Figure 1.
represents a simplified schematic of the problems that arise when decisions are made in the presence of salient but irrelevant distractors.
The lowest panel represents a quick overview of the relevant processing stages. The time-course of the sensory representation is determined by afferent delays, the time it takes for sensory evidence to develop stimulus-selective responses, and the time it takes selective attention to extract the relevant sensory information from salient distractors. The timing of the decision-making stage is determined by the onset of the integration process and its termination. The duration of the last stage of processing is determined by the efferent delays. For the current purposes, we assume that the afferent delays as well as the delay for stimulus-selective sensory information to reach the decision stage is hard-wired and cannot be changed through cognitive control. However, we test the possibility that the onset of the decision process may be under cognitive control. The middle panel provides a more detailed look at the mix of momentary sensory information available to the decision process as a function of time: the initial phase is dominated by random internal fluctuations or the first pulse of sensory information that is not stimulus selective (grey), the second phase is dominated by physically salient stimuli (orange) and the third phase is dominated by the task-relevant stimuli, regardless of salience (blue). The time of transition from the first to the second phase is determined by afferent delays, the transition from the second to the third phase depends on how quickly selective attention can be allocated to the target. The top panel examines the effects of adjusting decision onset: if the decision process is initiated early, it will integrate information from physically salient stimuli that may or may not be relevant for the current task. If decisions are initiated late, response latencies may be prolonged unnecessarily.
Figure 2.
(A) A cue instructed subjects to attend to one of two streams of coherently moving random dots and report its direction of motion by pressing a button with their left or right index finger. The relevant stream of target dots was either lighter than the background, in which case the distractor was black, or darker than the background, in which case the distractor was white. The distractor dots could move either in the same or opposite direction as the target (congruent and incongruent condition). In addition, the distractor could move upwards, i.e. orthogonal to the axis of motion of the target dots (neutral condition). (B/C) The task was performed in a cyclic manner requiring a response from the subject once every 2 seconds. The time of the intended response was indicated by an auditory click. Clicks were presented once every second to convey a stronger sense of rhythm, but responses were only required on every other click, following the cue and stimulus presentation. Stimulus duration was randomized between blocks of trials.
Figure 3.
The model combines two basic neural mechanisms: stimulus selection via biased competition between different luminance channels (layer 2) and the reduction of noise by integrating over time (layer 4) with a total of 6 free parameters: 3 describing biased competition, 1 describing the non-linear neural contrast response function, 1 scaling parameter, and 1 parameter describing non-decision related delays. (Layer 1) The input to the system consists of two boxcar functions that indicate the presence of the two moving random dot patterns with positive (white arrows) and negative luminance contrast (black arrows). In the first stage these inputs are filtered with a gamma-kernel to yield realistic temporal dynamics for the encoding of visual information by sensory neurons. (Layer 2) Biased competition between mutually inhibitory luminance channels is implemented as multiplicative weights s(t) for the target luminance and 1 - s(t) for the distractor where 0≤s(t)≤1. Prior to stimulus onset, the weights of both luminance channels are equal (i.e. 0.5). To simulate the development of the biased competition in favor of the target luminance (black-and-white inset in layer 2), the weight of the target stimulus increases from 0.5 to an asymptote. Three parameters determine the time course of the competition: the time of the transition (τ), the speed of the transition (σ), and the asymptote (α). (Layer 3) The information from each luminance channel is routed to both motion channels. A stimulus elicits activity only if it matches the preferred direction of the motion channel (i.e. leftward or rightward motion). Activity from both luminance channels is summed within each motion channel. To simulate physiological neuronal responses, the activity is passed through a non-linear, Naka-Rushton contrast response function. (Layer 4) The input to the integration stage is the difference of the output of the right and leftward motion channel. In addition, neural noise is added in the form of a continuous Gaussian white noise with a standard deviation of 1 arbitrary unit per second. The percentage of correct responses as a function of time is calculated as the mass of the integrator above zero. When modeling the free reaction time version of the task, upper and lower response thresholds are included in the model. A decision is aborted and considered correct/incorrect when the diffusion process reaches the positive/negative bound. In addition, the time of integration onset t0 is assumed to be variable and under cognitive control.
Table 1.
List of the 10 parameters of the biased competition model of decision-making.
Figure 4.
Processing times of all subjects as a function of stimulus duration, i.e. intended processing time in (A).
