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Maps on the mirror Heisenberg-Virasoro algebra, II 222 The first part of the
work is supported by the NNSF of China (Nos: 12001141,11971134) and NSF of Hei
Longjiang Province (No. JQ2020A002); FCT UIDB/00212/2020 and UIDP/00212/2020.
The second part of this work is supported by the Russian Science Foundation
under grant 22-71-10001.

Xuelian Guo333School of Mathematical Sciences, Harbin Normal University,
150025 Harbin, China<EMAIL_ADDRESS>Ivan Kaygorodov444CMA-UBI,
Universidade da Beira Interior, Covilhã, Portugal; Moscow Center for
Fundamental and Applied Mathematics, Russia; Saint Petersburg University,
Russia<EMAIL_ADDRESS>& Liming Tang555School of Mathematical
Sciences, Harbin Normal University, 150025 Harbin, China;
<EMAIL_ADDRESS>author

> Abstract: This is the second paper in our series of papers dedicated to the
> study of maps on the mirror Heisenberg-Virasoro algebra. The first paper is
> dedicated to the study of unary maps and the present paper is dedicated to
> the study of binary maps. Namely, we describe biderivations and left-
> symmetric algebra structures on the complex mirror Heisenberg-Virasoro
> algebra.
>
> Keywords: Lie algebra; left-symmetric algebra; biderivation
>
> Mathematics Subject Classification 2020: 17B40, 17B65, 17B68

## Introduction

One of the classical pure algebraic problems is the study of linear and
bilinear maps on a certain algebra (see, for example, [18, 24], survey [16],
and references therein). The most famous examples of binary maps on algebras
are biderivations. The study of biderivations on a Lie algebra can be traced
back to the study of commutative maps on associative algebras. In 2009, D.
Benkovič described all biderivations on triangular algebras [3]. Y. Q. Du and
Y. Wang determined biderivations on the generalized matrix algebra in 2013
[11]. Also in 2013, D. Y. Wang, and X. X. Yu characterized biderivations and
linear commuting maps on the Schrödinger-Virasoro algebra [29]. In 2017, X.
Cheng, M. Wang, J. Sun, and H. Zhang described biderivations and linear
commuting maps on the Lie algebra $\mathfrak{gca}$ [10]. In 2017,
biderivations, post-Lie algebra structures, and linear commutative maps on
$W$-algebras were studied by X. M. Tang [25]. In 2018, X. W. Liu, X. Q. Guo,
and K. M. Zhao described biderivations on the block Lie algebra [22]. In the
same year, X. M. Tang studied biderivations and commutative post-Lie algebra
structures on Lie algebras ${\mathcal{W}}(a,b)$ [26]. Later, X. M. Tang and X.
T. Li determined biderivations on the twisted Heisenberg-Virasoro algebra
[27]. Biderivations play an important role in describing Poisson structures on
associative algebras [17]. In 2024, H. J. Liu and Z. X. Chen determined all
biderivations on the centreless mirror Heisenberg-Virasoro algebra through the
action of the biderivations on the basis and then described the biderivations
on the mirror Heisenberg-Virasoro algebra. As an application, the description
of commutative linear maps was given [21]. In the present paper, by using the
isomorphism relationship between the mirror Heisenberg-Virasoro algebra and
${\mathcal{W}}(\frac{1}{2},0),$ all biderivations on the mirror Heisenberg-
Virasoro algebra are determined. Some applications of the biderivations are
given: commutative linear maps and commutative post-Lie algebra structures on
the mirror Heisenberg-Virasoro algebra, and biderivations on the graded mirror
Heisenberg-Virasoro left-symmetric algebra are described.

The left-symmetric algebras or LSAs for short, play an important role in
mathematics and physics. In 1857, A. Cayley first introduced left-symmetric
algebras in the context of the root algebra [7]. With the passage of time, E.
B. Vinberg, J.-L. Koszul, and M. Gerstenhaber again introduced left symmetric
algebras in the 1960s. LSAs also went by many different names, being called
Vinberg algebra, Koszul algebra, or pre-Lie algebra, among others. Later, LSAs
has been widely concerned in mathematics, and its research has made great
progress. It is closely related to vertex algebra, and it is also important in
the affine theory [28, 5, 6]. The left symmetric algebraic structure on a Lie
algebra originates from the theory of affine manifold [4]. In 2011, X. L.
Kong, H. J. Chen, and C. M. Bai determined the left-symmetric algebra
structure on Witt algebra and Virasoro algebra [19]. In 2012, H. J. Chen and
J. B. Li studied left-symmetric algebra structures on $W(2,2)$ [9]. They also
determined the left-symmetric algebra structure on the twisted Heisenberg-
Virasoro algebra in 2014 [8]. In 2022, C. K. Xu determined compatible left-
symmetric algebra structures on the high-rank Witt algebra and Virasoro
algebra (see [30]). In the present paper, we described compatible left-
symmetric algebra structures on the mirror Heisenberg-Virasoro algebra.

In this paper, we study biderivations and compatible left-symmetric algebra
structures on the complex mirror Heisenberg-Virasoro algebra (mHV algebra). It
is a Lie algebra with a basis $\\{d_{m},h_{r},{\bf c,l}\ |\
m\in\mathbb{Z},r\in\frac{1}{2}+\mathbb{Z}\\}$ and the following multiplication
table (as usual, all zero multiplications will be omitted):

$[d_{m},d_{n}]=(m-n)d_{m+n}+\frac{m^{3}-m}{12}\delta_{m+n},_{0}{\bf c},\
[d_{m},h_{r}]=-rh_{m+r},\ [h_{r},h_{s}]=r\delta_{r+s,0}{\bf l},$

where $m,n\in\mathbb{Z}$, $r,s\in\frac{1}{2}+\mathbb{Z}$. The mirror
Heisenberg-Virasoro algebra is a $\mathbb{Z}$-graded algebra. Virasoro algebra
is a subalgebra of mHV algebra. The mirror Heisenberg-Virasoro algebra, whose
structure is similar to that of the twisted Heisenberg-Virasoro algebra, is
the even part of mirror $N=2$ superconformal algebra [2]. As it is well known,
representation theory is essential in the research of Lie algebras. The
Whittaker modules and the tensor products of Whittaker modules on mHV algebra,
which are non-weight modules, are determined. Sufficient and necessary
conditions are given for these non-weight modules on mHV algebra to be
irreducible [13]. Also, the tensor product weight modules on mHV algebra are
studied, and some examples of irreducible weight modules on mHV algebra are
given in [14]. All Harish-Chandra modules on mHV algebra are classified in
[23]. Derivations, $2$-local derivations and $\frac{1}{2}$-derivations on mHV
algebra are classified in [31, 15].

The structure of this paper is as follows. Firstly, by using the known
biderivations on ${\mathcal{W}}(\frac{1}{2},0)$, all biderivations on the
mirror Heisenberg-Virasoro algebra are determined. Secondly, its applications
are given. One application is different from the method proved by M. Liang, J.
Lou, and S. L. Gao [20], the commutative linear map is determined by the
definition of the biderivation. Another application is to prove that post-Lie
algebra structures on the mirror Heisenberg-Virasoro algebra are trivial.
Third, the relationship between the coefficient functions of the graded
conditions required to satisfy the mirror Heisenberg-Virasoro algebra without
a center is determined. These relationships are used to determine concrete
expressions. Finally, the left-symmetric algebra structure on the mirror
Heisenberg-Virasoro algebra is determined. In the end, it is proved that
biderivations on the left-symmetric algebra of the mirror Heisenberg-Virasoro
algebra are trivial.

##  1\. Biderivations on mHV algebra

######  Definition 1.

Let $(\mathfrak{L},[\cdot,\cdot])$ be a Lie algebra and $f$ is a bilinear map
on $\mathfrak{L}$. If $f$ is a derivation of both components, that is, for
every $x\in\mathfrak{L}$, there exist linear maps $\phi_{x},\ \psi_{x}$ of
$\mathfrak{L}$ to itself, such that $\phi_{x}=f(x,\cdot),\
\psi_{x}=f(\cdot,x)$ are derivations on $\mathfrak{L}$, i.e.

$f([x,y],z)$ | $=$ | $[f(x,z),y]+[x,f(y,z)],$
---|---|---
$f(x,[y,z])$ | $=$ | $[f(x,y),z]+[y,f(x,z)].$

Then $f$ is called a biderivation of $\mathfrak{L}$. If $f(x,y)=\lambda[x,y]$
for an element $\lambda\in\mathbb{C}$, then $f$ is an inner biderivation.
Denote by ${\rm B}(\mathfrak{L})$ the set of all biderivations on
$\mathfrak{L}$.

######  Lemma 2.

(see, [27]) Let $\mathfrak{L}$ be a perfect Lie algebra and $f$ is a
biderivation of $\mathfrak{L}$. If $\alpha\in Z(\mathfrak{L})$, then
$f(x,\alpha)=f(\alpha,x)=0$ for all $x\in\mathfrak{L}$.

######  Lemma 3.

(see, [27]) Let $\mathfrak{L}$ be a perfect Lie algebra and
$Z^{\prime}(\mathfrak{L})\subset Z(\mathfrak{L})$, then a linear map
$\pi\colon{\rm B}(\mathfrak{L})\rightarrow{\rm
B}(\mathfrak{L}/Z^{\prime}(\mathfrak{L})),$ defined by

$\pi(f)(x+Z^{\prime}(\mathfrak{L}),y+Z^{\prime}(\mathfrak{L}))=f(x,y)+Z^{\prime}(\mathfrak{L})$

is injective. Moreover, $\pi$ is bijective if and only if a biderivation of
$\mathfrak{L}/Z^{\prime}(\mathfrak{L})$ can be extended to a biderivation of
$\mathfrak{L}$.

Let $\mathfrak{L}=\widetilde{\mathfrak{L}}\oplus Z^{\prime}(\mathfrak{L})$,
then $\widetilde{\mathfrak{L}}\cong\mathfrak{L}/Z^{\prime}(\mathfrak{L})$. As
follows from the two lemmas above, if $\mathfrak{L}$ is perfect, for
describing all biderivations on $\mathfrak{L}$, we need to draw all
biderivations on $\widetilde{\mathfrak{L}}$.

###  1.1. Biderivations on mHV algebra

Let $\mathfrak{D}$ be the complex mirror Heisenberg-Virasoro algebra (to
simplify, we will fix this notation for the rest of the paper). Then
$\mathfrak{D}$ is perfect, $\mathfrak{D}=\widetilde{\mathfrak{D}}\oplus
Z(\mathfrak{D})$, where $Z(\mathfrak{D})=\langle\bf c,\bf l\rangle$,
$\widetilde{\mathfrak{D}}$ is a Lie algebra with a basis
$\\{d_{n},h_{n+\frac{1}{2}}\ |\ n\in\mathbb{Z}\\}$ and the following
multiplication table:

$[d_{m},d_{n}]=(m-n)d_{m+n},\
[d_{m},h_{n+\frac{1}{2}}]=-(n+\frac{1}{2})h_{m+n+\frac{1}{2}}.$

By virtue of [12], $\widetilde{\mathfrak{D}}={\rm
W}^{\prime}(0,0)\cong{\mathcal{W}}(\frac{1}{2},0)$, where
${\mathcal{W}}(\frac{1}{2},0)=\langle d_{n},h_{n}\ |\ n\in\mathbb{Z}\rangle$,
and satisfies

$[d_{m},d_{n}]=(m-n)d_{m+n},\ [d_{m},h_{n}]=-(n+\frac{1}{2})h_{m+n}.$

######  Lemma 4.

(see, [26]) Let $f$ be a biderivation of ${\mathcal{W}}(\frac{1}{2},0)$, then
any $x,y\in{\mathcal{W}}(\frac{1}{2},0)$,
$f(x,y)=\lambda[x,y]+\widetilde{\Upsilon}_{\Omega}(x,y)$, and
$\widetilde{\Upsilon}_{\Omega}$ satisfies

$\widetilde{\Upsilon}_{\Omega}(d_{m},d_{n})$ | $=$ | $\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}h_{m+n+k},$
---|---|---
$\widetilde{\Upsilon}_{\Omega}(d_{m},h_{n})$ | $=$ | $\widetilde{\Upsilon}_{\Omega}(h_{m},d_{n})\ =\ \widetilde{\Upsilon}_{\Omega}(h_{m},h_{n})\ =\ 0,$

where $\lambda\in\mathbb{C}$ and $\Omega=(\mu_{k})_{k\in\mathbb{Z}}$ which
contains only finitely many nonzero entries.

The last observation trivially gives the key statement of the present section.

######  Lemma 5.

Let $\tilde{f}$ be a biderivation of $\widetilde{\mathfrak{D}}$, then we have
$\tilde{f}(x,y)=\lambda[x,y]+\Upsilon_{\Omega}(x,y)$, and $\Upsilon_{\Omega}$
satisfies

$\Upsilon_{\Omega}(d_{m},d_{n})$ | $=$ | $\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}h_{m+n+k+\frac{1}{2}},$
---|---|---
$\Upsilon_{\Omega}(d_{m},h_{n+\frac{1}{2}})$ | $=$ | $\Upsilon_{\Omega}(h_{m+\frac{1}{2}},d_{n})=\Upsilon_{\Omega}(h_{m+\frac{1}{2}},h_{n+\frac{1}{2}})=0,$

where $\lambda\in\mathbb{C}$ and $\Omega=(\mu_{k})_{k\in\mathbb{Z}}$ which
contains only finitely many nonzero entries.

