# The Price of Cognition and Replicator Equations in Parallel Neural Networks

Armen Bagdasaryan<EMAIL_ADDRESS>Antonios Kalampakas  Mansoor
Saburov

###### Abstract

In a broader context, the experimental investigation postulates that toxic
neuropeptides/neurotransmitters are causing damage to the functionality of
synapses during neurotransmission processes. In this paper, we are aiming to
propose a novel mathematical model that studies the dynamics of synaptic
damage in terms of concentrations of toxic neuropeptides/neurotransmitters
during neurotransmission processes. Our primary objective is to employ
Wardrop’s first and second principles within a neural network of the brain. In
order to comprehensively incorporate Wardrop’s first and second principles
into the neural network of the brain, we introduce two novel concepts:
neuropeptide’s (neurotransmitter’s) equilibrium and synapses optimum. The
neuropeptide/neurotransmitter equilibrium refers to a distribution of toxic
neuropeptides/neurotransmitters that leads to uniform damage across all
synaptic links. Meanwhile, synapses optimum is the most desirable distribution
of toxic neuropeptides/neurotransmitters that minimizes the cumulative damage
experienced by all synapses. In the context of a neural network within the
brain, an analogue of the price of anarchy is the price of cognition which is
the most unfavorable ratio between the overall impairment caused by toxic
neuropeptide’s (neurotransmitter’s) equilibrium in comparison to the optimal
state of synapses (synapses optimum). To put it differently, the price of
cognition measures the loss of cognitive ability resulting from increased
concentrations of toxic neuropeptides/neurotransmitters. Additionally, a
replicator equation is proposed within this framework that leads to the
establishment of the synapses optimum during the neurotransmission process. It
is important to note that our model serves as a high-level simplification and
abstraction of the natural neurotransmission process involving interactions
between two neurons. Nevertheless, we envision that this mathematically
abstract model can serve as a source of motivation to instigate novel
experimental, mathematical, and computational research avenues in the field of
contemporary neuroscience.

###### keywords:

neurotransmitter’s equilibrium , synapses optimum , price of cognition , Nash
equilibrium , replicator equation.

###### MSC:

[2020] 92B20 , 92C20 , 90B20 , 91A14 , 91A22

††journal: Journal

organization=College of Engineering and Technology, American University of the
Middle East, city=Egaila, postcode=54200, country=Kuwait

The connection between two neurons, established through unidirectional
synaptic links, is visualized as a parallel network between two nodes;

Our innovative approach involves the application of “Wardrop’s first and
second principles” in the examination of synaptic damage resulting from toxic
neuropeptides/neurotransmitters during neurotransmission processes;

This was achieved by introducing two novel concepts: neuropeptide’s
(neurotransmitter’s) equilibrium and synapses optimum;

The neuropeptide/neurotransmitter equilibrium refers to a distribution of
toxic neuropeptides/neurotransmitters that leads to uniform damage across all
synaptic links;

Synapses optimum is the most desirable distribution of toxic
neuropeptides/neurotransmitters that minimizes the cumulative damage
experienced by all synapses;

An analogous concept to the price of anarchy is the price of cognition that
represents the most unfavorable ratio between the overall impairment caused by
toxic neuropeptide’s (neurotransmitter’s) equilibrium in comparison to the
optimal state of synapses (synapses optimum);

To put it differently, the price of cognition measures the loss of cognitive
ability resulting from increased concentrations of toxic
neuropeptides/neurotransmitters;

Finally, a replicator equation is proposed within this framework that leads to
the establishment of the synapses optimum during the neurotransmission
process.

##  1 Introduction

The brain assumes responsibility for each cognitive process such as thoughts,
emotions, and physical actions. This role is facilitated through a fundamental
physiological mechanism known as neurotransmission. In essence,
neurotransmission is the intricate process through which brain cells establish
communication channels. Predominantly, this intricate exchange of information
transpires at a specialized inter-cellular junction known as the synapse. It
is now well-established within the domain of contemporary neuroscience that
the synapse plays a critical role in a variety of cognitive processes,
especially those involved with deep learning and memory. The term “synapse”
finds its etymological origins in the Greek words “syn” denoting “together,”
and “haptein” signifying to “fasten.” The synapse is the small pocket of space
situated between two neurons which is essential to the transmission of
electric nerve impulse from one neuron to another. In the context of a
chemical synapse, the membrane of the transmitting neuron, often referred to
as the presynaptic neuron, closely approximates the membrane of the recipient
neuron, known as the postsynaptic neuron. In both the presynaptic and
postsynaptic neurons, there exist comprehensive arrays of molecular machinery
that link the two membranes together and carry out the signal transmission
process. A single neuron has the capacity to house thousands of synaptic
links. The linkage space between a presynaptic fiber and a postsynaptic fiber
is called the synaptic cleft. The presynaptic neuron releases
neurotransmitters from synaptic vesicles into the synaptic cleft. Then those
neurotransmitters will be taken up by the membrane of receptors on the
postsynaptic neuron throughout the unidirectional synaptic links. In essence,
neurotransmitters are essential neurochemicals that maintain synaptic and
cognitive functions in humans by sending signals across presynaptic to
postsynaptic neurons throughout the unidirectional synaptic links. The reader
can refer to the monograph (see [1]) for the detailed discussion of elements
of contemporary neuroscience.

In the past, neuroscientists held the belief that all synapses remained
constant in their functionality, operating consistently at a uniform level.
However, contemporary understanding has evolved to acknowledge (see [2]) that
synaptic strength can be modified by activity or inactivity, leading to
strengthening or weakening of synapses or even damaging the functionality of
synapses in the brain. The augmentation of synaptic strength occurs in direct
proportion to its usage, thereby enhancing its capacity to exert a more
substantial influence over its adjacent postsynaptic neurons. Contemporary
neuroscientists believe (see [1, 2]) that this strengthening of synapses
constitutes a fundamental mechanism for facilitating the process of learning
and, as a result, the formation of memories.

Gradually, individuals afflicted with Alzheimer’s disease undergo a profound
and relentless decline in memory and cognitive abilities. These
transformations stem from the gradual deterioration and dysfunction of
synapses which are responsible for the encoding, retention, and processing of
information. In 2017, the issue 4 of volume 57 of the Journal of Alzheimer’s
Disease was specially dedicated to critically assess the current status of
synapses and neurotransmitters in Alzheimer’s disease. Experts in the domains
of synapses and neurotransmitters of Alzheimer’s disease provide a
comprehensive overview of the present state of fundamental biology of synapses
and neurotransmitters (see [3]). Additionally, they offer an update on the
current progress of clinical trials involving neurotransmitters for the
treatment of Alzheimer’s disease. There is a growing body of evidence (see [3]
and the references provided therein) indicating that impairments in the cortex
and hippocampus at the initial stages of Alzheimer’s disease are linked to
synaptic damage and deterioration which are attributed to the presence of
oligomeric forms of the neurotoxic amyloid-$\beta$ (A$\beta$) peptide acting
as neurotransmitter. According to 2016 statistics from the Alzheimer’s
Association (see [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]), recent molecular,
cellular, animal model, and postmortem brain studies have revealed that
amyloid-$\beta$ (A$\beta$) is a key factor that causes synaptic damage and
cognitive decline in patients with Alzheimer’s disease. In this context, it
has been postulated (see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]) that elevated
levels of amyloid-$\beta$ (A$\beta$) may induce early defects and damage in
synaptic function. This has been proposed as a potential contributing factor
to the manifestation of neuropsychiatric symptoms commonly observed during the
prodromal phase of Alzheimer’s disease. The reader may refer to the special
issue 57(4):(2017) of the Journal of Alzheimer’s Disease for a comprehensive
report of the present state of fundamental biology of synapses and
neurotransmitters in the field of Alzheimer’s disease (see also [15, 16, 17,
18, 19, 20, 21]).

