-1\right)!}$ . Substituting into the previous equality and multiplying by $\left(-1\right)^{k-1}$ , we obtain

$\begin{equation*} \quad \quad \quad \quad \frac{\left(-1\right)^{k-1}}{k!\left(k-1\right)!} = \frac{p\left(x_1, k\right)}{0!\left(k-1\right)!d^{k-1}}-\frac{p\left(x_2, k\right)}{1!\left(k-2\right)!d^{k-1}}+\dots+\left(-1\right)^{k-1}\frac{p\left(x_k, k\right)}{\left(k-1\right)!0!d^{k-1}}. \end{equation*}$

Multiplying the last equality with $\left(k-1\right)!d^{k-1}$ we obtain

$\begin{equation*} \left(-1\right)^{k-1}\frac{d^{k-1}}{k!} = \sum\limits_{i = 0}^{k-1}\left(-1\right)^i \binom{k-1}{i}p\left(j+i\cdot d, k\right), \end{equation*}$

which was to be proved. As these values are equal to each observed number of objects (1.2) within a class, the sum is invariant for any observed class.

All classes of the partition do not contain all the invariants we will list. This primarily refers to the classes from the beginning. Only the first invariant appears in all classes. The second invariant holds starting from the third class. The third invariant holds starting from the fifth class. Fourth, from the seventh class, etc. This coincides with the appearance of the common coefficients $\left\{a_k\right\}$ in quasi polynomials $p\left(n, k\right)$ , $k \in \mathbb{N}$ .

Theorem 2. Let $m$ , $j$ and $k$ be three positive integers, $k \ge 3$ and

$\begin{equation*} I_2\left(k, j, d\right) = \sum\limits_{i = 0}^{k-1}\left(-1\right)^{i+1}\left(j\left(k-1\right)+\left(\binom{k}{2}-i\right)d\right)\binom{k-1}{i}p\left(j+i\cdot d, k\right) \end{equation*}$

where $d = m\cdot LCM\left(2, 3, \dots, k\right)$ . Then $I_2\left(k, j, d\right) = \left(-1\right)^k\frac{\left(k-3\right)d^{k-1}}{4\left(k-2\right)!}$ and is independent of $j$ .

Remark. In the previous expression, we should not simplify as then the value for $k = 3$ cannot be obtained. However, the value for $k = 3$ exists and is equal to zero.

Proof. Analogously to Theorem 1, the fact that the sum does not depend on the parameter $j$ is a consequence of the periodicity per modulo $LCM\left(2, 3, \dots, k\right)$ using the same polynomial to calculate the partition class values.

In ^[2] it is shown how the system of linear equations can determine the other unknown coefficient of the polynomials which are calculated values of the partition classes. This coefficient is obtained from Cramer's Rule on system (1.4) and $a_2$ is given by

$\begin{equation} a_2 = \frac{-p\left(x_1, k\right)\Delta_{k}^{\left(1, 2\right)}+p\left(x_2, k\right)\Delta_{k}^{\left(2, 2\right)}-\dots\left(-1\right)^k p\left(x_k, k\right)\Delta_{k}^{\left(k, 2\right)} }{\Delta_k}. \end{equation}$

(2.2)

Considering (1.6), knowing that $\left\{x_i\right\}_{i = 1, 2, \dots, k}$ is an arithmetic progression, determinants $\Delta_{k}^{\left(a, 2\right)}$ can be written for $1 \le a \le k$ with

$\begin{multline*} \Delta_{k}^{\left(a, 2\right)} = \left(\sum\limits_{1\le i \le k}^{i\ne a}x_i \right) \Delta_{k-1} = \frac{\left(\left(k-1\right)j+\left(\binom{k}{2}-a+1\right)d\right)\Delta_k}{\prod\limits_{i = 1}^{a-1}\left(x_i-x_a\right)\prod\limits_{i = a+1}^{k}\left(x_a-x_i\right)}\\ \quad \quad \quad \quad \quad = \frac{\left(\left(k-1\right)j+\left(\binom{k}{2}-a+1\right)d\right)\Delta_k}{\left(-1\right)^{a-1}\left(a-1\right)!d^{a-1}\cdot \left(-1\right)^{k-a}\left(k-a\right)!d^{k-a}}\\ = \frac{\left(-1\right)^{k-1}\left(\left(k-1\right)j+\left(\binom{k}{2}-a+1\right)d\right)\Delta_k}{\left(a-1\right)!\left(k-a\right)!d^{k-1}}. \end{multline*}$

