\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 61, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE--2003/61\hfil Convergence and periodicity]
{Convergence and periodicity in a delayed network of neurons with
threshold nonlinearity}

\author[S. Guo, L. Huang, \& J. Wu \hfil EJDE--2003/61\hfilneg]
{Shangjiang Guo, Lihong Huang, \& Jianhong Wu}

\address{Shangjiang Guo  \hfill\break
College of Mathematics and Econometrics, Hunan
University,\hfill\break
 Changsha, Hunan 410082, China}
\email{shangjguo@etang.com}

\address{Lihong Huang \hfill\break
College of Mathematics and Econometrics, Hunan
University,\hfill\break
 Changsha, Hunan 410082, China}
\email{llhuang@hnu.net.cn}

\address{Jianhong Wu\hfill\break
Department of Mathematics and Statistics, York
University,\hfill\break Toronto, Ontario, M3J 1P3, Canada}
\email{wujh@mathstat.yorku.ca}

\date{}
\thanks{Submitted June 30, 2002. Revised January 23, 2003. Published May 26, 2003.}
\subjclass[2000]{34K25, 34K13, 92B20}
\keywords{Neural networks, feedback, McCulloch-Pitts nonlinearity,
\hfill\break\indent
one-dimensional map, convergence, periodic solution}

\begin{abstract}
  We consider an artificial neural network where the signal
  transmission is of a digital (McCulloch-Pitts) nature and is
  delayed due to the finite switching speed of neurons (amplifiers).
  The discontinuity of the signal transmission functions, however,
  makes it difficult to apply the existing dynamical systems theory
  which usually requires continuity and smoothness. Moreover,
  observe that the dynamics of the network completely depends on the
  connection weights, we distinguish several cases to discuss the
  behaviors of their solutions. We show that the dynamics of the
  model can be understood in terms of the iterations of a
  one-dimensional map. As, a result, we present a detailed analysis
  of the dynamics of the network starting from non-oscillatory
  states and show how the connection topology and synaptic weights
  determine the rich dynamics.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}
\maketitle

\section{Introduction}

    In this paper, we consider the following model for an
 artificial neural network of two neurons,
$$ \begin{gathered}
 \dot{x}=-\mu x+a_{11}f(x(t-\tau))+a_{12}f(y(t-\tau)),\\
 \dot{y}=-\mu y+a_{21}f(x(t-\tau))+a_{22}f(y(t-\tau)),
 \end{gathered}
 \eqno{(1.1)}
$$
where $\dot{x}={\rm d}x/{\rm d}t$, $x(t)$ and $y(t)$ denote the
activation of two neurons, $\mu>0$ is the decay rate, $\tau>0$ is
the synaptic transmission delay, $a_{ij}$ with $1\leq i,j\leq 2$
are the synaptic weights, $f:\mathbb{R}\to \mathbb{R}$ is the
activation function. Such a model describes the computational
performance of a Hopfield net \cite{Hopfield} where each neuron is
represented by a linear circuit consisting of a resistor and a
capacitor, and each neuron is connected to another via the
nonlinear activation function $f$ multiplied by the synaptic
weights $a_{ij}$ ($i\neq j$). We also allow that a neuron has
self-feedback and signal transmission is delayed due to the finite
switching speed of neurons.

Networks of two neurons have been used as prototypes for us to
understand the dynamics of large networks with delayed activation
functions, but much of the existing work has concentrated on the
case of a smooth activation function (see, for example,
\cite{Chen1, Chen2,Destexhe,Guo3,Olien,Pakdaman1,Wei}). In this
paper, we consider the McCulloch-Pitts activation function
$$
f(\xi)=\begin{cases}
-\delta, & \mbox{if $\xi>0$,}\\
\delta, & \mbox{if $\xi\leq 0$,}
\end{cases}\eqno{(1.2)}
$$
where $\delta\neq 0$ is a given constant. This case arises when
the signal transmission is of digital nature: a neuron is either
fully active or completely inactive. Very little has been done in
this case since results of the aforementioned references cannot be
applied as the dynamical systems theory heavily used in these
references usually requires the continuity and smoothness of the
nonlinear terms. In \cite{Guo1,Guo2,Huang,Huang2,Pakdaman2},
model equation (1.1) with a piecewise constant activation function
is studied when the synaptic connection topology satisfies
[$a_{11}=a_{22}=0$, $a_{21}=a_{12}=1$] or [$a_{11}=a_{22}=0$,
$a_{21}=-a_{12}=1$], and more generally,
 [$a_{11}+a_{12} =0$, $a_{11} >0$,
                  $a_{21} <0$, $a_{21}<a_{22}\leq -a_{21}$].

To simplify the presentation, we first rescale the variables by
$$
t^\ast=\mu t,\quad \tau^\ast=\mu\tau,\quad
x^\ast(t^\ast)=\frac{\mu}{\delta}x(t), \quad
y^\ast(t^\ast)=\frac{\mu}{\delta}y(t),\quad
f^\ast(\xi)=\frac{1}{\delta}f(\frac{\delta}{\mu}\xi), \eqno{(1.3)}
$$
 and then drop the $\ast$ to get
$$
\begin{gathered}
\dot{x}=-x+a_{11}f(x(t-\tau))+a_{12}f(y(t-\tau)),\\
\dot{y}=-y+a_{21}f(x(t-\tau))+a_{22}f(y(t-\tau))
\end{gathered} \eqno {(1.4)}
$$
 with
 $$
 f(\xi)= \begin{cases}
 -1,  & \mbox{if $\xi>0$,}\\
 1, & \mbox{if $\xi\leq 0$.}
 \end{cases} \eqno{(1.5)}
 $$
  Let
 $$
 a=a_{11}+a_{12},\quad b=a_{21}+a_{22},\quad
 c=a_{11}-a_{12},\quad d=a_{21}-a_{22}.
\eqno{(1.6)}
 $$
 We can rewrite (1.4) as
 $$
 \begin{gathered}
 \dot{x} = -x+\frac{a}{2}\left[f(x(t-\tau))+f(y(t-\tau))\right]
          +\frac{c}{2}\left[f(x(t-\tau))-f(y(t-\tau))\right],\\
 \dot{y} = -y+\frac{b}{2}\left[f(x(t-\tau))+f(y(t-\tau))\right]
          +\frac{d}{2}\left[f(x(t-\tau))-f(y(t-\tau))\right].
 \end{gathered} \eqno {(1.7)}
 $$

