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Edited by: Marius Usher, Tel-Aviv University, Israel

Reviewed by: Eddy J. Davelaar, Birkbeck College, UK; Don Van Ravenzwaaij, University of Amsterdam, Netherlands

*Correspondence: Stephanie Goldfarb, Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA. e-mail:

This article was submitted to Frontiers in Cognitive Science, a specialty of Frontiers in Psychology.

This is an open-access article distributed under the terms of the

We investigate human error dynamics in sequential two-alternative choice tasks. When subjects repeatedly discriminate between two stimuli, their error rates and reaction times (RTs) systematically depend on prior sequences of stimuli. We analyze these sequential effects on RTs, separating error and correct responses, and identify a sequential RT tradeoff: a sequence of stimuli which yields a relatively fast RT on error trials will produce a relatively slow RT on correct trials and vice versa. We reanalyze previous data and acquire and analyze new data in a choice task with stimulus sequences generated by a first-order Markov process having unequal probabilities of repetitions and alternations. We then show that relationships among these stimulus sequences and the corresponding RTs for correct trials, error trials, and averaged over all trials are significantly influenced by the probability of alternations; these relationships have not been captured by previous models. Finally, we show that simple, sequential updates to the initial condition and thresholds of a pure drift diffusion model can account for the trends in RT for correct and error trials. Our results suggest that error-based parameter adjustments are critical to modeling sequential effects.

Efforts to model and predict human behavior are informed by an understanding of the dynamics of error rates (ERs) and reaction times (RTs) in simple tasks. In particular, in two-alternative forced-choice (TAFC) tasks (e.g., Laming,

Patterns in RTs for individual trials are well documented in the literature. In particular, relative to their mean RTs on correct trials, subjects are known to respond faster on error trials and more slowly immediately following errors (Rabbitt,

Moreover, the characteristic patterns in speed and accuracy following sequences of repetitions and alternations are well documented only for tasks in which the stimuli are equally likely. While overall trends in speed and accuracy have received significant attention (Carpenter and Williams,

In a majority of TAFC studies, participants are either rewarded equally for overall participation or they are rewarded for each correct response. However, several studies (e.g., Corrado et al.,

When stimuli are equally probable and correct responses are equally rewarded, several effects are known. For small (<500 ms) response to stimulus intervals (RSIs), the behavior typically illustrates automatic facilitation (AF): mean RTs on the current trial are faster if the previous trial was a repetition, regardless of whether the current trial is a repetition or an alternation. For slow RSIs (≈1000 ms), mean RTs on the current trial after a series of alternations are faster if the current trial is a repetition and slower if the current trial is an alternation (Bertelson,

In this paper, we study sequential patterns in ERs as well as in RTs for error and correct responses independently in TAFC tasks in which stimuli are equally probable or strongly biased toward repetitions or alternations, focusing on sequences of three trials. Stimulus sequences can be biased by specifying stimulus probabilities (state orientation) or by specifying transition probabilities between states (transition orientation), and it is known that these processes produce distinct response patterns in RT (Brodersen et al., _{A}) to be unequal to the probability of a repetition (1 − _{A}). Transition probabilities _{A} and 1 − _{A} are held fixed over blocks of trials, and we use relatively long RSIs (800 and 1,000 ms mean), for which SE is most apparent. We reanalyze behavioral data from an equal-probability experiment (Cho et al., _{A} set to 10, 50, and 90%. We find significant transition probability effects on RTs for error and correct responses and on ERs.

_{A} and stimulus 1 (a repetition) with probability 1 − _{A}.