Knowing the value of the coefficient $a_2 = \frac{k-3}{4\left(k-1\right)!\left(k-2\right)!}$ ^[2] and substituting in (2.2), and after multiplication with $\left(-1\right)^k\left(k-1\right)!d^{k-1}$ we obtain

$\begin{multline*} \left(-1\right)^k\frac{k-3}{4\left(k-2\right)!}d^{k-1} = \left(\left(k-1\right)j+\binom{k}{2}d\right)\binom{k-1}{0}p\left(j, k\right)- \\ \left(\left(k-1\right)j+\left(\binom{k}{2}+1\right)d\right)\binom{k-1}{1}p\left(j+d, k\right)+\dots \\ = \sum\limits_{i = 0}^{k-1}\left(-1\right)^{i+1}\left(j\left(k-1\right)+\left(\binom{k}{2}-i\right)d \right)\binom{k-1}{i}p\left(j+i\cdot d, k\right). \square \end{multline*}$

2.2. The third partition invariant of classes

These invariants are in all classes starting from the fifth. For simplicity we denote them by

$\begin{multline*} R\left(i, j, k, d\right) = \frac{1}{2}\left(\left(\left(k-1\right)j+\left(\binom{k}{2}-i\right)d\right)^2-\left(k-1\right)j^2 \right.\\ \left.-\left(\frac{1}{6}k\left(k-1\right)\left(2k-1\right)-i^2\right)d^2-2d\cdot j\left(\binom{k}{2}-i\right)\right). \end{multline*}$

Theorem 3. Let $m, j$ and $k$ be three positive integers, $k \ge 5$ and

$\begin{equation*} I_3\left(k, j, d\right) = \sum\limits_{i = 0}^{k}\left(-1\right)^i R\left(i, j, k, d\right)\binom{k-1}{i}p\left(j+i\cdot d, k\right), \end{equation*}$

where $d = m\cdot LCM\left(2, 3, \dots, k\right)$ . Then $I_3\left(k, j, d\right) = \left(-1\right)^{k-1}\frac{9k^3-58k^2+75k-2}{288\left(k-3\right)!}d^{k-1}$ and is independent of $j$ .

Proof. For the third invariant we need the value of the third polynomial coefficient of $p\left(n, k\right)$ , and it is shown ^[2] that this is

$\begin{equation*} a_3 = \frac{9k^3-58k^2+75k-2}{288\left(k-1\right)!\left(k-3\right)!}, \enspace k \ge 5. \end{equation*}$

On the other hand, we have

$\begin{equation} a_3 = \frac{p\left(x_1, k\right)\Delta_{k}^{\left(1, 3\right)}-p\left(x_2, k\right)\Delta_{k}^{\left(2, 3\right)}+\dots+\left(-1\right)^{k-1}p\left(x_k, k\right)\Delta_{k}^{\left(k, 3\right)}}{\Delta_k} \end{equation}$

(2.3)

From formula (1.7) we find $\Delta_{k}^{\left(a, 3\right)}$ . The required sum $\sum\limits_{1 \le i < j\le k}x_i\cdot x_j$ is convenient to calculate from the equality

$\sum\limits_{1 \le i \lt j\le k}^{i, j\ne a}x_i\cdot x_j = \frac{1}{2}\left(\left(\sum\limits_{1 \le i \le k}^{i \ne a}x_i\right)^2-\sum\limits_{1 \le i\le k}^{i\ne a}x_i^2\right),$

where the sequence $\left\{x_i\right\}$ satisfies (1.1). Then, we should determine the quotient which can be simplified by reducing the following:

$\begin{equation*} \frac{\Delta_{k}^{\left(a, 3\right)}}{\Delta_k} = \frac{R\left(a-1, j, k, d\right)}{\prod\limits_{i = 1}^{a-1}\left(x_i-x_a\right)\prod\limits_{i = a+1}^{k}\left(x_a-x_i\right)}. \end{equation*}$

By multiplying (2.3) with $\left(-1\right)^{k-1}\left(k-1\right)!d^{k-1}$ and after shortening we obtain:

$\begin{equation*} I_3\left(k, j, d\right) = \sum\limits_{i = 0}^{k-1}\left(-1\right)^i\cdot R\left(i, j, k, d\right)\binom{k-1}{i}p\left(j+i\cdot d, k\right). \square \end{equation*}$

In every subsequent invariant, the proceedings become more complex. But, it is quite clear how further invariants can be calculated.

