Information Technology Reference
In-Depth Information
D
=
2
,
(3.145)
indicating that in a balanced steady state the feedback has twice the strength of the
fluctuations. In the present discussion there is no need to adopt this particular ratio for
the solution; any positive-definite ratio for
k
>
0 is sufficient. However, it is interesting
to note that when the strength of the fluctuations becomes sufficiently large that they
overwhelm the feedback control,
D
<
1
,
(3.146)
the first moment
∞
A
μ
−
k
P
ss
(
k
)
dk
=
k
=
(3.147)
2
1
diverges and the condition
μ<
2
(3.148)
is realized. The condition for non-ergodicity (
3.148
) is realized when the fluctuations
dominate dissipation in the multiplicative Langevin equation. This is another interesting
alternative to the perspective of the BA model of network growth [
5
].
3.3.2
The drift-diffusion model (DDM) of decision making
Suppose that an individual is asked to make a decision between two alternatives with
limited information and only one of which has a positive outcome. Such a situation is
called a two-alternative forced-choice (TAFC) task by psychologists. One of the most
often used models for deciding between two such choices is the drift-diffusion model
(DDM), which is based on the stochastic differential equation of the previous sections.
However, other important concepts enter into this model, such as the idea of optimality,
meaning that a decision of a specified accuracy is made in the shortest possible time.
The speed and accuracy of the decisions affect the cumulative reward obtained and
collectively are called the TAFC paradigm in the psychology literature.
As discussed in the excellent review by Bogacz
et al.
[
11
], TAFC task models typ-
ically make three assumptions: (1) the evidence for each choice is accumulated over
time; (2) the uncertainty in the evidence is modeled as random fluctuations in time; and
(3) a decision is triggered when the accumulated evidence for one of the alternatives first
exceeds a specified threshold. The accumulation of information for each choice can be
aggregated in a number of ways, most of which are found to be equivalent [
11
], but it
is often the difference in evidence that is integrated and it is this difference that trig-
gers the choice of one alternative over the other. This integrated difference is modeled
by reducing the number of variables to describe the dynamics from two, the evidence
or information for each of the alternatives separately, to one, which is the difference
between the two alternatives. It is often the case that this difference can be related
to neuronal models of inhibition and excitation, as we subsequently discuss. Another