To further study patterns in RT and ER we extend the pure drift diffusion model (DDM) to account for sequential patterns. The pure DDM describes choice between two alternatives by representing the noisy accumulation of the difference in evidence (logarithmic likelihood) from a given initial condition to one of two decision thresholds. This process is known to mimic aspects of neural integration (Carpenter and Williams,

Related TAFC models frequently involve a variant of the leaky competing accumulator (LCA; Usher and McClelland,

Physiological evidence suggests sources of systematic changes in behavior from trial to trial, providing some neurobiological basis for our proposed update mechanisms. An electroencephalogram (EEG) study has identified a SE pattern in the P300 response (Sommer et al.,

An understanding of the relationship between error correction and sequential biasing mechanisms may allow us to further differentiate between corresponding physiological processes. Such an understanding could have broad implications. Indeed, recent work suggests that the same mechanisms that account for sequential effects also account for sequence learning (Soetens et al.,

This paper is organized as follows. In Section

In this section, we describe the protocol followed for the two experiments presented in this paper. We then describe a general model of decision making, which accounts for choice behavior with two simple mechanistic adaptations to the pure drift diffusion model (DDM). Finally, we describe a procedure for fitting the model to match participant data in our adapted DDM.

The first experiment (reanalyzed from Cho et al.,

In the second experiment stimulus transition probabilities were varied from block to block, so that in a given block a participant would have a constant high, medium, or low probability of alternations. That is, given the current stimulus 1, a participant would next see the other stimulus 2 with probability _{A} and the same stimulus 1 again with probability 1 − _{A}, and the sequence of stimuli would be drawn from a transition-oriented Markov process, as shown in Figure

Sixteen adults (6 males) participated in exchange for a standard payment of $12 per session of 9 blocks. Participants were recruited from the Princeton University community via announcements posted online and on campus. The experiment was approved by the Institutional Review Panel for Human Subjects of Princeton University, and all participants provided their informed consent prior to participation.

Participants performed an RT version of a motion discrimination task. The visual stimulus consisted of a black screen showing a cloud of white moving dots with a red, stationary fixation dot at its center. The red dot had size 0.30° visual angle, and the white dots had size 0.15° each and moved within a circle of diameter 10° at a speed of 7°/s and a density of 20 dots/degree^{2}. On each trial 90% of the white dots would move coherently in a given, “correct” direction, and the remaining white dots would move randomly. The high coherence of motion was selected to ensure that some processing was necessary but that the difficulty of the task would remain low, consistent with other studies of sequential effects. A decision could be indicated with a left or right keypress at any point after dots appeared on the screen. Responses were collected via the standard Macintosh computer keyboard, with the “Z” key used to indicate leftward motion and the “M” key used to indicate rightward motion. The experiment was performed on a Macintosh computer using the Psychophysics Toolbox extension (Brainard,

The participants were instructed to fixate upon the red dot and then determine the direction of the moving dots. They were also instructed to complete the session as quickly and as accurately as possible. Each participant completed 1 session of approximately 60 min duration.

Each session consisted of 9 blocks of 200 trials each in which the _{A} remained fixed at 10, 50, or 90% (3 blocks for each condition). The order of the blocks was constrained to follow a Latin Square design. Participants were allowed a short break between blocks. To minimize anticipatory responding, response to stimulus intervals were drawn from a gamma distribution with a mean of 1 s for each trial, following the convention set in previous sequential RT tasks (Rabbitt,

During each block in the session, the subjects received the following feedback. Correct responses were denoted with a short beep sound, and error and premature, anticipatory responses were denoted with a buzz sound. In addition, on every fifth trial, the number of correct responses provided in that block so far was briefly flashed across the screen. This was the only feedback that was provided. Participants were seated at a viewing distance of approximately 60 cm from the screen. Our protocol in Experiment 2 is similar to others in the literature (e.g., Newsome and Pare,

To account for sequential effects and error effects, we consider a simple adaptation of the pure drift diffusion model (DDM) in which the initial condition and thresholds are updated sequentially following each trial (Ratcliff and Rouder,

Here ^{2}. The evidence thresholds are set at ±_{nd} such that RT = DT + _{nd} where DT is the decision time, it can be shown that the mean DT and ER are (Gardiner,

and

in which the parameters have been scaled so that

Given a non-zero initial condition

See the

The simplicity and analytical tractability of the DDM is a motivating factor in our decision to use it as a basis for our study. We note that the DDM is much simpler than the leaky competing accumulator (LCA) Model (Usher and McClelland,