To state our main results, we set
the phase space $X=C([-\tau,0];\mathbb{R}^2)$
 as the Banach space of continuous mappings from $[-\tau,0]$ to
 $\mathbb{R}^2$ equipped with the sup-norm, see \cite{Hale}.
 Note that for each given
 initial value $\Phi=(\varphi, \psi)^T\in X$,
 one can solve system (1.7)
 on intervals $[0,\tau]$, $[\tau,2\tau]$, $\cdots$ successively to
 obtain a unique mapping
 $\left (x^\Phi,y^\Phi\right)^T:[-\tau, \infty)\to \mathbb{R}^2$ such
 that $x^\Phi\mid_{[-\tau,0]}=\varphi$, $y^\Phi\mid_{[-\tau,0]}=\psi$,
 $\left(x^\Phi,y^\Phi\right)^T$ is continuous for all $t\ge -\tau$, almost
 differentiable and satisfies (1.7) for $t>0$. This gives a unique solution
 of (1.7) defined for all $t\ge -\tau$. In applications, a network usually starts
from a constant (or nearly constant) state. Therefore, in this
paper, we shall concentrate on the case where each component of
$\Phi$ has no sign change on $[-\tau,0]$. More precisely, we
consider
 $\Phi \in X^{+,+}\bigcup X^{+,-}\bigcup X^{-,+}\bigcup X^{-,-}=X_0$, where
 \begin{align*}
 C^{\pm}=&\big\{ \pm \varphi :
 \; \varphi:[-\tau,0]\to [0, \infty) \mbox{ is continuous and}\\
 &\quad\mbox{has only finitely many zeros on }[-\tau,0]\big\}
\end{align*}
 and
 $$
 X^{\pm,\pm}=\left \{\Phi \in X; \Phi =(\varphi, \psi)^T,
 \varphi \in C^\pm\mbox { and $\psi \in C^\pm$}\right \}.
 $$
 Clearly, all constant initial values (except $0$) are contained in $X_0$.
Our analysis shows that the semiflow defined by system (1.7) on
$X_0$ (in other words, the behavior of a solution $\left
(x^\Phi(t),y^\Phi(t)\right)^T$ of system (1.7) with initial value
$\Phi\in X_0$) is completely determined by the value $(\varphi(0),
\psi(0))^T$ and the synaptic connection topology.

Guo, Huang and Wu \cite{Guo1} showed that using form (1.2) and some
simple changes of variables, we can see that the semiflow defined
by the system
$$
\begin{gathered}
\dot{u}= -u+\frac{1}{2}f(u(t-\tau))-\frac{1}{2}f(v(t-\tau)),\\
\dot{v}= -v-\frac{1+B}{2}f(u(t-\tau))+\frac{1-B}{2}f(v(t-\tau))
\end{gathered} \eqno {(1.8)}
$$
with $B\geq 0$ is topologically equivalent to that of (1.7)-(1.5)
while one of the following four conditions is satisfied:
\begin{itemize}
\item[(A1)]  $a=0$, $b\leq 0$, $c>0$, $d<0$
\item[(A2)]  $a\leq 0$, $b= 0$, $c>0$, $d<0$
\item[(A3)]  $a>0$, $b >  0$, $c=0$, $d\geq 0$
\item[(A4)]  $a>0$, $b >  0$, $c\leq 0$, $d= 0$.
\end{itemize}

\begin{theorem}[\cite{Guo1}] \label{thmA} Let
$\omega = 2\ln (2e^{\tau}-1)$,
$M = (1-e^{-\tau})(e^{\tau}-\frac{B}{B+1})$,
$m = \frac{1-e^{-\tau}}{B+e^{-\tau}}$,
$\eta = (\varphi (0)+\psi (0))/(1-\varphi (0)) \geq 0$,
the  $\omega$-periodic function $q: \mathbb{R}\to\mathbb{R}$ be
$$
q(t)=\begin{cases}
 e^{-(t+\tau)}-1,  & \mbox{if }-\omega/2\leq t\leq 0,\\
 (e^{-\tau}-2)e^{-t}+1, & \mbox{if } 0<t\leq \omega/2,
 \end{cases}
 $$
and polynomials
\begin{align*}
h(B) &=B^3(e^{-\tau}-1-e^{\tau})+B^2(e^{2\tau}-3e^{\tau}+e^{-\tau}+
         e^{-2\tau}-3)\\
 & \quad  +B(2e^{2\tau}-e^{\tau}+e^{-2\tau}-4)+e^{2\tau}+e^{\tau}-e^{-\tau}-1,\\
g(x)&=(Be^{\tau}-B-1)x^2
    +[(1+3B)(e^{\tau}-1)+Be^{-\tau}-B(B+e^{-\tau})(e^{\tau}+1)]x\\
    & \quad  -B(B-1)(e^{\tau}-1).
\end{align*}
Then the behavior of the solution $\left(u(t),v(t)\right)^T$ of system
(1.8) with initial value $\Phi=(\varphi,\psi)^T\in X^{-,+}$
and $\varphi (0)+\psi (0)\geq 0$ is as follows:
\begin{itemize}
\item[(i)] Suppose that $B=2(1-e^{-\tau})$ and $\tau>\ln 2$. If $\eta\in [0,m]$,
then $\left(u(t),v(t)\right)^T$ is eventually periodic with
minimal period $\omega$; If $\eta\in (m,M)$, then
$\left(u(t),v(t)\right)^T$ approaches the periodic solution
corresponding to $\eta =m$ as $t\rightarrow \infty$; If $\eta\in
[M,\infty)$, then $\left(u(t),v(t)\right)^T$ tends
to $(0,B)^T$ as $t \to \infty$.

\item[(ii)] Suppose that $B=2(1-e^{-\tau})$ and $\tau<\ln 2$. If $\eta\in [0,m]$,
then $\left(u(t),v(t)\right)^T$ is eventually periodic with
minimal period $\omega$;
    If $\eta\in (m,\infty)$, then $\left(u(t),v(t)\right)^T$ tends
to $(0,B)^T$ as $t \to \infty$.

\item[(iii)] Suppose that $0\leq B<2(1-e^{-\tau})$ and $\tau\geq\ln 2$ or $0\leq B\leq B^{\ast}_1$
and $\tau<\ln 2$. Then $\left(u(t),v(t)\right)^T$  approaches the
periodic solution $\left(-q(t),q(t)\right)^T$ as $t \to \infty$, where $B^{\ast}_1$ is the unique positive zero of $h(B)$.\\
    (iv) Suppose that $B>2(1-e^{-\tau})$ and $\tau\leq\ln 2$
or $B\geq B^{\ast}_2$ and $\tau>\ln 2$. Then
$\left(u(t),v(t)\right)^T$ tends to $(0,B)^T$ as $t \to \infty$, where $B^{\ast}_2$ is the unique positive zero of $h(B)$.\\
    (v) Suppose that $B^{\ast}_1<B<2(1-e^{-\tau})$ and $\tau<\ln 2$. Then
there must exist $T_1\geq 0$ and $\Phi_1=(\varphi_1,\psi_1)^T\in
X^{-,+}$ with $\varphi_1 (0)+\psi_1 (0)> 0$ such that for $t\geq
T_1$, the solution $\left(u^1(t),v^1(t)\right)^T$ of (1.8) with
initial value $\Phi_1$ is periodic. Moreover, as $t \to \infty$,
every other solution $\left(u(t),v(t)\right)^T$ of system (1.8)
with initial value $\Phi=(\varphi,\psi)^T\in X^{-,+}$ and $\varphi
(0)+\psi (0)>0$ either tends to $(0,B)^T$ or
approaches the periodic solution $\left(-q(t),q(t)\right)^T$.

