The Stochastic Early Reaction, Inhibition, and late Action (SERIA) model for antisaccades
[['In the antisaccade task ( [1]; for reviews , see [2 , 3] ) , participants are required to saccade in the contralateral direction of a visual cue .', 'This behavior is thought to require both the inhibition of a reflexive saccadic response towards the cue and the initiation of a voluntary eye movement in the opposite direction .', 'A failure to inhibit the reflexive response leads to an erroneous saccade towards the cue ( i . e . , a prosaccade ) , which is often followed by a corrective eye movement in the opposite direction ( i . e . , an antisaccade ) .', 'As a probe of inhibitory capacity , the antisaccade task has been widely used to study psychiatric and neurological diseases [3] .', 'Notably , since the initial report [4] , studies have consistently found an increased number of errors in patients with schizophrenia when compared to healthy controls , independent of medication and clinical status [5–8] .', 'Moreover , there is evidence that an increased error rate constitutes an endophenotype of schizophrenia , as antisaccade deficits are also present in non-affected , first-degree relatives of diagnosed individuals ( for example [5 , 7]; but for negative findings see for example [9 , 10] ) .', 'Unfortunately , the exact nature of the antisaccade deficits and their biological origin in schizophrenia remain unclear .', 'One path to improve our understanding of these experimental findings is to develop generative models of their putative computational and/or neurophysiological causes [11] .', 'Generative models that capture the entire distribution of responses can reveal features of the data that are not apparent when only considering summary statistics such as mean error rate ( ER ) and reaction time ( RT ) [12–15] .', 'Additionally , this type of model can potentially relate behavioral findings in humans to their biological substrate .', 'Here , we apply a generative modeling approach to the antisaccade task .', 'First , we introduce a novel model of this paradigm based on previous proposals [16–20] .', 'For this , we formalize the ideas introduced by Noorani and Carpenter [17] and extend them into what we refer to as the Stochastic Early Reaction , Inhibition and late Action ( SERIA ) model .', 'Second , we apply both models to an experimental data set of three mixed blocks of pro- and antisaccades trials with different trial type probability .', 'More specifically , we compare several models using Bayesian model comparison .', 'Third , we use the parameter estimates from the best model to investigate the effects of our experimental manipulation .', 'We found that there was positive evidence in favor of the SERIA model when compared to our formalization of the model proposed in [17] .', 'Moreover , the parameters estimated through model inversion revealed the complexity of the decision processes underlying the antisaccade task that is not obvious from mean RT and ER .', 'This paper is organized as follows .', 'First , we formalize the model developed in [17] and introduce the SERIA model .', 'Second , we describe our experimental setup .', 'Third , we present our behavioral findings in terms of summary statistics ( mean RT and ER ) , the comparison between different models , and the parameter estimates .', 'Finally , we review our findings , discuss other recent models , potential future developments , and translational applications .'], ['All participants gave written informed consent before the study .', 'All experimental procedures were approved by the local ethics board ( Kantonale Ethikkomission Zürich , KEK-ZH-Nr . 2014-0246 ) .', 'In this section , we derive a formal description of the models evaluated in this paper .', 'We start with a formalized version of the model proposed by Noorani and Carpenter in [17] and proceed to extend it .', 'Their approach resembles the model originally proposed by Camalier and colleagues [21] to explain RT and ER in the double step and search step tasks , in which participants are either asked to saccade to successively presented targets or to saccade to a target after a distractor was shown .', 'Common to all these tasks is that subjects are required to inhibit a prepotent reaction to an initial stimulus and then to generate an action towards a secondary goal .', 'Briefly , Camalier and colleagues [21] extended the original ‘horse-race’ model [16] by including a secondary action in countermanding tasks .', 'In [17] , Noorani