3. Consideration of special cases

For each partitions class $k$ , $k\in \mathbb{N}$ we determine $d_0 = LCM\left(1, 2, 3, \dots, k\right)$ , and then form $d = m\cdot d_0$ , $\enspace m\in \mathbb{N}$ . In addition arbitrarily choose the natural number $j$ and than form sequences (1.1) and (1.2). Finally, we form an appropriate sum which is for the first invariant:

$\begin{equation} \sum\limits_{i = 0}^{k-1}\left(-1\right)^i\binom{k-1}{i}p\left(j+i\cdot d, k\right) = \sum\limits_{i = 0}^{k-1}\left(-1\right)^i\binom{k-1}{i}p\left(x_{i+1}, k\right), j\in \mathbb{N}. \end{equation}$

(3.1)

Sum (3.1) has a constant value in each partitions class and can be nominated as the first partitions class invariant.

3.1. The first partitions class invariant

For $k = 1$ , sum (3.1) has a constant value of 1.

For $k = 2$ , $d_0 = 2$ . If we choose some $m\in \mathbb{N}$ and set $d = 2m$ , the sum (3.1) has the form: $p\left(j, 2\right)-p\left(j+d, 2\right), j \in \mathbb{N}$ . According to ^[1], it is known that $p\left(n, 2\right) = \left[\frac{n}{2}\right]$ . Distinguishing between even and odd numbers of $j$ ( $j$ and $j+d$ have the same parity) and substituting into the sum, we obtain that the result, in both cases, is equal to $-\frac{d}{2} = -m$ .

For $k = 3$ , $d_0 = 6$ . If we choose some $m\in \mathbb{N}$ and set $d = 6m$ the sum (3.1) has the form:

$\begin{equation} p\left(j, 3\right)-2\cdot p\left(j+d, 3\right)+p\left(j+2d, 3\right), j\in \mathbb{N}. \end{equation}$

(3.2)

According to ^[1], it is known that:

$\begin{equation} p\left(n, 3\right) = \frac{n^2+\omega_i}{12}, \enspace i = n \enspace \mathrm{mod}\enspace 6, \enspace \omega_i \in \left\{0, -1, -4, 3, -4, -1\right\}. \end{equation}$

(3.3)

By replacing (3.3) in relation (3.2) we get

$\frac{j^2+w_{i_1}}{12}-2\frac{\left(j+d\right)^2+w_{i_2}}{12}+\frac{\left(j+2d\right)^2+w_{i_3}}{12}.$

Note that: $i_1 = j\enspace mod \enspace 6$ , $i_2 = \left(j+d\right)\enspace mod \enspace 6$ , $i_3 = \left(j+2d\right)\enspace mod\enspace 6$ and $w_{i_1} = w_{i_2} = w_{i_3}$ . Finally, we get the unique sum $6m^2$ .

For $k = 4$ , $d_0 = 12$ . If we choose some $m\in \mathbb{N}$ and set $d = 12m$ the sum (3.1) has the form:

$\begin{equation} p\left(j, 4\right)-3p\left(j+d, 4\right)+3p\left(j+2d, 4\right)-p\left(j+3d, 4\right), j\in \mathbb{N}. \end{equation}$

(3.4)

According to ^[1], it is known that:

$\begin{equation} p\left(n, 4\right) = \frac{1}{144}n^3+\frac{1}{48}n^2+ \begin{cases} \frac{w_i}{144}, \quad n\text{ even}, \\ -\frac{1}{16}n+\frac{w_i}{144}, \quad n\text{ odd}, \end{cases} i\equiv n\mathrm{\enspace mod\enspace}12, \end{equation}$

(3.5)

$w_i\in \left\{0, 5, -20, -27, 32, -11, -36, 5, 16, -27, -4, -11\right\}.$

Similar to case $k = 3$ , by distinguishing the even and odd $j$ and replacing (3.5) in relation (3.4) we obtain that the corresponding sums in both cases are equal to: $-72m^3$ . (Note that: ${i_1} = j\enspace mod \enspace 12$ , ${i_2} = \left(j+d\right) mod \enspace 12$ , ${i_3} = \left(j+2d\right) mod \enspace 12$ , ${i_4} = \left(j+3d\right) mod \enspace 12$ and $w_{i_1} = w_{i_2} = w_{i_3} = w_{i_4}$ .)

The number of invariants increases, when the class number increases. Starting with class three, another invariant can be observed.