\item[(vi)] Suppose that $2(1-e^{-\tau})<B<B^{\ast}_2$ and $\tau>\ln 2$. Then
there must exist $T_2\geq0$ and $\Phi_2=(\varphi_2,\psi_2)^T\in
X^{-,+}$ with $\varphi_2 (0)+\psi_2 (0)> 0$ such that for $t\geq
T_2$, the solution $\left(u^2(t),v^2(t)\right)^T$ of (1.8) with
initial value $\Phi_2$ is periodic and the minimal period is
$2\tau+\ln[(2-2e^{-\tau}-B)x^{\ast}_2
+(1-e^{-\tau})^2+3-2e^{-\tau}]$, where $x^*_2$ is the positive
zero of $g(x)$. Moreover, as $t \to \infty$, every solution
$\left(u(t),v(t)\right)^T$ of system (1.8) with initial value
$\Phi=(\varphi,\psi)^T\in X^{-,+}$ and $\varphi (0)+\psi (0)>0$
either tends to $(0,B)^T$
or approaches the periodic solution $\left(u^2(t),v^2(t)\right)^T$.

\item[(vii)] Suppose that $B=1$ and $\tau =\ln 2$. If $\eta\in [0,M)$, then
$\left(u(t),v(t)\right)^T$ is eventually periodic; If $\eta\in
[M,\infty)$, then $\left(u(t),v(t)\right)^T \to (0,B)^T$ as $t \to
\infty$.
\end{itemize}
\end{theorem}

Guo, Huang and Wu \cite{Guo2} also showed that the semiflow defined
by the system
$$
\begin{gathered}
\dot{u}= -u+\frac{1+m}{2}f(u(t-\tau))
       +\frac{1-m}{2}f(v(t-\tau)),\\
\dot{v}= -v+\frac{1+m}{2}f(u(t-\tau))
        +\frac{1-m}{2}f(v(t-\tau)).
\end{gathered}\eqno {(1.9)}
$$
with $m>0$ is topologically equivalent to
that of (1.7)-(1.5) when one of the following four conditions are
satisfied:\begin{itemize}
\item[(B1)]  $a>0$, $b> 0$, $c>0$, $d>0$, $ad=bc$
\item[(B2)]  $a>0$, $b< 0$, $c>0$, $d<0$, $ad=bc$
\item[(B3)]  $a<0$, $b> 0$, $c>0$, $d<0$, $ad=bc$
\item[(B4)]  $a>0$, $b> 0$, $c<0$, $d<0$, $ad=bc$.
\end{itemize}

\begin{theorem}[\cite{Guo2}] \label{thmB}
Every solution
$\left(u(t),v(t)\right)^T$ of system (1.9) with initial value
$\Phi=(\varphi,\psi)^T\in X_0$ is either eventually periodic with
minimal period $\omega$ or approaches the periodic solution
$\left(q(t),q(t)\right)^T$ as $t \to \infty$, where constant
$\omega$ and $\omega$-periodic function $q(t)$ are defined as in
Theorem \ref{thmA}.
\end{theorem}

In this paper, we consider the following cases:
\begin{itemize}
\item[(H1)]  $a\leq 0$, $b\leq 0$, $c\leq 0$, $d\geq 0$
\item[(H2)]  $a>0$, $b\leq 0$, $c\leq 0$, $d\geq 0$
\item[(H3)]  $a\leq 0$, $b>0$, $c\leq 0$, $d\geq 0$
\item[(H4)]  $a\leq 0$, $b\leq 0$, $c\leq 0$, $d<0$
\item[(H5)]  $a\leq 0$, $b\leq 0$, $c>0$, $d\geq 0$
\item[(H6)]  $a<0$, $b>0$, $c>0$, $d>0$
\item[(H7)]  $a>0$, $b<0$, $c<0$, $d<0$
\item[(H8)]  $a<0$, $b>0$, $c<0$, $d<0$
\item[(H9)]  $a>0$, $b<0$, $c>0$, $d>0$.
\end{itemize}
Let $\left (x(t),y(t)\right )^T$ be a solution  of (1.7) with
initial value  $\Phi \in X_0$. In this paper, we shall obtain the
following results:

\begin{theorem} \label{thm1}
Suppose that (H1) holds. Then as $t \to \infty$, $\left(x(t),y(t)\right)^T $
tends to  $(-a,-b)^T$
provided $\Phi \in X^{+,+}$, to $(c,d)^T$ provided $\Phi \in
X^{-,+}$, to $(a,b)^T$ provided $\Phi \in X^{-,-}$, and to
$(-c,-d)^T$ provided $\Phi \in X^{+,-}$.
\end{theorem}

\begin{theorem} \label{thm2} \begin{itemize}
\item[(i)] If (H2) holds, then as $t \to\infty$,
 $\left(x(t),y(t)\right)^T$ tends to $(c,d)^T$  provided $\Phi \in X^{+,+}\bigcup X^{-,+}$, and
  to $(-c,-d)^T$ provided $\Phi \in X^{+,-}\bigcup X^{-,-}$;
\item[(ii)] If (H3) holds, then
as $t \to\infty$,
 $\left(x(t),y(t)\right)^T$ tends to $(-c,-d)^T$ provided
$\Phi \in X^{+,+}\bigcup X^{+,-}$, and to $(c,d)^T$ provided
 $\Phi \in X^{-,+}\bigcup X^{-,-}$;
\item[(iii)] If (H4) holds, then as $t \to\infty$, $\left(x(t),y(t)\right)^T$ tends to $(-a,-b)^T$ provided $\Phi \in X^{+,+}\bigcup X^{+,-}$,
and to $(-a,-b)^T$ provided $\Phi \in X^{-,+}\bigcup X^{-,-}$;
\item[(iv)] If (H5) holds, then
as $t \to\infty$, $\left(x(t),y(t)\right)^T$ tends to $(-a,-b)^T$
provided $\Phi \in X^{+,+}\bigcup X^{-,+}$, and to $(a,b)^T$
provided $\Phi \in X^{+,-}\bigcup X^{-,-}$.
 \end{itemize}
\end{theorem}

Theorems \ref{thm1} and \ref{thm2} show that a simple network
described by (1.7) can be used as an associative memory device
because points representing the stored memories are locally stable
in some sense, and from any initial state close to one of these
attractors which represents partial knowledge of the memory stored
at the attractor, the trajectory is driven by the system to the
attractor, hence producing the full retrieval of the stored
memory. By Theorems \ref{thm1} and \ref{thm2}, system (1.7) has a
point as the global attractor if we further restrict the
parameters as follows:

\begin{corollary} \label{coro1}
Suppose that the parameters $a, b, c$ and $d$
satisfy one of the following conditions:
(1) $ab\leq 0$, $c=d=0$;
(2) $a=b=0$, $cd\geq 0$.
Then trajectories of system (1.7) starting from non-oscillatory
states converge to $(0,0)^T$.
\end{corollary}

We now consider the remaining cases.

\begin{theorem} \label{thm3}
If one of the two conditions (H6) and (H7)
holds, then there exist $\Phi_0 =(\varphi_0,\psi_0)^T \in X_0$ and
$T_0 \geq 0$ such that the solution
$(x^{\Phi_0}(t),y^{\Phi_0}(t))^T$ of (1.7) with initial value
$\Phi_0$ is periodic for $t \geq T_0$. Moreover,
$\lim_{t\rightarrow\infty} [x^\Phi(t)-x^{\Phi_0}(t)] = 0$ and
$\lim_{t\rightarrow\infty} [y^\Phi(t)-y^{\Phi_0}(t)] = 0$ for
every solution $(x^\Phi(t),y^\Phi(t))^T$ of (1.7) with $\Phi=
(\varphi,\psi)^T \in X_0$.
\end{theorem}