and Carpenter used a similar model in combination with the LATER model [22] in the context of the antisaccade task by postulating an endogenously generated inhibitory signal .', 'Note that this model , or variants of it , have been used in several experimental paradigms ( reviewed in [20] ) .', 'Here , we limit our discussion to the antisaccade task .', 'Following [17] , we assume that the RT and the type of saccade generated in a given trial are caused by the interaction of three competing processes or units .', 'The first unit up represents a command to perform a prosaccade , the second unit us represents an inhibitory command to stop a prosaccade , and the third unit ua represents a command to perform an antisaccade .', 'The time t required for unit ui to arrive at threshold si is given by:', 'si=rit ,', '( 1 )', 'siri=t ,', '( 2 )', 'where ri represents the slope or increase rate of unit ui , si represents the height of the threshold , and t represents time .', 'We assume that , on each trial , the increase rates are stochastic and independent of each other .', 'The time and order in which the units reach their thresholds si determines the action and RT in a trial .', 'If the prosaccade unit up reaches threshold before any other unit at time t , a prosaccade is elicited at t .', 'If the antisaccade unit arrives first , an antisaccade is elicited at t .', 'Finally , if the stop unit arrives before the prosaccade unit , an antisaccade is elicited at the time when the antisaccade unit reaches threshold .', 'It is worth mentioning that , although this model is motivated as a race-to-threshold model , actions and RTs depend only on the arrival times of each of the units and ultimately no explicit model of increase rates or thresholds is required .', 'Thus , for the sake of clarity , we refer to this approach as a ‘race’ model , in contrast to ‘race-to-threshold’ models that explicitly describe increase rates and thresholds .', 'Formally ( but in a slight abuse of language ) , the two random variables of interest , the reaction time T ∈ [0 , ∞[ and the type of action performed A ∈ {pro , anti} , depend only on three further random variables: the arrival times Up , Us , Ua ∈ [0 , ∞[ of each of the units .', 'The probability of performing a prosaccade at time t is given by the probability of the prosaccade unit arriving at time t , and the stop and antisaccade unit arriving afterwards:', 'p ( A=pro , T=t ) =p ( Up=t ) p ( Ua>t ) p ( Us>t ) .', '( 3 )', 'The probability of performing an antisaccade at time t is given by', 'p ( A=anti , T=t ) =p ( Ua=t ) p ( Up>t ) p ( Us>t ) +p ( Ua=t ) ∫0tp ( Us=τ ) p ( Up>τ ) dτ .', '( 4 )', 'The first term on the right side of Eq 4 corresponds to the unlikely case that the antisaccade unit arrives before the prosaccade and the stop units .', 'The second term describes trials in which the stop unit arrives before the prosaccade unit .', 'It can be decomposed into two terms:', 'p ( Ua=t ) ∫0tp ( Us=τ ) p ( Up>τ ) dτ=p ( Ua=t ) ( p ( Us<t ) p ( Up>t ) +∫0tp ( Us=τ ) p ( τ<Up<t ) dτ )', '( 5 )', '=p ( Ua=t ) ( p ( Us<t ) p ( Up>t ) +∫0tp ( Us<τ ) p ( Up=τ ) dτ )', '( 6 )', 'The term p ( Ua=t ) ∫0tp ( Us<τ ) p ( Up=τ ) dτ describes the condition in which the prosaccade unit is inhibited by the stop unit allowing for an antisaccade .', 'Note that if the prosaccade unit arrives later than the antisaccade unit , the arrival time of the stop unit is irrelevant .', 'That means that we can simplify Eq 4 to', 'p ( A=anti , T=t ) =p ( Ua=t ) ( p ( Up>t ) +∫0tp ( Us<τ ) p ( Up=τ ) dτ ) .', '( 7 )', 'Eqs 3 and 7 constitute the likelihood function of a single trial , and define the joint probability of an action and the corresponding RT .', 'We refer to this likelihood function as the PRO-Stop-Antisaccade ( PROSA ) model .', 'It shares the central assumptions of [17] , namely:', '( i ) the time to reach threshold of each of the units is assumed to depend linearly on the rate r ,', '( ii ) it includes a stop unit whose function is to inhibit prosaccades and', '( iii ) there is no lateral inhibition between the different units .', 'Finally ,', '( iv ) RTs are assumed to be equal to the arrive-at-threshold times .', 'Note that the RT distributions are different from the arrival time distributions because of the interactions between the units described above .', 'The main difference of this model compared to [17] is that we do not exclude a priori the possibility of the antisaccade unit arriving earlier than the other units .', 'Aside from this , both models are conceptually equivalent .', 'The PROSA model is characterized by a strict association between units and action types .', 'In other words , the unit up leads unequivocally to a prosaccade , whereas the unit ua always triggers an antisaccade .', 'This implies that if the distribution of the arrival times of the units is unimodal and strictly positive , the PROSA model cannot predict voluntary slow prosaccades with a late peak , or in simple words , the PROSA model cannot account for slow , voluntary prosaccades that have been postulated in the antisaccade task [23] .', 'Similarly , it has been argued that prosaccade RT can be described by the mixture of two distributions ( for example [2 , 22] ) .', 'To account for this , we introduce the Stochastic Early Reaction , Inhibition and Late Action ( SERIA ) model .', 'According to this model , and in analogy to the PROSA model , an early reaction takes place at time t if the early unit ue arrives before the late and inhibitory units , ul and ui , respectively .', 'If the inhibitory or late unit arrives before the early unit , a late response is triggered at the time the late unit reaches threshold .', 'Crucially , both early and late responses can trigger pro- and antisaccades with a certain probability .', 'Thus , in parallel to the race processes which determine RTs , an independent , secondary decision process is responsible for which reaction is generated .', 'Fig 1 shows the structure of the SERIA model .', 'To formalize the concept of early and late responses , we introduce a new unobservable random variable that represents the type of response R ∈ {early , late} .', 'The distribution of the RTs is analogous to the PROSA-model , such that , for instance , the probability of an early response at time t is given by', 'p ( R=early , T=t ) =p ( Ue=t ) p ( Ui>t ) p ( Ul>t )', '( 8 )', 'where Ue , Ui , and Ul represent the arrival times of the early , inhibitory , and late units , respectively .', 'The fundamental assumption of the SERIA model is that a secondary decision process , beyond the race between early , inhibitory , and late units , decides the action generated in a single trial .', 'An initial approach to model this secondary decision process is to assume that the action type ( pro- or antisaccade ) is conditionally independent of the RT given the response type ( early or late ) .', 'Hence , the distribution of RTs is not a priori coupled to the saccade type anymore; RT distributions for both pro- and antisaccades could in principle be bimodal , consisting of both fast reactive and slow voluntary saccades .', 'Formally , the conditional independency assumption can be written down as', 'p ( A , T|R ) =p ( A|R ) p ( T|R ) ,', '( 9 )', 'p ( A , T|R ) p ( R ) =p ( A|R ) p ( T|R ) p ( R ) ,', '( 10 )', 'p ( A , T , R ) =p ( A|R ) p ( T , R ) .', '( 11 )', 'The term p ( A|R ) is simply the probability of an action , given a response type .', 'We denote it as', 'p ( A=pro|R=early ) =πe∈[0 , 1] ,', '( 12 )', 'p ( A=anti|R=early ) =1−πe ,', '( 13 )', 'p ( A=pro|R=late ) =πl∈[0 , 1] ,', '( 14 )', 'p ( A=anti|R=late ) =1−πl .', '( 15 )', 'Since the type of response R is not observable , it is necessary to marginalize it out in Eq 11 to obtain the likelihood of the SERIA model:', 'p ( A , T ) =p ( A , T , R=early ) +p ( A , T , R=late ) .', '( 16 )', 'The complete likelihood of the model is given by substituting the terms in Eq 16', 'p ( A=pro , T=t ) =πep ( Ue=t ) p ( Ui>t ) p ( Ul>t ) +πlp ( Ul=t ) ( p ( Ue>t ) +∫0tp ( Ue=τ ) p ( Ui<τ ) dτ ) ,', '( 17 )', 'p ( A=anti , T=t ) = ( 1−πe ) p ( Ue=t ) p ( Ui>t ) p ( Ul>t ) + ( 1−πl ) p ( Ul=t ) ( p ( Ue>t ) +∫0t ( Ue=τ ) p ( Ui<τ ) dτ ) .', '( 18 )', 'It is worth noting here that the PROSA model is a special case of the SERIA model , namely , it corresponds to the assumption that πe = 1 and πl = 0 .', 'The SERIA model allows for bimodal distributions , as both early and late responses can be pro- and antisaccades .', 'Importantly , one prediction of the model is that late prosaccades have the same distribution as late antisaccades .', 'Until now , we have assumed that the competition that leads to late pro- and antisaccades does not depend on time in the sense that late actions are conditionally independent of RT .', 'This assumption can be weakened by postulating a secondary race between late responses; this leads us to a modified version of the SERIA model , that we refer to as the late race SERIA model ( SERIAlr ) .', 'The derivation proceeds similarly