Due to the importance of knowing damage and deterioration of synapses in the
brains of people with Alzheimer’s disease, in this paper, we are aiming to
propose a novel mathematical model that studies the dynamics of synaptic
damage in terms of concentrations of toxic neuropeptides/neurotransmitters
during neurotransmission processes. Our objective is to employ “Wardrop’s
first and second principles” in order to enhance our understanding of the
dynamics of toxic neuropeptides/neurotransmitters during the neurotransmission
process within a neural network of the brain. We first introduce the so-called
neuropeptide’s (neurotransmitter’s) equilibrium concept that encapsulates the
fundamental essence of Wardrop’s first principle. Namely, the
neuropeptide/neurotransmitter equilibrium refers to a distribution of toxic
neuropeptides/neurotransmitters that leads to uniform damage across all
synaptic links. Subsequently, we proceed to present a complete manifestation
of Wardrop’s second principle through the introduction of another so-called
synapses optimum concept. Synapses optimum is the most desirable distribution
of toxic neuropeptides/neurotransmitters that minimizes the cumulative damage
experienced by all synapses. Consequently, within the context of a neural
network of the brain, an analogue of the price of anarchy is the price of
cognition which is the most unfavorable ratio between the overall impairment
caused by toxic neuropeptide’s (neurotransmitter’s) equilibrium in comparison
to the optimal state of synapses (synapses optimum). To put it differently,
the price of cognition measures the loss of cognitive ability resulting from
increased concentrations of toxic neuropeptides/neurotransmitters. Finally, a
replicator equation is proposed within this framework that leads to the
establishment of the synapses optimum during the neurotransmission process.
Our model represents a sophisticated simplification and abstraction of the
natural neurotransmission process involving two neurons. Nevertheless, we
envision that this mathematically abstract model can serve as a catalyst and a
source of motivation to instigate novel experimental, mathematical, and
computational research avenues in the field of contemporary neuroscience.

##  2 The Price of Cognition Through Wardrop’s Equilibria Approaches

A brain neural network consists of a collection of neurons that are chemically
interconnected or functionally associated. Within the brain neural network,
synapses play a vital role in facilitating the transmission of nervous
impulses from one neuron to another. The fundamental types of connections
between neurons include both chemical and electrical synapses. An electrical
synapse’s primary advantage lies in its ability to swiftly transmit signals
from one neuron to the adjacent neuron. In the context of chemical synapses,
it’s important to note that there is a temporal delay in the neurotransmission
process. Therefore, a part of the brain neural network that relies on chemical
synapses for communication can be regarded as a transportation network. Within
this framework, the impairment, deterioration, degradation, or decline of the
functionality of synapses, occurring due to elevated levels of toxic
neuropeptides/neurotransmitters during the neurotransmission process, can be
linked to a traffic congestion problem in a transportation network. From a
theoretical standpoint, when investigating the dynamic relationship between
synaptic damage and the concentration of toxic neuropeptides/neurotransmitters
during the neurotransmission process, it is plausible to adapt “Wardrop’s
first and second principles”. Although these principles were originally
formulated to describe optimal flow distributions in transportation networks,
we believe that they can be adapted and applied effectively within the
framework of the neural network in the brain. This is a novel contribution of
this paper.

The network optimization problems have attracted much attention over the last
decade for their ubiquitous appearance in real-life applications and the
inherent mathematical challenges that they present, especially, in optimal
transportation theory and communication networks (see [22, 23, 24, 25, 26, 27,
28]). Back in 1952, J.G.Wardrop (see [22]) formulated two principles of
optimality of flows in networks that describe the circumstances of the user
equilibrium and the system optimum. The first Wardrop principle states that
the costs of all utilized links are equal and less than the costs of those
unutilized links for every fixed source-destination pair. Meanwhile, the
system optimum is the optimal distribution of the flow for which the total
cost of the system is minimal. The problems of finding the user equilibrium
and system optimum are the topics of active research both in theory and
practice. When making route choices in traffic networks, the network users
frequently display selfish behavior, that is the fundamental principle of the
first Wardrop principle, by selecting routes that minimize their individual
travel costs. It is widely recognized (see [23, 24, 25, 26]) that selfish
routing, in general, does not lead to a system optimum of the network that
minimizes the total travel cost. The so-called Price of Anarchy (see [29, 30])
is a quantitative measure of the inefficiency of the traffic network that was
caused by the selfish behavior of the network users. Namely, the price of
anarchy is the the worst-possible ratio between the total cost of a user
equilibrium in comparison to the system optimum.

The primary objective of this paper is to gain insight into specific
mechanisms and phenomena within the neural network of the brain by applying
methods originally designed for solving traffic assignment problems in
transportation networks. For the sake of simplicity, we focus on examining the
process of signal transmission between two neurons through chemical synapses.

Broadly speaking, there exist comprehensive arrays (links) of synapses between
presynaptic and postsynaptic neurons that carry out the signal transmission
process. In this scenario, let’s assume that the presynaptic neuron releases
one-unit of toxic neuropeptides/neurotransmitters. These toxic
neuropeptides/neurotransmitters are subsequently absorbed by receptors located
on the postsynaptic neuron across unidirectional synaptic links. We can
visualize two neurons connected by unidirectional synaptic links as a parallel
network between two nodes. Let $\mathbf{I}_{m}=\\{1,2,\dots,m\\}$ denote the
set of the synaptic links between two neurons. We denote by $x_{k}\geq 0$, the
concentration of toxic neuropeptides/neurotransmitters passing through the
synaptic link $k$ and $\mathbf{x}=(x_{1},x_{2},\ldots,x_{m})$ denote a
distribution of toxic neuropeptides/neurotransmitters over all synaptic links
$\mathbf{I}_{m}=\\{1,2,\dots,m\\}$, where $\sum_{i=1}^{m}x_{i}=1$.

$\ell_{1}(x_{1})$$\ell_{2}(x_{2})$$\ell_{m}(x_{m})$one unit
ofneurotransmitters Figure 1: The model of neurotransmission process between
two neurons

Within the framework of the neurotransmission process, increased levels of
toxic neuropeptides/neurotransmitters damage the synaptic links. To
investigate this correlation, we make the assumption that each synaptic link
$k\in\mathbf{I}_{m}$ is associated with a synaptic damage function
$\ell_{k}:[0,\infty)\to[0,\infty)$ dependent on the concentration of toxic
neuropeptides/neurotransmitters that measures the damage on the synaptic link
$k$. Throughout this paper, for each $k\in\mathbf{I}_{m}$ the synaptic damage
function $\ell_{k}$ will be assumed to be a convex, strictly increasing,
continuously differentiable function and $\ell_{k}(0)=0$. Let
$\mathbf{L}_{m}(\mathbf{x})=\left(\ell_{1}(x_{1}),\ldots,\ell_{m}(x_{m})\right)$
denote a synaptic damage vector function at $\mathbf{x}=(x_{1},\ldots,x_{m})$.
Hence, a synaptic damage between two neurons can be identified with the
synaptic damage vector function
$\mathbf{L}_{m}(\cdot)=(\ell_{1}(\cdot),\cdots,\ell_{m}(\cdot))$. In this
paper, we will interchangeably use a synaptic damage vector function and the
neural network involving two neurons. We denote by
$\mathbb{S}^{m-1}=\\{\mathbf{x}\in\mathbb{R}_{+}^{m}:\sum_{i=1}^{m}x_{i}=1\\}$
the standard simplex. Define
$\textup{\bf{supp}}(\mathbf{x}):=\\{i\in\mathbf{I}_{m}:x_{i}\neq 0\\}$ and
$\mathrm{int}\mathbb{S}^{m-1}=\\{x\in\mathbb{S}^{m-1}:\textup{\bf{supp}}(\mathbf{x})=\mathbf{I}_{m}\\}$.

A distribution $\mathbf{x}=(x_{1},\dots,x_{m})\in\mathbb{S}^{m-1}$ is called a
neuropeptide/neurotransmitter equilibrium if the synaptic damage is the same
across all synaptic links, i.e., $\ell_{i}(x_{i})=\ell_{j}(x_{j})$ for all
$i,j\in\mathbf{I}_{m}$. The cumulative damage experienced by synapses at a
distribution $\mathbf{x}=(x_{1},\dots,x_{m})\in\mathbb{S}^{m-1}$ is defined by
the sum $\sum_{i=1}^{m}x_{i}\ell_{i}(x_{i})$. Obviously, the
neuropeptide/neurotransmitter equilibrium that results in uniform damage
across all synaptic links does not generally minimize the cumulative damage
experienced by all synapses. The primary objective of experts in the domains
of synapses and neurotransmitters of Alzheimer’s disease is to reduce and
minimize the cumulative damage experienced by synapses in the brains of
individuals afflicted with Alzheimer’s disease. Therefore, a distribution
$\mathbf{x}=(x_{1},\dots,x_{m})\in\mathbb{S}^{m-1}$ is called the optimal
state of synapses (synapses optimum) if it minimizes the cumulative damage
$\sum_{i=1}^{m}x_{i}\ell_{i}(x_{i})$ experienced by all synapses.

It has been well-established (see [23, 25, 26]) that if for each
$k\in\mathbf{I}_{m}$ a synaptic damage function
$\ell_{k}:[0,\infty)\to[0,\infty)$ with $\ell_{k}(0)=0$ is a convex, strictly
increasing, continuously differentiable function then there always exists a
unique neuropeptide/neurotransmitter equilibrium
$\mathbf{x}^{(\mathbf{ne})}\in\mathrm{int}\mathbb{S}^{m-1}$, as well as a
unique synapses optimum
$\mathbf{x}^{(\mathbf{so})}\in\mathrm{int}\mathbb{S}^{m-1}$. In general, we
have that $\mathbf{x}^{(\mathbf{ne})}\neq\mathbf{x}^{(\mathbf{so})}$.