This theorem shows that when we restrict the initial value $\Phi$ to
$X_0$, then system (1.7) has a unique limit cycle which is the
global attractor. Note that this represents significant
improvement over a corresponding theorem in \cite{Huang2}. The
proof, elementary but technical, will be presented in Section 3.
The basic idea is to show that a typical trajectory of (1.7), when
described in the 2-dimensional Euclidean space (not the phase
space), is spiraling and rotates round the point $(0,0)$ (Section
2).

\begin{theorem} \label{thm4}
If one of the two conditions (H8) and (H9)
holds, then there exist $\Phi_0 =(\varphi_0,\psi_0)^T \in X_0$ and
$T_0 \geq 0$ such that the solution
$(x^{\Phi_0}(t),y^{\Phi_0}(t))^T$ of (1.7) with initial value
$\Phi_0$ is periodic for $t \geq T_0$, and
$(-x^{\Phi_0}(t),-y^{\Phi_0}(t))^T$ is also a solution of (1.7).
 Moreover, either $\lim_{t\rightarrow\infty}
[x^\Phi(t)-x^{\Phi_0}(t)] = 0$ and $\lim_{t\rightarrow\infty}
[y^\Phi(t)-y^{\Phi_0}(t)] = 0$ or
$\lim_{t\rightarrow\infty}[x^\Phi(t)+x^{\Phi_0}(t)] = 0$
and $\lim_{t\rightarrow\infty}[y^\Phi(t)+y^{\Phi_0}(t)] = 0$
for every solution
$(x^\Phi(t),y^\Phi(t))^T$ of (1.7) with $\Phi=
(\varphi,\psi)^T \in X_0$.
\end{theorem}

Therefore, system (1.7) may have two stable limit cycles.
Moreover, if we restrict initial value $\Phi$ to $X_0$, then every
solution of system (1.7) approaches one of the limit cycles as
$t\to \infty$. Theorems \ref{thm3} and \ref{thm4} show that a
simple two neuron model network is capable of producing and
sustaining periodic behaviors. It is worthy of noticing that
periodic sequences of neural impulses are of fundamental
significance for the control of dynamic functions of the human
body. Therefore,
 it is of great interest to understand various mechanisms
of neural networks which cause and sustain such periodic activities.


\section{Preliminary results}

 In this section, we establish several technical lemmas, which play
 important roles in the proof of our main results.
For the sake of simplicity, for the remaining part of this paper,
for a given $s\in [0, \infty)$ and a continuous function
$z: [-\tau, \infty)\rightarrow \mathbb{R}$, we define $z_s : [-\tau, 0]
\rightarrow \mathbb{R}$ by $z_s(\theta)=z(s+\theta)$ for $\theta
\in [-\tau, 0]$.

\begin{lemma} \label{lm2.1}
The semiflow defined by model (1.7) with
parameters $a, b, c$ and $d$ satisfying (H2) is topologically
equivalent to that defined by model (1.7) with parameters
 $a, b, c$ and $d$ satisfying any one of (H3), (H4) and (H5).
\end{lemma}

\begin{proof} We consider only the topological equivalence between
the semiflow defined by (1.7) under the condition (H2) and that
defined by model (1.7) under condition (H3). The remaining cases
can be dealt with analogously.

If (H2) holds, we can further redefine variables in (1.7) by
$$
x^\ast (t)=y(t),\quad  y^\ast (t)=x(t),\quad  a^\ast =b,\quad
b^\ast =a, \quad c^\ast =-d,\quad d^\ast =-c
$$
and then drop the $\ast$ to get (1.7) where the new parameters $a,
b, c$ and $d$ satisfy (H3). The converse holds true as well. This
justifies the claimed equivalence, according to the definition of
topological equivalence in \cite{Hale}. We complete the proof of
Lemma \ref{lm2.1}.
\end{proof}

Using similar arguments, we can also establish the following
lemmas.

\begin{lemma} \label{lm2.2}
The semiflow defined by the system
$$
\begin{gathered} \dot{x} =-x+\frac{1-A}{2}f(x(t-\tau))-\frac{1+A}{2}f(y(t-\tau)),\\
\dot{y} = -y+\frac{1+B}{2}f(x(t-\tau))+\frac{1-B}{2}f(y(t-\tau)),
\end{gathered} \eqno {(2.1)}
$$
with $A>0$ and $B>0$ is topologically equivalent to that defined
by model (1.7) with parameters $a, b, c$ and $d$ satisfying either
(H6) or (H7).
\end{lemma}

\begin{lemma} \label{lm2.3}
The semiflow defined by the system
$$
\begin{gathered}
\dot{x} =-x-\frac{1+M}{2}f(x(t-\tau))+\frac{1-M}{2}f(y(t-\tau)),\\
\dot{y} =-y-\frac{1-N}{2}f(x(t-\tau))+\frac{1+N}{2}f(y(t-\tau)),
\end{gathered}\eqno {(2.2)}
$$
with $M>0$ and $N>0$ is topologically equivalent to that defined
by model (1.7) with parameters
 $a, b, c$ and $d$ satisfying either (H8) or (H9).
\end{lemma}

\begin{lemma} \label{lm2.4}
If $(x(t),y(t))^T$ is the solution of system (1.7)(or (2.1), (2.2))
with initial value
$\Phi=(\varphi,\psi)^T\in X_0$,
 then the solution of (1.7)(respectively, (2.1), (2.2)) with initial value
 $\Phi=(-\varphi,-\psi)^T\in X_0$ is $(-x(t),-y(t))^T$.
\end{lemma}

We now describe the transition from one component of $X_0$ to another.

\begin{lemma} \label{lm2.5}
Suppose that  $(x(t),y(t))^T$ is a solution of (2.1) with
initial value in $X_0$. Then
\begin{itemize}

\item[(i)]  if there exists some $t_0\geq 0$ such that
$\left(x_{t_0},y_{t_0}\right)^T\in X^{+,+}$, then there exists some
$t^{\ast}_0\geq t_0$ such that
$\left(x_{t^{\ast}_0+\tau},y_{t^{\ast}_0+\tau}\right)^T\in X^{+,-}$

\item[(ii)]  if there exists some $t_0\geq 0$ such that
$\left(x_{t_0},y_{t_0}\right)^T\in X^{+,-}$, then there exists some
$t^{\ast}_0\geq t_0$ such that
$\left(x_{t^{\ast}_0+\tau},y_{t^{\ast}_0+\tau}\right)^T\in X^{-,-}$

\item[(iii)]  if there exists some $t_0\geq 0$ such that
$\left(x_{t_0},y_{t_0}\right)^T\in X^{-,-}$, then there exists some
$t^{\ast}_0\geq t_0$ such that
$\left(x_{t^{\ast}_0+\tau},y_{t^{\ast}_0+\tau}\right)^T\in X^{-,+}$