######  Theorem 6.

A bilinear map $f$ is a biderivation of $\mathfrak{D}$ if and only if there
exists $\lambda\in\mathbb{C}$, such that for any $x,y\in\mathfrak{D}$,
$f(x,y)=\lambda[x,y]+\Upsilon_{\Omega}(x,y)$, and $\Upsilon_{\Omega}$
satisfies

$\Upsilon_{\Omega}(d_{m},d_{n})$ | $=$ | $\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}h_{m+n+k+\frac{1}{2}},$
---|---|---
$\Upsilon_{\Omega}(d_{m},h_{n+\frac{1}{2}})$ | $=$ | $\Upsilon_{\Omega}(h_{m+\frac{1}{2}},d_{n})\ =\ \Upsilon_{\Omega}(h_{m+\frac{1}{2}},h_{n+\frac{1}{2}})\ =\ 0,$

where $\Omega=(\mu_{k})_{k\in\mathbb{Z}}$ which contains only finitely many
nonzero entries.

###### Proof.

Adequacy is clearly established, and necessity is proved below. Let $f$ be a
biderivation of $\mathfrak{D}$, we define a linear map $\pi\colon{\rm
B}(\mathfrak{D})\rightarrow{\rm B}(\mathfrak{D}/Z(\mathfrak{D})),$ that
satisfies

$\pi(f)(x+Z(\mathfrak{D}),y+Z(\mathfrak{D}))=f(x,y)+Z(\mathfrak{D}),$

where $Z(\mathfrak{D})=\langle{\bf c,l}\rangle$. Notice that
$\mathfrak{D}/Z(\mathfrak{D})\cong\widetilde{\mathfrak{D}}$, then by virtue of
Lemma 5, put

$f(x,y)=\lambda[x,y]+\Upsilon_{\Omega}(x,y),$

where $\lambda\in\mathbb{C}$ and $x,y\in\langle d_{i},h_{i+\frac{1}{2}}\ |\
i\in\mathbb{Z}\rangle$. According to lemma 2, for $\alpha,\beta\in\langle{\bf
c,l}\rangle,$ we have

$f(x+\alpha,y+\beta)$ | $=$ | $f(x,y)\ =\ \lambda[x,y]+\Upsilon_{\Omega}(x,y)\ =\ $
---|---|---
| $=$ | $\lambda[x+\alpha,y+\beta]+\Upsilon_{\Omega}(x+\alpha,y+\beta).$

Therefore, the conclusion holds. ∎

###  1.2. Applications of biderivations on mHV algebra

######  Definition 7.

A linear map $\phi$ on a Lie algebra $\mathfrak{L}$ is called a commuting map,
if $[\phi(x),x]=0$.

If $\phi$ is a linear commuting map, then $[\phi(x),y]=[x,\phi(y)]$. Put
$f(x,y)=[\phi(x),y]=[x,\phi(y)]$, then $f$ is a biderivation of
$\mathfrak{L}$.

######  Theorem 8.

A linear map $\phi$ on $\mathfrak{D}$ is commutative if and only if there
exists $\lambda\in\mathbb{C}$ and a linear map
$\tau\colon\mathfrak{D}\rightarrow Z(\mathfrak{D})$, such that
$\phi(x)=\lambda x+\tau(x).$

###### Proof.

Sufficiency: $[\phi(x),x]=[\lambda x+\tau(x),x]=0.$ Hence, $\phi$ is a linear
commuting map. Conversely, let $f(x,y)=[\phi(x),y]$, then $f$ is a
biderivation of $\mathfrak{D}$. According to Theorem 6, we have

$f(x,y)=[\phi(x),y]=\lambda[x,y]+\Upsilon_{\Omega}(x,y).$

Since $f(x,x)=0$, then $f(x,y)=-f(y,x)$, i.e.

$\lambda[x,y]+\Upsilon_{\Omega}(x,y)=-\lambda[y,x]-\Upsilon_{\Omega}(y,x).$

Therefore, $\Upsilon_{\Omega}(x,y)=0.$ So $[\phi(x)-\lambda x,y]=0.$ Hence,
$\phi(x)-\lambda x\in Z(\mathfrak{D})$. So there exists
$\tau\colon\mathfrak{D}\rightarrow Z(\mathfrak{D})$, such that
$\phi(x)=\lambda x+\tau(x).$ ∎

######  Definition 9.

Let $(\mathfrak{L},[\cdot,\cdot])$ be a Lie algebra. A commutative post-Lie
algebra structure on $\mathfrak{L}$ is a bilinear product $x\cdot y$ on
$\mathfrak{L}$ satisfying the following identities:

$x\cdot y$ | $=$ | $y\cdot x,$
---|---|---
$[x,y]\cdot z$ | $=$ | $x\cdot(y\cdot z)-y\cdot(x\cdot z),$
$x\cdot[y,z]$ | $=$ | $[x\cdot y,z]+[y,x\cdot z].$

A post-Lie algebra $(\mathfrak{L},[\cdot,\cdot],\cdot)$ is said to be trivial
if $x\cdot y=0$. If a bilinear map
$f\colon\mathfrak{L}\times\mathfrak{L}\rightarrow\mathfrak{L}$ is defined by
$f(x,y)=x\cdot y$, then $f$ is a biderivation of $\mathfrak{L}$.

######  Theorem 10.

Any commutative post-Lie algebra structure on $\mathfrak{D}$ is trivial.

###### Proof.

Suppose that $(\mathfrak{D},[\cdot,\cdot],\cdot)$ is a commutative post-Lie
algebra. By Lemma 6, there exists $\lambda\in\mathbb{C}$, such that

$f(x,y)=x\cdot y=\lambda[x,y]+\Upsilon_{\Omega}(x,y).$

Since $x\cdot y=y\cdot x$, then

$\lambda[x,y]+\Upsilon_{\Omega}(x,y)\ =\ \lambda[y,x]+\Upsilon_{\Omega}(y,x)$
and $2\lambda[x,y]=0.$ Therefore, $\lambda=0$.

If there is $\mu_{k}\in\Omega$ that makes $\mu_{k}\neq 0$, then for
$d_{1},d_{2},d_{3}$,

$[d_{2},d_{1}]\cdot d_{3}$ | $=$ | $d_{2}\cdot(d_{1}\cdot d_{3})-d_{1}\cdot(d_{2}\cdot d_{3})$
---|---|---
| $=$ | $d_{2}\cdot(\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}h_{k+\frac{9}{2}})-d_{1}\cdot(\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}h_{k+\frac{11}{2}})=0,$
$[d_{2},d_{1}]\cdot d_{3}$ | $=$ | $d_{3}\cdot d_{3}=\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}h_{k+\frac{13}{2}}.$

Comparing the obtained results, we have $\Upsilon_{\Omega}=0.$ Hence,
$f(x,y)=x\cdot y=0$. Therefore, the conclusion holds. ∎

##  2\. The left-symmetric algebra structures on mHV algebra

######  Definition 11.

An algebra ${\rm A}$ is called a left-symmetric algebra if it satisfies

$(xy)z-x(yz)=(yx)z-y(xz).$

It is known that the commutator multiplication $[x,y]=xy-yx,$ defined on a
left-symmetric algebra $\rm{A},$ gives a Lie algebra $\mathcal{L}(\rm{A})$,
which is called the sub-adjacent of $\rm{A}$ and $\rm{A}$ is called a
compatible left-symmetric algebra structure on the Lie algebra
$\mathcal{L}(\rm{A})$. A compatible left-symmetric algebra structure on
$\mathfrak{L}$ will be denoted by $\mathcal{A}(\mathfrak{L}).$ A compatible
left-symmetric algebra structure on the Virasoro algebra $\mathcal{V}=\langle
d_{n},{\bf c}\ |\ n\in\mathbb{Z}\rangle$ is said to have the natural grading
condition if nonzero multiplications of basis elements of
$\mathcal{A}(\mathcal{V})$ satisfies

$\displaystyle d_{m}d_{n}=f(m,n)d_{m+n}+\omega(m,n){\bf c}.$ (2.1)

######  Lemma 12.

(see, [19]) Any compatible left-symmetric algebra structure on the Virasoro
algebra $\mathcal{V}$ satisfying (2.1) is given by the multiplication table

$d_{m}d_{n}=-\frac{n(1+\epsilon
n)}{1+\epsilon(m+n)}d_{m+n}+\frac{1}{24}(m^{3}-m+(\epsilon-\epsilon^{-1})m^{2})\delta_{m+n,0}{\bf
c},$

where $\epsilon\neq 0$ and $\epsilon^{-1}\notin\mathbb{Z}.$

The mirror Heisenberg-Virasoro algebra $\mathfrak{D}$ contains twisted
Heisenberg subalgebra and Virasoro subalgebra and is also $\mathbb{Z}$-graded.
Let us remember the grading components of $\mathfrak{D}:$

$\mathfrak{D}_{0}$ | $=$ | $\langle d_{0},h_{\frac{1}{2}},{\bf c}\rangle,$ | $\mathfrak{D}_{-1}$ | $=$ | $\langle d_{-1},h_{-\frac{1}{2}},{\bf l}\rangle,$ | $\mathfrak{D}_{n}$ | $=$ | $\langle d_{n},h_{n+\frac{1}{2}}\ |\ n\in\mathbb{Z}\backslash\\{0,-1\\}\rangle.$
---|---|---|---|---|---|---|---|---

Naturally, we have the natural grading condition (in agreement with the
natural grading condition of the multiplication of $\mathcal{A}(\mathcal{V})$)
if nonzero multiplications of basis elements of $\mathcal{A}(\mathfrak{D})$
satisfies the following relations, which we will denote by ($\diamond$):

$d_{m}d_{n}$ | $=$ | $f(m,n)d_{m+n}+\omega(m,n){\bf c},$
---|---|---
$d_{m}h_{n+\frac{1}{2}}$ | $=$ | $g(m,n)h_{m+n+\frac{1}{2}},$
$h_{m+\frac{1}{2}}d_{n}$ | $=$ | $h(m,n)h_{m+n+\frac{1}{2}},$
$h_{m+\frac{1}{2}}h_{n+\frac{1}{2}}$ | $=$ | $a(m,n)d_{m+n}+b(m,n)h_{m+n+\frac{1}{2}}+\rho(m,n){\bf l}.$

The main aim of the present section is to prove the following theorem.

######  Theorem 13.

Any compatible left-symmetric algebra structure on the mirror Heisenberg-
Virasoro algebra satisfying relation $(\diamond)$ is given by the following
multiplication table:

$d_{m}d_{n}$ | $=$ | $-\frac{n(1+\epsilon n)}{1+\epsilon(m+n)}d_{m+n}+\frac{1}{24}(m^{3}-m+(\epsilon-\epsilon^{-1})m^{2})\delta_{m+n,0}{\bf c},$
---|---|---
$d_{m}h_{n+\frac{1}{2}}$ | $=$ | $-(n+\frac{1}{2})h_{m+n+\frac{1}{2}},$
$h_{m+\frac{1}{2}}h_{n+\frac{1}{2}}$ | $=$ | $\frac{1}{2}(n+\frac{1}{2})\delta_{n+m+1,0}{\bf l},$

where $\epsilon\neq 0$ and $\epsilon^{-1}\notin\mathbb{Z}.$

The algebra, obtained in the previous theorem will be called the graded mirror
Heisenberg-Virasoro left-symmetric algebra $\mathcal{A}$.

###  2.1. Left-symmetric algebra structures on mHV algebra.

######  Lemma 14.

A bilinear product induced from $(\diamond)$ gives a compatible left-symmetric
algebra structure on $\widetilde{\mathfrak{D}}$ if and only if

$f(m,n)-f(n,m)$ | $=$ | $m-n,$
---|---|---
$g(m,n)-h(n,m)$ | $=$ | $-(n+\frac{1}{2}),$
$a(m,n)$ | $=$ | $a(n,m),$
$b(m,n)$ | $=$ | $b(n,m),$
$f(m,k)f(n,m+k)-f(n,k)f(m,n+k)$ | $=$ | $(n-m)f(m+n,k),$
$g(m,k)g(n,m+k)-g(n,k)g(m,n+k)$ | $=$ | $(n-m)g(m+n,k),$
$h(m,k)g(n,m+k)-f(n,k)h(m,n+k)$ | $=$ | $-(m+\frac{1}{2})h(m+n,k),$
$a(m,k)f(n,m+k)-g(n,k)a(m,n+k)$ | $=$ | $-(m+\frac{1}{2})a(m+n,k),$
$b(m,k)g(n,m+k)-g(n,k)b(m,n+k)$ | $=$ | $-(m+\frac{1}{2})b(m+n,k),$
$h(m,k)a(a,m+k)-h(n,k)a(m,n+k)$ | $=$ | $0,$
$h(m,k)b(n,m+k)-h(n,k)b(m,n+k)$ | $=$ | $0,$
$b(m,k)a(n,m+k)-b(m,k)a(m,n+k)$ | $=$ | $0,$
$a(m,k)h(n,m+k)+b(m,k)b(n,m+k)$ | $=$ | $a(n,k)h(m,n+k)+b(n,k)b(m,n+k).$

The present relations will be denoted by $(\star)$.