Consequently, within the context of a neural network of the brain, an
analogous concept to the price of anarchy can be called as the price of
cognition. This represents the most unfavorable ratio between the overall
impairment caused by toxic neuropeptide’s (neurotransmitter’s) equilibrium in
comparison to the optimal state of synapses (synapses optimum). To put it
differently, the price of cognition measures the loss of cognitive ability
resulting from increased concentrations of toxic
neuropeptides/neurotransmitters. In terms of the mathematical formula, the
price of cognition is defined as follows

$\mathbf{PoC}({\mathbf{L}_{m}}):=\frac{\mathbf{CD}(\mathbf{x}^{(\mathbf{ne})})}{\mathbf{CD}(\mathbf{x}^{(\mathbf{so})})}$

where $\mathbf{x}^{(\mathbf{ne})}\in\mathrm{int}\mathbb{S}^{m-1}$ is the
unique neuropeptide/neurotransmitter equilibrium,
$\mathbf{x}^{(\mathbf{so})}\in\mathrm{int}\mathbb{S}^{m-1}$ is the unique
synapses optimum, and
$\mathbf{CD}(\mathbf{x})=\sum_{i=1}^{m}x_{i}\ell_{i}(x_{i})$ is the cumulative
damage experienced by all synapses at a distribution
$\mathbf{x}\in\mathbb{S}^{m-1}$.

Obviously, we always have that $\mathbf{PoC}({\mathbf{L}_{m}})\geq 1$. The
case $\mathbf{PoC}({\mathbf{L}_{m}})=1$ represents the most favorable and
desirable circumstance for individuals afflicted with Alzheimer’s disease.
Indeed, if $\mathbf{PoC}({\mathbf{L}_{m}})=1$ then we must have that
$\mathbf{x}^{(\mathbf{ne})}=\mathbf{x}^{(\mathbf{so})}$ due to the uniqueness
of the neuropeptide/neurotransmitter equilibrium as well as synapses optimum.
This means that the neuropeptide/neurotransmitter equilibrium that leads to
uniform damage across all synaptic links is also the optimal state of synapses
which minimizes the cumulative damage experiences by them. Under such
circumstances, when two equilibria coincide
$\mathbf{x}^{(\mathbf{ne})}=\mathbf{x}^{(\mathbf{so})}$ then it is referred to
as the optimal state of neurotransmitters-synapses (neurotransmitters-synapses
optimum).

Within the domain of transportation networks, the significance of achieving a
price of anarchy of $1$ in a transportation network was emphasized by S.
Dafermos as early as 1968, even before the concept of the price of anarchy
itself was formally introduced. Namely, the problem proposed by S. Dafermos in
her Ph.D. thesis [24] (see also [25]) seeks to identify the specific class of
cost functions for a given network that lead to the emergence of identical
equilibria according to Wardrop’s first and second principles. According to S.
Dafermos, “such networks are extremely desirable because the pattern created
by the individuals acting in their own self interests coincides with the
pattern most economical for the total society (see [24, 25]). She also
provided a solution to her problem by specifying a family of cost functions in
the form of monomial polynomials $\ell_{i}(x_{i})=a_{i}x_{i}^{k}$ where
$a_{i}>0$ and $k\in\mathbb{N}$. This family of cost functions is capable of
solving the problem for any given network (see [24, 25]).

Recent empirical studies in real-world networks (see [31]), as well as
theoretical studies (see [32, 33]), show that the actual value of the price of
anarchy is very close to $1$. The price of anarchy in proximity to $1$
indicates that the user equilibrium is approximately socially optimal, thereby
implying that the effects of selfish behavior are relatively benign. Moreover,
theoretical studies on real-world networks also indicate (see [32, 33]) that
the price of anarchy is approximately $1$ in both light and heavy traffic
conditions. In addition, the price of anarchy for a broad range of cost
functions, which includes all polynomials, tends to approach $1$ in both heavy
and light traffic, regardless of the network topology and the number of origin
and destination pairs within the network. Particularly, the comprehensive
asymptotic principle for polynomial cost functions states (see [32, 33]) that
in networks where costs are polynomial, the price of anarchy converges to $1$
under both light and heavy traffic. This implies that in light and heavy
traffic congestion conditions, a benevolent social planner with complete
authority over traffic assignment would not do any better than the selfish
behavior of users.

On the other hand, parallel networks hold a unique status among other network
types. Namely, parallel networks are nontrivial classes of network topologies
for which the price of anarchy is smaller than for any other networks (see
[34]). Among many other important results, one of the main results of the
paper (see [34]) state that under weak assumptions on the class of cost
functions, the price of anarchy for any multicommodity flow network is
achieved by a single-commodity instance within a network of parallel links.
Consequently, for a given class of cost functions, the price of anarchy cannot
be improved by any nontrivial constraint on the class of network topologies
and/or the number of commodities. Therefore, the lower value of the price of
anarchy can always be attained by the simplest parallel network (see [34]).

Despite its profound importance and practical applicability (see [31, 32,
33]), Dafermos’s problem did not receive adequate attention from experts until
more recently. All of these theoretical discoveries and empirical observations
(see [32, 33, 31]) served as the driving force behind the exploration of the
central and fundamental questions addressed in our recent studies(see [35, 36,
37]):

  * 1.

Can these observations be justified theoretically?

  * 2.

Can we theoretically describe all cost functions associated with parallel
networks in which the price of anarchy achieves its least value of 1?

Finally, Dafermos’s problem has been fully solved for parallel networks in the
papers (see [35, 36, 37]). For any prior given positive distribution
$\mathbf{p}=(p_{1},\ldots,p_{m})\in\mathrm{int}\mathbb{S}^{m-1}$ and for any
convex, strictly increasing, continuously differentiable function
$\ell:[0,\infty)\to[0,\infty)$ with $\ell(0)=0$, we define the following cost
vector function

$\mathbf{\Phi}_{m}(\mathbf{x})=\left(\ell\left(\frac{x_{1}}{p_{1}}\right),\ldots,\ell\left(\frac{x_{m}}{p_{m}}\right)\right).$

Among many other interesting results (see [35, 36, 37]), one of the main
results states that the distribution
$\mathbf{p}=(p_{1},\ldots,p_{m})\in\mathrm{int}\mathbb{S}^{m-1}$ is the user
equilibrium as well as the system optimum of the parallel network
$\mathbf{\Phi}_{m}$. In this case, the price of anarchy is equal to its least
value that is $1$. This shows that there are a vast class of parallel networks
for which the price of anarchy is always equal to its least value that is $1$.
Unlike Dafermos’s example, there are also another interesting classes of
parallel networks
$\mathbf{L}_{m}(\mathbf{x})=\left(\ell_{1}(x_{1}),\ldots,\ell_{m}(x_{m})\right)$
generated by different degrees of polynomials for which the price of anarchy
is always equal to $1$, where

$\displaystyle{\ell}_{1}(x_{1})=\frac{x_{1}}{p_{1}},$
$\displaystyle{\ell}_{2}(x_{2})=\frac{a_{22}}{2}\Big{(}\frac{x_{2}}{p_{2}}\Big{)}^{2}+\frac{a_{21}}{1}\Big{(}\frac{x_{2}}{p_{2}}\Big{)}+\frac{a_{22}}{2},$
$\displaystyle{\ell}_{3}(x_{3})=\frac{a_{33}}{3}\Big{(}\frac{x_{3}}{p_{3}}\Big{)}^{3}+\frac{a_{32}}{2}\Big{(}\frac{x_{3}}{p_{3}}\Big{)}^{2}+\frac{a_{31}}{1}\Big{(}\frac{x_{3}}{p_{3}}\Big{)}+\frac{2a_{33}}{3}+\frac{a_{32}}{2}$
$\displaystyle\vdots$
$\displaystyle\ell_{i}(x_{i})=\sum_{k=1}^{i}\frac{a_{ik}}{k}\Big{(}\frac{x_{i}}{p_{i}}\Big{)}^{k}+\sum_{k=2}^{i}\frac{k-1}{k}a_{ik}$
$\displaystyle\vdots$
$\displaystyle{\ell}_{m}(x_{m})=\sum_{k=1}^{m}\frac{a_{mk}}{k}\Big{(}\frac{x_{m}}{p_{m}}\Big{)}^{k}+\sum_{k=2}^{m}\frac{k-1}{k}a_{mk}.$

and

$\sum_{k=1}^{i}a_{ik}=1\quad\textup{and}\quad a_{ik}>0\quad\textup{for
all}\quad 1\leq i\leq m,\ 1\leq k\leq i.$

##  3 Replicator Equations in Parallel Neural Networks

Let us consider the following parallel neural network

$\mathbf{\Phi}_{m}(\mathbf{x})=\left(\ell\left(\frac{x_{1}}{p_{1}}\right),\ldots,\ell\left(\frac{x_{m}}{p_{m}}\right)\right).$

where $\ell:[0,\infty)\to[0,\infty)$ is a convex, strictly increasing,
continuously differentiable function such that $\ell(0)=0$. In this case, as
we already mentioned that
$\mathbf{p}=(p_{1},\ldots,p_{m})\in\mathrm{int}\mathbb{S}^{m-1}$ is the
neurotransmitters-synapses optimum, i.e., it is simultaneously the
neuropeptide/neurotransmitter equilibrium as well as the synapses optimum of
the parallel network $\mathbf{\Phi}_{m}$.