\item[(iv)]  if there exists some $t_0\geq 0$ such that
$\left(x_{t_0},y_{t_0}\right)^T\in X^{-,+}$, then there exists some
$t^{\ast}_0\geq t_0$ such that
$\left(x_{t^{\ast}_0+\tau},y_{t^{\ast}_0+\tau}\right)^T\in X^{+,+}$.
\end{itemize}
\end{lemma}

\begin{proof} We consider only the case where
$\left(x_{t_0},y_{t_0}\right)^T \in X^{+,+}$ for some $t_0\geq 0$,
the remaining cases can be dealt with analogously. In view of (2.1),
we have
$$
\begin{gathered}
\dot{x} =-x+A,\\
\dot{y}=-y-1
\end{gathered}\eqno{(2.3)}
$$
for $t\in [t_0, t_0+\tau]$ except at most finitely many $t$.
Therefore, the variation-of-constants formula and the continuity of
solutions yield
$$
\begin{gathered}
x(t)=[x(t_0)-A]e^{-(t-t_0)}+A,\\
y(t)=[y(t_0)+1]e^{-(t-t_0)}-1
\end{gathered}\eqno{(2.4)}
$$
for all $t\in \left[t_0, t_0+\tau\right]$. Let
$t_1$ be the first zero of $x(t)\cdot y(t)$ in
$(t_0, \infty)$. Then $(x(t),y(t))^T$ satisfies system (2.3)
for $t\in \left(t_0, t_1+\tau\right)$ except at most finitely many
$t$, and so (2.4) holds for $t\in \left[t_0, t_1+\tau\right]$.
It follows from (2.4) that
$$
t_1=t_0+\ln [y(t_0)+1],
$$
which implies that
\begin{gather*}
x_{t_1+\tau}(\theta) = \frac{x(t_0)-A}{y(t_0)+1}e^{-(\tau+\theta)}+A>0,\\
y_{t_1+\tau}(\theta) = e^{-(\tau+\theta)}-1<0
\end{gather*}
for $\theta \in (-\tau, 0],$ and so $(x_{t_1+\tau},y_{t_1+\tau})^T
\in X^{+,-}$. Thus, claim (i) holds with $t^{\ast}_0=t_1$.
\end{proof}

\begin{lemma} \label{lm2.6}
Suppose that $(x(t),y(t))^T$ is a solution of (2.2)
 with initial value in $X_0$. Then
\begin{itemize}
\item[(i)] if there exists some $t_0\geq 0$ such that
$\left(x_{t_0},y_{t_0}\right)^T\in X^{+,+}$, then there exists some
$t^{\ast}_0\geq t_0$ such that
$\left(x_{t^{\ast}_0+\tau},y_{t^{\ast}_0+\tau}\right)^T\in X^{+,-}$

\item[(ii)] if there exists some $t_0\geq 0$ such that
$\left(x_{t_0},y_{t_0}\right)^T\in X^{+,-}$, then there exists some
$t^{\ast}_0\geq t_0$ such that
$\left(x_{t^{\ast}_0+\tau},y_{t^{\ast}_0+\tau}\right)^T\in X^{+,+}$.
\end{itemize}
\end{lemma}

\begin{proof} (i) Using equations (2.2), (1.5) and the fact that $\left(x_{t_0},
y_{t_0}\right)^T \in X^{+,+}$, we have
$$
\begin{gathered}
\dot {x} =-x+M,\\
\dot{y}=-y-N
\end{gathered}\eqno{(2.5)}
$$
for $t\in \left[t_0, t_0+\tau\right]$ except at most finitely many $t$.
Therefore, the variation-of-constants formula and the continuity of
solutions yield
$$
\begin{gathered}
x(t)=[x(t_0)-M]e^{-(t-t_0)}+M,\\
y(t)=[y(t_0)+N]e^{-(t-t_0)}-N
\end{gathered}\eqno{(2.6)}
$$
for all $t\in \left[t_0, t_0+\tau\right]$. Let
$t_1$ be the first zero of $x(t)\cdot y(t)$ in
$(t_0, \infty)$. Then $(x(t),y(t))^T$ satisfies system (2.5)
for $t\in \left(t_0, t_1+\tau\right)$ except at most finitely many
$t$, and so (2.6) holds for $t\in \left[t_0, t_1+\tau\right]$.
It follows from (2.6) that
$$
t_1=t_0+\ln [y(t_0)+N]-\ln N,
$$
which implies that
\begin{gather*}
x_{t_1+\tau}(\theta)=\frac{x(t_0)-M}{y(t_0)+N}Ne^{-(\tau+\theta)}+M>0\\
y_{t_1+\tau}(\theta)=e^{-(\tau+\theta)}-1<0
\end{gather*}
for $\theta \in (-\tau, 0],$ and so
$(x_{t_1+\tau},y_{t_1+\tau})^T \in X^{+,-}$.

\noindent (ii) Using equations (2.1) and (1.5), as well as the fact that
$\left(x_{t_0}, y_{t_0}\right)^T \in X^{+,-}$, we have
$$
\begin{gathered}
\dot {x} =-x+1,\\
\dot{y}=-y+1
\end{gathered} \eqno{(2.7)}
$$
for $t\in \left[t_0, t_0+\tau\right]$ except at most finitely many $t$.
Therefore, the variation-of-constants formula and the continuity of
solutions yield
$$
\begin{gathered}
x(t)=[x(t_0)-1]e^{-(t-t_0)}+1,\\
y(t)=[y(t_0)-1]e^{-(t-t_0)}+1
\end{gathered} \eqno{(2.8)}
$$
for all $t\in \left[t_0, t_0+\tau\right]$. Let
$t_1$ be the first zero of $x(t)\cdot y(t)$ in
$(t_0, \infty)$. Then $(x(t),y(t))^T$ satisfies system (2.7)
for $t\in \left(t_0, t_1+\tau\right)$ except at most finitely many
$t$, and so (2.8) holds for $t\in \left[t_0, t_1+\tau\right]$.
It follows from (2.8) that
$$
t_1=t_0+\ln [1-y(t_0)],
$$
which implies
\begin{gather*}
x_{t_1+\tau}(\theta)=\frac{x(t_0)-1}{1-y(t_0)}e^{-(\tau+\theta)}+1>0,\\
y_{t_1+\tau}(\theta)=1-e^{-(\tau+\theta)}>0
\end{gather*}
for $\theta \in (-\tau, 0],$ and so
$(x_{t_1+\tau},y_{t_1+\tau})^T \in X^{+,+}$.
This completes the proof.
\end{proof}

   In what follows, we will need the following continuous functions:
\begin{gather}
f_1(x)=\frac{(A B+1-e^{-\tau})e^{-\tau}x+(A+1)(A
B+1)(1-e^{-\tau})}
{(B+1)e^{-\tau}x+(A+1)(B+1)(1-e^{-\tau})+2e^{-\tau}-e^{-2\tau}},
\tag{2.9}\\
T(x)=2\tau+\ln
\left[(B+1)e^{-\tau}x+(A+1)(B+1)(1-e^{-\tau})+2e^{-\tau}-e^{-2\tau}\right]
\tag{2.10},\\
f_2(x)=A+\frac{B(x-A)e^{-2\tau}+B(1-A)(1+B)(1-e^{-\tau})}
{[1+B(1-e^{-\tau})](1+B-e^{-\tau})} \tag{2.11}
\end{gather}
for $x\in [0, \infty)$.