###### Proof.

  * (i)

On the one hand $[d_{m},d_{n}]=(m-n)d_{m+n},$ and on the other hand

$[d_{m},d_{n}]\ =\ d_{m}d_{n}-d_{n}d_{m}\ =\ f(m,n)d_{m+n}-f(n,m)d_{m+n}.$

Comparing the two equations above, $f(m,n)-f(n,m)=m-n.$

  * (ii)

On the one hand
$[d_{m},h_{n+\frac{1}{2}}]=-(n+\frac{1}{2})h_{m+n+\frac{1}{2}},$ and on the
other hand

$[d_{m},h_{n+\frac{1}{2}}]=g(m,n)h_{m+n+\frac{1}{2}}-h(n,m)h_{m+n+\frac{1}{2}}.$

Comparing the two equations above, $g(m,n)-h(n,m)=-(n+\frac{1}{2}).$

  * (iii)

On the one hand $[h_{m+\frac{1}{2}},h_{n+\frac{1}{2}}]=0,$ and on the other
hand

$[h_{m+\frac{1}{2}},h_{n+\frac{1}{2}}]=a(m,n)d_{m+n}+b(m,n)h_{m+n+\frac{1}{2}}-a(n,m)d_{m+n}-b(n,m)h_{m+n+\frac{1}{2}}.$

Comparing the two equations above, $a(m,n)=a(n,m)$ and $b(m,n)=b(n,m).$

  * (iv)

By the left-symmetric identity $(d_{n},d_{m},d_{k})=(d_{m},d_{n},d_{k}),$ we
have

$f(n,m)f(n+m,k)d_{n+m+k}-f(m,k)f(n,m+k)d_{m+n+k}=$

$=f(m,n)f(n+m,k)d_{n+m+k}-f(n,k)f(m,n+k)d_{m+n+k}.$

Hence, $f(m,k)f(n,m+k)-f(n,k)f(m,n+k)=(n-m)f(m+n,k).$

By the left-symmetric identity
$(d_{n},d_{m},h_{k+\frac{1}{2}})=(d_{m},d_{n},h_{k+\frac{1}{2}})$, we have

$f(n,m)g(n+m,k)h_{m+n+k+\frac{1}{2}}-g(m,k)g(n,m+k)h_{m+n+k+\frac{1}{2}}=$

$=(m,n)g(n+m,k)h_{m+n+k+\frac{1}{2}}-g(n,k)g(m,n+k)h_{m+n+k+\frac{1}{2}}.$

Hence, $g(m,k)g(n,m+k)-g(n,k)g(m,n+k)=(n-m)g(m+n,k).$

By the left-symmetric identity
$(d_{n},h_{m+\frac{1}{2}},d_{k})=(h_{m+\frac{1}{2}},d_{n},d_{k})$, we have

$g(n,m)h(m+n,k)h_{m+n+k+\frac{1}{2}}-h(m,k)g(n,m+k)h_{m+n+k+\frac{1}{2}}=$

$=h(m,n)h(m+n,k)h_{m+n+k+\frac{1}{2}}-f(n,k)h(m,n+k)h_{m+n+k+\frac{1}{2}}.$

Hence, $h(m,k)g(n,m+k)-f(n,k)h(m,n+k)=-(m+\frac{1}{2})h(m+n,k).$

By the left-symmetric identity
$(d_{n},h_{m+\frac{1}{2}},h_{k+\frac{1}{2}})=(h_{m+\frac{1}{2}},d_{n},h_{k+\frac{1}{2}})$,
we have

$g(n,m)\big{(}a(n+m,k)d_{m+n+k}+b(n+m,k)h_{m+n+k+\frac{1}{2}}\big{)}$

$-a(m,k)f(n,m+k)d_{m+n+k}-b(m,k)g(n,m+k)h_{m+n+k+\frac{1}{2}}=$

$=h(m,n)\big{(}a(n+m,k)d_{m+n+k}+b(n+m,k)h_{m+n+k+\frac{1}{2}}\big{)}$

$-g(n,k)\big{(}a(m,n+k)d_{m+n+k}+b(m,n+k)h_{m+n+k+\frac{1}{2}}\big{)}.$

Hence,

$a(m,k)f(n,m+k)-g(n,k)a(m,n+k)$ | $=$ | $-(m+\frac{1}{2})a(m+n,k),$
---|---|---
$b(m,k)g(n,m+k)-g(n,k)b(m,n+k)$ | $=$ | $-(m+\frac{1}{2})b(m+n,k).$

By the left-symmetric identity
$(h_{n+\frac{1}{2}},h_{m+\frac{1}{2}},d_{k})=(h_{m+\frac{1}{2}},h_{n+\frac{1}{2}},d_{k})$,
we have

$h(m,k)\big{(}a(n,m+k)d_{m+n+k}+b(n,m+k)h_{m+n+k+\frac{1}{2}}\big{)}=$

$h(n,k)\big{(}a(m,n+k)d_{m+n+k}+b(m,n+k)h_{m+n+k+\frac{1}{2}}\big{)}.$

Hence, $h(m,k)a(a,m+k)=h(n,k)a(m,n+k)$ and $h(m,k)b(n,m+k)=h(n,k)b(m,n+k).$

By the left-symmetric identity
$(h_{n+\frac{1}{2}},h_{m+\frac{1}{2}},h_{k+\frac{1}{2}})=(h_{m+\frac{1}{2}},h_{n+\frac{1}{2}},h_{k+\frac{1}{2}})$,
we have

$a(m,k)h(n,m+k)h_{m+n+k+\frac{1}{2}}+b(m,k)\big{(}a(n,m+k)d_{m+n+k}+b(n,m+k)h_{m+n+k+\frac{1}{2}}\big{)}=$

$a(n,k)h(m,n+k)h_{m+n+k+\frac{1}{2}}+b(n,k)\big{(}a(m,n+k)d_{m+n+k}+b(m,n+k)h_{m+n+k+\frac{1}{2}}\big{)}.$

Hence,

$b(m,k)a(n,m+k)-b(m,k)a(m,n+k)$ | $=$ | $0,$
---|---|---
$a(m,k)h(n,m+k)+b(m,k)b(n,m+k)$ | $=$ | $a(n,k)h(m,n+k)+b(n,k)b(m,n+k).$

Therefore, the conclusion holds. ∎

######  Lemma 15.

A multiplication defined by relations $(\diamond)$ gives a compatible left-
symmetric algebra structure on $\mathfrak{D}$ if and only if $(\star)$ and the
following identities hold for all $m,n,k\in\mathbb{Z}$,

$\omega(m,n)-\omega(n,m)$ | $=$ | $\frac{m^{3}-m}{12}\delta_{m+n,0},$
---|---|---
$\rho(m,n)-\rho(n,m)$ | $=$ | $(m+\frac{1}{2})\delta_{m+n+1,0},$
$f(n,k)\omega(m,{n+k})-f(m,k)\omega(n,m+k)$ | $=$ | $(m-n)\omega(m+n,k),$
$g(m,k)\rho(n,m+k)-(n+\frac{1}{2})\rho(m+n,k)$ | $=$ | $0,$
$a(m,k)\omega(n,{m+k})$ | $=$ | $0,$
$h(n,k)\rho(m,n+k)$ | $=$ | $h(m,k)\rho(n,m+k),$
$b(m,k)\rho(n,m+k)$ | $=$ | $b(n,k)\rho(m,n+k).$

The present relations will be denoted by $(\ast)$.

###### Proof.

  1. (i)

On the one hand,
$[d_{m},d_{n}]=(m-n)d_{m+n}+\frac{m^{3}-m}{12}\delta_{m+n},_{0}{\bf c},$ and
on the other hand,

$[d_{m},d_{n}]=d_{m}d_{n}-d_{n}d_{m}=f(m,n)d_{m+n}+\omega(m,n){\bf
c}-f(n,m)d_{m+n}-\omega(n,m){\bf c}.$

Comparing the two above formulas, we obtain
$\omega(m,n)-\omega(n,m)=\frac{m^{3}-m}{12}\delta_{m+n,0}.$

  2. (ii)

On the one hand,
$[h_{m+\frac{1}{2}},h_{n+\frac{1}{2}}]=(m+\frac{1}{2})\delta_{m+n+1,0}{\bf
l},$ and on the other hand,

$[h_{m+\frac{1}{2}},h_{n+\frac{1}{2}}]=\rho(m,n){\bf l}-\rho(n,m){\bf l}.$

Comparing the two above formulas, we obtain

$\rho(m,n)-\rho(n,m)=(m+\frac{1}{2})\delta_{m+n+1,0}.$

  3. (iii)

By the left-symmetric identity $(d_{m},d_{n},d_{k})=(d_{n},d_{m},d_{k})$, we
have

$f(n,m)\big{(}f(n+m,k)d_{n+m+k}$ | $+$ | $\omega({n+m},k){\bf c}\big{)}-$
---|---|---
| $-$ | $f(m,k)\big{(}f(n,m+k)d_{m+n+k}+\omega(n,{m+k}){\bf c}\big{)}=$
$f(m,n)\big{(}f(n+m,k)d_{n+m+k}$ | $+$ | $\omega({n+m},k){\bf c}\big{)}-$
| $-$ | $f(n,k)\big{(}f(m,n+k)d_{m+n+k}+\omega(m,{n+k}){\bf c}\big{)}.$

Hence,

$f(n,k)\omega(m,{n+k})-f(m,k)\omega(n,{m+k})=(m-n)\omega({n+m},k).$

  4. (iv)

By the left-symmetric identity
$(d_{n},h_{m+\frac{1}{2}},h_{k+\frac{1}{2}})=(h_{m+\frac{1}{2}},d_{n},h_{k+\frac{1}{2}})$,
we have

$g(n,m)\big{(}a(n+m,k)d_{m+n+k}+b(n+m,k)h_{m+n+k+\frac{1}{2}}+\rho(m+n,k){\bf
l}\big{)}-$

$-a(m,k)\big{(}f(n,m+k)d_{m+n+k}+\omega(n,{m+k}){\bf
c}\big{)}-b(m,k)g(n,m+k)h_{m+n+k+\frac{1}{2}}\ =$

$h(m,n)\big{(}a(n+m,k)d_{m+n+k}+b(n+m,k)h_{m+n+k+\frac{1}{2}}+\rho(m+n,k){\bf
l}\big{)}-$

$-g(n,k)\big{(}a(m,n+k)d_{m+n+k}+b(m,n+k)h_{m+n+k+\frac{1}{2}}+\rho(m,n+k){\bf
l}\big{)}.$

Therefore,

$g(n,k)\rho(m,n+k)=(m+\frac{1}{2})\rho(m+n,k),$ and $a(m,k)\omega(n,{m+k})=0.$

  5. (v)

By the left-symmetric identity
$(h_{n+\frac{1}{2}},h_{m+\frac{1}{2}},d_{k})=(h_{m+\frac{1}{2}},h_{n+\frac{1}{2}},d_{k})$,
we obtain

$h(m,k)\big{(}a(n,m+k)d_{m+n+k}+b(n,m+k)h_{m+n+k+\frac{1}{2}}+\rho(n,m+k){\bf
l}\big{)}\ =$

$=\
h(n,k)\big{(}a(m,n+k)d_{m+n+k}+b(m,n+k)h_{m+n+k+\frac{1}{2}}+\rho(m,n+k){\bf
l}\big{)}.$

Therefore,

$h(n,k)\rho(m,n+k)=h(m,k)\rho(n,m+k).$

  6. (vi)

By the left-symmetric identity
$(h_{n+\frac{1}{2}},h_{m+\frac{1}{2}},h_{k+\frac{1}{2}})=(h_{m+\frac{1}{2}},h_{n+\frac{1}{2}},h_{k+\frac{1}{2}})$,
we have

$b(m,k)\big{(}a(n,m+k)d_{m+n+k}+b(n,m+k)h_{m+n+k+\frac{1}{2}}+\rho(n,m+k){\bf
l}\big{)}\ =$

$=\
b(n,k)\big{(}a(m,n+k)d_{m+n+k}+b(m,n+k)h_{m+n+k+\frac{1}{2}}+\rho(m,n+k){\bf
l}\big{)}.$

Hence, $b(n,k)\rho(m,n+k)=b(m,k)\rho(n,m+k).$

Therefore, the conclusion holds. ∎

By Theorem 12, in order to obtain a compatible left-symmetric algebra
structure on $\mathfrak{D}$ with the grading condition ($\diamond$), one can
suppose

$\displaystyle f(m,n)=-\frac{n(1+\epsilon n)}{1+\epsilon(m+n)}.$ (2.2)

Furthermore, the following theorem determines all compatible left-symmetric
algebraic structures on $\mathfrak{D}$ that satisfies $f(m,n)$ defined by
formula (2.2).

######  Theorem 16.

Any compatible left-symmetric algebraic structure on the mirror Heisenberg-
Virasoro algebra satisfying the relation $(\diamond)$ is given by the
following functions:

$f(m,n)$ | $=$ | $-\frac{n(1+\epsilon n)}{1+\epsilon(m+n)},$ | $a(m,n)$ | $=$ | $0,$
---|---|---|---|---|---
$g(m,n)$ | $=$ | $-(n+\frac{1}{2}),$ | $b(m,n)$ | $=$ | $0,$
$\rho(n,m)$ | $=$ | $\frac{1}{2}(n+\frac{1}{2})\delta_{n+m+1,0},$ | $h(m,n)$ | $=$ | $0,$
$\omega(m,n)$ | $=$ | $\frac{1}{24}(m^{3}-m+(\epsilon-\epsilon^{-1})m^{2})\delta_{m+n,0},$

where $\epsilon\neq 0$ and $\epsilon^{-1}\notin\mathbb{Z}.$

###### Proof.

For any $m,n\in\mathbb{Z}$, let

$G(m,n)=\frac{1+\epsilon(m+n)}{1+\epsilon n}g(m,n),\quad
H(n,m)=\frac{1+\epsilon(m+n)}{1+\epsilon m}h(n,m),$

then by ($\star$), we have

$\displaystyle(1+\epsilon n)G(m,n)-(1+\epsilon
m)H(n,m)=-(n+\frac{1}{2})(1+\epsilon(m+n)),$ (2.3) $\displaystyle
G(n,k)G(m,n+k)-G(m,k)G(n,m+k)=(m-n)G(m+n,k),$ (2.4) $\displaystyle
H(n,k)G(m,n+k)+kH(n,m+k)=-(n+\frac{1}{2})H(m+n,k).$ (2.5)

Taking $m=0$ in (2.3) and (2.5), there is

$(1+\epsilon n)G(0,n)-H(n,0)$ | $=$ | $-(n+\frac{1}{2})(1+\epsilon n),$
---|---|---
$H(n,k)G(0,n+k)+kH(n,k)$ | $=$ | $-(n+\frac{1}{2})H(n,k).$

Taking $n=n+k$ in the first formula, then

$G(0,n+k)=\frac{H(n+k,0)}{1+\epsilon(n+k)}-(n+k+\frac{1}{2}).$

Then it is plugged into the second formula, and we obtain

$\frac{H(n,k)H(n+k,0)}{1+\epsilon(n+k)}-(n+k+\frac{1}{2})H(n,k)+kH(n,k)=-(n+\frac{1}{2})H(n,k).$

Therefore, $H(n,k)H(n+k,0)=0$. Taking $k=0$, then $H(n,0)^{2}=0$, i.e.