Let
$\langle\mathbf{x},\mathbf{\Phi}_{m}(\frac{\mathbf{x}}{\mathbf{p}})\rangle=\sum_{k=1}^{m}x_{k}{\ell}\left(\frac{x_{k}}{p_{k}}\right)$
be the cumulative damage experienced by all synapses of the parallel neural
network $\mathbf{\Phi}_{m}$ at the distribution
$\mathbf{x}\in\mathbb{S}^{m-1}$.

We assume that the parallel neural network may dynamically change the
distribution of toxic neuropeptides/neurotransmitters over alternative
parallel synaptic links. Namely, if
$\mathbf{x}^{(n)}=(x_{1}^{(n)},\ldots,x_{m}^{(n)})\in\mathbb{S}^{m-1}$ is the
distribution at the step $n$ then the relative growth rate
$\frac{x^{(n+1)}_{k}-x^{(n)}_{k}}{x^{(n)}_{k}}$ of the distribution on the
synaptic link $k$ at the step $(n+1)$ is negatively proportional (with the
propositional coefficient $\varepsilon$) to the difference between the damage
${\ell}(\frac{x^{(n)}_{k}}{p_{k}})$ of the synaptic link $k$ at the step $n$
and the cumulative damage
$\langle\mathbf{x},\mathbf{\Phi}_{m}(\frac{\mathbf{x}}{\mathbf{p}})\rangle=\sum_{k=1}^{m}x_{k}{\ell}\left(\frac{x_{k}}{p_{k}}\right)$
of the parallel neural network as a whole at the step $n$. The key idea is
that the distribution of the synaptic link whose damage is smaller (larger)
than the cumulative damage of the neural network will increase (decrease) over
the neurotransmission process.

We now propose a novel mathematical model of distributions of toxic
neuropeptides/neurotransmitters by a discrete-time replicator equation
$\mathcal{R}:\mathbb{S}^{m-1}\to\mathbb{S}^{m-1}$

$\left(\mathcal{R}(\mathbf{x})\right)_{k}=x_{k}\left[1+\varepsilon\left(\ell\left(\frac{x_{k}}{p_{k}}\right)-\sum\limits_{i=1}^{m}x_{i}\ell\left(\frac{x_{i}}{p_{i}}\right)\right)\right],\quad\forall\
k\in\mathbf{I}_{m}$ (1)

where $\ell:[0,\bar{p}]\to[0,+\infty)$ is a convex, continuously
differentiable, and strictly increasing function,
$\mathbf{p}=(p_{1},\ldots,p_{m})\in\textup{int}\mathbb{S}^{m-1}$ is a positive
distribution, $\bar{p}:=\frac{1}{\min\limits_{i\in\mathbf{I}_{m}}{p_{i}}}$,
and $\varepsilon\in(-1,0)$. Some particular cases of equation (1) were studied
in the literature [38, 39].

######  Definition 1 (Nash Equilibrium).

A distribution $\mathbf{x}\in\mathbb{S}^{m-1}$ is called a Nash equilibrium of
the replicator equation given by (1) if one has
$\langle\mathbf{x},\varepsilon\mathbf{\Phi}_{m}(\frac{\mathbf{x}}{\mathbf{p}})\rangle\geq\langle\mathbf{y},\varepsilon\mathbf{\Phi}_{m}(\frac{\mathbf{x}}{\mathbf{p}})\rangle$
for any $\mathbf{y}\in\mathbb{S}^{m-1}$. A flow $\mathbf{x}$ is called a
strictly Nash equilibrium if one has
$\langle\mathbf{x},\varepsilon\mathbf{\Phi}_{m}(\frac{\mathbf{x}}{\mathbf{p}})\rangle>\langle\mathbf{y},\varepsilon\mathbf{\Phi}_{m}(\frac{\mathbf{x}}{\mathbf{p}})\rangle$
for any $\mathbf{y}\in\mathbb{S}^{m-1}$ with $\mathbf{y}\neq\mathbf{x}$.

We define

$\mu:=\max\limits_{z\in[0,\bar{p}]}\frac{d}{dz}\left(z\ell(z)\right),\qquad\mathbf{p}_{\alpha}:=\frac{1}{s_{\alpha}(\mathbf{p})}\sum_{i\in\alpha}p_{i}\mathbf{e}_{i}$

for all $\alpha\subset\mathbf{I}_{m}$, where
$s_{\alpha}(\mathbf{p})=\sum_{i\in\alpha}p_{i}$,
$\mathbf{p}=(p_{1},\ldots,p_{m})\in\textup{int}\mathbb{S}^{m-1}$, and
$\mathbf{e}_{i}$ is the vertex of the simplex ${\mathbb{S}}^{m-1}$ for all
$i\in\mathbf{I}_{m}$. Denote
$\textbf{Fix}(\mathcal{R})=\\{\mathbf{x}\in\mathbb{S}^{m-1}:\mathcal{R}(\mathbf{x})=\mathbf{x}\\}$
a set of fixed points of the replicator equation (1).

The next result is the main result of this paper that describes the dynamics
of distributions of toxic neuropeptides/neurotransmitters during the
neurotransmission process for sufficiently small $\varepsilon\in(-1,0)$.

######  Theorem A.

Let $\varepsilon\in(-\frac{1}{\mu},0)\cap(-1,0)$. Then the following
statements hold true:

$(i)$

One has
$\textbf{{Fix}}(\mathcal{R})=\bigcup\limits_{\alpha\subset\mathbf{I}_{m}}\\{\mathbf{p}_{\alpha}\\};$

$(ii)$

The unique neurotransmitters-synapses optimum $\mathbf{p}$ is the only Nash
equilibrium;

$(iii)$

The unique neurotransmitters-synapses optimum $\mathbf{p}$ is the only stable
fixed point;

$(iv)$

An interior orbit converges to the unique neurotransmitters-synapses optimum
$\mathbf{p}$.

###  3.1 Lyapunov Functions

We study the dynamics of the discrete-time replicator equation (1) by means of
a Lyapunov function. Namely, in order to study the stability of fixed points
of the replicator equation, we employ a Lyapunov function.

######  Definition 2 (Lyapunov Function).

A continuous function $\varphi:\mathbb{S}^{m-1}\to\mathbb{R}$ is called a
Lyapunov function if the number sequence

$\\{\varphi(\mathbf{x}),\varphi(\mathcal{R}(\mathbf{x})),\cdots,\varphi(\mathcal{R}^{(n)}(\mathbf{x})),\cdots\\}$

is a bounded monotone sequence for any initial point
$\mathbf{x}\in\mathbb{S}^{m-1}$.

######  Proposition 1.

Let $\varepsilon\in(-\frac{1}{\mu},0)\cap(-1,0)$. Then the following
statements hold true:

$(i)$

A continuous function
$\mathcal{M}_{\mathbf{p}:{k}}(\mathbf{x}):=\max\limits_{i\in\mathbf{I}_{m}}\\{\frac{x_{i}}{p_{i}}\\}-\frac{x_{k}}{p_{k}}$
for all $k\in\mathbf{I}_{m}$ ia a decreasing Lyapunov function for
$\mathcal{R}:\textup{int}\mathbb{S}^{m-1}\to\textup{int}\mathbb{S}^{m-1}$;

$(ii)$

A continuous function
$\mathcal{M}_{\mathbf{p}_{\alpha}:k}(\mathbf{x}):=\max\limits_{i\in\alpha}\\{\frac{x_{i}}{p_{i}}\\}-\frac{x_{k}}{p_{k}}$
for all $k\in\alpha\subset\mathbf{I}_{m}$ is a decreasing Lyapunov function
for
$\mathcal{R}:\textup{int}\mathbb{S}^{|\alpha|-1}\to\textup{int}\mathbb{S}^{|\alpha|-1}$.

