Two dimensional neuron models
Reduction to two dimensions
General approach
 Origin model: $\Sigma I_{k}=g_{Na}m^{3}h(uE_{Na})+g_{K}n^{4}(uE_{k})+g_{L}(uE_{L})$
It has 4arguments $m$ $h$ $n$ $u$
 In this chapter,we reduce it to 2 arguments: $u$ , $\omega$ $C\frac{du}{dt}=g_{Na}[m_0(u)]^{3}(b\omega)(uE_{Na})g_{k}(\frac{\omega}{a})^{4}(uE_{k})g_{L}(uE_{L})+I$
$\omega = b  h = an$
 What changes?
because of
m
changes fast: $m(u,t)$ > $m_{0}(u)$ because ofh
andn
seems to have linear relationship: $h$ > $b\omega$ $n$ > $\frac{\omega}{a}$
Morris–Lecar model
It seems that this model just change exponents of arguments to get a linear equation. This section also give a approximate equation for $m_{0}(u)$and $\omega_{0}(u)$
FitzHugh–Nagumo model
Phase plane analysis
On phase plane, point $(u(t),\omega(t))$ $\Delta{u}=\dot{u}\Delta{t}$ $\Delta{\omega}=\dot{\omega}\Delta{t}$ Use vector $(\dot{u},\dot{\omega})$plot a vector field on phase plane: For each point on the plane， we draw an arrow, the lenght of the arrow is proportional to the length of the vector, and the direction is same as the vection
Nullclines
 $u$nullcline: points with $\dot{u} = 0$. The direction of flow on the unullcline is in direction of $(0,\dot{\omega})$ vertical
 $\omega$nullcline: points with $\dot{\omega} = 0$ $(0,\dot{u})$ horizontal
 fixpoint: intersection of $u$nullcline and $\omega$nullcline on nullclines the direction of arrows change at fix points
Stability of fixed points

At fix point

Set $x(t) =\vec{e}e^{λt}$,we get eigenfunction:
$\lambda_{1}+\lambda_{2} = F_{u}+G_{\omega}$
$\lambda_{1}\lambda_{2} = F_{u}G_{\omega}F_{\omega}G_{u}$

Saddle point:
$\lambda_{1}>0$
$\lambda_{2}<0$

Stable points:
$\lambda_{1}<0$
$\lambda_{2}<0$

Unstable points:
$\lambda_{1}>0$
$\lambda_{2}>0$