Diffusion models to distinguish endogenous and exogenous temporal attention

4th Quarter: (NonContingent, NonContingent, Contingent, Contingent) + Cue, NoCue, Cue, NoCue)

2.2.3 Participants

Sixteen volunteers (17-30 years old, mean age = 22.8; 11 women) participated in the experiment. Part of them were students of the Higher School of Economics, part of them were recruited through advertising outside the department. All participants gave informed consent before the study. Only people with normal or corrected to normal vision participated in the experiment. Participants were paid 250 rubbles.

2.2.4 Procedure

Contingent condition (Figure 5). For each participant, there were four sessions. Each session consists of testing (15 min) and resting (5 min) parts. Each testing part consisted of practice and experimental blocks. The experimental block consisted of 40 experimental trials. This ratio is for each of 5 different SOA in Contingent condition.

So, first of all, participants are instructed to detect a change in the direction of dots that would warn them about a target that will be presented soon after. The task is to monitor moving dots and when at some moment the border of the circle changes the color (from gray to blue or orange), for 100ms), participant should respond by pressing the key on the keyboard (“d” and “k”, respectively) as soon and accurate as possible. Hands are counterbalanced between participants. After that new circle with gray borderline appears, a new trial begins. The feedback about mean RT was shown to the participant at the end of each trial to maintain the interest to the task.

NonContingent condition (Figure 6). The same procedure as for the Contingent condition with only difference in changing from varied to random SOA and instruction that tells participants to ignore the change in the direction of dots because it is random and it will not give them any cue about the following target.

Figure 5. Contingent condition

Figure 6. NonContingent condition.



Chapter 3. Results

3.1 Statistical tools

We used almost the same statistical plan, as was applied to Lawrence and Klein experiment because the main variables and interaction between comparisons of them are the same in our experiment.

So, calculation of evidence for the effects of interest (contingency, intensity, SOA, and interactions between) was achieved by comparison of generalized additive mixed models.

This class of models takes into account missing data, fewer assumptions. Also, we tend to think that effects associated with SOA might be nonlinear; that is why generalized additive mixed models were used for that comparisons to evaluate evidence for effects associated with it, but not GLMMs.

All behavioral analysis and visualization of data were coded in R (mgcv package for GAMMs, ggplot2, itsadug, mgcViz packages for plotting). Main effects were calculated for hypotheses 1 and 2. The interaction was calculated for hypothesis 3.

3.2 Empiricaal results

Trials with RTs faster than 200ms and slower than 1000ms, as well as trials with SOAs greater than 1s and less than -0.3s were excluded from the analysis (18%). Also, only accurate response trials were analyzed.

To build a model that accounts for all main effects and interactions, we decided to use a feedforward approach.

That means that we started from the simplest model with only one predictor and step by step added predictors and interactions thus increasing the complexity of the model. Akaike's information criterion (AIC) was used to compare models and choose the model that best describes our data, meanwhile is not so complex. The model on which we stopped looks like this:

Baseline <- gam(rt_ms ~ IntCont + s(soa) + s(soa, by=IntContO) + s(soa, subj, bs="fs", m=1), data = dt1[corrans == 1,], method = "ML")

The summary of the model "Baseline" is depicted in Figure 7. For this model, we created two new variables "IntCont" and "IntContO". Both of them represent the interaction between Intensity and Contingency (for more details about GAMMs model's comparison, see Wieling, 2018).

Figure 7. The summary of model «Baseline».

However, "IntCont" is just a factor, while "IntContO" is an ordered factor. "IntCont" variable is used for parametric coefficients. Levels of "IntContO" are used for smooth terms when they interact with SOAs.

So now we have four levels of each variable, instead of two variables with two levels each. In this model noCue in the Contingent condition (noCue.NC) plays the role of intercept both for parametric effects and for smooth terms.

The summary shows three contrasts with respect to the intercept and four smooths, one for each level of the new nominal variable "IntContO".

It is clear from the line for IntContnoCue.C (Full condition) and IntContnoCue.C (Endogenous condition) that both Contingent conditions differ significantly from the Baseline condition (IntContnoCue.NC) and it does not matter whether there was Cue or not. This main effect of Contingency is depicted in Figure 8. Also, the model shows that there is one significant interaction with SOA, namely Endogenous condition (noCue.C) significantly differs from Baseline condition

Figure 8. Main effect of Contingency. On the x-axis - four experimental conditions - Baseline, Exo, Endo and Full. On the y-axis - difference in reaction time in ms. Semitransparent bands represent 95% confidence intervals. Full and Endo conditions significantly differ from Exo and Baseline conditions.



