\documentclass[twoside]{article} \usepackage{amsfonts, amsthm, amsmath} \pagestyle{myheadings} \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)} \vspace{\bigskipamount} \\ % 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{} % \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}: $$\label{1.1} u(x,t)=\int_{-\infty}^t ds \int_{-\infty}^{\infty} dy h(t-s)k(x-y)S(u(y,s))$$ 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: $$\label{1.2} \partial u(x,t)/\partial t + u(x,t) = k*S(u)(x,t),$$ 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 $00$ 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 $$\label{2.3} u(\xi) = J_c*S(u)(\xi).$$ 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{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} 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 $$\label{2.5} -\alpha(c) \phi_c'(x)+\phi_c(x) -S(\phi_c)(x) >-\epsilon_0/2.$$ 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 \begin{thebibliography}{00} \frenchspacing \bibitem{kn:bc} P. W. Bates and F. Chen, \newblock Periodic travelling wave solutions of an integrodifferential model for phase transition, \newblock {\em Electronic J. Differential Equations}, {\bf 26} (1999) 1-19. \bibitem{kn:BC2} P. W. Bates and A. Chmaj, \newblock A discrete convolution model for phase transitions, \newblock {\em Arch. Rational Mech. Anal.}, {\bf 150} (1999), 281-305. \bibitem{kn:bfrw} P. W. Bates, P. C. Fife, X. Ren, and X. Wang, \newblock Traveling waves in a nonlocal model of phase transitions, \newblock {\em Arch. Rat. Mech. Anal.,} {\bf 138 }(1997), 105-136. \bibitem{kn:fchen} F. Chen, \newblock{Almost Periodic Traveling Waves of Nonlocal Evolution Equations} \newblock {\em J. Nonlinear Anal.},{\bf 50 } (2002), 807--838. \bibitem{kn:cem} Z. Chen, B. Ermentrout, B. McLeod, \newblock Traveling fronts for a class of non-local convolution differential equations. \newblock{\em Appl. Anal.} {\bf 64 } (1997), no. 3-4, 235--253. \bibitem{kn:em} B. Ermentrout, B. McLeod, \newblock Existence and uniqueness of travelling waves for a neural network. Proc. \newblock {\em Roy. Soc. Edinburgh Sect. A} {\bf 123 } (1993), no. 3, 461--478. \bibitem{kn:xchen} X. Chen, \newblock Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations, \newblock {\em Adv. Diff. Eqs., } \newblock {\bf 2 }(1997), 125-160. \bibitem{kn:dopt} A. de Masi, E. Orlandi, E. Presutti, and L. Triolo, \newblock Uniqueness of the instanton profile and global stability in nonlocal evolution equations, \newblock {\em Rend. Math., } \newblock {\bf 14 }(1994), 693-723. \bibitem{kn:ks} M. Katsoulakis and P. S. Souganidis, \newblock Interacting particle systems and generalized mean curvature evolution, \newblock {\em Arch. Rat Mech. Anal.,} {\bf 127 }(1994), 133-157. \end{thebibliography} \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}