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\markboth{\hfil Traveling-wave solutions \hfil}%
{\hfil Valentino Anthony Simpao \hfil}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc 15th Annual Conference on Applied Mathematics, Univ. of Central Oklahoma},
\newline
Electronic Journal of Differential Equations, Conference~02, 1999, pp. 133--136.
\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} \\
  Traveling-wave solutions of a modified Hodgkin-Huxley type neural model
via Novel analytical results for nonlinear transmission lines with arbitrary
$I(V)$ characteristics
\thanks{ {\em 1991 Mathematics Subject Classifications:} 
35L70, 92C20, 35K57.  \hfil\break\indent 
{\em Key words and phrases:}  Hodgkin-Huxley, 
hyperbolic quasilinear diffusion operator, \hfil\break\indent
non-linear transmission line, analytical solution. \hfil\break\indent
\copyright 2000 Southwest Texas State 
University  and University of North Texas. \hfil\break\indent 
Published January 21, 2000.} } 


\date{}
\author{Valentino Anthony Simpao}
\maketitle

\begin{abstract}
Herein an enhanced Hodgkin-Huxley (H-H) type model of neuron dynamics is
solved analytically via formal methods.  Our model is a variant of an earlier
one by M.A.  Mahrous and H.Y.  Alkahby [1].  Their modified model is
realized by a hyperbolic quasi-linear diffusion operator with time-delay
parameters; this compared to the original H-H model with standard parabolic
quasi-linear diffusion operator and no time-delay parameters.  Besides these
features, the present model also incorporates terms describing signal
dissipation into the background substrate (e.g., conductance to ground), making
it more experimentally amenable.  The solutions which results via the present
scheme are of traveling-wave profile, which agree qualitatively with those
observed in actual electro-physiological measurements made on the neural systems
originally studied by H-H These results confirm the physiological soundness of the
enhanced model and of the preliminary assumptions which motivated the present
solution strategy; the comparison of the present results with actual
electro-physiological data displays shall appear in later publications.
\end{abstract}


\section{Introduction}

Consider the nonlinear transmission-line model equation (viz. (3.8) in
Mahrous and Alkahby in \cite{m1})
\begin{equation} \label{1*}
(\partial_x^2 - \frac1{\theta^2} \partial_t^2) V
=\frac{2RC}{a} \partial_t V +\frac{2R}{a} I_i + \frac{2L}{a} \partial_t I_i\,.
\end{equation}
Where all the parameters are as defined in \cite{m1}, except for the
$J$-terms in their  section 5. In the present analysis,  only the
time-asymptotically stable expressions are being considered as $t\to\infty$
for the various   $J$-terms, particularly
$ J=\frac{\alpha_J(V)}{\alpha_J(V)+\beta_J(V)}\,.$
As a consequence of \cite{m1}   and the  present  $J$-term considerations,
the ionic current $I_i$ is here clearly a function of $V$
and the constant parameters of the system; the variation of $I_i$
with respect to  $(x,t)$ is implicit, being here determined exclusively by
$I_i(V(x,t))$.
Since traveling waves are physiologically useful constructs
[ubiquitous in natural phenomena], the present work is dedicated to obtaining
traveling-wave solutions to (\ref{1*}). Specifically sought are
solutions of form $V(x,t)=V(\mu_\pm)$ with $\mu_\pm=(x\pm \theta t)$.
Concerning  the empirically determined forms of the
$J$-terms in \cite{h1}, along  with the aforementioned stipulation about the
asymptotically stable terms, the particular  form of the ionic current
$I_i(V(x,t))$  is
\begin{eqnarray} \label{2*}
\lefteqn{ I_i(v) }\\
&=&\overline{G}_k \bigg( \frac{(0.1+0.01V)^4(V-V_k)}
{(e^{1+0.1V}-1)^4\big(0.125e^{v/80}+(0.1+0.01V)(e^{1+0.1V}-1)^{-1}\big)^4}
\bigg) \nonumber \\
&&+\overline{G}_{Na} \bigg( \frac{ 0.07e^{V/20}(2.5+0.1V)^3(V-V_{Na})}
{(e^{2.5+0.1V}-1)^3 \big(0.07e^{V/20}+ \frac 1{e^{3+0.1V}+1}\big)
\big( 4e^{V/18}+\frac{2.5+0.1v}{e^{2.5+0.1V}-1} \big)^3 }\bigg) \nonumber\\
&&+ \overline{G}_L(V-V_L)\,.  \nonumber
\end{eqnarray}
To solve (\ref{1*}), we consider an analytical result for a
general class of  Non Linear Transmission Line equations.