to the SERIA model , except that we postulate a fourth unit that generates late prosaccades instead of assuming that the late decision process is time insensitive .', 'This version of the SERIA model includes an early unit ue that , for simplicity , we assume produces only prosaccades , an inhibitory unit that stops early responses ui , a late unit that triggers antisaccades ua , and a further unit that triggers late prosaccades up .', 'As before , if the early unit reaches threshold before any other unit , a prosaccade is generated with probability', 'p ( Ue=t ) p ( Ui>t ) p ( Ua>t ) p ( Up>t ) .', '( 19 )', 'If any of the late units arrive first , the respective action is generated with probability:', 'Antisaccade:p ( Ua=t ) p ( Up>t ) p ( Ue>t ) p ( Ui>t ) .', '( 20 )', 'Prosaccade:p ( Up=t ) p ( Ua>t ) p ( Ue>t ) p ( Ui>t ) .', '( 21 )', 'Finally , if the inhibitory unit arrives first , either a late pro- or antisaccade is generated with probability', 'Antisaccades:p ( Ua=t ) p ( Up>t ) ( ∫0tp ( Ui=τ ) p ( Ue>τ ) dτ ) ,', '( 22 )', 'Prosaccades:p ( Up=t ) p ( Ua>t ) ( ∫0tp ( Ui=τ ) p ( Ue>τ ) dτ ) .', '( 23 )', 'Implicit in the last two terms is the competition between the late units , which are assumed again to be independent of each other .', 'Formally , this competition is expressed as the probability of , for example , the late antisaccade unit arriving before a late prosaccade p ( Ua = t ) p ( Up > t ) .', 'A schematic representation of the model is shown in Fig 2 .', 'This late race is similar to the Linear Ballistic Accumulation model proposed by [24] , although in that model decisions are seen as the result of a race of ballistic accumulation processes with fixed threshold but stochastic base line and increase rate .', 'Here we only assume that the late decision process is a GO-GO race [21] .', 'The likelihood of an action is given by summing over all possible outcomes that lead to that action:', 'p ( A=pro , T=t ) =p ( Ue=t ) p ( Ui>t ) p ( Ua>t ) p ( Up>t ) +p ( Up=t ) p ( Ua>t ) p ( Ui>t ) p ( Ue>t ) +p ( Up=t ) p ( Ua>t ) ( ∫0tp ( Ui=τ ) p ( Ue>τ ) dτ ) ,', '( 24 )', 'p ( A=anti , T=t ) =p ( Ua=t ) p ( Up>t ) p ( Ui>t ) p ( Ue>t ) +p ( Ua=t ) p ( Up>t ) ( ∫0tp ( Ui=τ ) p ( Ue>τ ) dτ ) .', '( 25 )', 'We have left out some possible simplifications in Eqs 24 and 25 for the sake of clarity .', 'The conditional probability of a late antisaccade is given by the interaction between the late units , such that', 'p ( Ua<Up ) =∫0∞p ( Ua=t ) p ( Up>t ) dt=1−p ( Up<Ua ) ,', '( 26 )', 'is analogous to the probability of a late antisaccade 1−πl in the SERIA model .', 'This observation shows that the main difference between the SERIA and SERIAlr model is that the former postulates that the distribution of late pro- and antisaccades are equal and conditionally independent of the action performed , whereas the latter constrains the probability of a late antisaccade to be a function of the arrival times of the late units .', 'The expected response time of late pro- and antisaccade actions is given by', '1p ( Up<Ua ) ∫0∞tp ( Up=t ) p ( Ua>t ) dt ,', '( 27 )', '1p ( Ua<Up ) ∫0∞tp ( Ua=t ) p ( Up>t ) dt .', '( 28 )', 'We will refer to these terms as the mean response time of pro- and antisaccade actions , in contrast to the mean arrival times , which are the expected value of any single unit .', 'The models above can be further finessed to account for non-decision times δ by transforming the RT t to tδ = t−δ .', 'The delay δ might be caused by , for example , conductance delays from the retina to the cortex .', 'In addition , the antisaccade or late units might include a constant delay δa , which is often referred to as the antisaccade cost [1] .', 'Note that the model is highly sensitive to δ because any RT below it has zero probability .', 'In order to relax this condition and to account for early outliers , we assumed that saccades could be generated before δ at a rate η ∈ [0 , 1] such that the marginal likelihood of an outlier is', 'p ( T<δ ) =p ( Tδ<0 ) =η .', '( 29 )', 'For simplicity , we assume that outliers are generated with uniform probability in the interval [0 , δ]:', 'p ( T=t ) =ηδift<δ .', '( 30 )', 'Furthermore , we assume that the probability of an early outlier being a prosaccade was approximately 100 times higher than being an antisaccade .', 'Because of the new parameter η , the distribution of saccades with a RT larger than δ needs to be renormalized by the factor 1−η .', 'In the case of the PROSA model , for example , this means that the