3.2. The second partitions class invariant

Form in the same way as in the previous section: $d_0$ , $d$ and the sequences (1.1) and (1.2) as well as the sum:

$\begin{equation*} \sum\limits_{i = 0}^{k-1}\left(-1\right)^i\left(j\left(k-1\right)+\left(\binom{k}{2}-i\right)d\right)\binom{k-1}{i}p\left(j+i\cdot d, k\right). \end{equation*}$

Previous sum has a constant value in each partitions class (starting from third class) and can be nominated as the second partitions class invariant.

For $k = 3$ , $d_0 = 6$ . If we choose some $m\in \mathbb{N}$ and set $d = 6m$ the general form of the second invariant in the third class can be written as

$\left(2j+3d\right)p\left(j, 3\right)-2\left(2j+2d\right)p\left(j+d, 3\right)+\left(2j+d\right)p\left(j+2d, 3\right), j\in \mathbb{N}$

The values $p\left(j, 3\right), p\left(j+d, 3\right)$ and $p\left(j+2d, 3\right)$ are calculated using the same polynomial (3.3). Using (3.3) in the last equality we have

$\begin{equation*} \left(2j+3d\right)\frac{j^2+w_{i_1}}{6}-2\left(2j+2d\right)\frac{\left(j+d\right)^2+w_{i_2}}{6}+\left(2j+d\right)\frac{\left(j+2d\right)^2+w_{i_3}}{6} \end{equation*}$

Note that: $i_1 = j\enspace mod \enspace 6$ , $i_2 = \left(j+d\right)mod \enspace 6$ , $i_3 = \left(j+2d\right)mod \enspace 6$ and $w_{i_1} = w_{i_2} = w_{i_3}$ . The last equality is identical to zero.

For $k = 4$ , $d_0 = 12$ . If we choose some $m\in \mathbb{N}$ and set $d = 12m$ the general form of the second invariant in the fourth class can be written as

$\begin{multline} \left(3j+6d\right)p\left(j, 4\right)-3\left(3j+5d\right)p\left(j+d, 4\right)+3\left(3j+4d\right)p\left(j+2d, 4\right)\\ -\left(3j+3d\right)p\left(j+3d, 4\right). \end{multline}$

(3.6)

The last equations can be verified in an analogous manner, by using the same form of the known polynomial for the fourth class given in (3.5). Note that: $i_1 = j\enspace mod \enspace12$ , $i_2 = \left(j+d\right)mod \enspace 12$ , $i_3 = \left(j+2d\right)mod \enspace 12$ , $i_4 = \left(j+3d\right)mod \enspace 12$ and $w_{i_1} = w_{i_2} = w_{i_3} = w_{i_4}$ . By distinguishing the even and odd $j$ and replacing (3.5) in relation (3.6) we obtain that the corresponding sums in both cases are equal to: $-216m^3$ .

3.3. The third partition invariants

Form in the same way as in the previous two section: $d_0$ , $d$ and the sequences (1.1) and (1.2) as well as the sum $I_3\left(k, j, d\right)$ (Theorem 3). For each class (starting from the fifth) $I_3\left(k, j, d\right)$ has constant values and can be nominated as the third partitions class invariant. It is known ^[1] that

$\begin{multline} p\left(n, 5\right) = \frac{1}{2880}n^4+\frac{1}{288}n^3+\frac{1}{288}n^2 + \begin{cases} -\frac{1}{24}n+\frac{w_i}{2880}, \quad n\text{ even}, \\ -\frac{1}{96}n+\frac{w_i}{2880}, \quad n\text{ odd}, \end{cases} i\equiv n\mathrm{\enspace mod\enspace}60, \end{multline}$

(3.7)

$w_i$ are following numeric respectively:

$\begin{align*} &0, 9,104, -351, -576,905, -216, -351, -256, 9,360, -31, -576, 9,104,225, \\ &-576,329, -216, -351,320, 9, -216, -31, -576,585,104, -351, -576,329,360, \\ &-351, -256, 9, -216,545, -576, 9,104, -351, 0,329, -216, -351, -256,585, \\ &-216, -31, -576, 9,680, -351, -576,329, -216,225, -256, 9, -216, -31.\\ \end{align*}$

For $k = 5$ , $d_0 = 60$ . If we choose some $m\in \mathbb{N}$ and set $d = 60m$ the invariant $I_3\left(k, j, d\right)$ can be written as:

$\begin{multline*} \frac{1}{2}\left(\left(4j+10d\right)^2-4j^2-20d\cdot j-30d^2\right) p\left(j, 5\right) -\frac{1}{2}\left(\left(4j+9d\right)^2-4j^2-18d\cdot j-29d^2\right)p\left(j+d, 5\right) \\ +\frac{1}{2}\left(\left(4j+8d\right)^2-4j^2-16d\cdot j-26d^2\right)p\left(j+2d, 5\right) -\frac{1}{2}\left(\left(4j+7d\right)^2-4j^2-14d\cdot j-21d^2\right)p\left(j+3d, 5\right) \\ +\frac{1}{2}\left(\left(4j+6d\right)^2-4j^2-12d\cdot j-14d^2\right)p\left(j+4d, 5\right). \end{multline*}$

Substituting (3.7) into the previous formula by distinguishing between even and odd $j$ , we obtain a unique value of $1080000m^4$ .

