# Inhibition theory

Inhibition theory is based on the basic assumption that during the performance of any mental task requiring a minimum of mental effort, the subject actually goes through a series of alternating latent states of distraction (non-work 0) and attention (work 1) which cannot be observed and are completely imperceptible to the subject.

Additionally, the concept of inhibition or reactive inhibition which is also latent, is introduced. The assumption is made that during states of attention inhibition linearly increases with a slope a1 and during states of distraction inhibition linearly decreases with a slope a0.According to this view the distraction states can be considered a sort of recovery state.

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. When 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, making the whole process of alternating distraction times and attention times a stochastic process.

## Theory

A non-negative continuous random variable T represents the time until an event will take place. The hazard rate λ(t) for that random variable is defined to be the limiting value of the probability that the event will occur 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:

${\displaystyle \lambda (t)=\lim _{\Delta t\to 0}{\frac {\Pr(t\leq T

The hazard rate λ(t) can also be written in terms of the density function or probability density function f(t) and the distribution function or cumulative distribution function F(t):

${\displaystyle \lambda (t)={\frac {f(t)}{1-F(t)}}}$

The transition rates λ1(t), from state 1 to state 0, and λ0(t), from state 0 to state 1, depend on inhibition Y(t): λ1(t) = 1(Y(t)) and λ0(t) = 0(Y(t)), where 1 is a non-decreasing function and 0 is a non-increasing function. Note, that 1 and l0 are dependent on Y, whereas Y is dependent on T. Specification of the functions l1 and l0 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 cannot be observed. It is, nevertheless, possible to estimate from the observable reaction times some properties of the latent process of distraction times and attention times, i.e., the average distraction time, the average attention time, and the ratio a1/a0. 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 1 and 0 are as follows:

${\displaystyle \ell _{1}(y)={\frac {c_{1}M}{M-y}}}$

and

${\displaystyle \ell _{0}(y)={\frac {c_{0}}{y}}}$

both with c0 > 0 and c1 > 0. Note that, according to the first assumption, as y goes to M (during an interval), 1(y) goes to infinity and this forces a transition to a state of rest before the inhibition can reach M. According to the second assumption, as y goes to zero (during a distraction), 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 t0 with inhibition level y0 = Y(t0) the transition rate at time t0 + t is given by λ1(t) = l1(y0 + a1t). For a non-work interval starting at t0 with inhibition level y0 = Y(t0) the transition rate is given by λ0(t) = 0(y0 − a0t). Therefore

${\displaystyle \lambda _{1}(t)={\frac {c_{1}M}{M-(y_{0}+a_{1}t)}}}$

and

${\displaystyle \lambda _{0}(t)={\frac {c_{0}}{y_{0}-a_{0}t}}}$

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 (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), for example, in the Attention Concentration Test, is referred to as A. The average stationary response time E(T) may be written as

${\displaystyle \operatorname {E} (T)=A+{\frac {a_{1}}{a_{0}}}A.}$

For M goes to infinity λ1(t) = c1. This model is known as the gamma − or Poisson inhibition − model (see Smit and van der Ven, 1995).

## Application

Inhibition theory has especially been developed to account for short-term oscillation as well as the long-term trend in the reaction time curves obtained in continuous response tasks such as the Attention Concentration Test (ACT). The ACT typically consists of an overlearned prolonged work task in which each response elicits the next. Several authors, among them Binet (1900), stressed the importance of the fluctuation in the reaction times suggesting the mean deviation as a measure of performance.

In this connection it is also worthwhile to mention a study by Hylan (1898). In his experiment B, he used, a 27 single digits addition task indicating the importance of the fluctuation of reaction times and was the first to report gradually increasing (marginally decreasing) reaction time curves (Hylan, 1898, page 15, figure 5).

Recently, the inhibition model has been also used to explain the phase durations in binocular rivalry experiments (van der Ven, Gremmen & Smit, 2005). The model is able to account for the statistical properties of alternating phase durations

T11, T01, T12, T02, T13, T03, ...,

representing the amount of time a person perceives the stimulus in one eye T1j and in the other eye T0j.