As with other sequential effects models (e.g., Cho et al.,

in which

The memory function is updated as follows:

where 0 < Δ < 1. The Δ parameter determines the dependence of behavior on previous trials, with higher values corresponding to the level of influence of trials further back in the sequence and lower values corresponding to dependence on only recent trials. A Δ value of 0.5 corresponds to a memory length of approximately four trials (Δ^{4} = 0.0625), after which history dependence goes below 5%. A single update parameter Δ can then account for responses to both R and A trials. In contrast, the Cho, Jones, and Gao models used a memory function

We also employ error-correction threshold modulation. Threshold modulation has been studied in the context of several sequential choice tasks (Bogacz et al.,

In the adapted DDM, the thresholds are adjusted after every trial and constrained to remain symmetric at

The range of

Sequential, error-correcting variations in the evidence thresholds _{nd}, for a total of eight parameters.

Fitted model parameters were used to validate the adapted DDM against data from the two experiments. Separate analysis and fitting was conducted for Experiments 1 and 2. In each case, the data were sorted by sequence, RT, and ER. Model behavior was computed for each parameter set and then sorted similarly. The model was run using the same stimulus sequences that each participant had encountered. Parameters were selected by attempting to minimize the sum of squares error between model prediction and participant data,

in which the elements _{i}_{A}. Time was considered in units of seconds and ERs in decimal fractions of trials, so that range of parameters for elements of

The search for parameters was conducted using a Trust-Region-Reflective Optimization (TRRO) algorithm (Coleman and Li,

Use of the analytical expressions of equations (

_{nd} |
Δ | |||||||
---|---|---|---|---|---|---|---|---|

Experiment 1 | 38.1747 | 0.2626 | 0.0943 | 0.0051 | 0.6860 | 0.0058 | 0.0348 | 0.2857 |

Experiment 2 | 19.3312 | 0.3359 | 0.1181 | 0.0034 | 0.6882 | 0.0034 | 0.1635 | 0.2062 |

A study of the differences between the two tasks can lend some insight into the different fit parameterizations for each of the experiments. We note that the choice tasks presented in Experiment 2 are more challenging than those of Experiment 1, in which stimuli were highly discernable. The difference in signal to noise ratios (

In order to better understand the relationship between sequential effects and error effects, data from the two experiments were sorted by stimulus sequence and response correctness and compared with model predictions. We first note several trends from this analysis in the Experiment 1 data. We then analyze data from Experiment 2, and we consider how error and sequential effects are influenced by the relative frequencies of repetition (R) and alternation (A) trials. At the same time, we validate our model fits by comparing them with the data from the two experiments.

In our analysis, we refer to RA and AR sequences as _{A} = 10, 90%) of Experiment 2, longer sequences of A’s, respectively, R’s, occur too rarely to yield sufficient data. The degrees of freedom for the

We consider sequential effects and error effects in Experiment 1 data (referred to as Cho Data), in which R and A trials were equally likely, and as has been customary, we initially average over all responses, correct and incorrect. We first discuss overall sequential effects in RT and ER, as shown in Figure ^{2} = 0.13] and ER [^{2} = 0.25] were significant in two one-way, within-groups ANOVAs. We consider also three published, generative models of the data in Experiment 1, which we refer to as the Cho et al. (

We next consider the data separated into correct and error trials, shown in Figure ^{2} = 0.35)], along with the interaction of response correctness and expectedness of a stimulus [^{2} = 0.19]. We note a slight asymmetry in the responses such that RTs for error and correct trials are closer for the R lines than for the A lines. Figures

Strikingly, we note that when plotted against each other as in Figure ^{2} = 0.995, ^{2} = 0.96,

The ordering of the tradeoffs is influenced by the nature of the task. However, in each task we see that an increase in time to respond correctly (or a bias toward the correct response) is correlated with a decrease in time to respond in error, and vice versa. Our proposed biasing mechanism achieves a similar effect.

Finally, we consider the RTs before, during, and after an error in Experiment 1, as shown in Figure ^{2} = 0.48]. We again compare the behavior with the adapted DDM and the three previous models. In the Cho Model, the RT after an error is slower than the RT on the error trial but faster than the trial immediately prior to the error. The Jones Model maintains the trends in the data but parameter values are skewed so that the range of RTs is larger. In the two Gao Models, mean RTs for trials immediately preceding and following an error are faster than those on the error trial itself: opposite to the data. The adapted DDM provides the best fit, with the RTs for error trials and post-error trials closely matching the data, although it underestimates RTs on the pre-error trial.