$H(n,0)=0$ and $G(0,n)=-(n+\frac{1}{2}).$

Taking $k=0,m=-n$ in (2.4), then

$\displaystyle G(n,0)G(-n,n)-G(-n,0)G(n,-n)=-2nG(0,0)=n.$ (2.6)

And by (2.3), we have

$\displaystyle H(n,m)=\frac{1+\epsilon n}{1+\epsilon
m}G(m,n)+\frac{(n+\frac{1}{2})(1+\epsilon(m+n))}{1+\epsilon m}.$ (2.7)

Taking $n=0,m=-k$ in (2.5), then $H(0,k)G(-k,k)+kH(0,0)=-\frac{1}{2}H(-k,k).$
Hence, $H(-k,k)=-2H(0,k)G(-k,k)$. On the one hand, by virtue of (2.7),

$H(-k,k)=\frac{1-\epsilon k}{1+\epsilon
k}G(k,-k)+\frac{-k+\frac{1}{2}}{1+\epsilon k}.$

On the other hand, by (2.6) and (2.7), we have

$G(k,0)G(-k,k)=G(-k,0)G(k,-k)+k,\quad H(0,k)=\frac{G(k,0)}{1+\epsilon
k}+\frac{1}{2}.$

So

$H(-k,k)$ | $=$ | $-2H(0,k)G(-k,k)\ =\ -2\left(\frac{G(k,0)}{1+\epsilon k}+\frac{1}{2}\right)G(-k,k)=$
---|---|---
| $=$ | $\frac{-2G(k,0)G(-k,k)}{1+\epsilon k}-G(-k,k)\ =\ \frac{-2(G(-k,0)G(k,-k)+k)}{1+\epsilon k}-G(-k,k)\ =$
| $=$ | $\frac{-2G(-k,0)G(k,-k)}{1+\epsilon k}-\frac{2k}{1+\epsilon k}-G(-k,k).$

Comparing the above two formulas, we obtain

$(1-\epsilon k)G(k,-k)+(-k+\frac{1}{2})=-2G(-k,0)G(k,-k)-2k-(1+\epsilon
k)G(-k,k).$

Therefore,

$\displaystyle(1+2g(-k,0))g(k,-k)+g(-k,k)=-(k+\frac{1}{2}).$ (2.8)

According to $G(0,n)=-(n+\frac{1}{2})$, $g(0,n)=-(n+\frac{1}{2})$. Therefore,
put

$g(m,n)=mg^{\prime}(m,n)-(n+\frac{1}{2}).$

Then

$h(n,m)$ | $=$ | $mg^{\prime}(m,n),$ | $g(-k,0)$ | $=$ | $-kg^{\prime}(-k,0)-\frac{1}{2},$
---|---|---|---|---|---
$g(k,-k)$ | $=$ | $k(g^{\prime}(k,-k)+1)-\frac{1}{2},$ | $g(-k,k)$ | $=$ | $-k(g^{\prime}(-k,k)+1)-\frac{1}{2}.$

Let $f(k)=g^{\prime}(k,-k)+1,$ then $g(k,-k)=kf(k)-\frac{1}{2}.$ It is plugged
into (2.8). Then

$kg^{\prime}(-k,0)(-2kf(k)+1)=k(f(-k)-1).$

Let us define two important subsets of $\mathbb{Z}:$

$A$ | $=$ | $\\{m\ |\ f(m)=\frac{1}{2m}\mbox{ \ and \ }f(-m)=1\\},$
---|---|---
$B$ | $=$ | $\\{m\ |\ f(m)=1\mbox{ \ and \ }f(m)\neq-\frac{1}{2m}\\}.$

It is easy to see that $A\cap-A=\varnothing.$ Let
$A=\\{m_{0},m_{1},\ldots,m_{i},\ldots\\}.$ Since $g(0,0)=-\frac{1}{2}$, then
$m_{i}\neq 0.$ Namely, if $m_{i}\in A\cap-A,$ then

$f(m_{i})=\frac{1}{2m_{i}},$ $f(-m_{i})=1,$ $f(-m_{i})=-\frac{1}{2m_{i}},$
$f(m_{i})=1.$

Obviously, this is a contradiction. In this case,

$g(m_{i},-m_{i})$ | $=$ | $m_{i}f(m_{i})-\frac{1}{2}$ | $=$ | $0,$
---|---|---|---|---
$g(-m_{i},m_{i})$ | $=$ | $-m_{i}f(-m_{i})-\frac{1}{2}$ | $=$ | $-(m_{i}+\frac{1}{2}),$

and

$h(m_{i},-m_{i})$ | $=$ | $0,$ |  $h(-m_{i},m_{i})$ | $=$ | $-m_{i}+\frac{1}{2}.$
---|---|---|---|---|---

Inside of the proof of our Theorem, we highlight two following lemmas, which
gives the characterization of sets $A$ and $B$. Namely, we have to prove that
$A=\varnothing$ and $B=\mathbb{Z}.$

######  Lemma 17.

$A=\varnothing.$

###### Proof.

Set $k\in\mathbb{Z}\backslash\\{-A\cup B\\}$, there is

$h(0,k)$ | $=$ | $\frac{k(f(k)-1)}{2kf(-k)+1},$ | $h(-k,k)$ | $=$ | $k(f(k)-1),$
---|---|---|---|---|---
$h(k,-k)$ | $=$ | $-k(f(-k)-1),$ | $h(k,0)$ | $=$ | $0,$

and

$g^{\prime}(k,0)$ | $=$ | $\frac{f(k)-1}{2kf(-k)+1},$ | $g^{\prime}(-k,k)$ | $=$ | $f(-k)-1,$ | $g^{\prime}(k,-k)$ | $=$ | $f(k)-1.$
---|---|---|---|---|---|---|---|---

Therefore, by the first formula, and then set

$g^{\prime}(m,n)=\frac{g^{\prime\prime}(m,n)(f(m)-1)}{2mf(-m)+1},\
m\in\mathbb{Z}\backslash\\{-A\cup B\\},$

in this case,

$g^{\prime\prime}(m,0)$ | $=$ | $1,$ | $g^{\prime}(k,-k)$ | $=$ | $\frac{g^{\prime\prime}(k,-k)(f(k)-1)}{2kf(-k)+1}.$
---|---|---|---|---|---

The last gives,

$g^{\prime\prime}(k,-k)=2kf(-k)+1,\ k\in\mathbb{Z}\backslash\\{-A\cup B\\}.$

Then by $g^{\prime\prime}(m,0)=1$, set
$g^{\prime\prime}(m,n)=ng^{\prime\prime\prime}(m,n)+1$. Therefore,

$g^{\prime\prime}(k,-k)=-kg^{\prime\prime\prime}(k,-k)+1.$

So $g^{\prime\prime\prime}(k,-k)=-2f(-k),$ and
$g^{\prime\prime\prime}(m_{i},-m_{i})=-2f(-m_{i})=-2.$ Hence, for
$m\in\mathbb{Z}$ and $n\in\mathbb{Z}\backslash\\{-A\cup B\\},$ we have

$h(m,n)=\frac{n(f(n)-1)(mg^{\prime\prime\prime}(n,m)+1)}{2nf(-n)+1}.$

Therefore,

$g(n,m)=\frac{n(f(n)-1)(mg^{\prime\prime\prime}(n,m)+1)}{2nf(-n)+1}-(m+\frac{1}{2}).$

If $mg^{\prime\prime\prime}(n,m)+1\equiv 0$, then
$g^{\prime\prime\prime}(n,m)=-\frac{1}{m}$. Therefore,
$g^{\prime\prime\prime}(m_{i},-m_{i})=\frac{1}{m}_{i}$, which is contradict to
$g^{\prime\prime\prime}(m_{i},-m_{i})=-2f(-m_{i})=-2$, then there are only two
cases:

  1. I.

“$mg^{\prime\prime\prime}(n,m)+1\equiv 0$” and $A=\varnothing$. Then
$h(m,n)=0,~{}g(m,n)=-(n+\frac{1}{2}).$

  2. II.

“$mg^{\prime\prime\prime}(n,m)+1$” is not constant to 0.

If $A\neq\varnothing$ , then at this point,
$g(m,n)=mg^{\prime}(m,n)-(n+\frac{1}{2}),m,n\in\mathbb{Z}$. Taking $m=0$ in
the fourth formula of ($\ast$), then there is

$(n+k+1)\rho(n,k)=0.$

When $n+k+1\neq 0$, $\rho(n,k)=0$. Let $\rho(n,k)=\theta(n)\delta_{n+k+1,0}$
and put it into the fourth formula of ($\ast$):

$g(m,k)\theta(n)\delta_{n+m+k+1,0}-(n+\frac{1}{2})\theta(m+n)\delta_{n+m+k+1,0}=0.$

When $m\in\mathbb{Z}\backslash\\{-A\cup B\\}$, according to the previous
proof,

$g(m,k)=\frac{m(f(m)-1)(kg^{\prime\prime\prime}(m,k)+1)}{2mf(-m)+1}-(k+\frac{1}{2}),$

then the above equation becomes

$\Big{(}\big{(}\frac{m(f(m)-1)(kg^{\prime\prime\prime}(m,k)+1)}{2mf(-m)+1}-(k+\frac{1}{2})\big{)}\theta(n)-(n+\frac{1}{2})\theta(m+n)\Big{)}\delta_{n+m+k+1,0}=0.$

Taking $m+n+k+1=0,$ where $m\in\mathbb{Z}\backslash\\{-A\cup B\\}$, then the
above equation becomes

$\big{(}\frac{m(f(m)-1)(kg^{\prime\prime\prime}(m,k)+1)}{2mf(-m)+1}-(k+\frac{1}{2})\big{)}\theta(n)=(n+\frac{1}{2})\theta(m+n).$

Making $n=0$, by $m+k+1=0,$ $k=-1-m$, where $m\in\mathbb{Z}\backslash\\{-A\cup
B\\}$, then the above equation becomes

$\frac{m(f(m)-1)}{2mf(-m)+1}\big{(}(-1-m)g^{\prime\prime\prime}(m,-1-m)+1\big{)}\theta(0)-\frac{1}{2}\theta(m)=-(m+\frac{1}{2})\theta(0).$

Therefore, when $m=m_{i}$, then the above equation becomes

$\displaystyle\frac{m_{i}(f(m_{i})-1)}{2m_{i}f(-m_{i})+1}\big{(}(-1-m_{i})g^{\prime\prime\prime}(m_{i},-1-m_{i})+1\big{)}\theta(0)-\frac{1}{2}\theta(m_{i})=-(m_{i}+\frac{1}{2})\theta(0).$
(2.9)

In the sixth formula of ($\ast$), set $k\in\mathbb{Z}\backslash\\{-A\cup
B\\}$, and from the previous proof, the sixth formula changes to:

$\frac{k(f(k)-1)(ng^{\prime\prime\prime}(k,n)+1)}{2kf(-k)+1}\theta(m)\delta_{m+n+k+1,0}=\frac{k(f(k)-1)(mg^{\prime\prime\prime}(k,m)+1)}{2kf(-k)+1}\theta(n)\delta_{m+n+k+1,0}.$

Taking $m+n+k+1=0$, then the above equation becomes

$(ng^{\prime\prime\prime}(k,n)+1)\theta(m)=(mg^{\prime\prime\prime}(k,m)+1)\theta(n).$

Let $n=0$. Then $k=-1-m$. Since $k\in\mathbb{Z}\backslash\\{-A\cup B\\}$, then

$\theta(m)=(mg^{\prime\prime\prime}(k,m)+1)\theta(0).$

Therefore, when $k=m_{i}$,

$\theta(-1-m_{i})=((-1-m_{i})g^{\prime\prime\prime}(m_{i},-1-m_{i})+1)\theta(0).$

Therefore, the equation (2.9) becomes

$\displaystyle\frac{m_{i}(f(m_{i})-1)}{2m_{i}f(-m_{i})+1}\theta(-1-m_{i})-\frac{1}{2}\theta(m_{i})=-(m_{i}+\frac{1}{2})\theta(0).$
(2.10)

Place $\rho(n,k)=\theta(n)\delta_{n+k+1,0}$ into the second formula of
($\ast$). Then there is

$\theta(m)\delta_{m+n+1,0}-\theta(n)\delta_{m+n+1,0}=(m+\frac{1}{2})\delta_{m+n+1,0},$

taking $m+n+1=0$, then $n=-1-m$, so the above equation becomes

$\displaystyle\theta(-1-m)-\theta(m)=-\Big{(}m+\frac{1}{2}\Big{)},\
m\in\mathbb{Z}.$ (2.11)

In particular, taking $m=m_{i}$ and $m=0$, then

$\displaystyle\theta(-1-m_{i})-\theta(m_{i})=-\Big{(}m_{i}+\frac{1}{2}\Big{)},$
(2.12)

and $\theta(-1)=\theta(0)-\frac{1}{2}.$

From the equation (2.10) and (2.12), there is

$\displaystyle\Big{(}\frac{m_{i}(f(m_{i})-1)}{2m_{i}f(-m_{i})+1}-\frac{1}{2}\Big{)}\theta(-1-m_{i})=\Big{(}\frac{1}{2}-\theta(0)\Big{)}\Big{(}m_{i}+\frac{1}{2}\Big{)}.$
(2.13)

By $f(m_{i})=\frac{1}{2m_{i}}$ $f(-m_{i})=1$, then

$\displaystyle\theta(-1-m_{i})=-{m_{i}^{-1}}{\Big{(}m_{i}+\frac{1}{2}\Big{)}^{2}}\Big{(}\frac{1}{2}-\theta(0)\Big{)}.$
(2.14)

From the fourth formula of ($\ast$), there is

$g(m,k)\theta(n)\delta_{m+n+k+1,0}-(n+\frac{1}{2})\theta(m+n)\delta_{m+n+k+1,0}=0.$

Taking $m+n+k+1=0$, then $g(m,k)\theta(n)=(n+\frac{1}{2})\theta(m+n),$ and by
$g(-m_{i},m_{i})=-m_{i}+\frac{1}{2}$, taking $m=-m_{i},k=m_{i},n=-1$, then the
above equation becomes

$g(-m_{i},m_{i})\theta(-1)=-\frac{1}{2}\theta(-1-m_{i}),$

that is

$-(m_{i}+\frac{1}{2})(\theta(0)-\frac{1}{2})=-\frac{1}{2}\theta(-1-m_{i}).$

Therefore,

$\displaystyle\theta(-1-m_{i})=-2\Big{(}m_{i}+\frac{1}{2}\Big{)}\Big{(}\frac{1}{2}-\theta(0)\Big{)}.$
(2.15)

If $\theta(0)=\frac{1}{2}$, then by (2.11), we have $\theta(-1)=0$. On the
other hand, in the fourth formula of ($\ast$), for $m=1,$ $k=1,n=-1$, we have
$\theta(0)=-2g(1,-1)\theta(-1)=0.$ The last gives a contradiction and
$\theta(0)\neq\frac{1}{2}.$ Then comparing the equation (2.14) with (2.15), we
have $m_{i}=\frac{1}{2}.$ But on the other hand, $m_{i}\in\mathbb{Z}.$ So
$A=\varnothing$.