joint distribution of action and RT is given by the conditional probability', 'p ( A=pro , T=tδ|tδ>0 ) =p ( Up=tδ ) p ( Ua>tδ−δa ) p ( Us>tδ ) ,', '( 31 )', 'p ( Ua<0 ) =0 ,', '( 32 )', 'p ( A=anti , T=tδ|tδ>0 ) =p ( Ua=tδ−δa ) ( p ( Up>tδ ) +∫0tδp ( Up=τ ) p ( Us<τ ) dτ ) .', '( 33 )', 'A similar expression holds for the SERIA models .', 'However , in the PROSA model a unit-specific delay is equal to an action-specific delay .', 'By contrast , in the SERIA model both early and late responses can generate pro- and antisaccades .', 'Thus , δa represents a delay in the late actions that affects both late pro- and antisaccades .', 'The models discussed in the previous sections can be defined independently of the distribution of the rate of each of the units .', 'In order to fit experimental data , we considered four parametric distributions with positive support for the rates: gamma [13] , inverse gamma , lognormal [25] and the truncated normal distribution ( similarly to [22] and [24] ) .', 'Table 1 and Fig 3 summarize these distributions , their parameters , and the corresponding arrival time densities .', 'We considered five different configurations: 1 ) all units were assigned inverse gamma distributed rates , 2 ) all units were assigned gamma distributed rates , 3 ) the increase rate of the prosaccade and stop units ( or early and the inhibitory units ) was gamma distributed but the antisaccade ( late ) unit’s increase rate was inverse gamma distributed , 4 ) all the units were assigned lognormal distributed rates or 5 ) all units were assigned truncated normal distributed rates .', 'All the parametric distributions considered here can be fully characterized by two parameters which we generically refer to as k and θ .', 'Hence , the PROSA model is characterized by the parameters for each unit kp , ka , ks , θp , θa , θs .', 'The SERIA model can be characterized by analogous parameters ke , kl , ki , θe , θl , θi and the probabilities of early and late prosaccades πe and πl .', 'In the case of the SERIAlr model , the probability of a late prosaccade is replaced by the parameters of a late prosaccade unit kp , θp .', 'In addition to the unit parameters , all models included the non-decision time δ , the antisaccade ( or late unit ) cost δa , and the marginal rate of early outliers η .', 'In this section , we describe the experimental procedures , statistical methods , and inference scheme used to invert the models above .', 'The data is from the placebo condition of a larger pharmacological study that will be reported elsewhere .', 'We aimed to answer three questions with the models considered here .', 'First , we investigated which model ( i . e . PROSA , SERIA or SERIAlr ) explained the experimental data best , and whether all important qualitative features of the data were captured by this model .', 'We did not have a strong hypothesis regarding the parametric distribution of the data and hence , comparisons of parametric distributions were only of secondary interest in our analysis .', 'Second , we investigated whether reduced models that kept certain parameters fixed across trial types were sufficient to model the data .', 'Third , we investigated how the probability of a trial type in a block affected the parameters of the model .', 'Inference on the model parameters was performed using the Metropolis-Hastings algorithm [31] .', 'To increase the efficiency of our sampling scheme , we iteratively modified the proposal distribution during an initial ‘burn-in’ phase as proposed by [32] .', 'Moreover , we extended this method by drawing from a set of chains at different temperatures and swapping samples across chains .', 'This method , called population MCMC or parallel tempering , increases the statistical efficiency of the Metropolis-Hasting algorithm [33] and has been used in similar contexts before [34] .', 'We simulated 16 chains with a 5-th order temperature schedule [35] .', 'For all but the models including a truncated normal distribution , we drew 4 . 1 × 104 samples per chain , from which the first 1 . 6 × 104 samples were discarded as part of the burn-in phase .', 'When a truncated normal distribution was included ( models m5 , m10 , and m15 ) , the total number of samples was increased to 6 × 104 , from which 2 × 104 were discarded .', 'The convergence of the algorithm was assessed using the Gelman-Rubin criterion [33 , 36] such that the R˜ statistic of the parameters of the model was aimed to be below 1 . 