We compare the adapted DDM with the other models using the Akaike Information Criterion (AIC; Akaike, _{c}; Hurvich and Tsai, _{c} values cannot be computed for the Gao model, because the number of means being compared is too close to the number of parameters used in the model itself.

Model | Total parameters | ^{2} |
AIC | AIC_{c} |
BIC |
---|---|---|---|---|---|

Adapted DDM | 8 | 0.996 | 84.7 | 115.1 | 108.2 |

Cho | 13 | 0.936 | 138.2 | 237.0 | 176.5 |

Gao 1 | 18 | 0.915 | 140.4 | – | 193.4 |

Gao 2 | 18 | 0.951 | 137.0 | – | 190.0 |

Jones | 16 | 0.943 | 146.9 | 450.9 | 194.1 |

We now consider the role that alternation frequency plays in sequential and error effects. We first address overall trends in RT and ER, as shown in Figures _{A} = 50% blocks match trends from Experiment 1 with longer RTs and higher ERs for unexpected trials, and shorter RTs and lower ERs for expected trials. Trends for the _{A} = 10% blocks and _{A} = 90% blocks are clearly distinguishable from the trends for _{A} = 50% blocks, notably in the magnitudes of the slopes of R and A lines. Further, there is an approximate symmetry between the _{A} = 10% case and the _{A} = 90% case.

_{A} in Experiment 2, averaged over correct and error trials_{A} is most apparent in the mean RT plot on expected trials (RR, AA) and in the mean ER on unexpected trials (AR, RA). Model fits for

Sequential effects in mean RTs are clearly influenced by the probability of alternations, with respect to both overall mean RTs and ERs (Figures _{A} conditions but there are significant differences in mean RTs for expected sequences (RR, AA). For the highest _{A}, RT is faster on AA trials than the corresponding sequence RTs for lower _{A}s, and for the lowest _{A}, the RT is faster on RR trials than the corresponding sequence RTs for higher _{A}s. As expected, we find that the effects of sequence [^{2} = 0.26] and its interaction with _{A} [^{2} = 0.26] on RT are both significant. Error rates are greatest for AR trials at the highest _{A} and RA trials at the lowest _{A}. The effects of sequence [^{2} = 0.31] and its interaction with _{A} [^{2} = 0.32] on ER are also both significant. The adapted DDM reproduces the qualitative patterns in the data, but overestimates RTs for expected sequences when their probabilities are low (RR, with _{A} = 90%; AA, with _{A} = 10%), and underestimates ERs for unexpected sequences (AR, RA): Figures

We also found that the overall sequential effects are influenced by the probability of alternations. The relationship between the time to respond to sequences ending in R versus A on the final sequence is known to indicate relative preference for R or A trials (Audley,

The green lines corresponding to _{A} = 50% in Figures _{A} = 10% blocks show a strong preference for R: the mean RT after an R is faster in the case of RR than it is for AR, whereas the RT after A is similar for both RA and AA. The blue lines corresponding to _{A} = 90% show a strong preference for A: the RT after an A is faster in the case of AA than it is for RA, whereas the RT after an R is similar for both RR and AR. For _{A} = 10%, the model predicts, as in the data, that the repetition RT is faster for RR than it is for AR, but the model predicts a slower alternation RT for AA than for RA, and it shows a symmetric trend for _{A} = 90%. In summary, both data and model exhibit increases in preference for A with increased probability of alternations, showing that relative preferences for R or A trials can be influenced by transition probabilities in addition to task properties such as RSI.

