Residual Information of Previous Decision Affects Evidence Accumulation in Current Decision

Bias in perceptual decisions can be generally defined as an effect which is controlled by factors other than the decision-relevant information (e.g., perceptual information in a perceptual task, when trials are independent). The literature on decision-making suggests two main hypotheses to account for this kind of bias: internal bias signals are derived from (a) the residual of motor signals generated to report a decision in the past, and (b) the residual of sensory information extracted from the stimulus in the past. Beside these hypotheses, this study suggests that making a decision in the past per se may bias the next decision. We demonstrate the validity of this assumption, first, by performing behavioral experiments based on the two-alternative forced-choice (TAFC) discrimination of motion direction paradigms and, then, we modified the pure drift-diffusion model (DDM) based on the accumulation-to-bound mechanism to account for the sequential effect. In both cases, the trace of the previous trial influences the current decision. Results indicate that the probability of being correct in the current decision increases if it is in line with the previously made decision even in the presence of feedback. Moreover, a modified model that keeps the previous decision information in the starting point of evidence accumulation provides a better fit to the behavioral data. Our findings suggest that the accumulated evidence in the decision-making process after crossing the bound in the previous decision can affect the parameters of information accumulation for the current decision in consecutive trials.

The mechanism of decision bias as one of the most pervasive biases across many domains of cognitive science, however, remains obscure (Glimcher, 2003;Hanks & Summerfield, 2017;Kim, Kabir, & Gold, 2017;Lauwereyns, 2010;Summerfield & Koechlin, 2010;White & Poldrack, 2014).Two main hypotheses have been proposed to explain the reasons of this bias, although to date, none of them have been adequately supported.According to the first view, the residual of the sensory information of the previous stimulus causes internal bias signals (Akaishi et al., 2014;Albright, 2012;Carnevale et al., 2012;Pearson & Brascamp, 2008).Therefore, a strong sensory signal in the previous trial raises the neural response in the brain sensory areas and the current decision is expected to be made under a larger bias (Akaishi et al., 2014).In the alternative view, the residual of motor response-related signals causes internal bias signals (Gold et al., 2008;Marcos et al., 2013); however, contrary to the first impression, the strength of the sensory signal in the previous trial does not seem to affect the decision-biasing.Akaishi et al. also suggest that in the absence of feedback, this bias is a mechanism to update the likelihood of a choice to be made (Akaishi et al., 2014).
Over and above the mentioned assumptions, the following hypothesis is proposed in this study as plausible: The residual evidence of the previous decision in the decision-related neuron can be informative for the current decision.We tested the validity of this claim using behavioral experiments based on the two-alternative forced-choice (TAFC) discrimination of motion direction and modeling approach.We revealed that firstly, the probability of being correct in the current decision increases if it is in line with the previous decision, showing a trace from the previous trial on the current one.Secondly, this effect is independent of the correctness of the previous decision and the feedback subjects receive.Thirdly, excluding the strong stimuli from our analysis amplifies the observed effect.This observation could refer to the repulsive adaptation effect of these strong stimuli (Kohn, 2007).These last two eliminate the effect of the previous stimuli and merely include the decision.

2.1.SUBJECTS
In this experiment, six adult subjects, three males and three females, with normal or correct-tonormal vision participated.All the subjects, except for two, were unfamiliar to the design of the experiment.They signed informed written consent before attending the study.All experimental protocols were approved by the Iran University of Medical Sciences.

2.2.VISUAL STIMULI
The stimuli consisted of known dynamic random dot motion used in a verity of perceptual decision making studies.These stimuli are movies in which some dots are randomly moving in different direction.In each frame, white dots (2×2 pixel, 0.088° per side) were displayed on a black background with a density of 16.7 dots/degree2/sec (Roitman & Shadlen, 2002;Shadlen & Newsome, 2001).The stimulus contained three interleaved sets of dots displayed on consecutive video frames.Each set was relocated, three frames (40 ms) later while a fraction of dots had a coherent continuous motion toward a direction and the rest of dots were resettled randomly.The stimulus strength was specified by the fraction of dots which moved coherently.Stimulus was presented using a psychophysics toolbox 3.0.12(Brainard, 1997;Pelli, 1997) for MATLAB R2013a (The MathWorks Inc, 2013) on a computer with the operating system of windows 7 (64-bit), Intel (R), Core (TM) i7, 16 GB internal storage, and NVIDIA Quadro K2000 GPU card.

