Biomedical Engineering Reference
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where the
W
k
are standard Wiener processes - see [61]. For example, consider the
Fitzhugh-Nagumo system
dX
=[
f
(
X
)
−
Y
+
I
]
dt
+
b
dW
dY
=
b
[
X
−
cY
]
dt
,
where
f
(
X
)=
kX
(
X
−
a
)(
1
−
X
)
. The means of
X
and
Y
, denoted by
m
1
and
m
2
respectively, satisfy the equations
f
(
dm
1
/
dt
=
f
(
m
1
)
−
m
2
+
m
1
)
S
1
/
2
+
I
(
t
)
cm
2
)
where
S
1
is the variance of
X
. Denoting the variance of
Y
by
S
2
and the covariance
of
X
and
Y
by
C
12
we also have
dm
2
/
dt
=
b
(
m
1
−
2
f
(
b
2
dS
1
/
dt
=
m
1
)
S
1
−
2
C
12
+
dS
2
/
dt
=
2
b
(
C
12
−
cS
2
)
and
f
(
]
.
This system of five ordinary differential equations may be easily solved and for small
b gives good agreement with moments from simulations (see [82]). The method can
also be used for small biological neuronal networks.
White noise perturbation of spatial nonlinear neuronal models
The analysis of spatial neuronal nonlinear model equations under the effects of
white noise perturbations has been performed for both scalar and vector forms of the
Fitzhugh-Nagumo model. In all cases a perturbation expansion was used to obtain
the moments of the dynamical variables. As a simple example consider the Fitzhugh-
Nagumo system without recovery driven by white noise of small amplitude:
dC
12
/
dt
=
bS
1
−
S
2
+
C
12
[
m
1
)
−
cb
u
t
=
u
xx
+
f
(
u
)+
e
(
a
+
b
W
xt
)
where
W
is a two-parameter Wiener process. An expansion in powers of e
•
Â
k
e
k
u
k
u
=
u
0
+
=
1
yields a recursive system of linear stochastic partial differential equations for the
u
k
. Solving the system recursively yields series expressions for the moments and
spectrum of the potential. These results, results on the full Fitzhugh-Nagumo system
of SPDEs and a general result on perturbation of a nonlinear PDE with white noise
are derived in [76, 78, 80].
A general approach to analyze dynamical systems with small perturbations has
been developed in recent years called
large deviation theory
. Basically, it is a gen-
eralization of the well-known Kramer's formula [60]. A general description is con-
tained in [27], see also [2, 14].
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