The Attentional Drift Diffusion Model of Simple Perceptual Decision-Making

Perceptual decisions requiring the comparison of spatially distributed stimuli that are fixated sequentially might be influenced by fluctuations in visual attention. We used two psychophysical tasks with human subjects to investigate the extent to which visual attention influences simple perceptual choices, and to test the extent to which the attentional Drift Diffusion Model (aDDM) provides a good computational description of how attention affects the underlying decision processes. We find evidence for sizable attentional choice biases and that the aDDM provides a reasonable quantitative description of the relationship between fluctuations in visual attention, choices and reaction times. We also find that exogenous manipulations of attention induce choice biases consistent with the predictions of the model.


A. Likelihood computation: details
We used the following simulation procedure to compute the likelihood of the experimental data under the aDDM. For each set of parameters, we calculated the likelihood of the observed choice and reaction time (RT), conditional on the fixations observed in the trial.
The simulations were carried out using a discretized state space: time evolved in steps of 10 ms, and the RDV was discretized into 21 bins of size 0.1. As illustrated in Figure S1, this transforms the state-space of the model into a two-dimensional table. Columns denote time, increasing from left to right. Rows denote the state of the RDV.
At any discretized time point, the simulation assigns a probability mass to each RDV bin, which measures the likelihood that the value of the RDV signal falls in that bin, at that time, conditional on a decision not having made yet (because a decision barrier has not been crossed). Importantly, note that the sum of these likelihoods in a particular RT column need not add to 1 if there is a positive probability that a decision can be made before that time.
The simulation starts at time zero (first column of the table) with the entire probability of the RDV signal placed in the zero-bin with probability 1. This reflects the assumption that in the aDDM the value of the RDV is zero at the beginning of every decision.
We then fill out the columns of the table from left to right, as illustrated in Figure S2. Let where ! " #→+ denotes the probability of transition from $ to &. This transition probability is easily computed from the equations describing the aDDM. The change in the RDV in a single time step is given by / 0, 2 , with µ = 0 before the first fixation to an item occurs, during non-item fixations and during inter-fixation transitions. During item fixations, 0 = 5(7 89:" − <7 =#>?" ) for fixations to the left option, and 0 = 5(<7 89:" −7 =#>?" ) for fixations to the right option. The probability of transition ! " #→+ is therefore given by the value of the probability density function / 0, 2 for A, where A is the difference in mean RDV values between bins & and $.
In order to ensure that the fixation state is fixed throughout the duration of each time bin, we take the duration of each fixation in milliseconds and divide it by the size of the time bin (10 ms). We then discard the remainder, such that each corrected fixation duration is now an exact multiple of the time bin size. By doing this, we simplify the computation by having all fixations terminate at the end of a time bin.
At every time step, we also calculate the probability of the RDV reaching each of the two barriers. In particular, the likelihood of the model crossing the upper barrier at time step % + 1 is given by where ! " #→BC denotes the probability of going from a value of RDV in bin $ to crossing the upper barrier at time step t, and is given by the probability of a draw from / 0, 2 that exceeds 1 − $. ! ")* DEFG is defined analogously.
For every trial, the model is simulated by filling out the The process is illustrated in Figure S2, in which the barrier crossing probabilities are depicted as additional top and bottom rows shaded in gray. In order to better understand the ability of the aDDM to explain the data from Experiment 1, and to better understand the shortcomings of the model, we carried out two additional tests.
Additional test 1 This test was designed to investigate the assumption that perceptual signals are integrated linearly in the aDDM. To do this, we fitted a variation of the aDDM with an additional free parameter J and allowed for the integration of non-linear perceptual signals of the form 7 89:" K and 7 =#>?" K .
Let LMN " denote the relative decision value at time t within the course of a single decision. At every time step " , its change is given by " + P " , where P " is i.i.d. zero mean white Gaussian noise with standard deviation 2, and is the deterministic change in the RDV over the time step.
= 0 until the first fixation to one of the two stimuli occurs, as well as during non-stimuli fixations and inter-fixation transitions, while 0 = 5(7 < is a parameter between 0 and 1 that measures the size of the attentional bias, 7 89:" and 7 =#>?" are the relative proximities of the left and right items shown in the trial, and J is the exponent for the relative proximities values. Figure S3 shows how different values of J affect the transformation from an item's angular distance to the target to its relative proximity value, which is used in the model fitting. We used the same grid search and MLE procedure described above and in the main text to find the optimal parameters for alternative model 1. We started with a search over the grid obtained from the cross product of 5 = {0.003, 0.004, 0.005} , 2 = {0.06, 0.07, 0.08}, < = {0.15, 0.25, 0.35} and J = {0, 0.5, 1}. We then zoomed into the optimal parameters at each step, making the grid finer, until reaching a stopping criterion (when the improvement in the likelihood of the proposed solution was less than 1%). The resulting solution obtained through this process was 5 = 0.0042, 2 = 0.063, < = 0.37 and J = 0.93. The similarity between this result and the one obtained for the simple aDDM (5 = 0.0041, 2 = 0.063, < = 0.36 and J = 1) suggests that the simpler model, with J = 1, can capture many of the patterns observed in the behavioral data without the need for non-linear perceptual signals. Figure S4 shows a comparison of some of these behavioral patterns in the data and in the simulations generated for alternative model 1.