2.3.BEHAVIORAL TASK
All the experiments were carried out in a semi-dark and sound-proof room.The subjects were seated in an adjustable chair at the distance of 57 cm from a cathode ray tube (CRT) display monitor (19 inch, with an 800×600 screen resolution and 75 Hz refresh rate,).An adjustable chinrest had been appropriated to support the subject's chin and forehead.Each trial started with a red fixation point (FP, 0.3° diameter) at the center of the screen and two red choice targets (0.5° diameter) on the right and left side of the fixation point (10° eccentricity).The subjects were asked to fix and maintain their gaze on the fixation point throughout the trial.After a 200 ms delay period, the random dots stimulus was displayed within a 5° circular aperture at the center of the screen for 120, 400, and 720 ms.The percentage of coherently moving dots was chosen from these following values: 0%, 3.2%, 6.4%, 12.8%, 25.6%, and 51.2%.At the end of the stimulus presentation, a 120 ms delay period occurred.After the delay period, the Go signal cued the subjects to respond by eliminating the fixation point.The subjects were asked to report their decision, about the direction of motion, within 1 s after the Go signal by pressing a left or right key.Distinctive auditory feedback was delivered for 100 ms for correct responses, error responses and missed trials.The type of feedback was chosen randomly for trials with 0% coherence.Trials have been separated by different gap durations: 0, 120, or 1200 ms (different gap durations were used to demonstrate their different effects on our results, but there was no significant difference between them, so we have pooled the data of the three gaps in all analysis).The arrangement of the motion direction, motion duration, gap duration, and the motion strength varied randomly from trial to trial (Figure 1).
All possible types of trials were randomly interleaved in blocks with 150 trials.The subjects were instructed to perform the experiments quickly and accurately to the possible extent.The overall probability of being correct was shown on the screen at the end of each block.Each subject performed one or two sessions (each session had four blocks) per day until 3600 trials were collected.The results were consistent across all participants but figures have collapsed the data across subjects.

Figure 1. Motion discrimination paradigm. A fixation point (FP) and two targets were presented for 200 ms.
After that, the motion stimulus was shown for 120 ms, 400 ms and 720 ms.The Go signal followed by a 120 ms delay period cued subjects to report their decision, within 1 second, by pressing two specific keys.Auditory feedback was played for 100 ms.The following trial began after a gap of 0-1.2s (See MATERIALS AND METHODS).

2.4.DATA ANALYSIS
For the purpose of this study, we focused our analysis on specific pairs of consecutive trials which will be explained along with their reasons in the following.In order to demonstrate the effect of previous stimulus strengths on the current decision, we picked out only the pair of trials in which the previous trial contains two groups of low (0%, 3.2%) and high (12.8%,51.2%) motion strengths.In pilot experiments, the results of 25.6%, and 51.2% were similar, so in the main experiment, we chose 12.8% and 51.2% as high motion strengths excerpts in previous trials.We also probed previous trials with three different motion duration to illustrate the effect of previous stimulus durations on the current decision.However, no significant difference was found and since then we have pooled the data of the three motion durations in other analysis.Current trials contained middle motion strength values (3.2%, 6.4%, and 12.8%) and 120 ms motion durations.
A variety of logistic regression models were used to characterize the effect of different parameters on the probability of correct choice.The following models are fitted by using the generalized linear model (GLM) with binomial error structure.We use Logit[P] as a short form of log( P 1−P ) and   as fitted coefficients.
The probability of a correct choice is defined by the following (to fit the psychometric function in Figure 2 and for the black curve in Figure 3): where C is motion strength.To evaluate the effect of the previous decision on the current choice accuracy, we fit the following: (to fit the psychometric function of same and different decision conditions in Figure 3): where C is the motion coherence of the current trials and I is an indicator variable for two successive decisions.The null hypothesis is that the current choice accuracy for same and different decision conditions are equal (H0: β1=0).
A modified version of Equation 2 was used to test whether the current choice accuracy was influenced by correctness and coherence level (low and high) of the previous trial: where C is the motion coherence of the current trials and I and A are the indicator variables for two successive decisions and correctness (or coherence level) of the previous trials, respectively.The null hypothesis is that the current choice accuracy does not depend on correctness and coherence level (low and high) of the previous trial (H0: β3=0).
To assess the impact of the motion strength of the previous trial on the current choice accuracy we used the following regression: where C1 and C2 are the motion coherence of the previous and current trials, respectively.The null hypothesis is that previous stimulus strength has no significant effect on current choice accuracy (H0: β1=0).
All statistical analyses were performed in R version 3.3.1 (The R Foundation for Statistical Computing, www.R-project.org).The statistical analyses outcomes are presented in the RESULTS section.