\begin{lemma} \label{lm2.7}
The function $f_1: [0,\infty) \rightarrow \mathbb{R}$ is continuous and has a unique 2-period point
$x^\ast_1$ which is stable (that is $\lim_{n\to
\infty}f^n_1(x)=x^\ast_1$ for all $x\in (0,\infty )$). Moreover,
$f_1$ is monotonically increasing provided that
$\max\{A,B\}<1-e^{-\tau}$ or $\min\{A,B\}>1-e^{-\tau}$, and is
monotonically decreasing provided that
$\min\{A,B\}<1-e^{-\tau}<\max\{A,B\}$.
\end{lemma}

\begin{proof} It is easy to see that $f_1$ is continuous and has a
unique fixed point $x^\ast_1\in [0,\infty)$. We first consider the
case where $\max\{A,B\}<1-e^{-\tau}$ or $\min\{A,B\}>1-e^{-\tau}$.
Then it is easy to verify that $f_1$ is monotonically increasing,
$f_1(x)>x$ for $x\in [0,x^\ast_1)$ and $f_1(x)<x$ for $x\in
(x^\ast_1, \infty)$. Thus, the fixed point $x^\ast_1$ is stable,
i.e., $\lim_{n\to \infty}f^n_1(x)=x^\ast_1$ for all $x\in
[0,\infty)$, where $f^n_1(x)=f_1\left(f^{n-1}_1(x)\right)$. It is
obvious that $x^\ast_1$ is also a 2-period point of $f_1(x)$,
i.e., the fixed point of $f^2_1(x)$. We can claim that $x^\ast_1$
is the unique 2-period point of $f_1(x)$. Suppose to the contrary,
let $u^\ast\in [0,\infty)$ be another 2-period point of $f_1(x)$,
namely, $f^2_1(u^\ast)=u^\ast\neq x^\ast_1$. Since $\lim_{n\to
\infty}f^n_1(u^\ast)=x^\ast_1$, let $n=2k$, then
$$
u^\ast=\lim_{k\to \infty}(f^2_1)^k(u^\ast)=
\lim_{n\to \infty}f^{2k}_1(u^\ast)=x^\ast_1,
$$
which is a contradiction. Therefore, The function $f_1$ has the
unique 2-period point $x^\ast_1$. Also since $f^2_1(x)>f_1(x)>x$
for $x\in [0,x^\ast_1)$ and $f^2_1(x)<f_1(x)<x$ for $x\in
(x^\ast_1, \infty)$, it is easy to see that $x^\ast_1$ is the
stable 2-period point of $f_1(x)$. On the other hand, if
$\min\{A,B\}<1-e^{-\tau}<\max\{A,B\}$, then $f_1$ is monotonically
decreasing. However, $f^2_1$ is monotonically increasing and has
one and only one unique $x^\ast_1$, which implies that $x^\ast_1$
is the stable 2-period point of $f_1(x)$. This completes the
proof.
\end{proof}

\begin{lemma} \label{lm2.8}
The function $f_2: [0,\infty) \rightarrow \mathbb{R}$ is continuous,
monotonically increasing and has a unique fixed point $x^\ast_2$ which
is stable.
\end{lemma}

The proof of Lemma \ref{lm2.8} is similar to that of Lemma \ref{lm2.7} and thus it is omitted.