Additional test 2
This test was designed to investigate if the best fitting parameters of the aDDM change within the course of a block of trials, as the memory of the target orientation dissipates.
To do this, we fitted the model separately for trials occurring immediately after target display (early trials) and trials occurring 4 trials after target display (late trials). Alternative model 2 is the same as the aDDM we describe in the main text. Let LMN " denote the relative decision value at time t within the course of a single decision. At every time step " , its change is given by " + P " , where P " is i.i.d. zero mean white Gaussian noise with standard deviation 2, and is the deterministic change in the RDV over the time step.
= 0 until the first fixation to one of the two stimuli occurs, as well as during non-stimuli fixations and inter-fixation transitions, while 0 = 5(7 89:" − <7 =#>?" ) during fixations to the left option, and 0 = 5(<7 89:" −7 =#>?" ) during fixations to the right option. 5 is a positive constant that controls the speed of integration, < is a parameter between 0 and 1 that measures the size of the attentional bias, and 7 89:" and 7 =#>?" are the relative proximities of the left and right items shown in the trial. We fit the model separately for two sets of trials, first using only trials occurring immediately after target display (early trials), in which the memory of the target is at its most recent for the subjects, then using only trials occurring 4 trials after target display (late trials), in which the memory decay of the target is at its peak. We used the same grid search and MLE procedure described above and in the main text to find the optimal parameters for these two instances of alternative model 2. Using early trials only, the optimal parameters obtained were 5 = 0.0042, 2 = 0.063 and < = 0.36; using late trials only, the optimal parameters were 5 = 0.004, 2 = 0.063 and < = 0.37. The similarity between these two results shows that we can use all trials in the same model fitting procedure, as the position of the trial in the timeline of the experiment does not affect the behavioral results in a way that significantly impacts the model fitting results (see also Figure S5 for a comparison between the behavioral results for early and late trials).

Figure S3.
Conversion from an item's angular distance to the target (Δ) to a relative proximity value (7) to be used in the model fitting, for different values of J. The aDDM fitted in the main paper uses J = 1, whereas other values of J were tested in alternative model 1, as discussed in these Supplementary Materials. (f) Probability that the first-seen item is chosen as a function of the duration of that first fixation. Subject data includes only even-numbered trials. Error bars show 95% confidence intervals for the subject's data pooled across all even trials, and across all simulated trials in the case of the model.  (f) Probability that the first-seen item is chosen as a function of the duration of that first fixation. Subject data includes only even-numbered trials. Error bars show 95% confidence intervals for the subject's data pooled across all even trials, and across all simulated trials in the case of the model.