2.5.MODELING
In order to investigate the mechanism of the last decision impact on the current decision, we used drift-diffusion model (DDM) (Ratcliff, 1978;Ratcliff & McKoon, 2008) implemented by Voss et al. in a computationally efficient, flexible and user-friendly program called fast-dm (Voss & Voss, 2007).Fast-dm estimated DDM's parameters using the partial differential equation (PDE) method through fast computations to calculate the cumulative distribution function (CDF) and the Chi-Square statistic (Lerche & Voss, 2017;Voss et al., 2013).

3.1.BEHAVIOR
Six human subjects reported the perceived direction of motion in trials with 120, 400 and 720 ms duration (Figure 1).Psychometric function for the subjects is shown in summarized in Appendix Figure A1).So, the result from these two TOST was that downward shifts in current motion strengths of 3.2% and 12.8% were statistically equivalent.Hence, in general it can be said that shifts are consistent in all the three motion strengths of current trials in Figure 3.This difference between psychometric functions of the same and different decision condition implies that the probability of being correct in a decision depends not only on the stimulus strength, but also on the previous decision (Eq.2, β1=0.25±0.09,p=5.8×10 -8 ).If the stimulus in the current trial has similar (different) direction with the chosen direction in the previous trial, the probability of being correct in this current trial increases (decreases).
Since the previous decision is itself correlated to the previous stimulus, one may conclude that this difference in performance is the effect of stimulus adaptation.Interestingly, the reported effect of the previous decision seems to be in contrast with the repulsive effect of adaptation.Taking the repulsive effect into account, we expect higher sensitivity for the perception of leftward (rightward) motion when it comes after a rightward (leftward) motion.As a result, the probability of being correct should be higher in the different decision condition than in the same one.Below, we tried to elaborate on these two probable contradictory effects through further analysis.
It is worth noting that in case there is an adaptation effect in our paradigm, it should be stronger when the stimulus of the previous trial has high motion coherence.We separated high and low coherence stimuli from the above analysis and calculated how same and different decision conditions differ in performance.Figure 4 illustrates the performance in current trials which includes motion strengths of 3.2%, 6.4% and 12.8%, when previous trials have low motion strengths of 0% and 3.2% (panel A, Eq. 2, β1=0.65±0.13,p=4.3×10 -22 ) and high motion strengths of 12.8% and 51.2% (panel B, Eq. 2, β1=-0.14±0.13,p=0.03).As shown in this figure, the subjects are significantly more probable to choose a correct decision in the different decision condition when the coherence of the previous trial is high, which is consistent with repulsive adaptation effect.
Whereas, the panel A in Figure 4 shows that a correct decision is more probable in the same decision condition when stimulus coherence in the previous trial is low (Eq.3, β3=0.28±0.09,p=1.5×10 -9 ).The observed effect is significant even when previous trails have 0% coherence.In this coherence, all dots move randomly, which minimizes the adaptation in any specific direction.
However, as illustrated in Figure 5, the probability of being correct is greater in the same decision condition than in the different decision condition, even when there is lack of evidence in the previous stimulus (Eq. 2, β1=0.98±0.19,p=2.2×10 -23 ).
A B As shown in Figure 4, decreasing the effect of stimulus adaptation by excluding current trials with high coherence stimulus seems to strengthen the effect of the previous decision presented in Figure 3.Although results show that probability of being correct in the current decision depends on the previous decision, there is another confounding factor which prevents making any conclusion about the source of this effect.As stated before, feedback signal is different in correct and incorrect trials and this signal may result in the observed effect.Here in Figure 6, by separating correct and incorrect previous trials in both the same and different decision conditions, we attempt to eliminate the effect of feedback.As illustrated in the figure, the correctness of the previous decision does not remove the effect explained above.As shown in Figure 6, similar decision trials are significantly more probable to be correct than different decision trials, regardless of the previous decision to be correct (panel A, Eq. 2, β1=0.37±0.17,p=3.3×10 -5 ) or incorrect (panel B, Eq. 2, β1=1.06±0.21,p=3.6×10 -22 ).Indeed, The observed effect in correct and incorrect previous decision conditions are qualitatively (not quantitatively) similar.Therefore, it is only the participant's decision that affects his/her decision in succeeding trials (Eq.3, β3=-0.07±0.1,p=0.19).