1 .', 'When a simulation did not satisfy this criterion , it was repeated until 99 . 5 percent of all simulations satisfied it .', 'Models were scored using their log marginal likelihood or log model evidence ( LME ) .', 'This is defined as the log probability of the data given a model after marginalizing out all its parameters .', 'When comparing different models , the LME corresponds to the log posterior probability of a model under a uniform prior on model identity .', 'Thus , for a single subject with data y , the posterior probability of model k , given models 1 to n is', 'p ( mk|y ) =p ( y|mk ) p ( mk ) ∑i=1np ( y|mi ) p ( mi ) =p ( y|mk ) ∑i=1np ( y|mi ) .', '( 35 )', 'Importantly , this method takes into account not only the accuracy of the model but also its complexity , such that overparameterized models are penalized [37] .', 'A widely used approximation to the LME is the Bayesian Information Criterion ( BIC ) which , although easy to compute , has limitations ( for discussion , see [38] ) .', 'Here , we computed the LME through thermodynamic integration [33 , 39] .', 'This method provides robust estimates and can be easily computed using samples obtained through population MCMC .', 'One important observation here is that the LME is sensitive to the prior distribution , and thus can be strongly influenced by it [40] .', 'We addressed this issue in two ways: On one hand and as mentioned above , we defined the prior distribution of the increase rates of all models in terms of the same mean and variance .', 'This implies that the priors were equal up to their first two moments , and hence all models were similarly calibrated .', 'On the other hand , we complemented our quantitative analysis with qualitative posterior checks [33] as shown in the results section .', 'Besides comparing the evidence of each model , we also performed a hierarchical or random effects analysis described in [38 , 41] .', 'This method can be understood as a form of soft clustering in which each subject is assigned to a model using the LME as assignment criterion .', 'Here , we report the expected probability of the model ri , which represents the percentage of subjects that is assigned to the cluster representing model i .', 'This hierarchical approach is robust to population heterogeneity and outliers , and complements reporting the group-level LME .', 'Finally , we compared families of models [42] based on the evidence of each model for each subject summed across conditions .', 'In addition to a Bayesian analysis of the data , we used classical statistics to investigate the effect of our experimental manipulation on behavioral variables ( mean RT and ER ) and the parameters of the models .', 'We have suggested previously [11 , 43 , 44] that generative models can be used to extract hidden features from experimental data that might not be directly captured by , for example , standard linear methods or purely data driven machine learning techniques .', 'In this sense , classical statistical inference can be boosted by extracting interpretable data features through Bayesian techniques .', 'Frequentist analyses of RT , ER , and parameter estimates were performed using a mixed effects generalized linear model with independent variables subject ( SUBJECT ) , prosaccade probability ( PP ) with levels PP20 , PP50 and PP80 , and when pro- and antisaccade trials were analyzed together , trial type ( TT ) .', 'The factor SUBJECT was always entered as a random effect , whereas PP and TT were treated as categorical fixed effects .', 'In the case of ER , we used the probit function as link function .', 'Analyses were conducted with the function fitglme . m in MATLAB 9 . 0 .', 'The significance threshold α was set to 0 . 05 .', 'All likelihood functions were implemented in the C programming language using the GSL numerical package ( v . 1 . 16 ) .', 'Integrals without an analytical form or well-known approximations were computed through numerical integration using the Gauss-Kronrod-Patterson algorithm [45] implemented in the function gsl_integration_qng .', 'The sampling routine was implemented in MATLAB ( v . 8 . 1 ) and is available as a module of the open source software package TAPAS ( www . translationalneuromodeling . org/tapas ) .'], ['Forty-seven subjects ( age: 23 . 8 ± 2 . 9 ) completed all blocks and were included in further analyses .', 'A total of 27072 trials were recorded , from which 569 trials ( 2% ) were excluded ( see Table 4 ) .', 'Both ER and RT showed a strong dependence on PP ( Fig 5 and Table 5 ) .', 'Individual data is included in the S1 Dataset and is displayed in S1 Fig . The mean RT of correct pro- and antisaccade trials was analyzed independently with two ANOVA tests with factors SUBJECT and PP .', 'We found that in both pro- ( F2 , 138 = 46 . 