Biomedical Engineering Reference
In-Depth Information
a reference solution or orbit, such as an equilibrium point or limit cycle. The funda-
mental methods employed for deterministic systems are called Lyapunov's first and
second methods. For stochastic dynamical systems, which have an extra dimension,
results on stability are naturally more difficult to obtain [5]. Here we consider the
effects (or methods of determining them) on some neuronal systems of perturbations
with small Gaussian white noise.
Firing time of a model neuron with small white noise
Consider an OUP model with threshold
V
thre
and stochastic equation
dV
=(
−
V
+
m
)
dt
+
s
dW
.
It should be noted that in the absence of noise and in the absence of a threshold, the
steady state potential is m.Ifm
≤
V
thre
the deterministic neuron never fires whereas
if m
>
V
thre
the firing time is
ln
a
a
T
=
T
R
+
,
−
1
where a
=
m
/
V
thre
and
T
R
is the refractory period. If we define the small noise pa-
rameter e
2
V
thre
then using perturbation techniques ([87] the mean and variance
of the firing time can be found to order e
2
as follows.
Steady state well above threshold.
When a
=
s
/
>>
e
+
1,
ln
e
2
4
a
a
1
a
2
E
[
T
]
≈
T
R
+
−
2
−
,
−
1
(
a
−
1
)
a
2
e
2
2
1
Var
[
T
]
≈
2
−
.
(
a
−
1
)
These results show clearly how small noise reduces the mean interspike interval.
Steady state well below threshold.
When a
<<
1
−
e, the expectation of the interspike interval is
e
√
p
1
a
ex p
(
2
1
−
a
)
E
[
T
]
≈
T
R
+
,
e
2
−
and the variance is
2
ex p
2
e
2
p
2
(
1
−
a
)
Var
[
T
]
≈
,
e
2
(
1
−
a
)
Many other results are given in the aforementioned reference, which includes an
exhaustive study of the dependence of the
coefficient of variation
of
T
on the input
parameters.
Differential equations for moments under Gaussian white noise perturbations
Ordinary differential equations have been derived for the asymptotic moments of
the dynamical variables in a general system of coupled (nonlinear) stochastic differ-
ential equations with white noise perturbations of the form
Â
k
dX
j
=
f
j
(
X
,
t
)
dt
+
g
jk
(
X
,
t
)
dW
k
=
1
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