Time-Window-of-Integration (TWIN) Model for Saccadic Reaction Time: Effect of Auditory Masker Level on Visual–Auditory Spatial Interaction in Elevation

Abstract

Saccadic reaction time (SRT) to a visual target tends to be shorter when auditory stimuli are presented in close temporal and spatial proximity, even when subjects are instructed to ignore the auditory non-target (focused attention paradigm). Previous studies using pairs of visual and auditory stimuli differing in both azimuth and vertical position suggest that the amount of SRT facilitation decreases not with the physical but with the perceivable distance between visual target and auditory non-target. Steenken et al. (Brain Res 1220:150–156, 2008) presented an additional white-noise masker background of three seconds duration. Increasing the masker level had a diametrical effect on SRTs in spatially coincident versus disparate stimulus configurations: saccadic responses to coincident visual–auditory stimuli are slowed down, whereas saccadic responses to disparate stimuli are speeded up. Here we show that the time-window-of-integration model accounts for this observation by variation of a perceivable-distance parameter in the second stage of the model whose value does not depend on stimulus onset asynchrony between target and non-target.

Appendix

The race in the first stage of the model is made explicit by assigning independent non-negative random variables V and A to the peripheral processing times for the visual target and auditory non-target stimulus, respectively. With τ as SOA value and ω as integration window width parameter, the time window of integration assumption is equivalent to the (stochastic) event I, say,

$$ I=\{A + \tau < V < A + \tau + \omega\}.$$

Thus, the probability of integration to occur, P(I), is a function of both τ and ω, and it can be determined numerically once the distribution functions of A and V have been specified.

The next step is to compute expected reaction time for the unimodal and crossmodal conditions. From the two-stage assumption, total reaction time in the crossmodal condition can be written as a sum of two random variables:

$$ RT_{crossmodal} = S_1 + S_2, $$

(5)

where S ₁ and S ₂ refer to the first and second stage processing time, respectively. For the expected saccadic reaction time in the crossmodal condition then follows:

$$ \begin{aligned} \hbox{E}[RT_{crossmodal}] = \quad & \hbox{E}[S_1] + \hbox{E}[S_2]\\ = \quad & \hbox{E}[S_1] + \hbox{E}[S_2|{\hbox {not}{\text -}}I]- P(I)\left(\hbox{E}[S_2|{\hbox {not}{\text -}}I]-\hbox{E}[S_2|I]\right), \end{aligned} $$

where E[S ₂|I] and E[S ₂|not-I] denote the expected second stage processing time conditioned on interaction occurring (I) or not occurring (not-I), respectively. Setting

$$ \Updelta \equiv \hbox{E}[S_2 |{\hbox {not}{\text -}}I] -\hbox{E}[S_2| I] $$

this becomes

$$ \hbox{E}[RT_{crossmodal}]= \hbox{E}[S_1] + \hbox{E}[S_2|{\hbox {not}{\text -}}I] - P(I)\cdot \Updelta. $$

(6)

In the unimodal condition, no integration is possible. Thus,

$$ \hbox{E}[RT_{unimodal}]= \hbox{E}[S_1] + \hbox{E}[S_2|{\hbox {not}{\text -}}I], $$

and we arrive at the simple product rule for expected crossmodal interaction (ECI)

$$ \hbox{ECI}\equiv \hbox{E}[RT_{unimodal}]-\hbox{E}[RT_{crossmodal}] = P(I)\cdot \Updelta. $$

(7)

Parameters were estimated by minimizing the Pearson χ² statistic

$$ \chi^2 = \sum\limits_{n=1}^{4}\sum\limits_{j=1}^{2} \left ({\frac{\overline{SRT}(j,n) - \widehat{SRT}(j,n)} {\sigma_{\overline{SRT}(j,n)}}}\right)^2 $$

(8)

using the FMINSEARCH routine of MATLAB. Here $\overline{SRT}(j,n)$ and $\widehat{SRT}(j,n)$ are, respectively, the observed and the fitted values of the mean SRT to visual–auditory stimuli) presented in spatial positions (coincident, j = 1; disparate, j = 2) with SOA (referred to by n = 1 to 4); $\sigma_{\overline{SRT}_{(j,n)}}$ are the respective standard errors.

For a more detailed formal presentation of the model we refer to Diederich and Colonius (2008a).