Two dimensional neuron models

Reduction to two dimensions

General approach

• Origin model: $\Sigma I_{k}=g_{Na}m^{3}h(u-E_{Na})+g_{K}n^{4}(u-E_{k})+g_{L}(u-E_{L})$

It has 4arguments $m$ $h$ $n$ $u$

• In this chapter,we reduce it to 2 arguments: $u$ , $\omega$ $C\frac{du}{dt}=-g_{Na}[m_0(u)]^{3}(b-\omega)(u-E_{Na})-g_{k}(\frac{\omega}{a})^{4}(u-E_{k})-g_{L}(u-E_{L})+I$

$\omega = b - h = an$

• What changes? because of m changes fast: $m(u,t)$ -> $m_{0}(u)$ because of h and n 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 u-nullcline is in direction of -$(0,\dot{\omega})$ -vertical
• $\omega$-nullcline: -points with $\dot{\omega} = 0$ -$(0,\dot{u})$ -horizontal
• fix-point: -intersection of $u$-nullcline and $\omega$-nullcline -on nullclines the direction of arrows change at fix points

Stability of fixed points

• At fix point

  $$\frac{d}{dt}\vec{x}=\left(\begin{array}{cc} F_{u} & F_{\omega} \ G_{u} & G_{\omega} \end{array} \right)\vec{x}$$

• Set $x(t) =\vec{e}e^{λt}$,we get eigenfunction: $$\lambda\vec{x}=\left(\begin{array}{cc}F_{u}&F_{\omega}\ G_{u}&G_{\omega}\end{array}\right)\vec{x}$$

  $\lambda_{1}+\lambda_{2} = F_{u}+G_{\omega}$

  $\lambda_{1}\lambda_{2} = F_{u}G_{\omega}-F_{\omega}G_{u}$

• Saddle point:

  $\lambda_{1}>0$

  $\lambda_{2}<0$

• Stable points:

  $\lambda_{1}<0$

  $\lambda_{2}<0$

• Unstable points:

  $\lambda_{1}>0$

  $\lambda_{2}>0$