Information Technology Reference
In-Depth Information
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
20
40
60
80
100
Time
Figure 3.5.
Two trajectories with the same initial condition are depicted. The random force is seen to drive
one to the lower decision and the other apparently to the upper decision, both from the same
initial state.
It should be pointed out that this kind of modeling has been used in neural science,
and is called the integrate-and-fire model. In that application it concerns the firing of
various neurons that contribute to the neuron of interest; when the voltage at the speci-
fied neuron exceeds a prescribed value that neuron, as well, fires and in turn contributes
to the firing of neurons to which it is connected.
One way to generalize the DDM for decision making is by assuming that the drift
is a random variable. In this situation the “constant” drift would not be a stochastic
variable in time like the random force, but would vary randomly from one realization
of the trajectory to another. It is often assumed that the constant drift is subject to the
law of frequency of errors so that its variation from realization to realization would have
normal statistics with a finite mean and variance.
Another generalization of the DDM is to introduce a control process that reduces
the deviation from the equilibrium state. This is a Langevin equation with dissipation,
called an Ornstein-Uhlenbeck (OU) process [
43
], which can be written as
dQ
(
t
)
=
C
−
λ
Q
(
t
)
+
ξ(
t
).
(3.160)
dt
In physics the sign of the parameter
is fixed because it is the physical dissipation in
the web of interest and therefore is positive definite. In a communications context
λ
is
a feedback control parameter and may be of either sign. In the absence of the random
force the fixed point for (
3.160
)isgivenby
λ
C
λ
,
Q
f
=
(3.161)
where the velocity vanishes. The deviation of the trajectory from the fixed point can be
determined from
Q
=
Q
f
+
δ
Q
such that
d
δ
Q
(
t
)
=−
λδ
Q
(
t
),
(3.162)
dt