\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{thm1}]
We consider only the case where $\Phi \in X^{+,+}$. The remaining cases can
be dealt analogously.
Using the definition of $f$ and the fact $\Phi \in X^{+,+}$,
$x(t)$ and $y(t)$ satisfy
$$
\begin{gathered}
\dot {x} =-x-a,\\
\dot{y}=-y-b
\end{gathered}\eqno{(3.1)}
$$
for $t \in [0,\tau]$. Therefore, for $t \in [t_0,t_0+\tau]$  we have
$$
\begin{gathered}
x(t)=(\varphi(0)+a)e^{-t}-a, \\
y(t)=(\psi(0)+b)e^{-t}-b,
\end{gathered}\eqno{(3.2)}
$$
which implies that $x_{\tau}(\theta)=x(\tau+\theta)>0$ and
$y_{\tau}(\theta)=y(\tau+\theta)>0$ for $\theta \in (-\tau,0)$,
and so $x_{\tau}\in C^+$ and $y_{\tau} \in C^+$. Repeating this
argument on $[\tau,2\tau]$, $[2\tau,3\tau]$, $\cdots$,
successively, we obtain that $x_t \in C^+$ and $y_t \in C^+$ for
all $t \geq 0$. Therefore, (3.1) holds for almost all $t>0$. It
follows that $(x(t),y(t))^T\to (-a,-b)^T$ as $t \to \infty$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
In view of Lemma \ref{lm2.1}, it suffices to
consider system (1.7) under the condition (H2). We distinguish
four cases in our discussions of the behaviors of a solution for
(1.7).
\\
{\bf Case 1.} $\Phi \in X^{-,+}$. Using a similar argument to that of
Theorem \ref{thm1}, we can show that $(x(t),y(t))^T\to (c,d)^T$
as $t \to \infty$.
\\
{\bf Case 2.} $\Phi \in X^{+,+}$. In view of the definition of
$f(\xi)$, $x(t)$ and $y(t)$ satisfy (3.1) for $t \in (0,\tau)$.
Therefore, (3.2) holds for $t \in [t_0,t_0+\tau]$. Let $t_1$ be
the first nonnegative zero of $x(t)\cdot y(t)$ on $[0, \infty)$.
Then for $t\in (0, t_1+\tau)$, (3.1) holds. Namely, (3.2) holds
for $t\in [0, t_1+\tau]$. In particular,
$$
t_1=\ln [\varphi(0)+a]-\ln a.
$$
It follows that
\begin{gather*}
x_{t_1+\tau}(\theta)=ae^{-(\tau+\theta)}-a<0, \\
y_{t_1+\tau}(\theta)=\frac{\psi(0)+b}{\varphi(0)+a}ae^{-(\tau+\theta)}-b>0
\end{gather*}
for $\theta \in (-\tau, 0],$ and so $(x_{t_1+\tau},
y_{t_1+\tau})^T \in X^{-,+}$. Then from the result for Case 1,
$(x(t),y(t))^T\to (c,d)^T$ as $t \to \infty$.
\\
{\bf Case 3} $\Phi \in X^{-,+}$. From the result for Case 1 and by
Lemma \ref{lm2.4}, it is easy to see that
$(x(t),y(t))^T\to (-c,-d)^T$ as $t \to \infty$.
\\
{\bf Case 4} $\Phi \in X^{-,-}$. From the result for Case 2 and by
Lemma \ref{lm2.4}, it is easy to see that
$(x(t),y(t))^T\to (-c,-d)^T$ as $t \to \infty$.
Thus the proof of Theorem \ref{thm2} is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
In view of Lemmas 2.2 and 2.5, it suffices to discuss the behavior of a
solution $(x(t),y(t))^T$ of
(2.1) with initial value $\Phi \in X^{++}$.  For the sake of
convenience, we introduce the parameter $u$:
$$
u=\frac{\varphi(0)+A\psi(0)}{1+\psi(0)}.
$$
We can show that the behavior of the solution $(x(t),y(t))^T$ as
$t \to \infty$ are completely determined by the value $u$. Let
$t_1$ be the first zero of $x(t)\cdot y(t)$ on $[0,\infty)$, then
from the proof of case (i) in Lemma \ref{lm2.5} it follows that
\begin{gather*}
t_1=\ln (1+\psi(0)), \quad
x(t_1)=\frac{\varphi(0)+A\psi(0)}{1+\psi(0)}=u\geq 0,\quad
y(t_1)=0,\\
x(t_1+\tau)=(u-A)e^{-\tau}+A>0,\quad
y(t_1+\tau)=e^{-\tau}-1<0.
\end{gather*}
Moreover, it is easy to see that $(x_{t_1+\tau},y_{t_1+\tau})^T \in
X^{+,-}$. Therefore, for $t\in (t_1+\tau,t_1+2\tau)$, we have $x(t-\tau)>0$,
$y(t-\tau)<0$ and $(x(t),y(t))^T$ satisfies
$$
\begin{gathered}
\dot {x} =-x-1,\\
\dot{y} =-y-B,
\end{gathered}\eqno{(3.3)}
$$
from which and the continuity of the solution it follows that
$$
\begin{aligned}
x(t) &=[x(t_1+\tau)+1]e^{t_1+\tau-t}-1 \\
     &=[(u-A)e^{-\tau}+A+1]e^{t_1+\tau-t}-1,\\
y(t) &=[y(t_1+\tau)+B]e^{t_1+\tau-t}-B   \\
     &=[B-1+e^{-\tau}]e^{t_1+\tau-t}-B
\end{aligned} \eqno{(3.4)}
$$
for $t\in [t_1+\tau,t_1+2\tau]$. Let $t_2$ be the second zero of
$x(t)\cdot y(t)$ on $[0,\infty)$. Then for $t\in
(t_1+\tau,t_2+\tau)$, $(x(t),y(t))^T$ satisfies (3.3). Thus,
(3.4) holds for $t\in [t_1+\tau,t_2+\tau]$. It follows from (3.4)
that
\begin{gather*}
t_2=t_1+\tau+\ln [(u-A)e^{-\tau}+A+1], \\
x(t_2+\tau)=e^{-\tau}-1<0,\\
y(t_2+\tau)=\frac{B-1+e^{-\tau}}{(u-A)e^{-\tau}+A+1}e^{-\tau}-B<0.
\end{gather*}
Moreover, it is easy to see that $(x_{t_2+\tau},y_{t_2+\tau})^T \in
X^{+,-}$. Therefore, for $t\in (t_2+\tau,t_2+2\tau)$, we have $x(t-\tau)>0$,
$y(t-\tau)<0$ and $(x(t),y(t))^T$ satisfies
$$
\begin{gathered}
\dot {x} =-x-A,\\
\dot{y}=-y+1.
\end{gathered}\eqno{(3.5)}
$$
Hence,
$$
\begin{aligned}
x(t) &=[x(t_2+\tau)+A]e^{t_2+\tau-t}-A\\
     &=[e^{-\tau}+A-1]e^{t_2+\tau-t}-A,\\
y(t) &=[y(t_2+\tau)-1]e^{t_2+\tau-t}+1  \\
     &=\big(\frac{B-1+e^{-\tau}}
     {(u-A)e^{-\tau}+A+1}e^{-\tau}-B-1\big)e^{t_2+\tau-t}+1
\end{aligned} \eqno{(3.6)}
$$
for $t\in [t_2+\tau,t_2+2\tau]$. Let $t_3$ be the third zero of $x(t)\cdot y(t)$ on
$[0,\infty)$. Then (3.5) holds for $t\in (t_2+\tau,t_3+\tau)$.
Namely, (3.6) holds for $t\in [t_2+\tau,t_3+\tau]$. Thus, it follows from (3.6) that
$$
\begin{aligned}
t_3 &= t_2+\tau+\ln (1-y(t_2+\tau)) \\
    &= t_1+2\tau+\ln\left[(B+1)e^{-\tau}x+(A+1)(B+1)(1-e^{-\tau})+2e^{-\tau}-e^{-2\tau}\right]  \\
    &= t_1+T(u)
\end{aligned}
$$
and
\begin{gather*}
\begin{aligned}
x(t_3) &= \frac{x(t_2+\tau)+A y(t_2+\tau)}{1-y(t_2+\tau)}\\
       &= \frac{(e^{-\tau}-1-A B)e^{-\tau}u+(A+1)(A B+1)(e^{-\tau}-1)}
           {(B+1)e^{-\tau}u+(A+1)(B+1)(1-e^{-\tau})+2e^{-\tau}-e^{-2\tau}}\\
       &= -f_1(u)<0,\\
\end{aligned}\\
y(t_3) = 0,
\end{gather*}
where the function $T$ and $f_1$ are defined as (2.10) and (2.9),
respectively. Let $t_4$ and $t_5$ be the next zeroes of $x(t)\cdot
y(t)$. Then from the above arguments and by Lemma \ref{lm2.1}, we have
$(x_{t_4+\tau},y_{t_4+\tau})^T \in X^{+,+}$ and
\begin{gather*}
t_5=t_3+T\left(f_1(u)\right)=t_1+T(u)+T\left(f_1(u)\right),\\
x(t_5)=f^2_1(u),\\
y(t_5)=0.
\end{gather*}
Thus, we can repeat the same argument to get a sequence
$$
u,\quad f_1(u),\quad f^2_1(u),\quad \cdots,\quad f^n_1(u),\quad
\cdots.
$$
In particular, the behavior of $(x(t),y(t))^T$ is determined by
the iteration $f_1$. By Lemma \ref{lm2.7}, the function $f_1$ has a
2-period point $x^\ast_1$. Namely, $f^2_1(x^\ast_1)=x^\ast_1$. Let
$(x^\ast(t),y^\ast(t))^T$ be a solution of (2.1) with the initial
value $\Phi^\ast=(\varphi^\ast,\psi^\ast)^T\in X^{+,+}$ satisfying
$[\varphi^\ast(0)+A\psi^\ast(0)]/[1+\psi^\ast(0)]=x^\ast_1$. Then
for $t\geq \ln (1+\psi^\ast(0))$, $(x^\ast(t),y^\ast(t))^T$ is
periodic with minimal period
$\omega=T(x^\ast_1)+T\left(f_1(x^\ast_1)\right)=2T(x^\ast_1)$.
Also since $x^\ast_1$ is the stable 2-period point, it is obvious
that the periodic solution $(x^\ast(t),y^\ast(t))^T$ is
attractive, i.e., every solution of initial value problem (2.1)
approaches $(x^\ast(t),y^\ast(t))^T$ as $t\to \infty$. Therefore,
it is a stable limit cycle and its uniqueness is guaranteed by the
uniqueness of the 2-period point of $f_1$. Thus, we complete the
proof of Theorem \ref{thm3}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4}]
In view of Lemmas 2.3 and 2.4, it
suffices to discuss the behavior of a solution $(x(t),y(t))^T$ of
(2.2) with initial value $\Phi \in X^{++}$.  For the sake of
convenience, we introduce the two parameters
\begin{gather*}
u = M+\frac{\varphi(0)-M}{\psi(0)+N}N,\\
\omega = 2\tau+\ln [1+N(1-e^{-\tau})]+\ln (1+N-e^{-\tau})-\ln N.
\end{gather*}
We show that the behavior of the solution
 $(x(t),y(t))^T$ as $t\to\infty$ is completely determined by the
value $u$.
Let $t_1$ be the first zero of $x(t)\cdot y(t)$ on $[0,\infty)$,
then from the proof of Lemma \ref{lm2.6} (i) we have
\begin{gather*}
t_1=\ln (\psi(0)+N)-\ln N, \\
x(t_1)=M+\frac{\varphi(0)-M}{\psi(0)+N}N=u\geq 0,\\
y(t_1)=0,\\
x(t_1+\tau)=(u-M)e^{-\tau}+M>0,\\
y(t_1+\tau)=Ne^{-\tau}-N<0.
\end{gather*}
Moreover, it is easy to see that $(x_{t_1+\tau},y_{t_1+\tau})^T \in
X^{+,-}$. This, together with the proof of Lemma \ref{lm2.6} (ii),
implies that the second zero of $x(t)\cdot y(t)$ is
$$
t_2=t_1+\tau+\ln [1-y(t_1+\tau)]=t_1+\tau+\ln [1+N(1-e^{-\tau})].
$$
Moreover,
$$
\begin{aligned}
x(t)&=[x(t_1+\tau)-1]e^{t_1+\tau-t}+1\\
    &=[(u-M)e^{-\tau}+M-1]e^{t_1+\tau-t}+1,\\
y(t)&=[y(t_1+\tau)-1]e^{t_1+\tau-t}+1 \\
    &=[Ne^{-\tau}-N-1]e^{t_1+\tau-t}+1
\end{aligned}\eqno{(3.7)}
$$
for all $t\in \left[t_1+\tau, t_2+\tau\right]$. It follows that
\begin{gather*}
x(t_2+\tau) = \frac{(u-M)e^{-\tau}+M-1}{1+N(1-e^{-\tau})}e^{-\tau}+1>0,\\
y(t_2+\tau) = 1-e^{-\tau}>0.
\end{gather*}
Moreover, it is easy to see that $(x_{t_2+\tau},y_{t_2+\tau})^T \in
X^{+,+}$. Again from the proof of Lemma \ref{lm2.6} (i),
the third zero of $x(t)\cdot y(t)$ is
\begin{align*}
t_3 &= t_2+\tau+\ln [y(t_2+\tau)+N]-\ln N\\
    &= t_1+2\tau+\ln [1+N(1-e^{-\tau})]+\ln (1+N-e^{-\tau})-\ln N\\
    &= t_1+\omega.
\end{align*}
Moreover,
\begin{gather*}
\begin{aligned}
x(t_3) &= M+\frac{x(t_2+\tau)-M)}{y(t_2+\tau)+N}N\\
       &= M+\frac{(u-M)e^{-2\tau}+(1+N)(1-M)(1-e^{-\tau})}
           {[1+N(1-e^{-\tau})](1+N-e^{-\tau})}N\\
       &= f_2(u)>0,
\end{aligned}\\
y(t_3) = 0,
\end{gather*}
where the function $f_2$ is defined as (2.11). Thus, we can repeat
the same argument to get a sequence
$$
u,\quad f_2(u),\quad f^2_2(u),\quad \cdots,\quad f^n_2(u),\quad
\cdots.
$$
Therefore, the behavior of $(x(t),y(t))^T$ as $t\rightarrow
\infty$ is determined by the iteration of the function $f_2$. By
Lemma \ref{lm2.8}, the function $f_2$ has a fixed point $x^\ast_2$.
Namely, $f_2(x^\ast_2)=x^\ast_2$. Let $(x^\ast(t),y^\ast(t))^T$ be
a solution of (2.2) with initial value
$\Phi^\ast=(\varphi^\ast,\psi^\ast)^T\in X^{+,+}$ satisfying
$M+N[\varphi^\ast(0)-M]/[\psi^\ast(0)+N]=x^\ast_2$. Then for
$t\geq \ln (\psi^\ast(0)+N)-\ln N$, $(x^\ast(t),y^\ast(t))^T$ is
periodic and is of the minimal period $\omega$. Also since
$x^\ast_2$ is the stable fixed point, the periodic solution
$(x^\ast(t),y^\ast(t))^T$ is attractive, i.e., every solution of
(2.2) with initial value $\Phi=(\varphi,\psi)^T\in X^{+,+}\bigcup
X^{+,-}$ approaches $(x^\ast(t),y^\ast(t))^T$ as $t\to \infty$. By
Lemma \ref{lm2.4}, every solution of (2.2) with initial value
$\Phi=(\varphi,\psi)^T\in X^{-,-}\bigcup X^{-,+}$ approaches the
solution $(-x^\ast(t),-y^\ast(t))^T$ as $t\to \infty$. Thus, we
complete the proof of Theorem \ref{thm3}.
\end{proof}

