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whose solution exponentially decays to zero for
λ>
0, and consequently the fixed point
is an attractor for positive
λ
.For
λ<
0 the fixed point is unstable and the solution to
(
3.162
) diverges to infinity.
The solution of the OU Langevin equation for
Q
(
0
)
=
0 is given by
t
C
λ
[
e
−
λ
(
t
)
ξ(
e
−
λ
t
t
−
t
)
dt
,
Q
(
t
)
=
1
−
]+
(3.163)
0
yielding the average value of the accumulated evidence
C
λ
[
e
−
λ
t
)
ξ
=
Q
(
t
1
−
]
(3.164)
with variance
D
λ
[
2
e
−
2
λ
t
σ
(
t
)
=
1
−
]
(3.165)
the same as that given by (
3.157
). The variance remains unchanged in functional form
because it is a measure of how much the trajectories diverge from one another over time.
This divergence is influenced not only by the random force but also by the dissipation,
so asymptotically the variance becomes constant. What is interesting is that at early
times,
t
1
/λ
, the exponential in the average evidence (
3.164
) can be expanded to
obtain
lim
t
0
Q
(
t
)
ξ
=
Ct
(3.166)
→
in agreement with the constant-drift model. At the other extreme the exponential in the
average evidence vanishes, giving rise to
C
λ
,
t
→∞
lim
Q
(
t
)
ξ
=
(3.167)
resulting in a time-independent average. The dissipation or negative feedback there-
fore prevents the average evidence from reaching an unrealistically high value, thereby
increasing the likelihood that an incorrect decision can be made under more realistic
circumstances.
The probability density can be obtained for the OU process using the characteristic
function as we did for the constant-drift case. In the OU case the characteristic function
can be written
e
ik
Q
(
t
)
ξ
e
ik
0
e
−
λ
(
dt
t
)
ξ
(
t
−
t
)
φ(
k
,
t
)
=
(3.168)
ξ
and again the properties of white noise can be used to evaluate the average. The inverse
Fourier transform of the characteristic function yields
exp
q
2
−
Q
(
t
)
ξ
1
p
(
q
,
t
)
=
4
−
,
(3.169)
2
2
σ
(
t
)
2
πσ
(
t
)
where the variance is given by (
3.165
) and the mean is given by (
3.164
).
It is worth reviewing that the sign of the control parameter
determines whether
the trajectory is attracted towards the fixed point as it would be for negative feedback
λ
