Biomedical Engineering Reference
In-Depth Information
l
I
a
I
and s
2
l
E
a
E
+
l
I
a
I
, it can be shown that
V
will reach
Putting m
=
l
E
a
E
−
=
V
thre
>
0; that is, the net excitatory
drive is greater or equal to the net inhibitory drive. This is not true for more realistic
models. When
V
(
0
)
with probability one if and only if
m
≥
≥
m
0 the probability density of the time for
V
to get from a value
(
)
<
V
0
V
thre
to threshold is the
inverse Gaussian
exp
2
)=
(
V
thre
−
V
(
0
))
−
(
V
thre
−
V
(
0
)
−
m
t
)
f
(
t
√
2ps
2
t
3
,
t
>
0
.
2s
2
t
Gluss model - Ornstein-Uhlenbeck process (OUP)
The jump process model with exponential decay given by (1.9) can be similarly
approximated by a diffusion model which is, for subthreshold
V
m
dt
g
+
dV
=
−
+
s
dW
.
(1.12)
This defines an
Ornstein-Uhlenbeck process
, which as a neuronal model was first
derived and analyzed by Gluss in 1967 [31]. Here
V
has continuous paths and, if
unrestricted, the same first and second moments as the jump process. Both processes
may get to the same values, such as a threshold for an action potential, at about
the same time, but in many cases this will be far from the truth, depending on the
values of the four parameters l
E
l
I
and
a
I
- see [81] for a complete discussion.
Thus
extreme caution
must be exercised if using a diffusion model such as (1.12) to
obtain input-output relations for neurons. For example, it was found in [27] that an
inhibitory input can drive the model to fire faster, with a fixed excitatory input, but
this is simply due to the error introduced by the diffusion approximation. Roy and
Smith [63] first solved the difficult problem of obtaining an exact expression for the
mean firing time in the case of a constant threshold. Even recently this model has
attracted much attention [18, 29, 44]. The OUP has also been shown to be a suitable
approximation for
channel noise
, see for example [75], Chapter 5 and
Chapter 6.
The general theory of diffusion processes is broad and often quite abstract. Such
fundamental matters as (appropriate) definition of stochastic integral and boundary
classifications are important but generally outside the domain of most computational
neuroscientists [38]. Fortunately such matters can be sidestepped as most modelling
is pragmatic and will involve trial and error
simulation methods
using software pack-
ages. However, it is useful to realize that diffusion processes, whether of one or
several dimensions, have an associated linear partial differential equation satisfied
by the transition probability function or its density. This is also true for Markov
jump processes, but the corresponding equations are more complicated and far less
studied.
Analytical theory for diffusion processes
Letting
X
,
a
E
,
beavectorwith
n
components, all neuronal ordinary differential
equation models in which the input current is approximated by white noise have the
general form
(
t
)
d
X
(
t
)=
f
(
X
(
t
)
,
t
)
dt
+
g
(
X
(
t
)
,
t
)
d
W
(
t
)
,
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