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\markboth{\hfil Travelling waves for a neural network \hfil EJDE--2003/??}
{EJDE--2003/??\hfil Fengxin Chen \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2003}(2003), No. 13, pp. 1--4. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
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  Travelling waves for a neural network
 %
\thanks{ {\em Mathematics Subject Classifications:} 35K55, 35Q99.
\hfil\break\indent 
{\em Key words:} Nonlocal phase transition,travelling waves, continuation. 
\hfil\break\indent 
\copyright 2003 Southwest Texas State University. \hfil\break\indent 
Submitted January 10, 2002. Published February 11, 2003.} }
\date{}
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\author{Fengxin Chen}
\maketitle

\begin{abstract}
 In this note, we give another proof of existence and uniqueness of
 travelling waves for a neural network equations and prove that
 all travelling waves are monotonic.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{rem}[theorem]{Remark}
\numberwithin{equation}{section}

\section{Introduction}

The following single-layer neural network over the real line was
introduced by  Ermentrout and Mcleod \cite{kn:em}:
\begin{equation}\label{1.1}
u(x,t)=\int_{-\infty}^t ds \int_{-\infty}^{\infty}
dy h(t-s)k(x-y)S(u(y,s))
\end{equation}
where $x\in {\mathbb R}$ and $t\in \mathbb R$; $u(x,t)$ is the
mean membrane potential of a patch of tissue at position $x$ and
at time $t$;  $S(u)$ is a nonlinear function and $S(u(x,t))$ is
the firing rate; $h$ and $k$ are nonnegative functions defined
$[0,\infty)$ and $\mathbb R$ respectively.  When $h(t)=e^{-t}$ for
$t>0$, then equation (\ref{1.1})  is equivalent to the following
differential equation:
\begin{equation}\label{1.2}
\partial u(x,t)/\partial t + u(x,t) = k*S(u)(x,t),
\end{equation}
where $ k*S(u)$ denotes the convolution of $k$ with $S(u)$, i.e.,
$k*S(u)(x,t)= \int_{-\infty}^{\infty}  k(x-y)S(u(y,t))dy$.

The existence and uniqueness of travelling waves of (\ref{1.1}) of
the form $u(x,t) = \phi(x-c t)$ satisfying $\phi(-\infty)=0$ and
$\phi(\infty)=1$ are established in \cite{kn:em}, where $\phi$ is a
smooth function, called  the wave profile,  and $c$ is a constant,
called the wave speed. A homotopy argument is employed to prove
the existence, which has fostered  other studies in similar topics
(see
\cite{kn:BC2, kn:bfrw, kn:fchen, kn:cem, kn:xchen,kn:dopt},
for example). This note serves to supplement the results obtained
in \cite{kn:em}, by applying results in  \cite{kn:xchen}, where a
comparison argument, together with constructions of appropriate
super- and sub- solutions, is used to study  travelling waves for
(\ref{1.2}).

First we state the conditions on $h$, $k$, and $S$. We assume that
\begin{enumerate}

\item[($\bf A1$)] $h\in C^1[0,\infty)$ is a positive function on
$[0,\infty)$ with $\int_{0}^\infty h(t)d t =1$ and
$\int_{0}^\infty  t h(t) d t<\infty$.

\item [($\bf A2$)]
$k$ is a nonnegative,
 continuous function on $\mathbb R$ with
$\int_{\mathbb R} k(x)d x =1$, $k'\in L^1({\mathbb R})$ and
$\mathop{\rm supp} J\bigcap (0,\infty)\neq \emptyset
 \neq \mathop{\rm supp} J\bigcap(-\infty,0)$.

\item [($\bf A3$)] $S\in C^1([0,1])$
satisfies that $S'(u)>0$, for $u\in [0,1]$,
 and that $f(u)=-u +S(u)$ has precisely
three zeros at $u=0, a, 1$ satisfying $f'(0)<0$ and $f'(1)<0$,
where $0<a<1$.
\end{enumerate}
Under the above assumptions, we can improve the results in
\cite{kn:em}:
\begin{theorem}\label{thm1.1}
Under the above assumptions on  $h$, $k$ and $S$, we have
\begin{enumerate}

\item[(a)] There exists a travelling wave solution
$u=\phi(x-c t)$ to (\ref{1.1}) satisfying $\phi\in C^1$,
$\phi(-\infty)=0$ and $\phi(\infty)=1$.

\item[(b)] Any travelling wave
solution to (\ref{1.1}) satisfying $\phi(-\infty)=0$ and
$\phi(\infty)=1$ is strictly  increasing.

\item [(c)] Traveling wave solution to (\ref{1.1}) is unique  module
spatial translation.
\end{enumerate}
\end{theorem}

\begin{rem}
\begin{enumerate} \rm
 \item[(a)] The monotonicity of travelling wave solutions to (\ref{1.1})
is established in \cite{kn:em} for special kernels $h$ and $k$ and
is conjectured for general case. Our result gives a positive
answer.

