Modelling Biological Neurons in Terms of Electrical Circuits - Advanced Models of Neural Networks - page 25

1.8.7 Calcium-Dependent Channels

Modelling of channels where there is transfer of Ca 2C is important because the
concentration of these ions affects several signaling mechanisms. The two most
important channels of this type are: (1) I K,Ca which is the calcium dependent
potassium channel, and (2) I can which is the inward calcium-dependent current. The
first current appears in several neurons and is responsible for low hyperpolarization
(AHP) and spike-frequency adaptation. It is known as AHP current.

1. Calcium-dependent potassium channel—the afterhyperpolarization
A typical model for the I K,Ca current comes from [ 140 ] and is given by

I K,Ca D g K,Ca m 2 .V E k /
dm
dt

D .m 1 .c/ m/= m .c/
m 1 .c/ D

(1.99)

c 2

Cc 2
m .c/ D max. min ; o =.1 C .c=k/ 2 //

k 2

Typical values for the parameters of the above model are k D 0:0025 mM, min D
0:1 ms, o variable. The firing rate f is affected by the value of the current I K,Ca
according to the relation

F 0 .Iagf/
1CagF 0 .I/agf/

df
dI

D (1.100)

and for large F 0 this relation is written approximately as

df
dI

1
ag

D (1.101)

A solution w hich has been also proposed about the firing rate is f.I/ D C

p 2

C A 2 I, where D A 2 ag=2. Moreover about the voltage spikes that appear
in the calcium-dependent potassium channel it holds

C d dt D I I fast g z .V E k /

dz
dt

(1.102)

D Œq.V/.1 z / z

2. Calcium-activated nonspecific cation current
This case is concerned with the inwards calcium current I CAN .Thisis
described by the relations

I CAN D g CAN m CAN .V E CAN / (1.103)

where the gate parameter m CAN varies according to the dynamic model

dm CAN
dt

D .g.c/.1 m CAN / m CAN /= CAN (1.104)