Figure 9. Response time as a function of SOA, Contingency and Intensity. Lines represent predictions of the model. Semitransparent bands represent 95% confidence intervals.

(noCue.NC). We can evaluate all three-way interaction between Contingency, Intensity, and SOA by plotting it to see non-linear patterns. The results of this model are shown in Figure 9. For more clear interpretation and in order to compare our results with Lawrence and Klein results, data were collapsed to the Contingency (Figure 10) and Intensity effects (Figure 11) on response time.

From Figure 10 we can see that in purely Endogenous condition (red line) there is facilitation effect within the range of SOAs from -300 to 265ms, which means that Contingency significantly improves performance within this period. However, when SOAs becomes greater than 330ms there is IOR effect. In Full condition, where there are both Contingency and cue (black line), Contingency slightly improves performance at early SOAs. This small difference is not proved by the model, although.



Figure 10. Contingency effect (Contingent minus Noncontingent; positive values reflect effect of IOR, negative - facilitation effect) on response time. Red horizontal line along x-axis represents significant difference for both curves.

Figure 11. Intensity effect (Cue minus noCue; positive values reflect effect of IOR, negative - facilitation effect) on response time. Red horizontal line along x-axis represent significant difference for both curves.



From Figure 11 we see that there is no Intensity effect for Exogenous conditions (red line). Moreover, performance is significantly worsened when target precedes cue in the contingent condition. Also, there is a short facilitation effect for Full condition for the late SOAs (black line) - a result that is also not significant according to the model.

3.3 Model building and fitting results

There are several packages that allow to fit your own diffusion model to data. Guided primarily by the educational goal, which was to implement the model in python, we decided to choose a drift-diffusion model simulator called PyDDM (https://github.com/mwshinn/PyDDM). This is a simulator and modeling framework for drift-diffusion models (DDM), with a focus on cognitive neuroscience. The source code is available under the MIT license. This framework allows using arbitrary functions for drift rate, noise, bounds, and initial position distribution - the main parameters that we are interested in. It also contains arbitrary loss function and method for parameter fitting. Source code for our model is available at (https://github.com/Asvarisch/DDM-temporal-attention).

Our idea was to fit the model to baseline sample (NC_noCue) and then, by changing only one parameter of this model, fit a new model to Endogenous and Exogenous samples. Below the whole process is described in detail.

The first step was to create four samples of experimental data that represent four different experimental conditions, namely C_Cue (Both Endogenous and Exogenous sample or Full sample), C_noCue (Endogenous sample), NC_Cue (Exogenous sample) and NC_noCue (Totally random sample or baseline sample).

The second step was to build the Fittable model. Fittable means that we create a model that contains special objects with all the parameters we are interested in to vary. In this model, we will focus on the following parameters: drift rate, noise, boundary separation and non-decision time.

The third step was to actually fit just created model with all parameters varied to baseline sample (NC_noCue). This fit returned the value of each model's parameter. We fitted model (model_baseline) to the actual experimental data using “LossSquaredError” as a loss function and “differential evolution” method to optimize the parameters. Loss function is a minimization function which is used for measuring how well a model predicts the expected outcome. Square error is one of the most commonly used regression loss functions. It represents the sum of squared distances between the target variable and predicted values. We also tested two other loss functions available in the package - “LossBIC” and “LossLikelihood” but visually they did not fit data well. Visual inspection was the main reason to choose squared error function. Although “LossSquaredError” function visually fitted Baseline's correct RTs very well, it did not fit error RTs at all (Figure 12).

Figure 12. On top: RT distribution for correct answers for baseline condition. Below: RT distribution for error answers for baseline condition. Blue curve stands for human data and red one - for model predictions.

We tend to explain this problem by assuming that loss function has to optimize correct and error distributions simultaneously. And this can be done by combining the two scores. However, then there is a problem of how to weight the two scores when combining them. So far, this is not fully resolved problem. In our case we can see that “LossSquaredError” loss function prioritizes correct distribution fit over the error fit.

Usually, as a measure of model fitness, chi-squared method or Kolmogorov-Smirnov (KS) statistic is used to compare human data with data predicted by the model (W. J. MacInnes, 2017; Ratcliff & Tuerlinckx, 2002). However, we decided to use the KS test as a distance between distributions, not as a statistic test. If the distance between distributions is big, the p-value will be very small. Also, if the distance is small, the p-value will be relatively big. The first reason why we use the KS test as a metric for evaluating the distance between distributions is that PyDDM package does not have the KS statistic as an optimization method.