Remark 1. From the Table 1, see ^[5], given at the end of the paper it is possible to check all of these explicitly with numerical values. For example:

Table 1. Partition classes values.

$d_0$	1	2	6	12	60	60	420	840	2520	2520
$n/k$	1	2	3	4	5	6	7	8	9	10	11	$\dots$	$p(n)$
1	1	0	0	0	0	0	0	0	0	0	0		1
2	1	1	0	0	0	0	0	0	0	0	0		2
3	1	1	1	0	0	0	0	0	0	0	0		3
4	1	2	1	1	0	0	0	0	0	0	0		5
5	1	2	2	1	1	0	0	0	0	0	0		7
6	1	3	3	2	1	1	0	0	0	0	0		11
7	1	3	4	3	2	1	1	0	0	0	0		15
8	1	4	5	5	3	2	1	1	0	0	0		22
9	1	4	7	6	5	3	2	1	1	0	0		30
10	1	5	8	9	7	5	3	2	1	1	0		42
11	1	5	10	11	10	7	5	3	2	1	1		56
12	1	6	12	15	13	11	7	5	3	2	1	$\dots$	77
13	1	6	14	18	18	14	11	7	5	3	2	$\dots$	101
14	1	7	16	23	23	20	15	11	7	5	3	$\dots$	135
15	1	7	19	27	30	26	21	15	11	7	5	$\dots$	176
16	1	8	21	34	37	35	28	22	15	11	7	$\dots$	231
17	1	8	24	39	47	44	38	29	22	15	11	$\dots$	297
18	1	9	27	47	57	58	49	40	30	22	15	$\dots$	385
19	1	9	30	54	70	71	65	52	41	30	22	$\dots$	490
20	1	10	33	64	84	90	82	70	54	42	30	$\dots$	627
21	1	10	37	72	101	110	105	89	73	55	43	$\dots$	792
22	1	11	40	84	119	136	131	116	94	75	56	$\dots$	1002
23	1	11	44	94	141	163	164	146	123	97	77	$\dots$	1255
24	1	12	48	108	164	199	201	186	157	128	100	$\dots$	1575
25	1	12	52	120	192	235	248	230	201	164	133	$\dots$	1958
26	1	13	56	136	221	282	300	288	252	212	171	$\dots$	2436
27	1	13	61	150	255	331	364	352	318	267	223	$\dots$	3010
28	1	14	65	169	291	391	436	434	393	340	282	$\dots$	3718
29	1	14	70	185	333	454	522	525	488	423	362	$\dots$	4565
30	1	15	75	206	377	532	618	638	598	530	453	$\dots$	5604
31	1	15	80	225	427	612	733	764	732	653	573	$\dots$	6842
32	1	16	85	249	480	709	860	919	887	807	709	$\dots$	8349
33	1	16	91	270	540	811	1009	1090	1076	984	884	$\dots$	10143
34	1	17	96	297	603	931	1175	1297	1291	1204	1084	$\dots$	12310
35	1	17	102	321	674	1057	1369	1527	1549	1455	1337	$\dots$	14883
36	1	18	108	351	748	1206	1579	1801	1845	1761	1626	$\dots$	17977
37	1	18	114	378	831	1360	1824	2104	2194	2112	1984	$\dots$	21637

| Show Table

DownLoad: CSV

1. Check the first invariant in the third class. Take $m = 2$ , $j = 5$ . The first invariant formula is

$p\left(5, 3\right)-2\cdot p\left(17, 3\right)+p\left(27, 3\right).$

From the Table we find: $p\left(5, 3\right) = 2$ , $p\left(17, 3\right) = 24$ , $p\left(29, 3\right) = 70$ . By substitution we find $2-2\cdot 24+70 = 24( = 6m^2)$ .

