Modelling Biological Neurons in Terms of Electrical Circuits - Advanced Models of Neural Networks - page 23

Voltage-Gated Potassium Channels K C 1.8.4

These are ion channels which open or close depending on the value of the
membrane's potential V .

1.8.4.1 A-Current

The transient sodium current and the delayed rectifier current are similar to those
of the Hodgkin-Huxley model, although in the present case they are faster. To this
model an additional potassium current is introduced.
An example about the voltage gated channel comes from a model which has the
form

I A D g A a 3 b.V E A / (1.92)

The activation variable a increases as current increases. The de-activation variable
b decreases as current increases. Steady-state current is almost zero.

I A;ss D g A ˛ 1 .V/ 3 b.1/.V/.V E A /

(1.93)

The conductivity variable g A affects the number of spikes that the membrane may
exhibit and is defined by the relation g A Cg K D g total D 67:7, where a typical value
for parameter g K is g K D 20.

1.8.4.2 M-Current

These are slow K C currents which may affect the firing rate (spiking) of the
membrane. The M current and the associated with it slow currents K C can inhibit
neurons' firing. Apart from slowing down the neuron's firing rate the M current
affects also the steady-state value of the neuron's voltage [ 140 ].
If there is no M-current (g m very small), then the model may exhibit bifurcations
on limit cycles. As g m grows then Hopf bifurcations may appear.
With reference to the dynamic model of Eq. ( 1.90 ) in the present model one
should consider to introduce the following input currents [ 16 ]

I M Dg M p.V E k /

dp
dt D . 1 .V/ /= p .V/
1 .V/ D

(1.94)

1
1Ce .V C 35/=100

mac
3:5Ce .V C 35=20/

p .V/ D

Ce .V C 35=20/