###### Proof.

Here, we only present the proof of the part $(i)$. The part $(ii)$ can be
similarly proved. Its proof is omitted here.

We show that
$\mathcal{M}_{\mathbf{p}:{k}}(\mathbf{x}):=\max\limits_{i\in\mathbf{I}_{m}}\\{\frac{x_{i}}{p_{i}}\\}-\frac{x_{k}}{p_{k}}$
for all $k\in\mathbf{I}_{m}$ is a decreasing Lyapunov function along an orbit
(trajectory) of the operator
$\mathcal{R}:\textup{int}\mathbb{S}^{m-1}\to\textup{int}\mathbb{S}^{m-1}$.

Let $\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$. It follows from (1) for any
$k,t\in\mathbf{I}_{m}$ that if $x_{t}=x_{k}$ then
$(\mathcal{R}(\mathbf{x}))_{t}=(\mathcal{R}(\mathbf{x}))_{k}$ and if
$x_{t}\neq x_{k}$ then

$\frac{(\mathcal{R}(\mathbf{x}))_{t}}{p_{t}}-\frac{(\mathcal{R}(\mathbf{x}))_{k}}{p_{k}}=\left(\frac{x_{t}}{p_{t}}-\frac{x_{k}}{p_{k}}\right)\left[1+\varepsilon\left(\frac{\frac{x_{t}}{p_{t}}\ell(\frac{x_{t}}{p_{t}})-\frac{x_{k}}{p_{k}}\ell(\frac{x_{k}}{p_{k}})}{\frac{x_{t}}{p_{t}}-\frac{x_{k}}{p_{k}}}-\sum\limits_{i=1}^{m}x_{i}\ell\left(\frac{x_{i}}{p_{i}}\right)\right)\right].$

Since $-\frac{1}{\mu}<\varepsilon<0$ and for all $t,k\in\mathbf{I}_{m}$

$\left|\frac{\frac{x_{t}}{p_{t}}\ell(\frac{x_{t}}{p_{t}})-\frac{x_{k}}{p_{k}}\ell(\frac{x_{k}}{p_{k}})}{\frac{x_{t}}{p_{t}}-\frac{x_{k}}{p_{k}}}\right|\leq\max\limits_{z\in[0,\bar{p}]}\frac{d}{dz}\left(z\ell(z)\right)=\mu,$

we obtain

$\mathsf{Sign}\left(\frac{(\mathcal{R}(\mathbf{x}))_{t}}{p_{t}}-\frac{(\mathcal{R}(\mathbf{x}))_{k}}{p_{k}}\right)=\mathsf{Sign}\left(\frac{x_{t}}{p_{t}}-\frac{x_{k}}{p_{k}}\right)$

for any $k,t\in\mathbf{I}_{m}$ and

$\mathsf{MaxInd}\left(\frac{\mathcal{R}(\mathbf{x})}{\mathbf{p}}\right)=\mathsf{MaxInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right),\quad\forall\
\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$

where
$\frac{\mathbf{u}}{\mathbf{v}}:=\left(\frac{u_{1}}{v_{1}},\cdots,\frac{u_{m}}{v_{m}}\right)$
for any two vectors $\mathbf{u}=(u_{1},\cdots,u_{m})$ and
$\mathbf{v}=(v_{1},\cdots,v_{m})>0$ and
$\mathsf{MaxInd}\left(\frac{\mathbf{u}}{\mathbf{v}}\right):=\\{k\in\mathbf{I}_{m}:\frac{u_{k}}{v_{k}}=\max\limits_{i\in\mathbf{I}_{m}}\\{\frac{u_{i}}{v_{i}}\\}\\}$.

Moreover, if
$t\in\mathsf{MaxInd}\left(\frac{\mathcal{R}(\mathbf{x})}{\mathbf{p}}\right)=\mathsf{MaxInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right)$
then for all $k\in\mathbf{I}_{m}$ we get

$\frac{(\mathcal{R}(\mathbf{x}))_{t}}{p_{t}}-\frac{(\mathcal{R}(\mathbf{x}))_{k}}{p_{k}}\\\
=\left(\frac{x_{t}}{p_{t}}-\frac{x_{k}}{p_{k}}\right)+\varepsilon\left(\frac{x_{t}}{p_{t}}-\frac{x_{k}}{p_{k}}\right)\left(\frac{\frac{x_{k}}{p_{k}}\left(\ell(\frac{x_{t}}{p_{t}})-\ell(\frac{x_{k}}{p_{k}})\right)}{\frac{x_{t}}{p_{t}}-\frac{x_{k}}{p_{k}}}+\sum\limits_{i=1}^{m}x_{i}\left(\ell(\frac{x_{t}}{p_{t}})-\ell(\frac{x_{i}}{p_{i}})\right)\right).$

Since $-1<\varepsilon<0$, we obtain that
$\mathcal{M}_{\mathbf{p}:{k}}\left(\mathcal{R}(\mathbf{x})\right)\leq\mathcal{M}_{\mathbf{p}:{k}}(\mathbf{x})$
for all $k\in\mathbf{I}_{m}$.

By repeating this process, we get for all $k\in\mathbf{I}_{m}$,
$n\in\mathbb{N}$, and $\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$ that

$\mathsf{MaxInd}\left(\frac{\mathcal{R}^{(n)}(\mathbf{x})}{\mathbf{p}}\right)=\mathsf{MaxInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right),\quad\mathcal{M}_{\mathbf{p}:{k}}\left(\mathcal{R}^{(n+1)}(\mathbf{x})\right)\leq\mathcal{M}_{\mathbf{p}:{k}}\left(\mathcal{R}^{(n)}(\mathbf{x})\right).$

This shows that $\mathcal{M}_{\mathbf{p}:{k}}(\mathbf{x})$ for
$k\in\mathbf{I}_{m}$ is a Lyapunov function over the set
$\textup{int}\mathbb{S}^{m-1}$. This completes the proof. ∎

###  3.2 Asymptotic Stability

We can describe the asymptotic stability of the discrete-time replicator
equation (1) all over the simplex $\mathbb{S}^{m-1}$.

######  Definition 3 (Stable and Attracting Points).

A fixed point $\mathbf{y}\in\mathbb{S}^{m-1}$ is called stable if for every
neighborhood $U(\mathbf{y})\subset\mathbb{S}^{m-1}$ of $\mathbf{y}$ there
exists a neighborhood $V(\mathbf{y})\subset\mathbb{S}^{m-1}$ of $\mathbf{y}$
such that an orbit
$\\{\mathbf{x},\mathcal{R}(\mathbf{x}),\cdots,\mathcal{R}^{(n)}(\mathbf{x}),\cdots\\}$
of any initial point $\mathbf{x}\in V(\mathbf{y})$ remains inside of the
neighborhood $U(\mathbf{y})$. A fixed point $\mathbf{y}\in\mathbb{S}^{m-1}$ is
called attracting if there exists a neighborhood
$V(\mathbf{y})\subset\mathbb{S}^{m-1}$ of $\mathbf{y}$ such that an orbit
$\\{\mathbf{x},\mathcal{R}(\mathbf{x}),\cdots,\mathcal{R}^{(n)}(\mathbf{x}),\cdots\\}$
of any initial point $\mathbf{x}\in V(\mathbf{y})$ converges to $\mathbf{y}$.
A fixed point $\mathbf{y}\in\mathbb{S}^{m-1}$ is called asymptotically stable
if it is both stable and attracting.

######  Proposition 2.

Let $\varepsilon\in(-\frac{1}{\mu},0)\cap(-1,0)$. Then an orbit of the
replicator equation $\mathcal{R}:\mathbb{S}^{m-1}\to\mathbb{S}^{m-1}$ starting
from any initial point $\mathbf{x}\in\mathbb{S}^{m-1}$ converges to the fixed
point $\mathbf{p}_{\textup{supp}(\mathbf{x})}$ in the interior of the face
$\mathbb{S}^{|\textup{supp}(\mathbf{x})|-1}$.

###### Proof.

Without loss of generality, we may assume that
$\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$. Otherwise, we choose a suitable
Lyapunov function given in the part $(ii)$ of Proposition 1 depending on an
initial point $\mathbf{x}\in\mathbb{S}^{m-1}$.

