Inhibition Theory
Inhibition Theory is based on the basic assumption,
that during the performance of any mental task, which requires
a minimum of mental effort, the subject actually goes through
a series of alternating states of distraction (non-work) and
attention (work).
These alternating states of distraction (state 0) and attention
(state 1) are latent states,
which cannot be observed and which are completely unaware to the subject.
Additionally, the concept of inhibition
is introduced, which is also latent. The assumption is made, that
- during states of attention inhibition linearly increases
with a certain slope \( a_1\), and
- during states of distraction linearly inhibition decreases
with a certain slope \( a_0\).
According to this view
the distraction states can be considered as a sort of recovery states.
It is further assumed, that
- when the inhibition increases during a state of attention,
depending on the amount of increase, the inclination
to switch to a distraction state also increases and
- when the inhibition decreases during a state of distraction,
depending on the amount of decrease, the inclination
to switch to an attention state increases.
The inclination to switch from one state to the other is mathematically
described as a transtion rate or hazard rate, which makes the whole
process of altenating distraction times and attention times a stochastic
process.
If one thinks of a non-negative continuous random variable \(T\)
as representing the time until some event will take place then the
hazard rate \(\lambda(t)\) for that random variable is
defined to be the limiting
value of the probability that the event will take place in a small
interval \([t, \, \Delta t]\) given the event has not occurred
before time \(T\) divided by \(\Delta t\) .
Formally, the hazard rate is defined by the following limit:
$$\lambda(t) = \lim_{\Delta t -> 0} {P(t \leq T < t + \Delta(t) \, | \, T \geq t) \over \Delta(t)}$$
The transition rates \(\lambda_1(t)\),
from state 1 to state 0, and
\(\lambda_0(t)\), from state 0 to state 1,
depend on inhibition \(Y(t)\):
\(\lambda_1 (t) = l_1(Y(T))\) and \(\lambda_0 (t) = l_0(Y(T))\),
where \( l_1\) is a non-decreasing function and
\( l_0\) is a non-increasing function.
Note, that \( l_1\) and \( l_0\) are
dependent on \( y\), whereas \( y\) is dependent on \(T\).
Specification of the functions \( l_1\)
and \( l_0\) leads to the various inhibition models.
What can be observed in the test are the actual reaction times.
A reaction time is the sum of a series of alternating distraction times
and attention times, which both cannot be observed. However, it is
never-the-less possible to estimate from the observable reaction times
some properties of the latent process of distraction times and attention
times, such as the average distraction time, the average attention
time and the
ratio \( a_1\)/\( a_0\).
In order to be able to simulate the consecutive reaction times,
inhibition theory has been specified into various inhibition models.
One is the so-called beta inhibition model.
In the beta-inhibition
model, it is assumed that the inhibition \(Y(t)\) oscillates between
two boundaries which are 0 and \(M\) (\(M\) for Maximum),
where \(M\) is positive. In this model \( l_1\)
and \( l_0\) are as follows:
$$l_1 (y) = {c_1 M \over M-y}, \, \, \, 0 < c_1, \, \, \, 0 < y < M$$
and
$$l_0 (y) = {c_0 \over y}, \, \, \, 0 < c_0$$
Note that, according to the first assumption,
as \( y\) goes to \(M\) (during a work interval),
\(l_1 (y)\) goes to infinity
and this forces a transition to a state of rest before
the inhibition can reach \(M\). Note further
that, according to the second assumption,
as \( y\) goes to zero (during a distraction),
\(l_0 (y)\) goes to infinity
and this forces a transition to a state of work before
the inhibition can reach zero.
For a work interval starting at \(t_0\)
with inhibition level \(y_0 = Y(t_0) \) the transition rate
at time \(t + t_0\) is given by \(\lambda_1(t) = l_1(y_0 + a_1 t) \).
For a non-work interval starting at \(t_0\)
with inhibition level \(y_0 = Y(t_0) \) the transition rate
at time \(t + t_0\) is given by \(\lambda_0(t) = l_0(y_0 - a_0 t) \).
Therefore
$$\lambda_1 (t) = {c_1 M \over M- (y_0 + a_1 t)}, \, \, \, 0 < c_1, \, \, \, 0 < y_0 < M$$
and
$$\lambda_0 (t) = {c_0 \over (y_0 - a_0 t) }, \, \, \, 0 < c_0, \,\,\, 0 < y_0 < M$$
The model has \( y\) fluctuating in the interval between 0
and \(M\). The stationary distribution of \(Y/M\) in this model
is a beta distribution (reason to call it the beta inhibition model).
A simulation program for the beta inhibition model is now available.
If you want to download this program, please,
click here.
Last update of this program was at October, 1, 2005.
The total real working time until the conclusion of the task
(or the task unit in case of a repetition of equivalent unit tasks,
such as is the case in the ACT)
is referred to as \(A\).
The average stationary response time \(E(T)\) may written as
$$E(T) = A + {a_1 \over a_0} \, A$$
For \(M\) goes to infinity \(\lambda_1(t) = c_1 \).
This model is known as the gamma - or Poisson
inhibition model (see Smit and van der Ven, 1995).
A simulation program for the gamma inhibition model is now also available.
If you want to download this program, please,
click here.
Last update of this program was at October, 1, 2005.
References
Smit, J.C. and van der Ven, A.H.G.S. (1995).
Inhibition in Speed and Concentration Tests: The Poisson Inhibition
Model. Journal of Mathematical Psychology, 39, 265-273.