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\begin{document}
\Large

Following is a summary of  two variable models of excitable media.
Two variable models that we describe here are always of the form
\bea
C_m{dv\over dt} &=& F(v,w)\label{eq:fhn1}\\
{dw\over dt} &=& G(v,w)\label{eq:fhn2}
\ena
Typically, $v$ is a ``fast'' variable, while $w$ is a ``slow''
variable.  The remaining sections are to specify the functions
$F(v,w)$ and $G(v,w)$. 

\section{FitzHugh Nagumo}
This is the model that started the whole business
 and is a variant of the van der Pol equation.
\bea
F(v,w) &=& Av(v-\alpha)(1-v) -w,\\
G(v,w) &=& \epsilon(v-\gamma w).
\ena
with $0 < \alpha < {1\over 2}$, and $\epsilon$ ``small''.  Also,
$\gamma$ is chosen so that $Ag(1-\alpha)^2 <4$, so that $A(v-\alpha)(1-v) -{1\over \gamma}= 0 $ has
no real roots, that is, so that there is only one steady state
solution.


\section{McKean}
McKean  suggested two models, both with 
$F(v,w) = f(v) - w$ and $G(v,w) =  \epsilon(v-\gamma w)$.  For the first,
\beq
f(v) = \left\{\begin{array}{cc}-v &v<{\alpha\over 2}\\
v-\alpha&{\alpha\over2} <v <{1+\alpha\over 2}\\1-v&v >{1+\alpha\over 2}
\end{array}\right.
\eeq
where $0<\alpha < {1\over 2}$.

The second model suggested by McKean had
\beq
f(v) = \left\{\begin{array}{cc}-v &v<\alpha\\
1-v&v >\alpha
\end{array}\right.
\eeq
and $\gamma = 0$.  This is an extremely popular model with hundreds of
related papers.

\section{Barkley}
A model devised to give very fast 2D computations (the code is known as EZspiral)
\bea
F(v,w)&=& v(1-v)(v-{w+b\over a}),\\
G(v,w)&=& \epsilon(v-w).
\ena

\section{Puschino}
A piecewise linear model devised to match cardiac restitution
properties 
\bea
F(v,w) &=& f(v) -w\\
G(v,w) &=& {1\over \tau(v)}(v-w)
\ena
where
\beq
f(v) = \left\{\begin{array}{cl}-30v,&v<v_1\\\gamma
v-0.12,&v_1<v<v_2,\\-30(v-1),&v>v_2\end{array}\right.,\qquad \tau(v) =
\left\{\begin{array}{cl}2&v<v_1\\
16.6&v>v_1\end{array}\right.,
\eeq
with $v_1={0.12\over 30+\gamma}$, $v_2 = {30.12\over 30+\gamma}$.

\section{Aliev}
For the Aliev model,
\bea
F(v,w) &=&g_a(v-\beta)(v-\alpha)(1-v) - vw\\
G(v,w) &=& - \epsilon(v,w)( w + g_s(v-\beta)(v-\alpha-1) 
\ena
where
$
      \epsilon(v,w)  = \epsilon_1+ \mu_1{w\over v+\mu_2}$.

Reasonable parameter values are
$\beta= 0.0001$
    , $\alpha = 0.05$,
    $g_a = 8.0$,
  $g_s = 8.0$,
  $\mu_1 = 0.05$,
  $\mu_2 = 0.3$,
  $\epsilon_1 = 0.03$,
 $\epsilon_2 = 0.0001$.
Parameter values that give spiral breakup are $\alpha = 0.1$,
$g_a=8.0$, $g_s = 8.0$, $\mu_1=0.07$, $\mu_2 = 0.3$, and $\epsilon_1 =
0.01$.