9 , p < 10−5 ) and antisaccade trials ( F2 , 138 = 37 . 3 , p < 10−5 ) the effect of PP was significant; with higher PP , prosaccade RT decreased , whereas the RT of correct antisaccades increased .', 'On a subject-by-subject basis , we found that between the PP20 and PP80 conditions , 91% of the participants showed increased RT in correct antisaccade trials , while 81% demonstrated the opposite effect ( a decrease in RT ) in correct prosaccade trials .', 'Similarly , there was a significant effect of PP on ER in both prosaccade ( F2 , 138 = 376 . 1 , p < 10−5 ) as well as in antisaccade ( F2 , 138 = 347 . 0 , p < 10−5 ) trials .', 'This effect was present in all but one participant in antisaccade trials and in all subjects in prosaccade trials .', 'Exemplary RT data of one subject in the PP50 condition is displayed in Fig 6 .', 'Finally , we investigated how some of the parameters of the model were related to each other across subjects .', 'Because it has been commonly reported that schizophrenia is related with higher ER , but also with increased antisaccade RT , an interesting question is whether higher late-action response times are correlated with the percentage of late errors and inhibition failures , i . e . , early saccades that are not stopped .', 'We found that the response time of late pro ( F1 , 135 = 13 . 6 , p < 0 . 001 ) and antisaccades ( F1 , 135 = 7 . 1 , p < 0 . 01 ) was negatively correlated with the probability of a late error ( Fig 13 ) , but no significant interaction between PP and response time was found ( pro: F2 , 135 = 1 . 7 , p = 0 . 19; anti: F2 , 135 = 0 . 3 , p = 0 . 76 ) .', 'Hence , late responders tended to make fewer late errors , suggesting a speed/accuracy trade-off in addition to the main effect of PP .', 'We further considered the question whether the percentage of inhibition failures was correlated with the expected arrival time of the late antisaccade unit in antisaccade trials ( Fig 13 right ) .', 'Note that the number of inhibition failures is the same in both trial types in a constrained model , but inhibition failures are errors in antisaccade trials and correct early reactions in prosaccade trials .', 'We found that these parameters were not significantly correlated ( F2 , 135 = 1 . 2 , p = 0 . 26 ) .', 'This was also the case when we considered the expected response time of late prosaccades in prosaccade trials ( not displayed; F2 , 135 = 0 . 0 , p = 0 . 98 ) .', 'Fig 14 illustrates the posterior distribution of late errors and inhibition failures of two representative subjects as estimated using MCMC .', 'Clearly , PP induced strong differences in the percentage of inhibition failures and late errors in prosaccade trials in both subjects .', 'The effect of PP is less pronounced in late errors in antisaccade trials .', 'The posterior distributions also illustrate how the SERIAlr model can capture individual differences: For example , the percentage of late prosaccade errors in the PP80 condition and the percentage of inhibition failures across all conditions are clearly different in each subject .'], ['Our results show that both RT and ER depend on PP .', 'While this was a highly significant factor in our study , there are mixed findings in previous reports .', 'ER in antisaccade trials was found to be correlated with TT probability in several studies [29 , 46 , 47] .', 'However , this effect might depend on the exact implementation of the task [47 , 48] .', 'Changes in prosaccade ER similar to our study have been reported by [29] and [48] .', 'Studies in which the type of saccade was signaled at fixation prior to the presentation of the peripheral cue do not always show this effect [47] .', 'The results on RTs are less consistent in the literature .', 'Our findings of increased anti- and decreased prosaccade RTs with higher PP are in line with the overall trend in [29] , and with studies in which the cue was presented centrally [47] .', 'Often , there is an additional increase in RT in the PP50 condition [29 , 47] , which was visible in our data as a slight increase in RT in the PP50 condition on top of the linear effect of PP .', 'Overall , RTs in our study were relatively slow compared to studies in which the TT cue was separated from the spatial cue [46 , 47] .', 