## Spike amplitude

The basics of how to develop, test, and use models.
Kim Ha
Posts: 18
Joined: Sun Jun 09, 2013 11:26 pm

### Spike amplitude

I have a question about the action potential of neurons. As I know, the amplitude of an action potential is always around 40mV and it does not depend on the intensity of the stimuli. However when I build the model of fish, the voltage graph shows the action potential of some firt peaks are around 40mV but the other peaks are under 30mV. I also see that pattern in your hybrid network model in document page, the peaks of the action potentials of M cell are smaller than 40mV. There is a contradiction here.
Can you explain me?
ted
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Joined: Wed May 18, 2005 4:50 pm
Location: Yale University School of Medicine
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### Re: Spike amplitude

Although my answer assumes we're dealing with a spherical cell whose membrane contains squid axon ion channels, with slight modification it also applies to cells with other topologies (branching patterns) and other kinds of ion channels.

For a spherical cell, conservation of charge requires that
sum of membrane ionic currents = membrane capacitive current
In squid axon, this equation is
ina + ik + ileak = cm * dv/dt
where v is membrane potential and cm is membrane capacitance.

At the peak of the spike dv/dt is 0, so the equation becomes
ina + ik + ileak = 0
Rewriting this in terms of ionic conductances and reversal potentials
gna*(v-ena) + gk*(v-ek) + ileak*(v-eleak) = 0

Solving this for v gives
v at peak of spike = (gna*ena + gk*ek + g_leak*eleak)/(gna + gk + gleak)
At the peak of the spike the sodium and potassium conductances are much larger than gleak, so the terms that involve gleak can be ignored and we have
v at peak of spike ~ (gna*ena + gk*ek)/(gna + gk)

So v at the peak of the spike depends on the balance between gna and gk. That is, v at the spike peak depends on the relative abundance of open sodium and potassium channels. If gna = gk at the spike peak, v will be halfway between ena and ek.

Now remember that any depolarization of a neuron inactivates some sodium channels and activates some potassium channels. If two spikes are elicited in a short time, the second spike will have smaller amplitude than the first because the first spike inactivates sodium channels and activates potassium channels. This means the second spike will have smaller gna and larger gk, so the second spike's peak will not be as close to ena as the peak of the first spike was.