\item[(b)] For the existence and uniqueness in \cite{kn:em},  that $k$
is even and $h$ is monotonically decreasing is assumed.  While it
is natural, we can relax these restrictions.
\end{enumerate}
\end{rem}



\section{Proof of Theorem \ref{thm1.1}}

First we need the following result:

\begin{lemma}\cite{kn:xchen}\label{lem2.1} For any $k$ and  $S$
satisfying  ($\bf A2$) and ($\bf A3$) respectively, there exists
one and only one (modulo spatial translation) travelling wave
solution $u(x,t) = \phi(x-ct)$ to (\ref{1.2}) satisfying
$\phi(-\infty)=0$ and $\phi(\infty)=1$. Moreover, $\phi'>0$ for all
$x\in \mathbb R$.
\end{lemma}

For any $c\in \mathbb R$, let $J_c(\cdot) = \int_{0}^\infty
h(s)k(\cdot+cs)d s$. Then $J_c$ satisfies ($\bf A2$).  For each
$c\in \mathbb R$, by Lemma \ref{lem2.1}, there is a travelling
wave solution  $\phi_c(x- \alpha(c)t)$ to the equation (\ref{1.2})
with $k$ replaced by $J_c$, where $\phi_c$ is the profile and
$\alpha(c)$ is the wave speed, depending on $c$. Let $\xi =x-ct$.
Then the pair $(\phi_c, \alpha(c))$ satisfies the following
equations:
\begin{gather}
 -\alpha(c)\phi_c'(\xi) + \phi_c(\xi) - J_c*S(\phi_c)(\xi)=0, \label{2.1}\\
\phi(-\infty)=0, \mbox{   and   } \phi(\infty)=1.\label{2.2}
\end{gather}
On the other hand, a travelling wave solution $u=u(x-ct)$  to
(\ref{1.1}) satisfies
\begin{equation}\label{2.3}
u(\xi) = J_c*S(u)(\xi).
\end{equation}
Therefore, if $(u,c)$ is a travelling wave solution to (\ref{1.1}),
$(u,0)$ is a travelling wave solution to (\ref{1.2}) corresponding
to $k(x) = J_c(x)$. Similarly, if $(\phi_c, 0)$ is a travelling wave
solution to (\ref{1.2}) with $k(x) = J_c(x)$, then $(\phi_c,c)$ is a
travelling wave solution to (\ref{1.1}). Therefore to prove the
existence of a travelling wave, we only need to prove that there is
a $c\in \mathbb R$ such that $\alpha(c)=0$. To that end, we need:

\begin{lemma}
The wave speed $\alpha(\cdot)$ is a continuous function on $\mathbb
R$.
\end{lemma}

\begin{proof}
Let $c_0\in \mathbb R$ and $(\phi_{c_0},\alpha(c_0))$ be a travelling
wave solution to (\ref{1.2}) corresponding to $k=J_{c_0}$. Then,
$\phi_c'>0$ for all $x\in \mathbb R$ and $(\phi_c,\alpha(c))$ can be
obtained as a solution to (\ref{2.1}) by the Implicit Function
Theorem, applying in the neighborhood of $c_0$ (see \cite{kn:em},
for example). Therefore, $\phi(c)$ is indeed continuously
differentiable.
\end{proof}

\begin{lemma}
 $\alpha(c)<0$ for $c$ positively sufficiently large and  $\alpha(c)>0$ for
  $c$ negatively sufficiently large.
\end{lemma}

\begin{proof}
We only prove the lemma when  $c$ is positive. The other case can
be proved similarly. We can choose $z_0\in (0,1)$ such that
$\epsilon_0= S(z_0)-z_0>0$. For this $\epsilon_0$, we can choose
two positive constants $A=A(\epsilon_0)$ and $B=B(\epsilon_0)$
such that $(\int_0^A+\int_B^\infty) h(s) ds<\epsilon_0/8$ and
$(\int_{-\infty}^{-B}+\int_B^\infty) k(s) ds<\epsilon_0/8$. Since
$(\phi_c,\alpha(c))$ satisfies (\ref{2.1}), we have
\begin{equation}
\begin{aligned}
  \mbox{}&-\alpha(c)\phi_c'(x) + \phi_c(x) -S(\phi_c)(x) \\
 &=  \int_{0}^\infty h(s)\int_{-\infty}^\infty
k(x+cs -y)\{S(\phi_c(y))-S(\phi_c(x))\} dy\, ds \\
&\ge \int_{A}^B h(s)\int_{x+cs-B}^{x+cs+B} k(x+cs
-y)\{S(\phi_c(y))-S(\phi_c(x))\} dy\, ds -\epsilon_0/2
\end{aligned}\label{2.4}
\end{equation}
 where we have used the fact that $S(u(x))\le 1$. If $c\ge
 A^{-1}B$, then $y>x$  for $y$ in the range of the integration on the right
 of (\ref{2.4}). Therefore the integral on the right side of
 (\ref{2.4})
 is positive and
 \begin{equation}\label{2.5}
 -\alpha(c) \phi_c'(x)+\phi_c(x) -S(\phi_c)(x) >-\epsilon_0/2.
 \end{equation}
 Since $\phi_c(-\infty)=0$,  and $ \phi_c(\infty)=1$,
  we choose $x_0$ such that $\phi_c(x_0)=z_0$, Then we
 deduce from (\ref{2.5}) that $\alpha(c)\phi_c'(x_0)<0$. Therefore,
  $\alpha(c)<0$ since $\phi_c'(x_0)>0$.
\end{proof}



{\bf Proof of Theorem 1.1} By lemma 2.2 and 2.3, there is constant
$c$ such that $\alpha(c)=0$. The pair $(\phi_c,c)$ is the travelling wave
solution to (\ref{1.1}). By lemma 2.1, $\phi_c'>0$ for all $x$.  The
uniqueness is established  in \cite{kn:em}, where  uniqueness for
monotonic travelling waves is proved. \qed

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\noindent\textsc{Fengxin Chen}\\
Department of Applied Mathematics,\\
University of Texas at San Antonio, \\
San Antonio, TX 78249, USA\\
e-mail: feng@sphere.math.utsa.edu

\end{document}