\section{Reduced HH}

\bea
F(v,w)&=& g_Kw^4(v-v_K)
+g_{Na}m_\infty^3(v)(0.8-w)(v-v_{Na})\nonumber
\\&+&g_l(v-v_l)\\ G(v,w) &=&
\alpha_w(v)(1-h)-\beta_w(v)w
\ena
where
\beq
\alpha_w = 0.01{10-v\over \exp({10-v\over 10})-1},\qquad \beta_w =
0.125\exp({-v\over 80})
\eeq
Typical parameter values are $g_{Na} = 120$, $g_k = 36$, $g_l = 0.3$,
$v_{Na} = 115$, $v_K = -12$, and $v_l  = -10.5988$.  The high
precision of $V_l$ is to insure that $v=0$ is the rest state.

\section{Morris-Lecar}
This model was devised for barnacle muscle fiber.
  \bea
    F(v,w) &=& -g_{ca}m_\infty(v)(v-v_{ca})-g_kw(v-v_k)\nonumber\\
&-&g_l(v-v_l)
+I_{app}\\
    G(v,w) &=& \phi \cosh({1\over 2}{v-v_3\over v_4})(w_\infty-w),
\ena
where
\bea 
 m_\infty(v) &=& {1\over 2}+{1\over 2}\tanh({v-v_1\over v_2}),\\
w_\infty &=& (1+\tanh({v-v_3)\over 2v_4})).
\ena
Typical parameter values are
 $C_m = 20$, 
      $g_{ca} = 4.4$,
      $g_l = 2$, 
      $g_k = 4.5$,
      $v_2 = 15$,
      $v_4 = 15$,
      $v_1 = -1$,
     $v_3 = 0$, 
      $v_k = -70$, 
      $v_{ca} = 100$, 
      $v_l = -50$,
      $\phi = 0.02$,
	  $I_{app}=0.1$. 

\section{Mitchell-Schaeffer}
Mitchell-Schaeffer two-variable model  (also in a
slightly different but equivalent form by Karma)
\bea
{dv\over dt} &=& {1\over \tau_{in}}hv^2(1-v) -{v\over \tau_{out}},\\
{dh\over dt} &=&\left\{\begin{array}{cc}{1\over
\tau_{open}}(1-h)&v<v_{gate}\\
-{h\over
\tau_{close}}&v>v_{gate}
\end{array}
\right.
\ena

\section{Mitchell-Schaeffer Revised}
To make the Mitchell-Schaeffer look more like a cardiac ionic model, take
\bea
C_m{dv\over dt}&=&g_{Na}hm^2(V_{Na}-v) +g_K(V_K-v),\\
{dh\over dt} &=&a_h(v)(1- h) - b_h(v)h
\ena
where 
\beq
m(v) = \left\{\begin{array}{cc} 0,
&v<0\\v,&0<v<1\\1,&v>1\end{array}\right.
\eeq

To match the M-S model, we take
\beq
{g_{Na}\over C_m} = {1\over \tau_{in}} = {1\over 0.3} = 3.333/{\rm ms}
\qquad {g_K\over C_m} = {1\over \tau_{out}} = {1\over 6} = 0.16 /{\rm ms}
\eeq
We also take 
\beq
a(v) = {1-f\over\tau_{open}+(\tau_{close}-\tau_{open})f},\qquad b(v) =
{f\over\tau_{open}+(\tau_{close}-\tau_{open})f}
\eeq
where 
\beq f(v) = {1\over 2}(1+\tanh(\kappa(v-v_{gate})),
\eeq
so that $h_\infty = 1-f(v)$, and $\tau_h =
{\tau_{open}+(\tau_{close}-\tau_{open})f}$.   If $\kappa \rightarrow
\infty$, this reduces exactly to the M-S model.  In the M-S model,
$\tau_{open} = 120$ms, $\tau_{close} = 150$ms, and
$v_{gate} = 0.13$. 


\section{Tyson-Fife}
These dynamics describe the oxidation-reduction of malonic acid.   For
this system,
\bea
F(v,w) &=& v-v^2-(fw+\phi_0){v-q\over v+q}\\
G(v,w) &=&\epsilon(v-w)
\ena
with typical parameter values
$\epsilon = 0.05$, $q=0.002$, $f=3.5$, $\phi_0 = 0.01$. 

\end{document}