_{A}, the R lines overlap, and for high _{A}, the A lines overlap, resulting in an approximate reflectional symmetry between data for the high and low _{A} blocks, so that sometimes the mean time for an error trial is slower than for a correct trial. The error bars in plots

In Figures _{A}. A two-way within-groups ANOVA shows that the effects of correctness [^{2} = 0.80], whether or not the trial was expected [^{2} = 0.44], and the interaction of these two factors [^{2} = 0.62] are all significant. For unbiased sequences (_{A} = 50%), sequential effects are again similar to those for correct and error trials in Experiment 1 (Figure _{A} blocks, the orientations of the R and A lines are maintained, with correct R lines sloping upwards from RR to AR and correct A lines sloping down from RA to AA. For _{A} = 50%, the slopes of the R and A lines for incorrect responses are nearly opposite the slopes of the R and A lines for correct responses. For _{A} = 10% blocks, the R lines cross and the A lines are further apart than in the _{A} = 50% blocks. For _{A} = 90% blocks, we see a mirrored trend, in which the A lines cross and the R lines are further apart than in the _{A} = 50% block. We also note a striking asymmetry for the biased stimuli: for _{A} = 10% R lines are, on average, closer together than A lines, and for _{A} = 90% this relationship is mirrored, so that A lines are closer than R lines. However, the mirroring is not perfect: the degree of overlap in R lines is greater for _{A} = 10% than the corresponding overlap in A lines for _{A} = 90%.

The trends in correct and error trial RTs, including the crossover of the R and A lines, are generally captured by the adapted DDM, as shown in Figures _{A} = 90% (10%), due to overestimation of the RR (AA) RTs.

Next, we note that the sequential RT tradeoff between correct and error responses is also observed in Experiment 2, as shown in Figure ^{2} = 0.75, ^{2} = 0.74, _{A}^{’}s of 10, 50, and 90% is not quite as strongly correlated. Differences in order can be expected because the sequential effects for each probability of alternation are influenced by the probability of alternation.

_{A} for both

Finally, we note that post-error slowing occurs for all _{A} blocks with the same trend: the error trial itself incurs a slightly faster RT than the trial which precedes it, and the post-error trial incurs an RT significantly slower than RTs for the preceding two trials, as shown in Figure ^{2} = 0.57] and on _{A} [^{2} = 0.07] are both significant, but the effect of their interaction is not significant. Thus, in Experiment 2, pre- and post-error RTs share the pattern of RTs in Experiment 1, and this pattern is preserved over all three values of _{A}. The bottom panel shows that our model both qualitatively and quantitatively captures the post-error slowing in Experiment 2. However, as in Experiment 1 (Figure

_{A}.

In this paper, we propose priming and error-correcting mechanisms to account for sequential effects and post-error slowing, respectively. Each mechanism, on its own, is commonplace in models of decision making. Indeed, various priming mechanisms have been previously proposed to account for sequential effects (Cho et al.,

Our model is informed by previous work: the initial conditions are varied according to a priming function similar to those in other models (Cho et al.,

Our adaptation of the pure drift diffusion model has multiple advantages. The pure DDM is analytically simple, and explicit expressions exist for both RT distributions and accuracy, and separate and closed-form expressions for mean RTs can be derived for correct and error responses, as shown in the Appendix. With non-zero initial conditions, the pure DDM can also account for RT distributions for correct and error trials. Moreover, the priming and error-correction mechanisms that we have proposed are conceptually straightforward. With the error-correction mechanism, our model accounts for post-error slowing: the RT for the trial which immediately follows an error trial is not only significantly slower than the error trial but also slower than the RT for the trial immediately preceding the error. We show that when thresholds are systematically adjusted to account for error and correct responses and priming is implemented, sequential patterns in error and correct response trial RTs emerge and are consistent with participant behavior, as shown in Figures

Indeed, our adapted DDM predicts the characteristic trends in mean RTs for sequences ending in correct or incorrect responses whereas several other models do not. We show experimentally and for the first time that unexpected trials (AR or RA) result in relatively slow correct responses and fast errors, whereas expected trials (RR or AA) result in relatively fast correct responses and slow errors as shown again in Figures

The relationship between RTs for correct and error trials is central to our model: biasing the initial conditions toward expected sequences automatically biases them

Significantly, we also identify a sequential RT tradeoff, in which the correlation between the mean RTs for error and correct trials for each of the sequences (RR, AR, RA, AA) is quite strong: a faster RT on an error response corresponds to a slower RT on a correct response. The correlation between mean RTs for correct and error trials is captured by our model, as shown in Figures