\section{Conclusions}
The model equation (1.1) with the McCulloch-Pitts nonlinearity (1.2)
describes a combination of analog and digital signal processing in
a network of two neurons with delayed feedback. For the sake of
convenience, we can transform system (1.1)--(1.2) to the form
(1.4)--(1.5) by the appropriate change of variables (1.3). Observe
that the dynamics of the network completely depends on the
connection weights, we distinguish several cases and discuss the
behaviors of solutions of (1.4). We show that the dynamics of the
model (1.4) can be understood in terms of the iterations of a
one-dimensional map. As a result, we obtain the convergence of
solutions as well as the existence, multiplicity and attractivity
of periodic solutions. Throughout the paper, we only consider the
case where the initial value $\Phi=(\varphi,\psi)^T\in X$ does not
change sign at the initial time interval. Moreover, the digital
nature of the sigmoid function allows us to relate equation (1.4)
to four systems of simple linear nonhomogeneous ordinary
differential equations. In future work, we shall describe the
dynamics of solutions of (1.1)--(1.2) with initial data in
$X\setminus X_0$ (i.e., solutions whose initial states oscillate
around $0$ with high frequencies).

\subsection*{Acknowledgments}
S. Guo and L. Huang's research was partially supported by the
Science Foundation of Hunan University, by the National Natural
Science Foundation of P. R. China (10071016), by the Foundation
for University Excellent Teacher by the Ministry of Education, and
by the Key Project of Chinese Ministry of Education (No [2002]78).
J. Wu's research was partially supported by the Natural Sciences
and Engineering Research Council of Canada, by the Network of
Centers of Excellence "Mathematics for Information Technology and
Complex Systems", and by Canada Research Chairs Program. The
authors express their sincere gratitude to the anonymous referee
who carefully read the manuscript and made remarks leading to
improvements on the presentation of this paper.

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