\section{Main result}

Consider the class of Non Linear Transmission Line (NLTL) equations, which
arise in the context of transmission line models for systems with
1-configuration space variable $x$ degree of freedom (the longitudinal axis
of the cable), a single time variable
$t$, and a specified but otherwise arbitrary dependence of the line current
$I(x,t)$ upon the line voltage $V(x,t)$,  i.e., $I(V(x,t))$.
Then
\begin{eqnarray} \label{1}
&\partial_x V(x,t) = -RI(x,t)-L\partial_t I(x,t)& \\
&\partial_x I(x,t) = -GV(x,t)-C\partial_t V(x,t)&  \nonumber
\end{eqnarray}
where  $R$, $G$, $L$, $C$  are the
constant resistance per unit of length, constant conductance (leakage loss
to ground) per unit of length, constant inductance per unit of length
and constant capacitance per unit of length.  Re-arranging (\ref{1}), we obtain
\begin{eqnarray}   \label{2}
\lefteqn{ \big[(\partial_xV(x,t))^2-LC(\partial_tV(x,t))^2\big]
D_{V(x,t)}^2I(V(x,t)) }  \\
\lefteqn{ +\big[\partial_x^2V(x,t)-LC\partial_t^2V(x,t)
-(GL+RC)\partial_tV(x,t)\big]
D_{V(x,t)} I(V(x,t)) }\nonumber \\
&=& GRI(V(x,t))\,, \hspace{7cm} \nonumber
\end{eqnarray}
where $x$ and $t$ are real variables and $G,L,R,C$ are constants.

Now define the characteristic variable as $\mu_\pm=x\pm t/\sqrt{LC}$.
Substituting the characteristic variable as the particular $V(x,t)=V(\mu_\pm)$,
the functional  dependence in (2) yields
\begin{eqnarray}  \label{3}
\lefteqn{ \big[(D_{\mu_\pm}V(\mu_\pm))^2-\frac{LC}{LC}(D_{\mu_\pm}V(\mu_\pm))^2\big]
D_{V(\mu_\pm)}^2I(V(\mu_\pm)) } \\
\lefteqn{ +\big[D_{\mu_\pm}^2V(\mu_\pm)-\frac{LC}{LC}D_{\mu_\pm}^2V(\mu_\pm)
-\frac{GL+RC}{\pm\sqrt{LC}} D_{\mu_\pm}V(\mu_\pm)\big]
D_{V(\mu_\pm)} I(V(\mu_\pm)) }\nonumber \\
&=& GRI(V(\mu_\pm))\,. \hspace{7.5cm} \nonumber
\end{eqnarray}
Simplifying this equation, we obtain
\begin{equation} \label{4}
-\frac{GL+RC}{\pm\sqrt{LC}} D_{V(\mu_\pm)} I(V(\mu_\pm))
D_{V(\mu_\pm)} V(\mu_\pm) = GRI(V(\mu_\pm))\,.
\end{equation}

Since $I(V(x,t))$  is specified but otherwise arbitrary, (\ref{4})
has an  analytical implicit solution given by
$$\int \frac{D_{V(\mu_\pm)} I(V(\mu_\pm))}{I(V(\mu_\pm))}\,dV(\mu_\pm)
= \int -\frac{\pm GR\sqrt{LC}}{GL+RC}D_{V(\mu_\pm)}\mu_\pm(V(\mu_\pm))
\,dV(\mu_\pm)
$$
Therefore,
$\ln(I(V))= -\pm GR\sqrt{LC}(\mu_\pm+\mu_{\mbox{\scriptsize const.}})/
(GL+RC)$ and
\begin{equation} \label{5}
I(V)=\exp\big(-\frac{\pm GR\sqrt{LC}}{GL+RC}
(\mu_\pm +\mu_{\mbox{\scriptsize const.}}) \big)\,.
\end{equation}

By the inversion theorem on power series \cite{a1}, the explicit analytical
form of $V$ ascends
\begin{equation} \label{6}
V(\mu_\pm)=\sum_{n=1}^\infty \frac 1{n!} D_V^{n-1}
\big(\frac V{I(V)}\big)^n \bigg|_{V=0}
\exp\big(-\frac{\pm GR\sqrt{LC}}{GL+RC}
(\mu_\pm+\mu_{\mbox{\scriptsize const.}}) \big)\,.
\end{equation}
Regarding the arbitrary constant $\mu_{\mbox{\scriptsize const.}}$, it
may be used to designate advances or delays in the time and/or space domains of the
solution.