Consequently, it is not appropriate to fit a model using one method and compare distributions using another method. The second reason is the fact that the sample size of generated data dramatically affects the result of the KS statistic. For example, if there is little data (e.g., n = 20), the p-value will be close to 1, which would mean that distributions are the same. However, if there are many data (e.g., n = 1500, which is close to human data trials), the p-value will be very low, testifying that distributions are different and there is no fit at all. So, further, p-values will be used and interpreted only in relation to each other.

The fourth step was to actually get this relative metric of the goodness of fit of the baseline condition of human data and distribution of the model. Our metric for this comparison was (KS = 0.074, p-value = 1.45*10^-8). Again, here we do not interpret the actual score of the p-value. For now, this value is a kind of baseline for other conditions.

The fifth step was to take the same model distribution (model_baseline) and compare it with the Endogenous sample (human data) without changing any parameters. We supposed that if model_baseline that was trained on baseline sample (NC_noCue) will be applied to Endogenous sample (C_noCue), it would give us terrible fit because Contingent and NonContingent conditions theoretically and experimentally are very different, which is also testified by our behavioral analysis. That would be reflected in very low p-value. Indeed, in this case statistic was (KS = 0.170, p-value = 3.25*10^-75). The visual fit is depicted in Fig 13 (a). Now, we have the range of our metric from 1.45*10^-8 (the best fit) to 3.25*10^-75 (the worse fit). We do the same procedure for the Exogenous sample. In this case statistic was (KS = 0.077, p-value = 3.69*10^-9). The visual fit is depicted in Fig 13 (b). This result supports the behavioral analysis results and reflects the absence of difference between Cue and noCue conditions or absence of Intensity effect. In other words, the model trained on the baseline sample almost perfectly fits the Exogenous sample, even without changing any parameter of the model.

a b

Figure 13. a) model_baseline fits RT distribution for correct answers for Endogenous (C_noCue) condition. Below: model_baseline fits RT distribution for error answers for Endogenous (C_noCue) condition. b) model_baseline fits RT distribution for correct answers for Exogenous (NC_Cue) condition. Below: model_baseline fits RT distribution for error answers for Exogenous (NC_Cue) condition. Blue curve stands for human data and red one - for model predictions.

The sixth step was to create two new models («model_to_End» and «model_to_Ex»), where 3 out of 4 parameters would be constant, and one parameter would be variable. Then these two models were fitted to Endogenous and Exogenous samples. Constant parameters were taken from the previous model (model_fit) and worked as input parameters for new models. The main idea behind the creation of new models is the following. Will we get a good model fit for Exogenous and Endogenous conditions by changing only one parameter of the initially (baseline) trained model? If the answer is affirmative and models fit Endogenous and/or Exogenous conditions well with a simple change of one parameter, we believe that the model might simulate similar processes. If the answer is negative, then we come to the conclusion that Endogenous and Exogenous modes of attention have different model structures and, consequently, different mechanisms. attention temporary exogenous behavior

Because we could not gain any strict predictions from literature about parameters that would be mainly responsible for Endogenous and Exogenous modes of attention, we could not formulate any hypothesis and tried all combinations of three constant versus one Fittable parameter. Learned parameters for the baseline model are depicted in table 2.

Table 2. Baseline model's parameters.

Parameter	Value
Drift rate	11.45
Noise	2.15
Boundary separation	3.70
Non-decision time	0.13

We got the best result when noise, bound and overlay (non-decision) were constant parameters and drift rate was variable parameter. When we fitted «model_to_Ex» to Exogenous sample, drift rate value decreased and became 2.31. When we fitted «model_to_End» to Endogenous sample, drift rate value became 3.79. In both cases lower drift rate means that there is a performance decrease in those conditions, compared to baseline condition. A reduction in drift rate for Endogenous condition might be interpreted as late IOR. The result that is supported by behavioral analysis. However, we cannot interpret decreased drift rate for Exogenous condition, since this result contradicts behavioral analysis result, which showed that there is no Intensity effect.

By fitting new models to Endogenous and Exogenous samples, we have got the following results. The visual fit of both models to corresponding samples are depicted in Figure 14.

a b

Figure 14. a) «model_to_End «fits RT distribution for correct answers for Endogenous (C_noCue) condition. Below: «model_to_End «fits RT distribution for error answers for Endogenous (C_noCue) condition. b) «model_to_Ex» fits RT distribution for correct answers for Exogenous (NC_Cue) condition. Below: «model_to_Ex» fits RT distribution for error answers for Exogenous (NC_Cue) condition. Blue curve stands for human data and red one - for model predictions.