The box and whiskers indicate mean ±one and two standard deviations, respectively. The brackets over two adjacent stimulus durations indicate significant differences at p<0.01. An additional asterisk above the bracket indicates a p-value below 0.001. In nearly all cases, a 16.7 ms increase in intended processing time resulted in a significant difference in observed processing time. (B) Percent correct responses are plotted as a function of mean processing time and stimulus congruency. Each dot describes a block of trials with a particular intended processing time and congruency (green-congruent; black-neutral; red-incongruent). The x-value of the dots corresponds to the mean observed processing time and the y-value corresponds to the mean proportion of correct responses. In the congruent and neutral conditions response accuracy increases monotonically as a function of RT. For the incongruent condition, in contrast, there is an initial dip with accuracy decreasing significantly below chance. The solid lines correspond to the maximum-likelihood fit of the data with the biased competition model. The left-hand vertical dotted lines correspond to the total non-decision times in the system. The right-hand dotted vertical line corresponds to the time at which the stimulus selection process has finished. (C) Time-resolved estimates of drift rate that give rise to the model predictions in (B). For all three conditions, drift rate converges to the same asymptote. This feature is not hard-coded into the model, but arises from the fit to the data. It indicates that attention selects the target stimulus in a winner-take-all fashion (α = 1), such that in the steady state, the identity of the distractor no longer affects drift rate. Drift rate in (C) is determined by four parameters of the biased-competition model.
Table 2.
Timing accuracy in the cyclic deadline task.
Table 3.
Determinants of processing time.
Table 4.
Trial history effects: processing time.
Table 5.
Trial history effects: accuracy.
Table 6.
Model fit of the biased competition model to the cyclic deadline task.
Figure 5.
Properties of the onset mechanism.
In the biased competition model, speed and accuracy can be manipulated in two independent ways — adjusting response threshold and/or decision onset. The simulations are based on the fit of the biased competition model to the data of the CD paradigm combined across all 7 subjects (Figure 3). Simulated response accuracy (A), response latency (B) and rate of information transfer (C) are depicted as functions of response threshold on the x-axis and onset of the decision process on the y-axis. The dotted line at 0 ms in each panel indicates the average non-decision time across all subjects. Our null-hypothesis states that decision onset is fixed and coincides with this time-point. Delaying decision onset increases response accuracy and RT. Increasing response threshold has the same effects, but the effect on response accuracy is weaker while the effect on RT is stronger. P1 denotes the threshold that is necessary to achieve 99% response accuracy if decision onset is fixed at 0 ms. P2 denotes a second set of parameters that leads to the same accuracy of 99% but allows decision onset to deviate from 0. Note that P1 is associated with mean reaction times around 570 ms. In contrast, P2 achieves the same accuracy with a mean RT of 370 ms. This shows that decision onset can be a very effective way to trade speed for accuracy.
Figure 6.
Fit of the threshold and non-decision time (TN_22) model to the data from the RT paradigm in (A–D).
The first three panels represent RT distributions as a function of accuracy from the congruent, neutral and incongruent condition. Note that the x-axis covers different ranges in the incongruent panel in order to include the error trials. The empirical results are represented as 95% confidence ellipses around 5 different RT quantiles as a function of accuracy (green/red: correct/error trials speed instruction, blue: correct trials accuracy instruction). The model fit for the same quantiles is represented as the axes of the 95% confidence intervals. The rightmost panel displays the model parameters (green: accuracy instruction; red: speed instruction). Data-points from each subject are connected by a line. Results of a paired t-test are indicated above the data (bracket: ns; one star: p<0.05; two stars: p<0.01; three starts: p<0.001). The current model used 4 parameters (bound and non-decision time two parameters each) to fit the data. Note that the model fails to capture some key properties of the data. (E–H) Fit of the threshold, decision-onset and non-decision time (TON_222) model to the data from the RT paradigm. Conventions as in panels A–D. Note that the model provides a much better fit to the data. Speed-accuracy tradeoff is mediated through response threshold and decision onset. There are no systematic differences of non-decision time between the two conditions.
Figure 7.
Fit of the threshold, onset, starting-point variability and non-decision time (TONS_2221) model to the data from the RT paradigm in (A–D).
Figure 8.
Fit of the threshold, selection-speed and non-decision time (TAN_222) model to the data from the RT paradigm in (A–D).
(E–H) Fit of the threshold, non-decision time and starting-point variability model (TNS_222).
Figure 9.
Fits of one of the “From-Scratch” models (CD_TON_222) to the data from the RT paradigm.
Allowing the parameters of the CD model to vary provided a significant improvement to the fits. Three of the CD parameters were not significantly different from the values recovered from the CD paradigm. Two parameters (stimulus selection speed and stimulus selection onset) did show significant differences. Nevertheless, decision onset was needed to explain the speed-accuracy tradeoff.
Figure 10.
Parameter estimates for decision onset and response threshold.
(A) Time dependent drift rate as estimated from the CD paradigm. The arrows indicate decision onset in the speed and accuracy condition as estimated from the TONS_2221 model. (B) Rate of information transfer as a function of response threshold on the x-axis and onset of the decision process on the y-axis. Overlaid are the parameter estimates for decision onset and response threshold from the TONS_2221 model. The lower left point on each line represents the parameters in the speed condition, the upper right point represents the parameters in the accuracy condition. All subjects delayed decision onset and increased response threshold in the accuracy condition. However, none of the subjects performed optimally. Information transfer in both conditions could have improved by further delaying the onset of the decision and lowering response threshold.