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**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 transition rate or hazard rate, which makes the whole process of alternating distraction times and attention times a stochastic process.

## Theory[]

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 λ(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,t+Δt] given the event has not occurred before time t. divided by Δt. Formally, the hazard rate is defined by the following limit:

λ(t) = lim 1/Δt P(t ≤ T < t+Δt | T ≥ t) Δt->0

The transition rates λ[1](t), from state 1 to state 0, and λ[0](t), from state 0 to state 1, depend on inhibition Y(t): λ[1](t) = l[1](Y(t)) and λ[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:

c[1]M l[1](y) = _____ with c[1] > 0 M - y

and

c[0] l[0](y) = ____ with c[0] > 0. y

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[0]+t is given by λ[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 is given by λ[0](t) = l[0](y[0]-a[0]t). Therefore

c[1]M λ[1](t) = ________________ with c[1] > 0 M - (y[0]+a[1]t)

and

c[0] λ[0](t) = __________ with c[0] > 0. y[0]-a[0]t

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).

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]/a[0])A.

For M goes to infinity λ[1](t) = c[1]. This model is known as the gamma - or Poisson inhibition model (see Smit and van der Ven, 1995).

## 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.

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