3.2.MODEL FITS
As mentioned above, to investigate the underlying mechanism of the previous decision's effect on the probability of being correct in the current choice, we used the drift-diffusion model (DDM).
Dependence of the model parameters on the previous decision gave us the chance to examine the effect of the previous decision on each parameter.To do so, besides of the pure DDM, we ran three three current stimulus coherences (3.2%, 6.4%, and 12.8%) and two conditions (same and different).Regarding the parameters of the third model in Table 3, there are two different starting points for the same and different decision conditions.As presented in Table 4, modela has two different decision threshold related to two different decision conditions.
As we expected from behavioral results (which indicated the current decision had higher accuracy when the direction of current stimulus was similar to the chosen direction in the previous trial compared to when the direction of current stimulus was different from the chosen direction in the previous trial), when the drift-rate and starting point are dependent on the previous decision, they obtain bigger values in the same decision condition in comparison to the different decision condition and conversely, the decision threshold in the same decision condition is smaller than its value in the different decision condition.Models have compared using the Bayes Information Criterion (BIC) (Kass & Wasserman, 1995;Liddle, 2007;Smith & Spiegelhalter, 1980) for the different model fits are exposed in Table 5 (mean±sd across subjects).For details of subjective scores, see Appendix Tables A5-A8.
As shown in Table 5, the overall quality of the fits was good (R 2 > 0.83).Furthermore, the modified DDM with the dependent starting point (modelz) received the smallest BIC compared to the modelp (p=5.6×10 - ) and modela (p=3.2×10 - ).These significant differences strongly suggested that the modelz is the best one among these three models.Whereas, two subjects' (1 and 2) BIC values in modelv and modelz were close to each other and caused the overall BIC scores were not significantly different in these two models (see Tables A6 and A7).Since these two models yielded nearly equivalent BIC, the simpler model with fewer free parameters confirmed as the preferred model.Eventually, we chose the modelz with the best explanation for how the current choice accuracy is influenced by previous decision.In the latest step, we investigated the difference of the dependent parameter in different conditions for the winner model (modelz).As stated before, starting point gained higher value in the same decision condition (z_s) compared to the different decision condition (z_d) and it's consistent across all participants except subject 3 (for subjects' details, see Appendix Tables A3).

