Biomedical Engineering Reference
In-Depth Information
where h
is called a
white noise
process. It can be defined heuristically as a random
variable taking values
(
t
)
1
√
dt
)
h
(
t
)=
dt
→
0
N
lim
(
0
,
(15.11)
for all
t
independently, where we have defined
N
is a Gaussian random vari-
able of mean a and variance b
2
. The mean and two-point correlation function of
the white noise process are therefore,
(
a
,
b
)
t
)
>
=
t
)
<
h
(
t
)
>
=
0and
<
h
(
t
)
h
(
d
(
t
−
respectively. In effect, we are now replacing Equation (15.1) by
C
m
dV
)
dt
=
−
(
t
g
L
(
V
(
t
)
−
V
L
)+
m
C
+
s
C
h
(
t
)
,
(15.12)
or
s
V
√
t
m
h
t
m
dV
)
dt
=
−
(
(
t
V
(
t
)
−
V
ss
)+
(
t
)
.
(15.13)
This is called the white-noise form of the Langevin equation of the process
V
.
It has the appeal that it is written as a conventional differential equation so that the
dynamics of
V
(
t
)
is described in terms of its sample paths, rather than in terms of the
temporal evolution of its probability distribution, as in the Fokker-Planck Equation
(15.9). In general, the practical use of the Langevin equation is that it provides a
recipe for the numerical simulation of the sample paths of the associated process.
Developing Equation (15.13) to first order one obtains
(
t
)
s
V
d
t
m
N
d
t
m
)
V
ss
d
t
m
+
V
(
t
+
dt
)=(
1
−
V
(
t
)+
(
0
,
1
)
.
(15.14)
Assuming that
dt
t
m
is small but finite, Equation (15.14) provides an iterative pro-
cedure which gives an
approximate
description of the temporal evolution of
V
/
.
This scheme is general and can be used for any diffusion process. For the O-U pro-
cess in particular, in the absence of threshold Equation (15.9) can be solved exactly.
The population density of this process is a Gaussian random variable with a time-
dependent mean and variance [97, 123], so that
(
t
)
N
V
ss
+(
t
−
t
0
t
m
s
V
√
2
r
(
V
,
t
|
V
0
,
t
0
)=
V
0
−
V
ss
)
exp
(
−
)
,
1
/
2
1
(15.15)
2
(
t
−
t
0
)
−
exp
(
−
)
.
t
m
Using this result one can find an exact iterative procedure for the numerical simula-
tion of the process. Assuming
V
0
is the value of the depolarization in the sample path
at time
t
,e.g.,
V
0
=
V
(
t
)
, the depolarization at a latter time
t
+
D
t
will be
D
t
m
)
(
+
)=
+(
(
)
−
)
(
−
V
t
D
t
V
ss
V
t
V
ss
exp
1
1
/
2
(15.16)
s
V
2D
t
+
−
exp
(
−
t
m
)
N
(
0
,
1
)
.
√
2
This update rule is exact for all D
t
[54].
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