2. Check the second invariant in the forth class. Take $m = 1$ , $j = 3$ . The second invariant formula is

$81p\left(3, 4\right)-3\cdot 69p\left(15, 4\right)+3\cdot 57p\left(27, 4\right) -45p\left(39, 4\right).$

From the Table we find: $p\left(3, 4\right) = 0$ , $p\left(15, 4\right) = 27$ , $p\left(27, 4\right) = 150$ , $p\left(39, 4\right) = 441$ . By substitution we find:

$81\cdot0-3\cdot69\cdot27+3\cdot57\cdot150-45\cdot441 = -216( = -216m^3) .$

3. Check the third invariant in the fifth class. Take $m = 1$ , $j = 1$ . The third invariant formula is

$\begin{multline*} 127806\cdot p\left(1, 5\right) - 380904\cdot p\left(61, 5\right)\\+419076\cdot p\left(121, 5\right)-206664\cdot p\left(181, 5\right) +40686\cdot p\left(241, 5\right) = 1080000. \end{multline*}$

Using formulas from (3.7), we find that: $p\left(1, 5\right) = 0$ , $p\left(61, 5\right) = 5608$ , $p\left(121, 5\right) = 80631$ , $p\left(181, 5\right) = 393369$ and $p\left(241, 5\right) = 1220122$ , and so by checking we are assured of the accuracy.

Remark 2. Obviously, $p\left(n, k\right)$ define values only for $n \ge k$ . The invariants determine very precisely that values for $n < k$ should be taken as zero.

4. Conclusions

In this paper, authors have demonstrated a new approach to partitions class invariants, as a way of proving the relevance and accuracy of all formulas given in ^[1,2]. Also, it I can be considered to be another way to obtain some of the formulas in ^[2]. The quasi polynomials $p\left(n, k\right)$ needed to calculate the number of partitions of a number $n$ to exactly $k$ parts consists of at most $LCM\left(1, 2, \dots, k\right)$ different polynomials. The invariants claim that the more different polynomials in one quasi polynomial, the more invariable sizes connect them.

Acknowledgments

The author thank to The Academy of Applied Technical Studies Belgrade for partial funding of this paper.