D I S C U S S I O N
Our results showed in sequential perceptual decisions, the probability of being correct in the current choice increases if it is similar to the previous one and conversely decreases when they are different.Against to the views that claim sequential effects (Falmagne, 1965(Falmagne, , 1968;;Gold et al., 2008;Goldfarb et al., 2012;Remington, 1969) on decision processes are due to the motor response bias or sensory bias (Albright, 2012;Carnevale et al., 2012;Gold et al., 2008;Marcos et al., 2013;Pearson & Brascamp, 2008), this decision history effect cannot be defined through these biases (Akaishi et al., 2014).We hypothesize the state of decision variable, which accumulates information to reach any of the two alternative bounds (Gold & Shadlen, 2007;Kiani et al., 2008), is not reset as soon as making the decision.Even it seems this bound crossing in the previous decision provides information for the subsequent decision and can bias it (Bogacz et al., 2006;Diederich & Busemeyer, 2006;Link & Heath, 1975;Ratcliff, 1985;Voss et al., 2004;Wagenmakers et al., 2008).
To verify this assertion, we presented the results of a behavioral study of decision making using 2AFC paradigm based on randomly moving dots with fixed duration and short interval time, focusing on sequences of two trials.To further study variations in the probability of correct, we extend the pure drift-diffusion model (DDM) (Bogacz et al., 2006;Ratcliff, 1978Ratcliff, , 2002;;Ratcliff & Tuerlinckx, 2002) to account for sequential effects (Falmagne, 1965(Falmagne, , 1968;;Luce, 1986;Ratcliff, 1985;Ratcliff & Smith, 2004;Ratcliff et al., 1999;Remington, 1969).In a modified version of DDM, we proposed a simple mechanism for the dependence of baseline to the previous decision.
We indicated how our modified DDM can account for the observed changes in subjects' performance for the same and different decision conditions.
Since in RT tasks which have long Go signals, the time between two decisions in different sequential trials has a lot of fluctuations and might not be short, the sequential effect seems will be diminished.In the current study, to avoid increasing the time between consecutive decisions, we utilized fixed duration task which had fixed period for each part of a trial and limited Go signal.
Nevertheless, we recorded the reaction time (time elapsed from Go signal onset to a hand key-press) besides the choice accuracy in our experiment.Although there was no significant difference in reaction times in different decision conditions due to fixed duration task, reaction time decreased with increasing strength of motion (Link, 1992;Ratcliff & Smith, 2004;Roitman & Shadlen, 2002).
That's why we used reaction times as the input data of the model in addition to the accuracy, current stimulus strength, and previous decision.
we did not apply the previous stimulus coherency as the model input considering the behavioral results that provided further support this idea, the beheld sequential effect cannot be caused by the sensory bias.Specifically, β1 in Equation 4 is close to zero (Eq.4, β1=-0.003±0.002,p=0.017) which implies that the strength of the stimulus in the previous trial does not affect the probability of being correct in the current decision.Though as shown in Figure 4, two different types of motion coherence level (low and high) in the previous trial exert completely opposite effects (decision bias and repulsive adaptation) on the current decision.Indeed, the integration of these two types of previous stimulus coherency in variable C1 in Equation 4 led them to neutralize the effect of each other.Similar to what Akaishi et al. (2014) showed the choice repetition probability was significantly higher when low coherence motion was presented on the previous trial than when high coherence motion was presented on the previous trial (Akaishi et al., 2014).
However, our work is principally different from theirs since they indicated the impact of the previous decision on the choice repetition probability (Akaishi et al., 2014) while our purpose is to investigate this effect on the probability of correct in current decision.Another difference is they did not use feedback in their experiments and declared that the mechanism is associated with making an incorrect choice rather than recognition of an error is responsible for the decision bias.
Whereas we claim that the decision, independent of the correctness and having positive or negative feedback, affects the probability of being correct in the next decision (as shown in Figure 6).
Moreover, to support this statement we did another analysis by separating correct and incorrect previous trials with 0% motion strength in both the same and different conditions.Actually, we duplicated Figure 6 only for 0% coherent motion of previous trials (See the Appendix Contents, Figure A2).In these trials, all dots had random movements which prevented the sensory bias in any particular direction and feedback was given randomly to the subjects, independently of Figure 2. Psychometric function of current trials separated in the three conditions is plotted in Figure 3.The first conditionor the so called same decision condition, blue data points, shows the performance of current trials in which the participants have taken a decision similar to the previous trial.In the second condition or different decision condition, red data points, the participants' decisions in current trials are different from those in the previous trials.The third condition, black data points, is the performance of all current trials, independent of the decision in previous trials.Considering the black data points as a reference, an upward and a downward shift is obvious in the psychometric function of the same and different decision conditions, respectively.Shifts in all the three motion strengths were compared by using equivalence test based on Welch's t-test: two one-sided test (TOST)(Berger & Hsu, 1996 ;Gruman, Cribbie, & Arpin-Cribbie, 2007).Applying TOST procedure to test data against equivalence bounds of  = 0.65 and  = −0.65,revealed that upward shifts in current motion strengths of 3.2% and 6.4% were statistically equivalent, (5832.581)= 21.84555, = 0.5 × 10 −101 .The mean difference was −0.002, 90% CI [-0.004,-0.001].As well as upward shifts in current motion strengths of 6.4% and 12.8% were statistically equivalent, (5637.74)= 3.332844,  = 0.4 × 10 −3 .The mean difference was −0.015, 90% CI [-0.016,-0.014].Therefore, the conclusion from these two TOST was that upward shifts in current motion strengths of 3.2% and 12.8% were statistically equivalent.Furthermore, performing TOST procedure to test data against equivalence bounds of  = 0.68 and  = −0.68,determined that downward shifts in current motion strengths of 3.2% and 6.4% were statistically equivalent, (5826.239)= 25.56059, = 0.6 × 10 −136 .The mean difference was −0.001, 90% CI [-0.002,0].Also downward shifts in current motion strengths of 6.4% and 12.8% were statistically equivalent, (5663.589)= 2.587412,  = 0.005.The mean difference was −0.016, 90% CI [-0.017,-0.015](results are