∎

######  Lemma 18.

$B=\mathbb{Z}.$

###### Proof.

If $\mathbb{Z}\neq B$ and $A=\varnothing$, then for $m\in\mathbb{Z}\backslash
B$, the formula (2.10), (2.12) and (2.13) hold. For formula (2.13), there is

$\Big{(}\frac{m(f(m)-1)}{2mf(-m)+1}-\frac{1}{2}\Big{)}\theta(-1-m)=\Big{(}\frac{1}{2}-\theta(0)\Big{)}\Big{(}m+\frac{1}{2}\Big{)}.$

Then

$\displaystyle\theta(-1-m)=\frac{(m+\frac{1}{2})(2mf(-m)+1)}{m(f(m)-f(-m))-(m+\frac{1}{2})}\Big{(}\frac{1}{2}-\theta(0)\Big{)},\
\forall~{}m\in\mathbb{Z}\backslash B.$ (2.16)

By the fourth formula of ($\ast$), there is

$g(m,k)\theta(n)\delta_{m+n+k+1,0}-\Big{(}n+\frac{1}{2}\Big{)}\theta(m+n)\delta_{m+n+k+1,0}=0.$

Taking $m+n+k+1=0$, and taking $m=-m,k=m,n=-1$, then

$g(-m,m)\theta(-1)=-\frac{1}{2}\theta(-1-m).$

By $g(-m,m)=-mf(-m)-\frac{1}{2}$, then the above equation becomes

$\displaystyle\theta(-1-m)=-(2mf(-m)+1)\Big{(}\frac{1}{2}-\theta(0)\Big{)},\
\forall~{}m\in\mathbb{Z}\backslash B.$ (2.17)

Comparing the equation (2.16) and (2.17), there is

${m+\frac{1}{2}}{}=-(m(f(m)-f(-m))-(m+\frac{1}{2})),\
\forall~{}m\in\mathbb{Z}\backslash B.$

Obviously, if $m\in\mathbb{Z}\backslash B,$ $f(m)\neq 1$, there must be
$f(m)=f(-m)$. It is also easy to see that for all $m\in\mathbb{Z},$ we have
$f(m)=f(-m)$. Then

$\theta(-1-m)=-(2mf(m)+1)\Big{(}\frac{1}{2}-\theta(0)\Big{)},\
\forall~{}m\in\mathbb{Z}\backslash B.$

Similarly to the previous discussion of $\theta(0)$, we have that
$\theta(0)\neq\frac{1}{2}$. Then

$\theta(-1-m)=(2mf(m)+1)\theta(-1),\ \forall~{}m\in\mathbb{Z}\backslash B.$

At this time, for $n\in\mathbb{Z}\backslash B,$ we have

$h(-n,n)$ | $=$ | $n(f(n)-1),$ | $g(n,-n)$ | $=$ | $nf(n)-\frac{1}{2},$ | $g(n,0)$ | $=$ | $-\frac{n+\frac{1}{2}}{2nf(n)+1}.$
---|---|---|---|---|---|---|---|---

In the seventh formula of ($\star$), taking $m=n,$ $k=-n,$ $n=n,$
$n\in\mathbb{Z}\backslash B$, then

$h(n,-n)g(n,0)-f(n,-n)h(n,0)=-\Big{(}n+\frac{1}{2}\Big{)}h(2n,-n).$

Therefore,

$h(2n,-n)$ | $=$ | $-\frac{n(f(n)-1)}{2nf(n)+1},$
---|---|---
$g(-n,2n)$ | $=$ | $-\frac{n(f(n)-1)}{2nf(n)+1}-(2n+\frac{1}{2}).$

In the seventh formula of ($\star$), taking
$m=2n,k=-n,n=-n,n\in\mathbb{Z}\backslash B$, then

$h(2n,-n)g(-n,n)-f(-n,-n)h(2n,-2n)=-(2n+\frac{1}{2})h(n,-n).$

By calculating,

$-\frac{1-\epsilon n}{1-2\epsilon n}h(2n,-2n)=2n(f(n)-1).$

If $2n\in B$, then $h(2n,-2n)=0$. While $f(n)\neq 1$, so the two sides of the
above formula are not equal. Therefore, $2n\in\mathbb{Z}\backslash B$. Then
the above formula becomes

$\displaystyle\frac{(f(2n)-1)(1-\epsilon n)}{1-2\epsilon n}=f(n)-1.$ (2.18)

From this formula, $f(n)\neq 0$. In the fourth formula of ($\ast$), taking
$m=-n,$ $k=2n$ and $n=-1-n,$ then

$g(-n,2n)\theta(-1-n)=-\Big{(}n+\frac{1}{2}\Big{)}\theta(-1-2n).$

By calculating,

$\displaystyle\text{the left end of
equation}=\Big{(}n+\frac{1}{2}\Big{)}(4nf(n)+1),$ $\displaystyle\text{the
right end of equation}=\Big{(}n+\frac{1}{2}\Big{)}(4nf(2n)+1).$

Therefore, $f(n)=f(2n).$ And by $\epsilon\neq 0$, then the two sides of
equation (2.18) are not equal. Therefore, for $m\in\mathbb{Z},f(m)$ can only
be equal to 1.

∎

Continuation of the proof of Theorem 16. Now we have $A=\varnothing$ and
$B=\mathbb{Z}.$ At this point, for $k\in\mathbb{Z}$, there is

$g(-k,k)=-(k+\frac{1}{2})$ and $h(-k,k)=0.$

In the seventh formula of ($\star$), taking $n=-m-k$, then

$h(m,k)g(-m-k,m+k)-f(-m-k,k)h(m,-m)=-(m+\frac{1}{2})h(-k,k).$

Therefore, for $m,k\in\mathbb{Z}$, there is $h(m,k)=0$ and
$g(k,m)=-(m+\frac{1}{2}).$

Nextly, the expressions for $\rho(m,n)~{}(m,n\in\mathbb{Z})$ are determined.
Taking $m=0$ in the fourth equation of ($\ast$), we have $(n+k+1)\rho(n,k)=0.$
When $n+k+1\neq 0$, $\rho(n,k)=0$. It is plugged into the fifth equation of
($\ast$):

$-\Big{(}k+\frac{1}{2}\Big{)}\theta(n)\delta_{m+n+k+1,0}-\Big{(}n+\frac{1}{2}\Big{)}\theta(m+n)\delta_{m+n+k+1,0}=0.$

Taking $m+n+k+1=0$, we have

$(m+n+\frac{1}{2})\theta(n)=(n+\frac{1}{2})\theta(m+n).$

Let $n=1$, we have

$\theta(m+1)=(\frac{2}{3}m+1)\theta(1)=(\frac{2}{3}(m+1)+\frac{1}{3})\theta(1).$

Taking $m$ into $m-1$, there is

$\theta(m)=\theta((m-1)+1)=\Big{(}\frac{2}{3}((m-1)+1)+\frac{1}{3}\Big{)}\theta(1)=(\frac{2}{3}m+\frac{1}{3})\theta(1).$

It is plugged into the second equation of ($\ast$): taking $m+n+1=0$, we have

$\Big{(}\frac{2}{3}m+\frac{1}{3}\Big{)}\theta(1)-\Big{(}\frac{2}{3}n+\frac{1}{3}\Big{)}\theta(1)=m+\frac{1}{2}.$

Hence, $\theta(1)=\frac{3}{4}.$ Therefore,
$\theta(n)=\frac{1}{2}(n+\frac{1}{2}).$ So

$\rho(n,k)=\frac{1}{2}(n+\frac{1}{2})\delta_{n+k+1,0}.$

At last, the expressions for $a(m,n)$ and $b(m,n)$ are determined. By the
eighth equation of ($\star$), taking $n=0$, there is

$-(m-\frac{1}{2})a(m,k)=-(m+\frac{1}{2})a(m,k).$

Therefore, $a(m,k)=0.$ Again, by the ninth equation of ($\star$), taking
$n=0$, we have

$-mb(m,k)=-(m+\frac{1}{2})b(m,k).$

Therefore, $b(m,k)=0.$

In turn, it is easy to verify that the function of the Theorem 13 satisfies
($\star$) and ($\ast$).

∎

###  2.2. Applications

Let us remember a well-known statement about biderivations of sub-adjacent and
original algebras.

######  Lemma 19.

Let $\mathcal{A}$ be an algebra, and $\mathfrak{L}(\mathcal{A})$ be a sub-
adjacent Lie algebra of $\mathcal{A}$. Then every biderivation of
$\mathcal{A}$ is also a biderivation.

######  Theorem 20.

Any biderivations $f$ of the graded mirror Heisenberg-Virasoro left-symmetric
algebra $\mathcal{A}$ is trivial, i.e. $f=0.$

###### Proof.

First, the mirror Heisenberg-Virasoro algebra $\mathfrak{D}$ is the sub-
adjacent Lie algebra of the left-symmetric algebra $\mathcal{A}$. Let $f$ be a
biderivation of graded mirror Heisenberg-Virasoro left-symmetric algebra
$\mathcal{A}$. Then by Lemma 19, $f$ is also a biderivation of $\mathfrak{D}$.
According to Theorem 6, there exists $\lambda\in\mathbb{C}$, such that
$f(x,y)=\lambda[x,y]+\Upsilon_{\Omega}(x,y)$. Hence,

$f(d_{2}d_{1},d_{3})$ | $=$ | $\lambda[d_{2}d_{1},d_{3}]+\Upsilon_{\Omega}(d_{2}d_{1},d_{3})\ =\ $
---|---|---
| $=$ | $\lambda\left[-\frac{1+\epsilon}{1+3\epsilon}d_{3},d_{3}\right]-\Upsilon_{\Omega}(\frac{1+\epsilon}{1+3\epsilon}d_{3},d_{3})\ =\ -\frac{1+\epsilon}{1+3\epsilon}\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}h_{k+\frac{13}{2}}.$

On the other hand,

$f(d_{2}d_{1},d_{3})$ | $=$ | $d_{2}f(d_{1},d_{3})+f(d_{2},d_{3})d_{1}\ =$
---|---|---
| $=$ | $d_{2}(\lambda[d_{1},d_{3}]+\Upsilon_{\Omega}(d_{1},d_{3}))+\big{(}\lambda[d_{2},d_{3}]+\Upsilon_{\Omega}(d_{2},d_{3})\big{)}d_{1}\ =\ $
| $=$ | $-2\lambda d_{2}d_{4}+d_{2}\Upsilon_{\Omega}(d_{1},d_{3})-\lambda d_{5}d_{1}+\Upsilon_{\Omega}(d_{2},d_{3})d_{1}\ =\ $
| $=$ | $-\lambda(2d_{2}d_{4}+d_{5}d_{1})+\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}d_{2}h_{k+\frac{9}{2}}+\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})\mu_{k}h_{k+\frac{9}{2}}d_{1}\ =\ $
| $=$ | $\lambda\left(\frac{8(1+3\epsilon)}{1+6\epsilon}+\frac{1+\epsilon}{1+6\epsilon}\right)d_{6}-\sum_{k\in\mathbb{Z}}(k+\frac{1}{2})(k+5)\mu_{k}h_{k+\frac{13}{2}}.$

Comparing the above two formulas, we have $\lambda=0$ and $\mu_{k}=0,\ {\rm
i.e.}\ \Upsilon_{\Omega}=0.$ Hence, $f=0.$ ∎

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# Synchronization in a small-world network of non-identical Chialvo neurons

Javier Used<EMAIL_ADDRESS>Nonlinear Dynamics, Chaos and Complex Systems
Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933
Móstoles, Madrid, Spain  Jesús M. Seoane  Nonlinear Dynamics, Chaos and
Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos,
Tulipán s/n, 28933 Móstoles, Madrid, Spain  Irina Bashkirtseva  Institute of
Natural Sciences and Mathematics, Ural Federal University, 620000, Lenina, 51,
Ekaterinburg, Russia  Lev Ryashko  Institute of Natural Sciences and
Mathematics, Ural Federal University, 620000, Lenina, 51, Ekaterinburg, Russia
Miguel A.F. Sanjuán  Nonlinear Dynamics, Chaos and Complex Systems Group,
Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933
Móstoles, Madrid, Spain

###### Abstract

Synchronization dynamics is a phenomenon of great interest in many fields of
science. One of the most important fields is neuron dynamics, as
synchronization in certain regions of the brain is related to some of the most
common mental illnesses. In this work, we study synchronization in a small-
world network of non-identical Chialvo neurons that are electrically coupled.
We introduce a mismatch in one of the model parameters to construct non-
identical neurons. Our study examines the effects of this parameter mismatch,
the noise intensity in the stochastic model, and the coupling strength between
neurons on synchronization and firing frequency. We have identified critical
values of noise intensity, parameter mismatch, and rewiring probability that
facilitate effective synchronization within the network. Furthermore, we
observe that the balance between excitatory and inhibitory connections plays a
crucial role in achieving global synchronization. Our findings offer insights
into the mechanisms driving synchronization dynamics in complex neuron
networks.

