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(
λ>
0
)
or repelled away from the fixed point as it would be for positive feedback (
λ<
0
).
Bogacz
et al.
[
11
] interpret this behavior in terms of decision making as follows:
In the stable case, all solutions approach and tend to remain near the fixed point, which lies nearer
the correct threshold, so they typically slow down before crossing it, corresponding to conserva-
tive behavior. The unstable case corresponds to riskier behavior. Solutions on the
correct
side of
the fixed point accelerate toward the correct threshold, giving faster responses, but solutions on
the
incorrect
side accelerate toward the incorrect threshold, possibly producing more errors.
From this interpretation of the trajectories it is possible to propose that rewards for
correct responses should increase the magnitude of the control parameter, whereas pun-
ishments for errors should decrease it, as suggested by Busemeyer and Townsend [
12
].
These authors noted the decay effect of the control parameter for
λ>
0, incorporating
the fact that earlier input does not influence the trajectory (accumulated evidence) as
much as does later input. By the same token, there is a primacy effect of the control
parameter for
0, namely that the earlier input influences the trajectory more than
later inputs. This earlier influence is manifest in the divergence of the trajectory.
λ<
3.3.3
First-passage-time distributions
Consider a web characterized by a continuous state variable
q
and assume that it is ini-
tially in the state
q
0
.
There exist numerous situations such that when a specific state
q
1
is achieved for the first time the web changes its nature or perhaps activates a device to
change the character of another web that is controlled by the first. An example would
be the climate-control system in your home that has two temperature settings. When the
temperature falls below a certain temperature
T
l
the heating system is turned on, warm-
ing the rooms, and when the temperature reaches
T
l
from below the heating system is
turned off. If the temperature continues to climb to the level
T
u
then the air conditioning
comes on, cooling the rooms, and when the temperature falls to
T
u
from above the air
conditioning is turned off. In the statistics literature the distribution that describes this
behavior is called the first-passage time. The earliest work on the subject dates back to
Daniel Bernoulli (1700-1782) in his discussion of the gambler's ruin. Ruin was identi-
fied as the consequence of the vanishing of the gambler's fortune “for the first time” in
a succession of gambling trials.
Much of our previous discussion was concerned with determining the statistical prop-
erties of the stochastic variable
Q
. Now we are concerned with a more restricted
question associated with when the dynamical variable takes on certain values for the
first time. The DDM model just discussed is of this form. The dynamical variable was
the accumulated evidence for making a decision, either positive or negative, and had the
form of determining when a diffusive trajectory crossed symmetric barriers. Consider
the case when the variable of interest is in the interval
q
1
<
(
t
)
q
2
and we consider a
class of trajectories connecting
q
1
with
q
2
that must proceed by going through the point
q
; that is, the process cannot jump over any point. This transition can be decomposed
into two independent stages, namely the first taking the trajectory from
q
1
to the inter-
mediate point
q
for the first time in the interval
q
<
τ
, and the second taking the trajectory
