Equilibrium Potential and Hodgkin-Huxley Model

Equilibrium Potential

Nerst Potential

For ions with positive charge

• high potential -> more energy -> low density

• low potential -> less energy -> high density

The reverse,

• low density -> more energy -> high potential

• high density -> less energy -> low potential

Nerst Potential:

The voltage generates by concentration difference.

Reverse Potential

In the cell membrane:

• ion pumps(ions go single direction)

• ion channels(ions go both direction)

Nernst potential $E_{Na}=+50mV$

• $\Delta{u}<E_{Na}$ : $Na^{+}$ flow into cell

• $\Delta{u}>E_{Na}$ : $Na^{+}$ flow out of cell

Rest Potential

$u_{rest}\approx-65mV$

$E_{K} < u_{rest} < E_{Na}$

at rest potential:

• potassium flow out of cell

• sodium flow into cell

• ion pumps balance these flows

Hodgkin-Huxley Model

Definition

$C\frac{du}{dt}=-\Sigma I_{k}(t)+I(t)$

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

$\dot{m}=\alpha_{m}(u)(1-m)-\beta_{m}(u)m$

$\dot{n}=\alpha_{n}(u)(1-n)-\beta_{n}(u)n$

$\dot{h}=\alpha_{h}(u)(1-h)-\beta_{h}(u)h$

sodium

m:activate

h:inactivate

another form: $\dot{x}=-\frac{1}{\tau_{x}(u)}[x-x_{0}(u)]$

$x_{0}(u)=\frac{\alpha_{x}(u)}{\alpha_{x}(u)+\beta_{x}(u)}$

$\tau_{x}(u)=\frac{1}{\alpha_{x}(u)+\beta_{x}(u)}$

Dynamics

spike generation

external input ->

membrane voltage rise ->

m increase ->

sodium into cell ->

membrane potential rise ->

action potential

fig 2.3

$\tau(h)>\tau(m)$

h:channel close

m:channel open

close is more slowly then open

then, potassium sets in->

lower potential

mean firing rates and gain function

$I_{0}>I_{\theta}$spike train

step current input

$\Delta{I}$ -> generate single spike $I_{2}$ -> generate repeat spikes

inhibitory rebound:

• $I_{2}=0$

• $\Delta{I}$ is large enough

refractoriness

• hyperpolarizing -> needing more stimulation
• more channel open -> resistance is lower, stimulation decay faster

The Zoo of ion channels

Sodium channels

$$I_{NaP}=\bar{g}_{NaP}m(u-E_{Na})$$

does not have h, Noninactivating

result: larger depolarization