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to close, ˇ is the transition rate from closed to open. Moreover it is assumed that the
number of open channels in the membrane is o.
Then N o is the number of closed channels at time instant t, r
1
D ˛.N o/ is
the total rate of transitions from open to closed, and r
2
D ˇo is the total number of
transitions from closed to open. The aggregate rate for all events is r D r
1
C r
2
.
Denoting X
1
2.0;1/ the probability of appearance of an event until time instant
t
new
(under exponential waiting time) one has
1
r
ln.X
1
/
X
1
D e
rt
new
(7.28)
)t
new
D
Regarding the Morris-Lecar neuron model the following may hold: (1) opening of
a calcium channel, (2) closing of a calcium channel, (3) opening of a potassium
channel, (4) closing of a potassium channel. It has been shown that the rate of
opening or closing of the channels is associated with the voltage of the membrane
which is described by the relation
C
n
d
dt
D I g
l
.V E
l
/ .g
k
w
=N
w
/.V
E
k
/ g
Ca
.M=N
m
/.V E
Ca
/ (7.29)
where W is the total number of open ion channels K
C
, M is the total number of
total ion channels Ca
2C
. Equation (
7.29
) is written also as
dV
dt
(7.30)
D .V
1
V/g
where V
1
and g are functions of the parameters W and M.
When introducing stochastic models to describe the change of the membrane's
voltage then either a Fokker-Planck or a Langevin equation follows, that is
(7.31)
dx
D Œ˛.1 x/ C ˇx
dt
C
x
dW
.t/
˛.1
x/
C
ˇx
N
where the standard deviation is approximated by
x
D
(standard
deviation).
7.5
Fokker-Planck Equation and the Integrate-and-Fire
Neuron Model
7.5.1
The Integrate-and-Fire Neuron Model
A simple model of the neuron's dynamics (apart from the elaborated model of
the FitzHugh-Nagumo, Hodgkin-Huxley, or Morris-Lecar neuron) is that of the
integrate and fire neuron, and this is given by
dV
D .I aV/
dt
C
dW
t
(7.32)