Figure 2 .
Figure 2. Psychometric function of all the trials; each data point presents the performance of pooled data for all the three durations and two directions.The curve is the fit of a logistic regression to the data (Eq.1).Error bars indicate SE (Standard Error).

Figure 3 .
Figure 3. Psychometric function of the current trials.Red and blue data points depict performance of subjects for the different and same decision conditions, respectively.Black data points are pooled from these two conditions.Curves are the fit of the logistic regression to the data (Eq. 1 for black curve and Eq. 2 for red and blue curves).Error bars indicate SE (Standard Error).

Figure 4 .
Figure 4.The performance of the current trials with motion strengths of 3.2%, 6.4%, and 12.8%.Panel A shows performance in the current trials when previous trials have low motion strengths of 0% and 3.2%.Panel B illustrates performance in the current trials when previous trials have high motion strengths of 12.8% and 51.2%.Overall, the same decision condition resulted in greater accuracy than the different decision condition with the low motion strength in the previous trials.Error bars indicate SE (Standard Error).* p<0.05, *** p<1E-3

Figure 6 .
Figure 6.The performance of the current trials which includes motion strengths of 3.2%, 6.4%, and 12.8%.Panel A is the performance of the current trials when their previous trials are correct with low motion strengths (0% and 3.2%).Panel B is the performance of the current trials when their previous trials are incorrect with low motion strengths (0% and 3.2%).The figure shows that the accuracy of the current trials is higher in the same decision condition compared to the different decision condition for both correct and incorrect previous trials.Error bars indicate SE (Standard Error).*** p<1E-3 whether they pressed the left or the right key.So, the subjects received positive feedback on 50% of the trials (FigureA2, panel A) and negative feedback on the other 50% of the trials (FigureA2, panel B).As demonstrated in this figure, similar decision trials are significantly more probable to be correct than different decision trials regardless the kind of previously received feedback.A C K N O W L E D G E M E N T SThiswork was partially supported by the Cognitive Sciences and Technologies Council, Institute for Research in Fundamental Sciences (IPM)-School of Cognitive Sciences (SCS), and Shahid Rajaee Teacher Training University [grant number 29602].We are thankful to Amirhossein Farzmahdi, Hamed Nili and Abdolhossein Vahabie for helpful discussions.has contributed to the conception and study design, data collection, data analysis and interpretation, statistical analysis, modeling and drafting.M.T.M has contributed to the study design, interpretation of data and drafting.S.Z has contributed to the study design, drafting, modeling, data analysis and interpretation.R.E has contributed to the design of the work and interpretation of data.All authors have approved this final version of the manuscript to be published.

Table 5 .
Model performance comparison via BIC and R2 metrics (mean±sd across subjects).