We fix $t_{0}\in\mathsf{MaxInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right)$. As
shown in Proposition 1, we obtain for any $n\in\mathbb{N}$ that

$\mathsf{MaxInd}\left(\frac{\mathcal{R}^{(n)}(\mathbf{x})}{\mathbf{p}}\right)=\mathsf{MaxInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right).$

Since the sequence
$\left\\{\mathcal{M}_{\mathbf{p}:{k}}\left(\mathcal{R}^{(n)}(\mathbf{x})\right)\right\\}_{n=0}^{\infty}$
where

$\mathcal{M}_{\mathbf{p}:{k}}\left(\mathcal{R}^{(n)}(\mathbf{x})\right)=\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{t_{0}}-\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{k}$

is convergent for each $k\in\mathbf{I}_{m}$, the sequence
$\left\\{\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{t_{0}}\right\\}_{n=0}^{\infty}$
where

$\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{t_{0}}=\frac{1}{m}\left(1+\sum_{k=1}^{m}\mathcal{M}_{\mathbf{p}:{k}}\left(\mathcal{R}^{(n)}(\mathbf{x})\right)\right)$

is also convergent ($m$ is the number of total links
$\mathbf{I}_{m}=\\{1,2,\cdots m\\}$).

Hence, the sequence
$\left\\{\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{k}\right\\}_{n=0}^{\infty}$
for each $k\in\mathbf{I}_{m}$ where

$\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{k}=\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{t_{0}}-\mathcal{M}_{\mathbf{p}:{k}}\left(\mathcal{R}^{(n)}(\mathbf{x})\right)$

is also convergent. This means that the orbit (trajectory)
$\left\\{\mathcal{R}^{(n)}(\mathbf{x})\right\\}_{n=0}^{\infty}$ of the
discrete-time replicator equation
$\mathcal{R}:\textup{int}\mathbb{S}^{m-1}\to\textup{int}\mathbb{S}^{m-1}$
starting from any initial point $\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$ is
convergent and its omega limiting point is some fixed point $\mathbf{x}^{*}$.

Moreover, we now show that the orbit (trajectory)
$\left\\{\mathcal{R}^{(n)}(\mathbf{x})\right\\}_{n=0}^{\infty}$ is separated
from the boundary $\partial\mathbb{S}^{m-1}$ of the simplex
$\mathbb{S}^{m-1}$, i.e., for some $\delta_{0}>0$ we have

$\min\limits_{k\in\mathbf{I}_{m}}\frac{\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{k}}{p_{k}}\geq\delta_{0},\quad\forall\
n\in\mathbb{N}.$

Indeed, as we already showed in the part $(i)$ of Proposition 1 that

$\mathsf{Sign}\left(\frac{(\mathcal{R}(\mathbf{x}))_{t}}{p_{t}}-\frac{(\mathcal{R}(\mathbf{x}))_{k}}{p_{k}}\right)=\mathsf{Sign}\left(\frac{x_{t}}{p_{t}}-\frac{x_{k}}{p_{k}}\right)$

for any $k,t\in\mathbf{I}_{m}$, we obtain

$\mathsf{MinInd}\left(\frac{\mathcal{R}(\mathbf{x})}{\mathbf{p}}\right)=\mathsf{MinInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right),\quad\forall\
\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$

Since $-1<\varepsilon<0$, it follows from (1) and
$k\in\mathsf{MinInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right)$ that

$\frac{\left(\mathcal{R}(\mathbf{x})\right)_{k}}{p_{k}}=\frac{x_{k}}{p_{k}}\left[1-\varepsilon\sum\limits_{i=1}^{m}x_{i}\left(\ell\left(\frac{x_{i}}{p_{i}}\right)-\ell\left(\frac{x_{k}}{p_{k}}\right)\right)\right]\geq\frac{x_{k}}{p_{k}}.$

This means that
$\min\limits_{k\in\mathbf{I}_{m}}\frac{\left(\mathcal{R}(\mathbf{x})\right)_{k}}{p_{k}}\geq\delta_{0}$
for $\delta_{0}:=\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{p_{k}}>0$ (since
$\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$).

By repeating this process, we obtain that

$\mathsf{MinInd}\left(\frac{\mathcal{R}^{(n)}(\mathbf{x})}{\mathbf{p}}\right)=\mathsf{MinInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right),\quad\min\limits_{k\in\mathbf{I}_{m}}\frac{\left(\mathcal{R}^{(n)}(\mathbf{x})\right)_{k}}{p_{k}}\geq\delta_{0},\quad\forall\
n\in\mathbb{N}.$

Since the trajectory
$\left\\{\mathcal{R}^{(n)}(\mathbf{x})\right\\}_{n=0}^{\infty}$ of the
operator
$\mathcal{R}:\textup{int}\mathbb{S}^{m-1}\to\textup{int}\mathbb{S}^{m-1}$ is
separated from the boundary $\partial\mathbb{S}^{m-1}$ of the simplex
$\mathbb{S}^{m-1}$, its limiting point $\mathbf{x}^{*}$ belongs to the
interior $\textup{int}\mathbb{S}^{m-1}$ of the simplex $\mathbb{S}^{m-1}$.
Then the only fixed point in the interior $\textup{int}\mathbb{S}^{m-1}$ of
the simplex $\mathbb{S}^{m-1}$ is the Wardrop optimal flow $\mathbf{p}$.
Hence, the omega limiting point of the trajectory starting from any initial
point $\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$ is the Wardrop optimal flow
$\mathbf{p}\in\textup{int}\mathbb{S}^{m-1}$. This completes the proof. ∎

We need the following result in the proof of the main result Theorem A.

######  Proposition 3.

Let $\mathbf{x},\mathbf{y}\in\mathbb{S}^{m-1}$ be two distinct elements of the
simplex $\mathbb{S}^{m-1}$ such that $\mathbf{y}>0$. Let
$\frac{\mathbf{x}}{\mathbf{y}}:=(\frac{x_{1}}{y_{1}},\cdots,\frac{x_{m}}{y_{m}})$,
$\bar{y}:=\min\limits_{k\in\mathbf{I}_{m}}y_{k}>0$ and
$\|\mathbf{x}-\mathbf{y}\|_{1}:=\sum_{k\in\mathbf{I}_{m}}|x_{k}-y_{k}|$. Then
the following statements hold true:

$(i)$

$\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}<1<\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}$

$(ii)$

$\bar{y}\left[\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\right]\leq\|\mathbf{x}-\mathbf{y}\|_{1}\leq
m\left[\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\right]$

###### Proof.

Since $\mathbf{x}\neq\mathbf{y}$ and
$\mathbf{x},\mathbf{y}\in\mathbb{S}^{m-1}$, we have that $x_{k_{0}}\neq
y_{k_{0}}$ for some $k_{0}\in\mathbf{I}_{m}$ and

$\sum_{k\in\mathbf{I}_{m}}x_{k}=\sum_{k\in\mathbf{I}_{m}}y_{k}=1.$

$(i)$ Let us first assume that
$\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\leq 1$. Then
$\frac{x_{k}}{y_{k}}\leq 1$ for all $k\in\mathbf{I}_{m}$ or equivalently
${x_{k}}\leq{y_{k}}$ for all $k\in\mathbf{I}_{m}$. Since $x_{k_{0}}\neq
y_{k_{0}}$ for some $k_{0}\in\mathbf{I}_{m}$, we have that
${x_{k_{0}}}<{y_{k_{0}}}$. Consequently, we have

$1=\sum_{k\in\mathbf{I}_{m}}x_{k}<\sum_{k\in\mathbf{I}_{m}}y_{k}=1$

However, this is a contradiction. So, we must have
$\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}>1$.

Let us now assume that
$\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\geq 1$. Then
$\frac{x_{k}}{y_{k}}\geq 1$ for all $k\in\mathbf{I}_{m}$ or equivalently
${x_{k}}\geq{y_{k}}$ for all $k\in\mathbf{I}_{m}$. Since $x_{k_{0}}\neq
y_{k_{0}}$ for some $k_{0}\in\mathbf{I}_{m}$, we have that
${x_{k_{0}}}>{y_{k_{0}}}$. Consequently, we have

$1=\sum_{k\in\mathbf{I}_{m}}x_{k}>\sum_{k\in\mathbf{I}_{m}}y_{k}=1$

However, this is a contradiction. So, we must have
$\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}<1$.

Hence, we always have
$\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}<1<\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}$.

