Biomedical Engineering Reference
In-Depth Information
Spatial diffusion process models - SPDEs
If the postsynaptic potentials are not too large and fairly frequent, a diffusion ap-
proximation for a spatial model such as (1.11) may be employed. Linear models
of this kind may involve distributed one-parameter white noises representing each
synaptic input or group of synaptic inputs
l
E
,
i
dW
E
,
i
dt
n
Â
i
∂
2
V
∂
x
2
+(
x
E
,
i
)
a
E
,
i
l
E
,
i
+
|
∂
∂
t
=
−
V
+
V
−
V
E
)
d
(
x
−
a
E
,
i
|
=
1
x
I
,
j
)
a
I
,
j
l
I
,
j
+
|
a
I
,
j
|
l
I
,
j
dW
I
,
j
dt
.
Â
n
I
+(
V
−
V
I
)
d
(
x
−
j
=
1
Simplified versions of this and similar models were analyzed in [83, 84]. Alterna-
tively, if the synapses are very densely distributed, a two-parameter white noise may
be employed as an approximation:
∂
2
V
∂
x
2
+
∂
2
W
∂
t
∂
x
,
∂
∂
t
=
−
V
+
f
(
x
,
t
)+
g
(
x
,
t
)
where
W
(
t
,
x
)
is a standard two-parameter Wiener process. For details see [84].
1.3.3
Jump-diffusion models
It is possible that some inputs to a neuron, including channel noise are frequent and
of small amplitudes whereas others are less frequent and large amplitude, such as
occur at certain large and critically placed synapses or groups of synapses of the
same type. Such a model was introduced in [74] and in its simplest form has the
stochastic equation
dV
=
−
Vdt
+
a
E
dN
E
+
a
I
dN
I
+
s
dW
,
(1.13)
where the unit of time is the time constant. The corresponding equations for the
n
-th
moments of the firing time for an initial potential
x
can be obtained by solving:
s
2
2
d
2
M
n
dx
2
−
x
dM
n
dx
+
l
E
M
n
(
x
+
a
E
)+
l
I
M
n
(
x
−
a
I
)
−
(
l
E
+
l
I
)
M
n
(
x
)
=
−
nM
n
−
1
(
x
)
,
n
l
I
are the mean frequencies of excitation and
inhibition, respectively. However, for particular parameter values, solutions can be
readily obtained by simulating the Poisson and white noise inputs. Neuron models
with correlated inputs can be exactly written as Equation (1.13) [19, 68, 90].
=
1
,...,
with
M
0
=
1.
Here
l
E
,
1.3.4
Perturbation of deterministic dynamical systems
One of the key topics addressed in the theory of differential equations or dynamical
systems is the asymptotic (large time) effect of a small disturbance or perturbation on
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