We then show that sequential effects in mean RTs overall, as well as in mean RTs for correct and error trials, are significantly influenced by the probability of alternations. Our data reveals remarkable near-mirror-symmetry between RT patterns for alternations when the probability of alternations is low and repetitions when the probability of alternations is high: incorrect responses are fast and correct responses are significantly slower. Sequential effects in ER also vary with the probability of alternations. Our model captures this near-symmetry in Figures

Moreover, we have shown, both in our data and in our model, that an increase in the likelihood of alternations corresponds to an increase in relative preference for alternations. This can be inferred from the RT versus sequence plots in Figure

The sequential effects in RT and ER for various probabilities of alternation are of particular interest due to their relevance to prior physiological studies. In particular, previous work has shown that the anterior cingulate cortex (ACC) is sensitive to alternations in a sequence of stimuli and identified corresponding neural signals (e.g., Botvinick et al.,

Additional directions for future work include a consideration of alternative error-correction and priming mechanisms. For example, the magnitude of adjustments made due to our priming mechanism varies from trial to trial, while adjustments from the error-correction mechanism are consistent. Alternate models in which different update schemes are employed are worthy of consideration. Such a study could allow for further model simplification and provide a stronger account of behavior in choice tasks. Moreover, sufficient data should be gathered so that sequential and error effects can be studied and described for individual participants, by fitting RT distributions for different stimulus sequences and individual participants. Finally, a consideration of human behavior in more difficult tasks, such as those with low or variable stimulus discriminability, or tasks in which the probability of alternations varies during blocks of trials, can build upon our work.

In this paper, we have presented a neurally plausible and conceptually straightforward account of sequential effects and post-error slowing by developing a simple repetition-based priming mechanism, coupled with an error-correction mechanism. We implemented these mechanisms within the context of a pure DDM, so the behavior can be described analytically and in closed form. Despite its simplicity, our implementation of the DDM accounts for nuances in behavior which are not found in previous models. In particular, we identified in our data, and our model accounted for, sequential effects for correct and error trials, as well as for trials during blocks with high and low probabilities of alternations. This suggests that an error-correction process, such as a simple adjustment of response thresholds after each trial, plays an instrumental role in sequential patterns in RT. Future work may further clarify the implementation of the error-correction process and its implications for perceptual decision making tasks.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

In this section, we derive the mean reaction time for the drift diffusion model (DDM) conditioned on hitting either the upper _{u} or lower −_{l} boundaries, and for a general initial condition _{0} ∈ (−_{l}, _{u}).

Suppose that

in which μ is the deterministic drift of the particle, _{0} is the starting position, and σ_{u} or a lower boundary _{l} where _{u} and −_{l} are given by

where

To obtain the conditional densities, one must divide the above equations by the probability of hitting that particular boundary, i.e., g(t|x(T) = _{u}) = g(t,x(T) = _{u})/P[x(T) = _{u}].) These probabilities are (Feller,

Thus, the mean reaction time conditioned on hitting the upper boundary is given by

Fortunately, a closed-form expression exists for the sum of the infinite series (Prudnikov et al.,

We set _{u} − _{0}). After some algebra, we arrive at a closed form for the mean decision time conditioned on hitting the upper boundary:

In a similar fashion we obtain the mean decision time conditioned on hitting the lower boundary:

This research was partially supported by the Air Force Office of Scientific Research under grants FA 9550-07-1-0528 and FA 9550-07-1-0537, Multi-disciplinary University Research Initiatives. The authors would like to thank Fuat Balci, Jonathan D. Cohen, Juan Gao, Patrick Simen, Marieke van Vugt, and Robert C. Wilson for helpful comments, as well as the reviewers of this paper and Raymond Cho, Leigh Nystrom, and Ines Jentzsch for sharing their data. Stephanie Goldfarb is supported by a National Science Foundation Graduate Research Fellowship and was previously supported by a National Defense Science and Engineering Graduate Fellowship. Michael Schwemmer is currently supported by NIH grant T32-MH065214-1 through the Princeton Neuroscience Institute.