$(ii)$ On the one hand, we have

$\|\mathbf{x}-\mathbf{y}\|_{1}:=\sum_{k\in\mathbf{I}_{m}}|x_{k}-y_{k}|=\sum_{k\in\mathbf{I}_{m}}y_{k}\left|\frac{x_{k}}{y_{k}}-1\right|\geq\sum_{k\in\mathbf{I}_{m}}\bar{y}\left|\frac{x_{k}}{y_{k}}-1\right|\\\
\geq\bar{y}\left|\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-1\right|+\bar{y}\left|\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-1\right|=\bar{y}\left[\left(\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-1\right)+\left(1-\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\right)\right]\\\
=\bar{y}\left[\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\right]$

On the other hand, since $1$ and $\frac{x_{k}}{y_{k}}$ belong to the segment
$\left[\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}},\
\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\right]$ we have

$\left|\frac{x_{k}}{y_{k}}-1\right|\leq\left[\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\right],\quad\forall\
k\in\mathbf{I}_{m}.$

Consequently, by means of $0<y_{k}<1$, we derive

$\|\mathbf{x}-\mathbf{y}\|_{1}:=\sum_{k\in\mathbf{I}_{m}}|x_{k}-y_{k}|=\sum_{k\in\mathbf{I}_{m}}y_{k}\left|\frac{x_{k}}{y_{k}}-1\right|\leq\sum_{k\in\mathbf{I}_{m}}\left|\frac{x_{k}}{y_{k}}-1\right|\\\
\leq\sum_{k\in\mathbf{I}_{m}}\left[\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\right]=m\left[\max\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}-\min\limits_{k\in\mathbf{I}_{m}}\frac{x_{k}}{y_{k}}\right]$

This completes the proof. ∎

###  3.3 The Proof of Theorem A

We now provide the proof of Theorem A.

$(i)$ We first show
$\textbf{Fix}(\mathcal{R})=\bigcup\limits_{\alpha\subset\mathbf{I}_{m}}\\{\mathbf{p}_{\alpha}\\}$.
It is obvious that
$\bigcup\limits_{\alpha\subset\mathbf{I}_{m}}\\{\mathbf{p}_{\alpha}\\}\subset\textbf{Fix}(\mathcal{R})$.
Let $\mathbf{x}\in\textbf{Fix}(\mathcal{R})$ be any fixed point. We set
$\alpha:=\textup{supp}(\mathbf{x})$. It follows from (1) that

$\ell\left(\frac{x_{k}}{p_{k}}\right)=\sum\limits_{i\in\alpha}x_{i}\ell\left(\frac{x_{i}}{p_{i}}\right),\quad\forall\
k\in\alpha.$

This means that
$\ell\left(\frac{x_{k_{1}}}{p_{k_{1}}}\right)=\ell\left(\frac{x_{k_{2}}}{p_{k_{2}}}\right)$
for any distinct $k_{1},k_{2}\in\alpha$. Since $\ell$ is strictly increasing,
we obtain $\frac{x_{k_{1}}}{p_{k_{1}}}=\frac{x_{k_{2}}}{p_{k_{2}}}$ for any
$k_{1},k_{2}\in\alpha$. Hence, we get $\mathbf{x}=\mathbf{p}_{\alpha}$. This
shows
$\textbf{Fix}(\mathcal{R})=\bigcup\limits_{\alpha\subset\mathbf{I}_{m}}\\{\mathbf{p}_{\alpha}\\}$.

$(ii)$ We now describe the set $\textbf{NE}(\mathcal{R})$ of all Nash
equilibria of the discrete-time replicator equation. Namely, we now show
$\textbf{NE}(\mathcal{R})=\\{\mathbf{p}\\}$. For the sake of convenience, we
set
$\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{y},\mathbf{x}):=\langle\mathbf{y},\varepsilon\mathbf{\Phi}_{m}(\frac{\mathbf{x}}{\mathbf{p}})\rangle$
for all $\mathbf{x},\mathbf{y}\in\mathbb{S}^{m-1}$. For any
$\mathbf{y}\in\mathbb{S}^{m-1}$, it is obvious that

$\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{p},\mathbf{p})=\varepsilon\sum\limits_{i\in\mathbf{I}_{m}}p_{i}\ell\left(\frac{p_{i}}{p_{i}}\right)=\varepsilon\ell(1)=\varepsilon\sum\limits_{i\in\mathbf{I}_{m}}y_{i}\ell\left(\frac{p_{i}}{p_{i}}\right)=\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{y},\mathbf{p}),$

which means that $\mathbf{p}\in\textbf{NE}(\mathcal{R})$.

Let $\mathbf{x}\in\textbf{NE}(\mathcal{R})$ be any Nash equilibrium where
$\alpha:=\textup{supp}(\mathbf{x})$. We then want to prove that
$\alpha=\mathbf{I}_{m}$. We assume the contrary, i.e.,
$\alpha\neq\mathbf{I}_{m}$. Let $k\in\mathbf{I}_{m}\setminus\alpha$ (since
$\mathbf{I}_{m}\setminus\alpha\neq\emptyset$). Since $-1<\varepsilon<0$ and
$\mathbf{x}\in\textbf{NE}(\mathcal{R})$, we face the contradiction

$0=\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{e}_{k},\mathbf{x})\leq\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{x},\mathbf{x})=\varepsilon\sum_{i\in\alpha}x_{i}\ell\left(\frac{x_{i}}{p_{i}}\right)<0.$

This shows $\alpha=\mathbf{I}_{m}$. In this case, we have for any
$\mathbf{y}\in\mathbb{S}^{m-1}$

$\varepsilon\max\limits_{i\in\mathbf{I}_{m}}\ell\left(\frac{x_{i}}{p_{i}}\right)\leq\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{y},\mathbf{x})=\varepsilon\sum_{i\in\mathbf{I}_{m}}y_{i}\ell\left(\frac{x_{i}}{p_{i}}\right)\leq\varepsilon\min\limits_{i\in\mathbf{I}_{m}}\ell\left(\frac{x_{i}}{p_{i}}\right).$

Since $\mathbf{x}\in\textbf{NE}(\mathcal{R})$, we must have

$\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{x},\mathbf{x})=\max\limits_{\mathbf{y}\in\mathbb{S}^{m-1}}\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{y},\mathbf{x})=\varepsilon\min\limits_{i\in\mathbf{I}_{m}}\ell\left(\frac{x_{i}}{p_{i}}\right).$

On the other hand, we have

$\mathcal{E}_{\varepsilon\mathbf{\Phi}_{m}}(\mathbf{x},\mathbf{x})=\varepsilon\sum_{i\in\mathbf{I}_{m}}x_{i}\ell\left(\frac{x_{i}}{p_{i}}\right)\leq\varepsilon\min\limits_{i\in\mathbf{I}_{m}}\ell\left(\frac{x_{i}}{p_{i}}\right).$

Consequently, we obtain
$\ell\left(\frac{x_{k}}{p_{k}}\right)=\min\limits_{i\in\mathbf{I}_{m}}\ell\left(\frac{x_{i}}{p_{i}}\right)$
for any $k\in\mathbf{I}_{m}$. Since $\ell$ is strictly increasing, we obtain
that $\frac{x_{k_{1}}}{p_{k_{1}}}=\frac{x_{k_{2}}}{p_{k_{2}}}$ for any
$k_{1},k_{2}\in\mathbf{I}_{m}$ or equivalently $\mathbf{x}=\mathbf{p}$. This
shows $\textbf{NE}(\mathcal{R})=\\{\mathbf{p}\\}$, i.e., the unique Wardrop
optimal flow $\mathbf{p}$ is the only Nash equilibrium.

$(iii)$ We now show that the only stable fixed point is the Wardrop optimal
flow $\mathbf{p}=(p_{1},\cdots,p_{m})$ which is also the only Nash
equilibrium.

On the one hand, according to Proposition 2, if $|\alpha|<m$ for
$\alpha\subset\mathbf{I}_{m}$ (which means $\alpha\neq\mathbf{I}_{m}$) then
the fixed point $\mathbf{p}_{\alpha}$ of the face $\mathbb{S}^{|\alpha|-1}$
(which means different from $\mathbf{p}$) is not stable. Indeed, for any small
neighborhood $U(\mathbf{p}_{\alpha})\subset\mathbb{S}^{m-1}$ of the fixed
point $\mathbf{p}_{\alpha}$ there is a point $\mathbf{x}\in
U(\mathbf{p}_{\alpha})\cap\textup{int}\mathbb{S}^{m-1}$ such that an orbit of
the interior point $\mathbf{x}$ converges to the Wardrop optimal flow
$\mathbf{p}$. Therefore, the fixed point $\mathbf{p}_{\alpha}$ of the face
$\mathbb{S}^{|\alpha|-1}$ is not stable whenever $|\alpha|<m$ for any
$\alpha\subset\mathbf{I}_{m}$ (which means $\alpha\neq\mathbf{I}_{m}$).