Synchronization, Chialvo neuron map model, Noise-induced effects, Neuron
network, Small world.

###### pacs:

05.45.-a, 05.90.+m, 46.40.Ff, 87.19.II

##  I Introduction

The phenomenon of collective synchronization occurs when a system of coupled
oscillators spontaneously locks to a common behavior, independently of their
individual characteristics. This phenomenon has been observed in many
different scientific fields such as biology Peskin (1975); Liu et al. (1997);
Aldridge and Pye (1999), engineering Kourtchatov et al. (1995); York and
Compton (1991), and ecology Buck and Buck (1976); Walker (1969). In addition
to all these examples, the synchronization of brain neurons is acquiring
special interest in neuroscience because it is a fundamental neural mechanism
Engel et al. (2001); Uhlaas (2006). Different types of synchronous behaviors
between neurons have been described, including complete synchronization, phase
synchronization, and lag synchronization Boccaletti et al. (2018). These
behaviors have been related to various brain functions. For instance, the
transition between in-phase and anti-phase synchronization is associated with
locomotor brain rhythms Jia et al. (2018); Song and Xu (2012), while phase
synchronization is related to neural integration and working memory Hu and
Zhou (2000); Batista et al. (2007); Varela et al. (2001). On the other hand,
pathological cases of synchronization may result in serious brain disorders,
such as essential tremor, Parkinson’s disease, and epilepsy Paré et al.
(1990). Beyond synchronization in the brain network, the firing frequency of
neurons in the brain is of particular interest because it relates to neural
coding. Recent studies have shown that the timing of spikes plays a critical
role in encoding, representing, and processing knowledge and events in the
brain Fujii et al. (1996); MacLeod and Laurent (1996). With all these
considerations in mind, various models have been proposed to understand the
synchronization of coupled neurons. Among them are continuous neuron models
based on ordinary differential equations, such as Hodgkin-Huxley Hodgkin and
Huxley (1952a); Hodgkin and Huxley (1952b), FitzHugh-Nagumo FitzHugh (1961),
Morris-Lecar Morris and Lecar (1981), and Hindmarsh-Rose Hindmarsh and Rose
(1982); Shilnikov and Kolomiets (2008), which provide a realistic depiction of
the electrophysiological processes within neurons. Conversely, discrete neuron
models like Rulkov, Izhikevich, and Courbage-Nekorkin-Vdovin Rulkov (2001);
Ibarz et al. (2011); Girardi-Schappo et al. (2013); Kinouchi and Tragtenberg
(1996); Courbage et al. (2007) offer computational ease due to their
simplicity, thereby reducing the computational workload. In our work, we
propose using the stochastic version of the Chialvo map Chialvo (1995) to
understand synchronization in a neuron network. The effect of noise in
excitable systems has been extensively studied for both white and colored
noise Zambrano et al. (2010); Lindner et al. (2004), and it has been found
that noise intensity is related to changes in the particular behavior of
neuron models Bashkirtseva et al. (2023); Bashkirtseva et al. (2018a, b).
Moreover, it is also known that factors such as coupling strength Sun et al.
(2015); Shen et al. (2008), coupling forms Wang et al. (2011); Sriram et al.
(2023), and diversity Tang et al. (2011) play key roles in the global
synchronization of the network. For all these reasons, in this work, we
represent the diversity in the network by introducing a parameter mismatch in
the neuron model and consider different types of coupling, either excitatory
or inhibitory. The global synchronization of the network and the firing
frequency will be studied using numerical methods for different
characteristics of the neurons and the network itself. We have found critical
values of noise intensity that enhance the synchronization of the neuron
network. We observed that the presence of inhibitory couplings desynchronizes
the system, while an increase in the rewiring probability improves the
stability of the neuron network. All these features are explained in detail
throughout the manuscript. The organization of this paper is as follows. In
Section II, we describe the stochastic Chialvo model. In Section III, we
analyze the synchronization and the firing frequency in the small-world neuron
network as a function of the mismatch between the neurons in one of the
parameters of the model, the noise intensity an the proportion between the
excitatory and the inhibitory couplings. The main conclusions of our results
are provided in Section IV.

##  II The stochastic Chialvo neuron model

To construct the neuron network, we will use the stochastic version of the
map-based Chialvo model Used et al. (2024). This model is based in the
deterministic Chialvo model Chialvo (1995) where the variable $x$ is related
to an instantaneous membrane potential of the neuron, while the variable $y$
stands for the recovery current. In this case a random disturbance
$\varepsilon\xi_{t}$ is added in the parameter $I$ of the acting ion current
injected into the neuron. The parameter $\varepsilon$ represents the intensity
of the noise, while $\xi_{t}$ is uncorrelated white Gaussian noise with
parameters $\langle\xi_{t}\rangle=0,\;\langle\xi^{2}_{t}\rangle=1$. The
parameter $a$ is related to the time of recovery ($a<1$), the activation-
dependence of the recovery process is defined by the parameter $b$ ($b<1$)
with the offset value $c$. All this information is summarized in Eq. 1:

$\begin{array}[]{l}x_{t+1}=x_{t}^{2}\exp(y_{t}-x_{t})+I+\varepsilon\xi_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{t+1}=ay_{t}-bx_{t}+c.\end{array}$ (1)

This neuron model is capable of reproducing some of the basic features behind
the firing dynamics of the neurons Chialvo (1995); Hoppensteadt (1986). For
example, by modifying the parameter $b$, one could obtain different dynamical
behaviors: fixed point, periodic, quasi-periodic or chaotic dynamics. These
basic dynamics of the neuron can be appreciated in Fig. 1(a-d) where the
potential action, $x_{t}$, is represented for some typical values of the model
parameters ($a$, $c$ and $I$) and for different values of the parameter $b$.

Figure 1: Time series for the stochastic Chialvo neuron model. Action
potential $x_{t}$ is represented for different values of the parameter $b$:
(a) $b$ = 0.18, (b) $b$ = 0.20, (c) $b$ = 0.22, (d) $b$ = 0.35. The rest of
the parameters of the model remain fixed, $a=0.89$, $c=0.28$, $I=0.03$ and
$\varepsilon=1.5\cdot 10^{-3}$.

To understand the influence of different parameters in the firing frequency of
the neurons, we define the inter-spike interval, $ISI$, as the time between
two consecutive spikes in the neuron time series. The dependence of the $ISI$
as a function of the parameter $b$ can be appreciated in Fig. 2, where we
represent the mean value of the $ISI$ as a function of $b$ and the error bars
represent the standard deviation of the $ISI$. Every value of the plot is
obtained as the mean value of $50$ simulations, each one starting from random
conditions. It can be appreciated clearly three different regions: Region I:
$0.18\leq b\leq 0.206$, Region II: $0.206\leq b\leq 0.23$, and Region III:
$0.23\leq b$. In Region I, the value of $ISI$ increases as the parameter $b$
increases. In this region, small changes in the parameter $b$ imply big
changes in the value of $ISI$. It can also be observed that the standard
deviation of the $ISI$ is quite significant, which means that the $ISI$ is not
very stable. In Region II, $b$ and $ISI$ have an inverse relationship, that
is, when $b$ increases the mean value of $ISI$ decreases. Finally, in Region
III the relationship between $b$ and $ISI$ is again direct and an increase in
$b$ involves an increase in $ISI$. In the last two regions, it can be observed
that the standard deviation of the parameter $ISI$ is negligible, being zero
in many cases, that is the firing frequency of the neuron is almost constant.
This means that, in spite of the differences in the amplitude in the signal
that the model reproduces, the dynamical system is very stable in the
frequency domain.

Figure 2: Inter-spike interval ($ISI$) as a function of the parameter $b$. The
values of the parameters are: $a=0.89$, $c=0.28$, $I=0.03,$ and the noise
intensity is $\varepsilon=1.5\cdot 10^{-3}$. The three regions where the
relation between the parameters $b$ and $ISI$ is different are clearly
appreciated and denoted in the plot.

The diversity of these regimes can be also analyzed in Fig. 3 where the
largest Lyapunov exponent $\Lambda(b)$ is plotted for the case
$\varepsilon=0.003$. Here, we have fixed $a=0.89,\;I=0.03,\;c=0.28$ and
consider $b$ as the bifurcation parameter. As can be seen, a variation of $b$
implies crucial changes in the system (Eq. 1) dynamics with order-chaos
transitions. Comparing Fig. 3 with Fig. 2, it can be appreciated how the
region (I) corresponds to the highest values of $\Lambda$, that is when the
system has a chaotic behavior. Regions (II) and (III) corresponds to low
positive and negative values of $\Lambda$.

Figure 3: Largest Lyapunov exponent for the Chialvo neuron model versus the
parameter $b$ for $\varepsilon=0.003$. The rest of the parameters of the model
are fixed to $a=0.89,\;I=0.03,$ and $\;c=0.28$.

##  III Small-world neuron network

In a previous work Used et al. (2024), the effect of noise intensity and the
mismatch in the parameter $b$ on synchronization and the firing frequency was
anlyzed in a system constructed by connecting only two non-identical neurons.
In this simple model we also considered two different types of couplings
between the neurons: excitatory and inhibitory. To generalize the results
obtained in this previous work we propose a similar study but in a small-world
neuron network. With this purpose in mind, the first step is to construct the
neuron network, that is, to connect mathematically a group of neurons. In our
case, we connect the neurons in the well-known small-world network Watts and
Strogatz (1998). To build a small-world network of $N$ neurons, the first step
is to order the $N$ neurons, and then connect each neuron with its $l$ first
neighbors in a cyclical way. In this sense, the first node should be connected
with the $l$ latest nodes. Finally, each node has a probability $p$ to rewire
one of these edges with any other node. Thus, this kind of graph is
characterized by three parameters: (i) $N$, the number of nodes; (ii) $l$, the
number of first neighbors connected with each neuron; and (iii) $p$, the
probability of rewiring an edge. This last parameter indicates the random
nature of the graph, i.e., as $p$ increases the graph will be more random
Watts and Strogatz (1998). After the random rewiring of the network
connections, each neuron will have a total number of connections $N_{i}$ that
can change from one neuron to another. This parameter, $N_{i}$, plays an
important role in the network, because the connection strength between the
neuron $i$ and the neuron $j$ will be done by ($k/N_{i}$).

The randomness of the graph implies that, for the same choice of parameters,
there might be different degrees of synchronization that could depend on the
specific structure of the graph. This phenomenon makes sense in the real
brain, since all brains have the same macroscopic structure but from a
microscopic point of view the connection patterns are different for every
person. For this reason, all throughout the paper, we will repeat the
calculations $50$ times for every set of parameters of the network and we will
represent the mean value of the parameter we are interested in.

Figure 4: Sketch of the neuron network. In order to clarify the sketch we
consider that initially each neuron is connected to its two closest neighbors
($l=1$). Solid arrows mark the excitatory couplings while the dashed arrows
mark the inhibitory ones.

As previously mentioned, the connections between nodes in the graph represent
the electrical couplings between the neurons. Mathematically, the coupling is
modeled by the addition of a term to our equations that represents the voltage
difference between the two connected neurons. Besides, we have to consider if
the coupling between every pair of neurons is excitatory or inhibitory. With
all this information in mind, we can construct the mathematical representation
of all the couplings for the neuron $i$:

$\dfrac{k}{N_{i}}\sum_{j}\delta_{ij}(x_{j,t-1}-x_{i,t-1}),$ (2)

where $i$ is the evaluated neuron, $j$ the remaining neurons, $k$ is the
coupling strength and $\delta_{ij}$ is a coupling constant between neurons $i$
and $j$. If the connection between the neuron $i$ and the neuron $j$ is
excitatory the parameter $\delta_{ij}=1$. Conversely, if the coupling is
inhibitory $\delta_{ij}=-1$. This coupling constant would be equal to zero if
neuron $i$ and neuron $j$ are not connected.

Summarizing all this previous information, see Fig. 4, we proceed to build the
complete model of the network. In this case, we will fix the same value for
the coupling strength for all neurons, so that the model can be written as:

$\begin{array}[]{l}x_{i,t+1}=x_{i,t}^{2}\exp(y_{i,t}-x_{i,t})+I+\dfrac{k}{N_{i}}\sum_{j}\delta_{ij}(x_{j,t-1}-x_{i,t-1})+\varepsilon\xi_{i,t}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{i,t+1}=ay_{i,t}-b_{i}x_{i,t}+c.\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \end{array}.$ (3)

Once we have developed the mathematical model for the neuron network (Eq. 3),
we proceed to the numerical simulations. In our case, we will set the mean
value of the parameter $b$ at $b=0.35$ and will introduce a random mismatch in
the parameter $b$ of some neurons. The maximum value of this mismatch would be
($\pm 0.01b$). We introduce some small differences between the neurons using
this value, while still considering them similar.