Statistic for «model_to_End» was (KS = 0.085, p-value = 4.35*10^-21). From this result, we see that by fitting «model_to_End» to Endogenous sample, we got no so bad fit, as if the drift rate was set constant. However, it is worse than the initial fit to the baseline sample.

The difference between baseline condition and Endogenous condition, therefore, is reflected by our metric as 10^-13. Additional verification for the previous step would be the following. We created another model «model_to_Full» that was fitted to C_Cue sample with only drift rate Fittable. Since there is no difference whether there is cue or not, that we already know from behavioral analysis, conditions C_noCue and C_Cue should be very close to each other. Consequently, «model_to_End» and «model_to_Full» should also return very similar predictions. Statistic for «model_to_Full» was (KS = 0.092, p-value = 1.65*10^-22). Since statistic for both compared models was almost identical, this hypothesis was confirmed.

Statistic for «model_to_Ex» was (KS = 0.077, p-value = 3.69*10^-9). From this result, we may conclude that there is no difference whether we change the drift rate or not. The fit of «model_to_Ex» with Exogenous sample is almost identical to fit of model_baseline with Exogenous sample. That again justify that there is no difference between baseline condition and Exogenous condition or, in other, words, there is no Intensity effect caused by the cue.



Conclusion

In order to study modes of temporal attention separately and within their interaction within the visual domain, we applied a combination of two methodologies to the two-alternative speeded choice task (2ASC). One of them was Rescorla`s “Truly random control” paradigm, intended for minimizing the engagement of Endogenous temporal attention.

Another one was new signal stimuli manipulation, adopted from Lawrence and Klein study (Lawrence & Klein, 2013b), used for minimizing the engagement of Exogenous temporal attention. Based on the results of our GAMMs analysis, we have got a statistically significant effect of Contingency, as the main effect.

That means that temporal predictability of the target's appearance actually affects the performance.

Previous studies showed that temporal patterns or contingencies might be extracted and then used to predict when to allocate attentional resources, which is then reflected in improved performance.

However, our results show that Endogenous attention actually worsens the performance, compared to baseline condition.

This main effect of Contingency or Endogenous orienting of attention might be explained by later IOR effect, that we can clearly see in Figure 10. Main effect of Contingency is also well explained by our drift diffusion model, which shows that decreased drift rate is reflected in pure performance drop.

There was no main effect of Intensity or Exogenous orienting of attention. That means that there was not any difference between cue and no cue conditions. This result is not unusual.

A number of papers in our lab do not show exogenous orienting of attention (J. W. MacInnes & Bhatnagar, 2017; Malevich, Ardasheva, Krьger, & MacInnes, 2018). It is, therefore, possible that early facilitation effect caused by exogenous cue is not so automatic and might be affected by attentional set.

The absence of early facilitation for the exogenous condition could also be because of the current experimental design.

There also was significant three-way interaction. Endogenous orienting of attention differed significantly from baseline condition over SOAs. So, by looking at Figure 10 we can see that Contingency improved performance at early SOAs, providing facilitation effect.

However, there was an IOR effect at later SOAs. This result is partly consistent with Lawrence and Klein result in terms of the presence of a biphasic cueing effect.

In their experiment Endogenous mode of attention served to enhance the speed of response with optimal performance at 400ms SOAs. In our case we observed earlier facilitation and, as a consequence, early onset of IOR. We did not get three-way interaction for Exogenous condition.

Thus, in our study, we replicated and adjusted the methodology used in Lawrence and Klein's study by applying this methodology to exclusively visual stimuli, unlike the original experiment.

Even though we did not get any match between Lawrence and Klein results and ours, therefore, none of our hypothesis was confirmed, we still believe that closer match might be obtained by increasing statistical power of the experiment and by increasing luminance of cues.

The second goal of our study was to improve our understanding of which information processing components are speeded when there is temporal Contingency between cues and targets, which leads to decreased RTs.

For that purpose, we applied the drift-diffusion model to our behavioral data. With the drift-diffusion model we reinforced that there was no effect of cueing with the model parameter.

That means that we cannot link any model parameter to Exogenous orienting of attention since baseline and Exogenous conditions did not differ. Nevertheless, for Endogenous orienting of attention, we have got improvements in the fit by changing the drift rate parameter.

Drift rate, therefore, served here as the best explanation of Contingency effect. Reduction in drift rate that we observed for Endogenous orienting of attention might be interpreted as late IOR.

So, the best interpretation we may propose now based on our model is that the drift rate change is responsible for Contingency effect. If the original model had better fit on error and RT distributions, then there may have been a different result on the best fit changing parameter. This reasoning should be considered in subsequent studies.



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