On the other hand, according to Proposition 2, the Wardrop optimal flow
$\mathbf{p}\in\textup{int}\mathbb{S}^{m-1}$ is stable. Namely, for any ball
$U_{R}(\mathbf{p})\Subset\textup{int}\mathbb{S}^{m-1}$ of radius $R$ there
exists a (small) ball $U_{r}(\mathbf{p})\Subset\textup{int}\mathbb{S}^{m-1}$
of radius $r$ (we may choose $r=\frac{\bar{p}R}{m}$) such that
$\|\mathcal{R}^{(n)}(\mathbf{x})-\mathbf{p}\|_{1}\leq R$ for any
$n\in\mathbb{R}$ whenever $\|\mathbf{x}-\mathbf{p}\|_{1}\leq r$.

Indeed, it follows from Proposition 3

$\bar{p}\left[\max\limits_{i\in\mathbf{I}_{m}}\frac{x_{i}}{p_{i}}-\min\limits_{i\in\mathbf{I}_{m}}\frac{x_{i}}{p_{i}}\right]\leq\|\mathbf{x}-\mathbf{p}\|_{1}\leq
r$

Since $-1<\varepsilon<0$, for
$k_{1}\in\mathsf{MinInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right)$ and
$k_{2}\in\mathsf{MaxInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right)$ it follows
from (1) that

$\displaystyle\frac{\left(\mathcal{R}(\mathbf{x})\right)_{k_{1}}}{p_{k_{1}}}=\frac{x_{k_{1}}}{p_{k_{1}}}\left[1-\varepsilon\sum\limits_{i=1}^{m}x_{i}\left(\ell\left(\frac{x_{i}}{p_{i}}\right)-\ell\left(\frac{x_{k_{1}}}{p_{k_{1}}}\right)\right)\right]\geq\frac{x_{k_{1}}}{p_{k_{1}}},$
$\displaystyle\frac{\left(\mathcal{R}(\mathbf{x})\right)_{k_{2}}}{p_{k_{2}}}=\frac{x_{k_{2}}}{p_{k_{2}}}\left[1+\varepsilon\sum\limits_{i=1}^{m}x_{i}\left(\ell\left(\frac{x_{k_{2}}}{p_{k_{2}}}\right)-\ell\left(\frac{x_{i}}{p_{i}}\right)\right)\right]\leq\frac{x_{k_{2}}}{p_{k_{2}}}.$

Since

$\mathsf{MinInd}\left(\frac{\mathcal{R}^{(n)}(\mathbf{x})}{\mathbf{p}}\right)=\mathsf{MinInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right)\quad\textup{and}\quad\mathsf{MaxInd}\left(\frac{\mathcal{R}^{(n)}(\mathbf{x})}{\mathbf{p}}\right)=\mathsf{MaxInd}\left(\frac{\mathbf{x}}{\mathbf{p}}\right),$

by repeating this process, we obtain

$\displaystyle\min_{i\in\mathbf{I}_{m}}\frac{x_{i}}{p_{i}}\leq\cdots\leq\min_{i\in\mathbf{I}_{m}}\frac{(\mathcal{R}^{(n)}(\mathbf{x}))_{i}}{p_{i}}\leq\cdots\leq\max_{i\in\mathbf{I}_{m}}\frac{(\mathcal{R}^{(n)}(\mathbf{x}))_{i}}{p_{i}}\leq\cdots\leq\max_{i\in\mathbf{I}_{m}}\frac{x_{i}}{p_{i}}$

Hence, we have

$\displaystyle\left[\max_{i\in\mathbf{I}_{m}}\frac{(\mathcal{R}^{(n)}(\mathbf{x}))_{i}}{p_{i}}-\min_{i\in\mathbf{I}_{m}}\frac{(\mathcal{R}^{(n)}(\mathbf{x}))_{i}}{p_{i}}\right]\leq\cdots\leq\left[\max_{i\in\mathbf{I}_{m}}\frac{x_{i}}{p_{i}}-\min_{i\in\mathbf{I}_{m}}\frac{x_{i}}{p_{i}}\right]\leq\frac{r}{\bar{p}}.$

Consequently, it follows from Proposition 3

$\displaystyle\|\mathcal{R}^{(n)}(\mathbf{x})-\mathbf{p}\|_{1}\leq
m\left[\max_{i\in\mathbf{I}_{m}}\frac{(\mathcal{R}^{(n)}(\mathbf{x}))_{i}}{p_{i}}-\min_{i\in\mathbf{I}_{m}}\frac{(\mathcal{R}^{(n)}(\mathbf{x}))_{i}}{p_{i}}\right]\leq
m\cdot\frac{r}{\bar{p}}=R.$

for any $n\in\mathbb{R}$ whenever $\|\mathbf{x}-\mathbf{p}\|_{1}\leq r$. This
means that the Wardrop optimal flow $\mathbf{p}=(p_{1},\cdots,p_{m})$ is the
stable fixed point.

$(iv)$ Due to Proposition 2, any interior orbit starting from any interior
initial point $\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$ converges to the
Wardrop optimal flow $\mathbf{p}\in\textup{int}\mathbb{S}^{m-1}$. This
completes the proof of Theorem A.

##  4 Conclusion

A part of the brain neural network that relies on chemical synapses for
communication can be regarded as a transportation network. The primary
objective of this paper was to gain insight into specific mechanisms and
phenomena within the neural network of the brain by applying methods
originally designed for solving traffic assignment problems in transportation
networks. In this regard, the impairment, deterioration, degradation, or
decline of the functionality of synapses, occurring due to elevated levels of
toxic neuropeptides/neurotransmitters during the neurotransmission process,
can be linked to a traffic congestion problem in a transportation network.
Therefore, when investigating the dynamic relationship between synaptic damage
and the concentration of toxic neuropeptides/neurotransmitters during the
neurotransmission process, it is plausible to adapt “Wardrop’s first and
second principles”. Although these principles were originally formulated to
describe optimal flow distributions in transportation networks, we believed
that they could be applied effectively within the framework of the neural
network in the brain. This was a novel approach adopted in this paper.

For the sake of simplicity, our primary focus was on the investigation of the
signal transmission process between two neurons through chemical synapses. The
connection between two neurons, established through unidirectional synaptic
links, was visualized as a parallel network between two nodes. Within the
framework of the neurotransmission process, increased levels of toxic
neuropeptides/neurotransmitters damage the synaptic links. To investigate this
correlation, we made an assumption that each synaptic link is associated with
a synaptic damage function. This synaptic damage function depends on a
concentration of toxic neuropeptides/neurotransmitters on a synaptic link that
measures its damage. In this context, we presented a complete manifestation of
“Wardrop’s first and second principles” within a neural network of the brain.
This was achieved by introducing two novel concepts: neuropeptide’s
(neurotransmitter’s) equilibrium and synapses optimum. By definition, the
neuropeptide/neurotransmitter equilibrium refers to a distribution of one-unit
neuropeptides/neurotransmitters that leads to uniform damage across all
synaptic links. Obviously, a such kind of distributions that leads to uniform
damage across all synaptic links does not generally minimize the cumulative
damage experienced by all synapses. The primary objective of experts in the
domains of synapses and neurotransmitters of Alzheimer’s disease is to reduce
and minimize the cumulative damage experienced by all synapses in the brains
of individuals afflicted with Alzheimer’s disease. Therefore, a distribution
of one-unit neuropeptides/neurotransmitters is called the optimal state of
synapses (synapses optimum) if it minimizes the cumulative damage experienced
by all synapses. Consequently, within the context of a neural network of the
brain, an analogous concept to the price of anarchy can be called as the price
of cognition. This represents the most unfavorable ratio between the overall
impairment caused by toxic neuropeptide’s (neurotransmitter’s) equilibrium in
comparison to the optimal state of synapses (synapses optimum). To put it
differently, the price of cognition measures the loss of cognitive ability
resulting from increased concentrations of toxic
neuropeptides/neurotransmitters. Finally, a replicator equation was proposed
within this framework that leads to the establishment of the synapses optimum
during the neurotransmission process. Our model represents a high-level
simplification and abstraction of the natural neurotransmission process
involving two neurons. Nevertheless, we envisioned that this mathematically
abstract model could serve as a source of motivation to instigate novel
experimental, mathematical, and computational research avenues in the field of
contemporary neuroscience.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Declaration of data availability statement

The authors declare that data sharing is not applicable to this article as no
new data were created or analyzed in this study.

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We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface $\Sigma$ on itself by proving that:

* All conjugacy classes of $\MCG(\Sigma)$ are meager for every $\Sigma$,

* $\MCG(\Sigma)$ has a somewhere dense conjugacy class if and only if $\Sigma$ has at most two maximal ends and no non-displaceable finite-type subsurfaces,

* $\MCG(\Sigma)$ has a dense conjugacy class if and only if $\Sigma$ has a unique maximal end and no non-displaceable finite-type subsurfaces.

Our techniques are based on model-theoretic methods developed by Kechris, Rosendal and Truss in [9, 17].

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