It would be senseless to consider neurons located in the same brain region and
part of the same local network as very different. Introducing a mismatch in
the parameter $b$ for every neuron and the fact that some of the links between
the neurons are inhibitory connections introduce some instability in the
neuron network, significantly affecting the synchronization of the network.
Previous simulations have led us to reject values of $b$ corresponding to the
Region I of the Fig. 2, because in this case, the neuron is highly sensitive
to the value of $b$ and even the small mismatch we propose makes almost
impossible to have a great synchronization in the network without considering
a very high coupling strength between the neurons, which would be unrealistic.
We also discard values from the Region II because the results are quite
similar to those obtained in the case we propose ($b=0.35$).

With these features in mind, we proceed to simulate the behavior of the
previously described small-world network . In our case, we make a network of
$50$ neurons ($N=50$), where each neuron is initially connected to its four
nearest neighbours ($l=2$). We will increase the rewiring probability $p$ to
analyze its effect on the global behavior of the network. Finally, the model
parameters of the neuron are fixed at $a=0.89$, $c=0.28$ and $I=0.03$. These
parameters will remain fixed all throughout the paper.

The objective of the numerical simulations is to see how variations in the
rest of the model parameters affect the global synchronization of the network
and to the firing frequency of the neurons. To analize the synchronization
between the neurons of the network, we introduce the order parameter $R$ as in
Ref. García-Ojalvo et al. (2004). This parameter allows us to measure the
synchronization between different signals. In our case, we will apply this
parameter to study the synchronization between the variable $x$ of each neuron
from the network. Its mathematical expression is given by:

$R(x)=\frac{\left<\overline{x}^{2}\right>-\left<\overline{x}\right>^{2}}{\overline{\left<x^{2}\right>-\left<x\right>^{2}}},$

where $\overline{x}$ is the mean average over all neurons, and
$\left<x\right>$ is the mean in time, i.e., the mean of the variable $x$ of
each neuron. Accordingly, $R(x)\in\left[0,1\right]$, where $R(x)=0$ means
total de-synchronization and $R(x)=1$ means total synchronization of the time
series $x$.

As an example, we have plotted Fig. 5, which shows the times series of the
variable $x$ from $10$ neurons under two different working conditions. In this
figure, the relationship between the synchronization of the neurons and the
value of the order parameter $R$ can be clearly appreciated.

Figure 5: The time series of the variable $x_{i,t}$ for the neurons of the
network. In both figures, the model parameters and the noise intensity are the
same ($a=0.89$, $c=0.28$, $I=0.03$ and $\varepsilon=0.003$) and the values of
$b$ for the neurons are randomly chosen from the range $0.3464\leq b\leq
0.3535$ ($b\pm 0.01b\xi_{b}$), where $\xi_{b}$ is uncorrelated white Gaussian
noise with parameters $\langle\xi_{b}\rangle=0,\;\langle\xi^{2}_{b}\rangle=1$,
but the coupling strength and the properties of the network are different. The
order parameter $R$ is calculated for both cases: in case (a) where the
neurons are very well synchronized ($R=0.9716$) and in case (b) where the
neurons are unsynchronized ($R=0.210$). (For clarity of the figure only the
time series from 10 out of the 50 neurons that are in the network are
plotted).

In our case, we will calculate the value of the order parameter $R$ as a
function of the noise intensity $\varepsilon$, coupling strength $k$ and the
probability of rewiring in the network $p$. Additionally, we will evaluate the
parameter $R$ as a function of the ratio between excitatory and inhibitory
connections. The number of neurons with a mismatch in the parameter $b$ also
plays a crucial role in the synchronization of the network. For this reason,
the number of neurons with mismatch in parameter $b$ will be gradually
increased.

Every result of this work is obtained as the mean value of $50$ simulations
for the same set of parameters values, starting from different initial
conditions. Note that in each figure the rewiring probability of the
connections in the network, $p$, increases. We start with a network with no
rewiring, $p=0$, and finish with a network where the probability of rewiring
is $p=0.25$.

###  III.0.1 Excitatory coupling in the small-world network

We will start considering an homogeneous neuron network where all the neurons
take the same value for the parameter $b=0.35$ and then gradually increase the
number of neurons with different values for the parameter $b$. Besides, the
parameter $R$ will be calculated for different values of the rewiring
probability, $p$, that will allow us to see how the randomness of the network
affects the global synchronization of the system.

Figure 6: Plot of the synchronization order parameter R. Representation of the
parameter $R$ for a network where all neurons have the same values for the
parameters, $b=0.35$, $a=0.89$, $c=0.28$ and $I=0.03$.

In Fig. 6, where all the neurons are identical, the network is synchronized
for almost all the points of the parameter plane $(k,\varepsilon)$ and only
for low values of the cloupling strength $k$ and high values of noise
intensity $\varepsilon$ does the neuron network become desynchronized. This
result is quite expected because the noise intensity makes the signal of the
neurons more chaotic, and low values of the coupling strength makes a weak
interaction between the neurons of the network, so the influence between them
does not affect the synchronization. It can also be appreciated that as the
probability of rewiring of the network $p$ increases, the synchronization of
the global system also increases. An interesting effect observed in Fig. 6 is
that for all cases of $p$ there are two bands centered around
$\varepsilon=0.8\cdot 10^{-3}$ and $\varepsilon=2\cdot 10^{-3}$ where the
value or $R$ reaches its highest values regardless of the value of the
coupling strength, that is, there is a critical value of the noise intensity
that optimizes the global synchronization of the neuron network.

In Fig. 7, $25$ out of the $50$ neurons have a mismatch in the parameter $b$
and in Fig. 8, all the neurons have this mismatch, that is, all neurons are
different. In these cases we can observe similar phenomena to those observed
in the homogenous case: increasing the rewiring probability of the network $p$
enhances the synchronization, and the bands with the highest synchronization
are also reproduced, centered around the same values of the $\varepsilon$.
Finally, it can be appreciated that increasing the heterogeneity of the
neurons worsens the network synchronization: the region with low values of the
parameter $R$ increases from Fig. 6 to Fig. 8.

Figure 7: Plot of the synchronization order parameter R. Representation of the
parameter $R$ for a network where 25 out the 50 neurons have a random value
for the parameter $b$. The mean value of the parameter $b=0.35$ and the
mismatch is randomly selected with maximum value $\Delta b=0.01b$. The rest of
the parameters are fixed for all the neurons, $a=0.89$, $c=0.28$ and $I=0.03$.
Figure 8: Plot of the synchronization order parameter R. Representation of the
order parameter $R$ for a network where all neurons of the network have a
random value for the parameter $b$. The mean value of the parameter $b=0.35$
and the mismatch is randomly selected with maximum value $\Delta b=0.01b$. The
rest of the parameters are fixed for all the neurons, $a=0.89$, $c=0.28$ and
$I=0.03$.

As mentioned in the introduction, the firing frequency is of special interest
because it is related to neural coding. For this reason, the value of the
inter-spike interval has been studied as a function of the noise intensity and
the coupling strength. When we refer to parameter $ISI$, we are representing
the mean value of the inter-spike interval of every neuron in the network. The
$ISI$ is also calculated for different values of the rewiring probability of
the network $p$. As in the previous study of the order parameter $R$, the
value of the parameter $ISI$ has been obtained for three different cases
depending on the number of neurons with a mismatch in the parameter $b$. In
Fig. 9 all the neurons are identical, in Fig. 10, $25$ out the $50$ neurons
have mismatch in the parameter $b$ and finally in Fig. 11, all the neurons
have a different value for the parameter $b$.

Figure 9: Plot of the synchronization order parameter R. Representation of the
mean value of the $ISI$ for a network where all neurons have the same values
for the parameters, $b=0.35$, $a=0.89$, $c=0.28$ and $I=0.03$.  Figure 10:
Plot of the synchronization order parameter R. Representation of the mean
value of the $ISI$ for a network where 25 out of the 50 neurons have a random
value for the parameter $b$. The mean value of the parameter $b=0.35$ and the
mismatch is randomly selected with maximum value $\Delta b=0.01b$. The rest of
the parameters are equal for all the neurons, $a=0.89$, $c=0.28$ and $I=0.03$.
Figure 11: Plot of the synchronization order parameter R. Representation of
the mean value of the $ISI$ for a network where all the 50 neurons of the
network have a random value for the parameter $b$. The mean value of the
parameter $b=0.35$ and the mismatch is randomly selected with maximum value
$\Delta b=0.01b$.

If we compare the figures obtained, we can observe that all the figures are
quite similar, in fact, the differences between Fig. 9 and Fig. 10 are almost
negligible, and for Fig. 11 what we observe is that the range of values of the
$ISI$ is slightly reduced compared to the one obtained for Figs. 9-10. In all
cases, it can be observed how the $ISI$ decreases as the noise intensity
$\varepsilon$ increases and that there is no direct relationship between the
$ISI$ and the coupling strength $k$ or the rewiring probability in the network
$p$. In all cases, we have also calculated the standard deviation in the $ISI$
and we have obtained that its value is very close to zero, even in the regions
of the parameter plane $(k,\varepsilon)$ where the parameter $R$ takes its
maximum values, we can assure that is zero. That means that all the neurons
have the same firing frequency, and the neuron network is very stable in the
frequency domain.

###  III.0.2 Inhibitory coupling in the small-world network

Normally, when two neurons are connected by inhibitory coupling, their
synchronization worsens. But can this observation be extended to a neural
network? Can we assume that the introduction of some inhibitory connections
must desynchronize the system? To answer these questions, we propose to
introduce a percentage of inhibitory connections in the network and analyze
its effect on global synchronization. The percentage of inhibitory coupling
has a great dependence on the area of the brain analyzed and on the type of
neurons connected Gulyás et al. (1999), and its effect is quite important in
the global behavior of the neuron network Börgers and Kopell (2003, 2005). As
in the previous section, the figures obtained for the cases corresponding to
values of $b$ corresponding to the Regions II and III remarked in Fig. 1 are
quite similar, so we focus our simulations on the case $b=0.35$. The procedure
is similar to the excitatory case: we will start by considering that all the
neurons have the same value for the parameter $b$ and gradually increase the
number of neurons with a mismatch in this parameter.

We will start considering that a $1\%$ of the total links in the network are
inhibitory connections.

Figure 12: Plot of the synchronization order parameter R. Representation of
the order parameter $R$ for a network where all neurons have the same values
for the parameters, $b=0.35$, $a=0.89$, $c=0.28$ and $I=0.03$. In the network
an $1\%$ of the connections are inhibitory couplings. Figure 13: Plot of the
synchronization order parameter R. Representation of the order parameter $R$
for a network where 25 out of the 50 neurons have a random value for the
parameter $b$. The mean value of the parameter $b=0.35$ and the mismatch is
randomly selected with maximum value $\Delta b=0.01b$. The rest of the
parameters are equal for all the neurons, $a=0.89$, $c=0.28$ and $I=0.03$. In
the network an $1\%$ of the connections are inhibitory couplings. Figure 14:
Plot of the synchronization order parameter R. Representation of the order
parameter $R$ for a network where all the 50 neurons from the network have a
random value for the parameter $b$. The mean value of the parameter $b=0.35$
and the mismatch is randomly selected with maximum value $\Delta b=0.01b$. The
rest of the parameters are equal for all the neurons, $a=0.89$, $c=0.28$ and
$I=0.03$. In the network an $1\%$ of the connections are inhibitory couplings.

The two main results that we can extract from these figures are: (i) In all
cases, from the homogeneous case (Fig. 12) to the case where all the neurons
have a mismatch in the parameter $b$, (Fig. 14), the increase of the rewiring
probability $p$ produces better synchronization of the network. It can be
clearly appreciated how the area with high values for the parameter $R$
increases as $p$ increases. Additionally, the two bands centered around
$\varepsilon=0.8\cdot 10^{-3}$ and $\varepsilon=2\cdot 10^{-3}$, observed for
the case where all connections are excitatory and where the network reached
its maximum values of synchronization, are also observable for these cases.
(ii) The increase of the number of neurons with mismatch in the parameter $b$
results in worse synchronization of the network. This can be observed by
comparing the evolution from Fig. 12 to Fig. 14.

As in the previous case where all the connections were excitatory, we also
represent the mean value of the $ISI$ for three different cases depending on
the number of neurons with mismatch in the parameter $b$.

Figure 15: Plot of the synchronization order parameter R. Representation of
the mean value of the $ISI$ for a network where all neurons have the same
values for the parameters, $b=0.35$, $a=0.89$, $c=0.28$ and $I=0.03$. In the
network an $1\%$ of the connections are inhibitory couplings. Figure 16: Plot
of the synchronization order parameter R. Representation of the mean value of
the $ISI$ for a network where 25 out of the 50 neurons have a random value for
the parameter $b$. The mean value of the parameter $b=0.35$ and the mismatch
is randomly selected with maximum value $\Delta b=0.01b$. The rest of the
parameters are equal for all the neurons, $a=0.89$, $c=0.28$ and $I=0.03$. In
the network an $1\%$ of the connections are inhibitory couplings. Figure 17:
Plot of the synchronization order parameter R. Representation of the mean
value of the $ISI$ for a network where all the 50 neurons from the network
have a random value for the parameter $b$. The mean value of the parameter
$b=0.35$ and the mismatch is randomly selected with maximum value $\Delta
b=0.01b$. The rest of the parameters are equal for all the neurons, $a=0.89$,
$c=0.28$ and $I=0.03$. In the network an $1\%$ of the connections are
inhibitory couplings.

In general, upon comparing Figs. 15-17 we observe that the results are quite
similar. In each of these figures, the value of the $ISI$ decreases as the
noise intensity $\varepsilon$ increases, but remains constant for all the
values of the coupling strength $k$. It can also be appreciated that the range
of values of the $ISI$ is slightly lower as the number of neurons with
mismatch in the parameter $b$ increases. Finally, we observe that as the
number of neurons with a mismatch increases and the rewiring probability in
the network $p$ increases, the value of the $ISI$ becomes less stable. That
is, in the subplots of the Figs. 15 -17 corresponding to $p=0.1$, $p=0.15$ and
$p=0.25$, the value of the $ISI$ for some points changes significantly
compared to the values obtained in the surrounding points. Since the value of
the $ISI$ is obtained as the mean value of 50 simulations where the inhibitory
links are chosen randomly from all the connections in the network, this leads
us to think that there could be a relationship between the $ISI$ and the
internal structure of the network.