Conflict of interest

Authors declare no conflicts of interest in this paper.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Lloyd C, Smith J, Weinger K (2005) Stress and Diabetes: A Review of the Links. Diabetes Spectrum 18: 121-127. https://doi.org/10.2337/diaspect.18.2.121
[2]	Manni L, Fausto VD, Fiore M, et al. (2008) Repeated Restraint and Nerve Growth Factor Administration in Male and Female Mice: Effect on Sympathetic and Cardiovascular Mediators of the Stress Response. Curr Neurovasc Res 5: 1-12. https://doi.org/10.2174/156720208783565654
[3]	Wirtz PH, Redwine LS, Baertschi C, et al. (2008) Coagulation activity before and after acute psychosocial stress increases with age. Psychosom Med 70: 476-481. https://doi.org/10.1097/PSY.0b013e31816e03a5
[4]	Quick SK, Shields PG, Nie J, et al. (2008) Effect modification by catalase genotype suggests a role for oxidative stress in the association of hormone replacement therapy with postmenopausal breast cancer risk. Cancer Epidemiol Biomarkers Prev 17: 1082-1087. https://doi.org/10.1158/1055-9965.EPI-07-2755
[5]	Chaplin TM, Hong K, Bergquist K, et al. (2008) Gender differences in response to emotional stress: an assessment across subjective, behavioral, and physiological domains and relations to alcohol craving. Alcohol Clin Exp Res 32: 1242-1250. https://doi.org/10.1111/j.1530-0277.2008.00679.x
[6]	Harris RBS, Mitchell TD, Simpson J, et al. (2002) Weight loss in rats exposed to repeated acute restraint stress is independent of energy or leptin status. Am J Physiol Regulatory Integrative Comp Physiol 282: R77-R88. https://doi.org/10.1152/ajpregu.2002.282.1.R77
[7]	Hamer M, Stamatakis E (2008) Inflammation as an intermediate pathway in the association between psychosocial stress and obesity. Physiol Behav 94: 536-539. https://doi.org/10.1016/j.physbeh.2008.03.010
[8]	Vallès A, Martí O, García A, et al. (2000) Single exposure to stressors causes long-lasting, stress-dependent reduction of food intake in rats. Am J Physiol Regulatory Integrative Comp Physiol 279: R1138-R1144. https://doi.org/10.1152/ajpregu.2000.279.3.R1138
[9]	Charrier C, Chigr F, Tardivel C, et al. (2006) BDNF regulation in the rat dorsal vagal complex during stress-induced anorexia. Brain Res 1107: 52-57. https://doi.org/10.1016/j.brainres.2006.05.099
[10]	Adam TC, Epel ES (2007) Stress, eating and the reward system. Physiol Behav 91: 449-458. https://doi.org/10.1016/j.physbeh.2007.04.011
[11]	Laurent L, Jean A, Manrique C (2013) Anorexia and drugs of abuse abnormally suppress appetite, the result of a shared molecular signal foul-up. Animal Models of Eating Disorders . Totowa: Humana Press 319-331. https://doi.org/10.1007/978-1-62703-104-2_19
[12]	Gu H, Tang C, Yang Y (2012) Psychological stress, immune response, and atherosclerosis. Atheroscler 223: 69-77. https://doi.org/10.1016/j.atherosclerosis.2012.01.021
[13]	Paskitti ME, McCreary BJ, Herman JP (2000) Stress regulation of adrenocorticosteroid receptor gene transcription and mRNA expression in rat hippocampus: time-course analysis. Mol Brain Res 80: 142-152. https://doi.org/10.1016/S0169-328X(00)00121-2
[14]	Johnson JD, Barnard DF, Kulp AC, et al. (2019) Neuroendocrine regulation of brain cytokines after psychological stress. J Endocr Soc 3: 1302-1320. https://doi.org/10.1210/js.2019-00053
[15]	Chigr F, Rachidi F, Tardivel C, et al. (2014) Modulation of orexigenic and anorexigenic peptides gene expression in the rat DVC and hypothalamus by acute immobilization stress. Front Cell Neurosci 8: 198. https://doi.org/10.3389/fncel.2014.00198
[16]	Hayashi T (2014) Conversion of psychological stress into cellular stress response: Roles of the sigma-1 receptor in the process. Psychiatry Clin Neurosci 69: 179-191. https://doi.org/10.1111/pcn.12262
[17]	Chaudhari N, Talwar P, Parimisetty A, et al. (2014) A molecular web: endoplasmic reticulum stress, inflammation, and oxidative stress. Front Cell Neurosci 8: 213. https://doi.org/10.3389/fncel.2014.00213
[18]	Santos CXC, Tanaka LY, Wosniak JJ, et al. (2009) Mechanisms and implications of reactive oxygen species generation during the unfolded protein response: roles of endoplasmic reticulum oxidoreductases, mitochondrial electron transport, and NADPH oxidase. Antioxid Redox Signal 11: 2409-2427. https://doi.org/10.1089/ars.2009.2625
[19]	Bernasconi R, Molinari M (2011) ERAD and ERAD tuning: disposal of cargo and of ERAD regulators from the mammalian ER. Curr Opin Cell Biol 23: 176-183. https://doi.org/10.1016/j.ceb.2010.10.002
[20]	Chaudhuri O, Koshy ST, Da Cunha CB, et al. (2014) Extracellular matrix stiffness and composition jointly regulate the induction of malignant phenotypes in mammary epithelium. Nat Mater 13: 970-8. https://doi.org/10.1038/NMAT4009