'However , a study with a similar design and added visual search reported even slower RTs in both pro- and antisaccades [29] .', 'Formal comparison of generative models can offer insight into the mechanisms underlying eye movement behavior [11] and might be relevant in translational neuromodeling applications , such as computational psychiatry [49–53] .', 'Here , we have presented what is , to our knowledge , the first formal statistical comparison of models of the antisaccade task .', 'For this , we formalized the model introduced in [17] and proceeded to develop a novel model that relaxes the one-to-one association of early and late responses with pro- and antisaccades , respectively .', 'All models and estimation techniques presented here are openly available under the GPLv3 . 0 license as part of the open source package TAPAS ( www . translationalneuromodeling . org/tapas ) .', 'Bayesian model comparison yielded four conclusions at the family level .', 'First , the SERIA models were clearly favored when compared to the PROSA models .',
['Introduction', 'Materials and methods', 'Results', 'Discussion']
['The antisaccade task is a classic paradigm used to study the voluntary control of eye movements .', 'It requires participants to suppress a reactive eye movement to a visual target and to concurrently initiate a saccade in the opposite direction .', 'Although several models have been proposed to explain error rates and reaction times in this task , no formal model comparison has yet been performed .', 'Here , we describe a Bayesian modeling approach to the antisaccade task that allows us to formally compare different models on the basis of their evidence .', 'First , we provide a formal likelihood function of actions ( pro- and antisaccades ) and reaction times based on previously published models .', 'Second , we introduce the Stochastic Early Reaction , Inhibition , and late Action model ( SERIA ) , a novel model postulating two different mechanisms that interact in the antisaccade task: an early GO/NO-GO race decision process and a late GO/GO decision process .', 'Third , we apply these models to a data set from an experiment with three mixed blocks of pro- and antisaccade trials .', 'Bayesian model comparison demonstrates that the SERIA model explains the data better than competing models that do not incorporate a late decision process .', 'Moreover , we show that the early decision process postulated by the SERIA model is , to a large extent , insensitive to the cue presented in a single trial .', 'Finally , we use parameter estimates to demonstrate that changes in reaction time and error rate due to the probability of a trial type ( pro- or antisaccade ) are best explained by faster or slower inhibition and the probability of generating late voluntary prosaccades .']
['One widely replicated finding in schizophrenia research is that patients tend to make more errors than healthy controls in the antisaccade task , a psychometric paradigm in which participants are required to look in the opposite direction of a visual cue .', 'This deficit has been suggested to be an endophenotype of schizophrenia , as first order relatives of patients tend to show similar but milder deficits .', 'Currently , most models applied to experimental findings in this task are limited to fit average reaction times and error rates .', 'Here , we propose a novel statistical model that fits experimental data from the antisaccade task , beyond summary statistics .', 'The model is inspired by the hypothesis that antisaccades are the result of several competing decision processes that interact nonlinearly with each other .', 'In applying this model to a relatively large experimental data set , we show that mean reaction times and error rates do not fully reflect the complexity of the processes that are likely to underlie experimental findings .', 'In the future , our model could help to understand the nature of the deficits observed in schizophrenia by providing a statistical tool to study their biological underpinnings .']
['medicine and health sciences', 'ballistics', 'classical mechanics', 'reaction time', 'random variables', 'neuroscience', 'cognitive neuroscience', 'probability distribution', 'mathematics', 'sensory physiology', 'behavior', 'schizophrenia', 'mental health and psychiatry', 'probability theory', 'physics', 'visual system', 'eye movements', 'normal distribution', 'physiology', 'biology and life sciences', 'sensory systems', 'physical sciences', 'cognitive science']