03.Equilibrium Potential and Hodgkin-Huxley Model

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

does not have h, Noninactivating

result: larger depolarization

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