[21]	Ellman GL, Courtney KD, Andres JV, et al. (1961) A new and rapid colorimetric determination of acetylcholinesterase activity. Biochem Pharmacol 7: 88-95. https://doi.org/10.1016/0006-2952(61)90145-9
[22]	Buege JA, Aust SD (1978) Microsomal lipid peroxidation. Methods Enzymol 105: 302-310. https://doi.org/10.1016/S0076-6879(78)52032-6
[23]	Aebi H (1974) Catalase. Methods Enzym Anal 2: 673-684. https://doi.org/10.1016/B978-0-12-091302-2.50032-3
[24]	Flohe L, Gunzler WA (1984) Assays of glutathione peroxidase. Methods Enzymol 105: 114-121. https://doi.org/10.1016/S0076-6879(84)05015-1
[25]	Habig WH, Pabst MJ, Jakoby WB (1974) Glutathione-S-transferase the first step in mercapturic acid formation. J Biol Chem 249: 7130-9.
[26]	Asada K, Takahashi M, Nagate M (1974) Assay and inhibitors of spinach superoxide dismutase. Agric Biol Chem 38: 471-473. https://doiorg/10.1080/00021369.1974.10861178
[27]	Lowry OH, Rosebrough NJ, Farr AL, et al. (1951) Protein measurement with the Folin phenol reagent. J Biol Chem 193: 265-275.
[28]	Kovács LÁ, Schiessl JA, Nafz AE, et al. (2018) Both basal and acute restraint stress-induced c-Fos expression is influenced by age in the extended amygdala and brainstem stress centers in male rats. Front Aging Neurosci 10: 248. https://doi.org/10.3389/fnagi.2018.00248
[29]	Bland ST, Schmid MJ, Der-Avakian A, et al. (2005) Expression of c-fos and BDNF mRNA in subregions of the prefrontal cortex of male and female rats after acute uncontrollable stress. Brain Res 1051: 90-99. https://doi.org/10.1016/j.brainres.2005.05.065
[30]	Migdal C, Serres M (2011) Espèces réactives de l'oxygène et stress oxydant. Med Sci 27: 405-412. https://doi.org/10.1051/medsci/2011274017
[31]	Cherian DA, Peter T, Narayanan A, et al. (2019) Malondialdehyde as a marker of oxidative stress in periodontitis patients. J Pharm Bioallied Sci 11: S297-S300. https://doi.org/10.4103/JPBS.JPBS_17_19
[32]	Tirani MM, Haghjou MM (2019) Reactive oxygen species (ROS), total antioxidant capacity (AOC) and malondialdehyde (MDA) make a triangle in evaluation of zinc stress extension. J Anim Plant Sci 29: 1100-1111.
[33]	Čolović M, Krstić D, Petrović S, et al. (2010) Toxic effects of diazinon and its photodegradation products. Toxicol Lett 193: 9-18. https://doi.org/10.1016/j.toxlet.2009.11.022
[34]	Akhgari M, Abdollahi M, Kebryaeezadeh A, et al. (2003) Biochemical evidence for free radicalinduced lipid peroxidation as a mechanism for subchronic toxicity of malathion in blood and liver of rats. Hum Exp Toxicol 22: 205-11. https://doi.org/10.1191/0960327103ht346oa
[35]	Abdollahi M, Ranjbar A, Shadnia S, et al. (2004) Pesticides and oxidative stress: a review. Med Sci Monit 10: 141-7.
[36]	Dal-Zotto S, Martí O, Delgado R, et al. (2004) Potentiation of glucocorticoid release does not modify the long-term effects of a single exposure to immobilization stress. Psychopharmacology 177: 230-237. https://doi.org/10.1007/s00213-004-1939-y
[37]	Akkerman S, Blokland A, Reneerkens O, et al. (2012) Object recognition testing: Methodological considerations on exploration and discrimination measures. Behav Brain Res 232: 335-347. https://doi.org/10.1016/j.bbr.2012.03.022
[38]	Parent MB, Baxter MG (2004) Septohippocampal Acetylcholine: Involved in but not Necessary for Learning and Memory?. Learn Mem 11: 9-20. https://doi.org/10.1101/lm.69104
[39]	Micheau J, Marighetto A (2011) Acetylcholine and memory: A long, complex and chaotic but still living relationship. Behav Brain Res 221: 424-429. https://doi.org/10.1016/j.bbr.2010.11.052
[40]	Deschamps R, Moulignier A (2005) La mémoire et ses troubles Memory and related disorders. EMC- Neurol 2: 505-525. https://doi.org/10.1016/j.emcn.2005.07.003
[41]	Hasselmo ME (2006) The role of acetylcholine in learning and memory. Curr Opin Neurobiol 6: 710-715. https://doi.org/10.1016/j.conb.2006.09.002
[42]	Picciotto MR, Higley MJ, Mineur YS (2012) Acetylcholine as a neuromodulator: cholinergic signaling shapes nervous system function and behavior. Neuron 76: 116-129. https://doi.org/10.1016/j.neuron.2012.08.036
[43]	Haam J, Yakel JL (2017) Cholinergic modulation of the hippocampal region and memory function. J Neurochem 142: 111-121. https://doi.org/10.1111/jnc.14052
[44]	Newman EL, Gupta K, Climer JR, et al. (2012) Cholinergic modulation of cognitive processing: insights drawn from computational models. Front Behav Neurosci 6: 24. https://doi.org/10.3389/fnbeh.2012.00024
[45]	Torres LF, Duchen LW (1987) The mutant mdx: inherited myopathy in the mouse: morphological studies of nerves, muscles and end-plates. Brain 110: 269-99. https://doi.org/10.1093/brain/110.2.269
[46]	Hedner M, Larsson M, Arnold N, et al. (2010) Cognitive factors in odor detection, odor discrimination, and odor identification tasks. J Clin Exp Neuropsychol 32: 1062-7. https://doi.org/10.1080/13803391003683070

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