With these results in place, consider the fundamental system of coupled
partial differential equations (\ref{1}) defining the transmission line equation
(\ref{1*}),
$$ \displaylines{
\partial_x v(x,t) =-ri_a(x,t) - l\partial_t i_a(x,t) \cr
\partial_x i_a(x,t) = -i_i(x,t)-c_a\partial_tv(x,t)\,. \cr
}$$
Identifying $v=V$, $i_a=I$, $l=L$, $c_a=C$, $r=R$, and
$i_i(x,t)= 2\pi a(C_m\partial_tV(x,t)+I_i(V(x,t))=-GV(x,t)\,,$
terms in (\ref{1}) with terms in \cite{m1}
(with the $(x,t)$ dependence suppressed for notational simplicity)
yields
$$
2\pi a(C_m\partial_tV(x,t)+I_i(V(x,t))=-GV(x,t)
$$
\begin{eqnarray*}
\lefteqn{ I_i(v) }\\
&=&\overline{G}_k \bigg( \frac{(0.1+0.01V)^4(V-V_k)}
{(e^{1+0.1V}-1)^4\big(0.125e^{v/80}+(0.1+0.01V)(e^{1+0.1V}-1)^{-1}\big)^4}
\bigg)  \\
&&+\overline{G}_{Na} \bigg( \frac{ 0.07e^{V/20}(2.5+0.1V)^3(V-V_{Na})}
{(e^{2.5+0.1V}-1)^3 \big(0.07e^{V/20}+ \frac 1{e^{3+0.1V}+1}\big)
\big( 4e^{V/18}+\frac{2.5+0.1v}{e^{2.5+0.1V}-1} \big)^3 }\bigg) \\
&&+ \overline{G}_L(V-V_L)\,.
\end{eqnarray*}
So the line current, defined in terms of the ionic current $I_i(V)$,
and $V(x,t)$ are given by
$$ \displaylines{
I(V)=\exp\big(-\frac{\pm GR\sqrt{LC}}{GL+RC}
(x\pm \frac {t}{\sqrt{LC}}+\mu_{\mbox{\scriptsize const.}}) \big)
\cr
V(\mu_\pm)=\sum_{n=1}^\infty \frac 1{n!} D_V^{n-1}
\big(\frac V{I(V)}\big)^n \bigg|_{V=0}
\exp\big(-\frac{\pm GR\sqrt{LC}}{GL+RC}
(x\pm \frac t{\sqrt{LC}}+\mu_{\mbox{\scriptsize const.}}) \big)\,.
}$$

Explicit calculation of the above formula with numerical values for the system
parameters indicates that the functional form, $V(x\pm t/\sqrt{LC})$,
 theoretically-predicted traveling-wave potential
solution matches the experimentally observed action potential of the neuron;
these results shall appear in later reports.

\begin{thebibliography}{0}

\bibitem{a1} G. Arfken, {\em Mathematical Methods for Physicists},
Academic Press, 1970, 2nd Ed, pp. 366-367.

\bibitem{h1} A. L. Hodgkin and A. F. Huxley, {\em Quantitative Description of
Membrane Current and its Application to Conduction and Excitation in
Nerve}, J. Physiol. London, 117 (1952), pp. 500-544.

\bibitem{m1} M. A. Mahrous and H.Y. Alkahby,
{\em Mathematical Model for Signal Transmission on Nerve
Axon with Time Delay}, Proceedings of the 14th Annual Conference on
Applied Mathematics, University of Central Oklahoma, 1998.

\end{thebibliography}

\noindent{\sc Valentino Anthony Simpao} \\
Mathematical Consultant Services \\
108 Hopkinsville St. \\
Greenville, Kentucky 42345 USA \\
e-mail: mcs007@muhlon.com Tel.: 502-338-5543

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