Finally, we increase the percentage inhibitory connections in the network to a
$5\%$. Under this condition, we study the synchronization of the network and
the value of the $ISI$ with a similar procedure to the one we did previously.

Figure 18: Plot of the synchronization order parameter R. Representation of
the order parameter $R$ for a network where all neurons have the same values
for the parameters, $b=0.35$, $a=0.89$, $c=0.28$ and $I=0.03$. In the network
an $5\%$ of the connections are inhibitory couplings. Figure 19: Plot of the
synchronization order parameter R. Representation of the order parameter $R$
for a network where 25 out of the 50 neurons have a random value for the
parameter $b$. The mean value of the parameter $b=0.35$ and the mismatch is
randomly selected with maximum value $\Delta b=0.01b$. The rest of the
parameters are equal for all the neurons, $a=0.89$, $c=0.28$ and $I=0.03$. In
the network an $5\%$ of the connections are inhibitory couplings. Figure 20:
Plot of the synchronization order parameter R. Representation of the order
parameter $R$ for a network where all the 50 neurons from the network have a
random value for the parameter $b$. The mean value of the parameter $b=0.35$
and the mismatch is randomly selected with maximum value $\Delta b=0.01b$. The
rest of the parameters are equal for all the neurons, $a=0.89$, $c=0.28$ and
$I=0.03$. In the network a $5\%$ of the connections are inhibitory couplings.

The results of the study of the synchronization of the network are shown in
Figs. 18, 19 and 20. The first observation is that the value of the order
parameter $R$ is significantly lower than that obtained in the previous cases.
It is evident how the increase in the percentage of inhibitory connections
negatively affects the global synchronization of the network. In all cases, it
is noticeable how the increase in the rewiring probability enhances the global
synchronization of the network. This effect warrants detailed analysis and
could be a promising avenue for further investigation.

Lastly, the study of the $ISI$ is depicted in Figs. 21 to 23. Unlike the cases
where all connections were excitatory or where only $1\%$ of the connections
were inhibitory, the stability of the $ISI$ is worse in this scenario. Here,
the range of $ISI$ values is noticeably wider.

Figure 21: Plot of the synchronization order parameter R. Representation of
the mean value of the $ISI$ for a network where all neurons have the same
values for the parameters, $b=0.35$, $a=0.89$, $c=0.28$ and $I=0.03$. In the
network a $5\%$ of the connections are inhibitory couplings. Figure 22: Plot
of the synchronization order parameter R. Representation of the mean value of
the $ISI$ for a network where 25 out of the 50 neurons have a random value for
the parameter $b$. The mean value of the parameter $b=0.35$ and the mismatch
is randomly selected with maximum value $\Delta b=0.01b$. The rest of the
parameters are equal for all the neurons, $a=0.89$, $c=0.28$ and $I=0.03$. In
the network a $5\%$ of the connections are inhibitory couplings. Figure 23:
Plot of the synchronization order parameter R. Representation of the mean
value of the $ISI$ for a network where all the 50 neurons from the network
have a random value for the parameter $b$. The mean value of the parameter
$b=0.35$ and the mismatch is randomly selected with maximum value $\Delta
b=0.01b$. The rest of the parameters are equal for all the neurons, $a=0.89$,
$c=0.28$ and $I=0.03$. In the network a $5\%$ of the connections are
inhibitory couplings.

##  IV Conclusions

We have investigated the global behavior of a neuron network organized in the
well-known small-world pattern. We used a stochastic version of the map-based
Chialvo neuron model, introducing a random mismatch in the parameter $b$ in
some neurons to induce inhomogeneity within the network. Under these
conditions, we analyzed the synchronization of the network and the inter-spike
interval ($ISI$) as a function of the noise intensity $\varepsilon$ introduced
in the stochastic model, the coupling strength between neurons $k$, and the
mismatch in the parameter $b$. Additionally, we examined the network’s global
behavior based on its structure by increasing the rewiring probability of
network connections $p$ and adjusting the proportion between excitatory and
inhibitory connections. In general, synchronization is more pronounced in
homogeneous networks compared to inhomogeneous ones. As the number of neurons
with parameter mismatches increases, the system becomes more desynchronized.
When the network consists solely of excitatory connections, synchronization is
high and further increases with coupling strength. However, transitioning some
connections to inhibitory decreases the region of high synchronization in the
parameter plane ($k$,$\varepsilon$) as the proportion of inhibitory
connections rises. We have observed that increasing the rewiring probability
$p$ enhances network synchronization, indicating that randomness in network
connections adds stability to the system. Moreover, certain values of noise
intensity optimize network synchronization, remaining almost constant
regardless of coupling strength and the proportion of excitatory/inhibitory
connections.

The inter-spike interval ($ISI$) of neurons remains relatively constant across
all figures, regardless of coupling strength. We noted a decrease in $ISI$ as
noise intensity increases, which is consistent for cases with all excitatory
connections and a lower percentage of inhibitory connections (e.g., $1\%$).
However, when the proportion of inhibitory connections increases (e.g.,
$5\%$), this trend is slightly modified and becomes more noticeable as the
rewiring probability of network connections increases.

Finally, we observed that $ISI$ values become less homogeneous in cases with
inhibitory connections and high rewiring probability. This phenomenon may be
attributed to the random selection of inhibitory connections from the network,
leading to internal structures that significantly affect the $ISI$ of neurons.
These findings provide valuable insights for future studies investigating
network synchronization and $ISI$ as functions of the network structure.

Acknowledgments JU, JMS and MAFS acknowledge financial support from the
Spanish State Research Agency (AEI) and the European Regional Development Fund
(ERDF, EU) under Project No. PID2019-105554GB-I00
(MCIN/AEI/10.13039/501100011033). IB and LR acknowledge financial support from
the Ministry of Science and Higher Education of the Russian Federation
(project ”Ural Mathematical Center” N 075-02-2024-1428).

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# A convenient category to study asymptotic primes and related questions

Tony J. Puthenpurakal  Department of Mathematics, IIT Bombay, Powai, Mumbai
400 076<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a Noetherian ring and let $\mathcal{R}=\bigoplus_{n\geq
0}\mathcal{R}_{n}$ be a standard graded ring with $\mathcal{R}_{0}=A$. We
define a category $\mathfrak{A}(\mathcal{R})$ of graded $\mathcal{R}$-modules
(not necessarily finitely generated) with the following properties: if
$X=\bigoplus_{n\in\mathbb{Z}}X_{n}\in\mathfrak{A}(\mathcal{R})$ then

  1. (1)

$X_{i}$ is finitely generated $A$-module for all $i\in\mathbb{Z}$ and
$X_{i}=0$ for $i\ll 0$.

  2. (2)

There exists $n_{0}$ such that
$\operatorname{Ass}_{A}X_{n}=\operatorname{Ass}_{A}X_{n_{0}}$ for all $n\geq
n_{0}$.

  3. (3)

If $X_{n}$ has finite length as an $A$-module for all $n$ then there exists
$P_{X}(z)\in\mathbb{Q}[z]$ such that $P_{X}(n)=\ell_{A}(X_{n})$ for all $n\gg
0$.

  4. (4)

If $F$ is a coherent functor on the category of finitely generated $A$-modules
then $F(X)=\bigoplus_{n\in\mathbb{Z}}F(X_{n})\in\mathfrak{A}(\mathcal{R})$.

  5. (5)

For an ideal $J$ in $A$, there exists $c_{J}^{X}$ such that
$\operatorname{grade}(J,X_{n})=\operatorname{grade}(J,X_{c_{J}^{X}})$ for all
$n\geq c_{J}^{X}$.

We give a unified proof of several results in theory of associate primes and
related areas.

###### Key words and phrases:

Associate primes, coherent functors, Rees Algebras

###### 1991 Mathematics Subject Classification:

Primary 13A30; Secondary 13E05, 18A22

##  1\. introduction

The theory of asymptotic prime divisors was initiated by Ratliff [R]. Brodmann
[Br, p. 16] showed that when $M$ is a finite module over a Noetherian
commutative ring $A$ and $I$ is an ideal of $A$, then the sequences of sets of
associated prime ideals $\operatorname{Ass}_{A}(I^{n}M/I^{n+1}M)$ and
$\operatorname{Ass}_{A}(M/I^{n}M)$, $n=1,2,\ldots$ are ultimately constant.
Brodmann’s result has many applications and has been generalized in various
forms. In [ME, Proposition 2], McAdam and Eakin proved that if $R$ is a
standard graded Noetherian ring with $R_{0}=A$ and $M$ is a finitely generated
graded $A$-module, then the set of primes $\operatorname{Ass}_{A}(M_{n})$ is
stable for all large $n$. They also showed that Brodmann’s result followed
from their result concerning the stability of primes associated to homogeneous
components of graded modules. Later Brodmann, [Br2, Theorem 2(i) and 12(i)]
proved that if $J$ is an ideal in $A$ then
$\operatorname{grade}(J,I^{n}M/I^{n+1}M)$ and
$\operatorname{grade}(J,M/I^{n}M)$ stabilize. He used this result to give a
refinement of an inequality due to Burch, [Br2, Theorem 2(ii)]. For a nice
introduction to the theory of asymptotic primes see [Mc].

In [MS, Theorem 1] the authors prove that for each $i\geq 0$ the sets
$\operatorname{Ass}_{A}\operatorname{Tor}^{A}_{i}(M,A/I^{n})$ and
$\operatorname{Tor}^{A}_{i}(M,I^{n}/I^{n+1})$ stabilize. Later in [KW, 3.5]
the authors proved that for finitely generated $A$-modules $M,N$ the sets
$\operatorname{Ass}_{A}\operatorname{Tor}^{A}_{i}(M,N/I^{n}N)$ and
$\operatorname{Ass}_{A}\operatorname{Ext}_{A}^{i}(M,N/I^{n}N)$ are stable.

In [KV, Theorem 2] it was shown that if $A$ is local and $\ell(M\otimes_{A}N)$
is finite then the functions
$n\rightarrow\ell(\operatorname{Tor}^{A}_{i}(M,N/I^{n}N))$ and
$n\rightarrow\ell(\operatorname{Ext}^{i}_{A}(M,N/I^{n}N))$ are of polynomial
type. In [T, Corollary 4] the result was generalized by just assuming that
$\operatorname{Tor}^{A}_{i}(M,N/I^{n}N)$ is of finite length for all $n$ then
the function $n\rightarrow\ell(\operatorname{Tor}_{i}^{A}(M,N/I^{n}N))$ is of
polynomial type (similar result holds for
$\operatorname{Ext}^{i}_{A}(M,N/I^{n}N)$).

Recently in a far-reaching generalization T.Se, [Se, 2.4, 2.7, 2.13], showed
that if $F$ is a coherent covariant $A$-linear functor from the category of
finitely generated $A$-modules to itself the sets
$\operatorname{Ass}_{A}(F(I^{n}M/I^{n+1}M))$,
$\operatorname{Ass}_{A}(F(M/I^{n}M))$,and the values
$\operatorname{grade}(J,F(I^{n}M/I^{n+1}M))$ and
$\operatorname{grade}(J,F(M/I^{n}M))$ become independent of $n$ for large $n$.
We note that if $N$ is a finitely generated $A$-module then the functors
$\operatorname{Tor}^{A}_{i}(N,-)$ and $\operatorname{Ext}^{i}_{A}(N,-)$ are
coherent for all $i\geq 0$; see [H, 2.4, 2.5]. In [BM, 3.3] it is proved that
if $F(M/I^{n}M)$ has finite length for all $n$ then the function
$n\rightarrow\ell(F(M/I^{n}M))$ is of polynomial type.

In [KN, 2.1] the authors proved that if $\mathcal{R}$ is a standard graded
subring of a polynomial ring $\mathcal{S}=A[X_{1},\ldots,X_{m}]$ with
$\mathcal{R}_{0}=A$ then the set
$\operatorname{Ass}_{A}\mathcal{S}_{n}/\mathcal{R}_{n}$ is stable for all
$n\gg 0$. In [HF, 1.1], this result was generalized as follows: Let
$\mathcal{R},\mathcal{S}$ be standard graded rings with
$\mathcal{R}_{0}=\mathcal{S}_{0}=A$ and we have an inclusion of graded rings
$\mathcal{R}\subseteq\mathcal{S}$. Assume $M$ is a finitely generated graded
$\mathcal{R}$-module and $N$ is a finitely generated $\mathcal{S}$ module. If
$M\subseteq N$ as graded $\mathcal{R}$-modules then
$\operatorname{Ass}_{A}N_{n}/M_{n}$ is stable for $n\gg 0$. Furthermore if $J$
is an ideal of $A$ then $\operatorname{grade}(J,N_{n}/M_{n})$ is constant for
$n\gg 0$, see [HF, 3.1].

######  1.1.

In view of [Se] we may ask that if $F$ is a coherent functor and $M,N$ as
above then

  1. (1)

is $\operatorname{Ass}_{A}F(N_{n}/M_{n})$ stable for $n\gg 0$ ?

  2. (2)

if $J$ is an ideal in $A$ then is $\operatorname{grade}(J,F(N_{n}/M_{n}))$ is
constant for $n\gg 0$ ?

  3. (3)

if $F(N_{n}/M_{n})$ has finite length for all $n$ then is the function
$n\rightarrow\ell(F(N_{n}/M_{n}))$ of polynomial type?

The techniques in [Se], [BM] and [HF] do not generalize to deal with this
case. In this paper we show all the above questions have a positive answer.