Response inhibition is the ability to override a planned or an already initiated response. It is the hallmark of executive control as its deficits favour impulsive behaviours, which may be detrimental to an individual's life. This article reviews behavioural and computational guises of response inhibition. It focuses only on inhibition of oculomotor responses. It first reviews behavioural paradigms of response inhibition in eye movement research, namely the countermanding and antisaccade paradigms, both proven to be useful tools for the study of response inhibition in cognitive neuroscience and psychopathology. Then, it briefly reviews the neural mechanisms of response inhibition in these two behavioural paradigms. Computational models that embody a hypothesis and/or a theory of mechanisms underlying performance in both behavioural paradigms as well as provide a critical analysis of strengths and weaknesses of these models are discussed. All models assume the race of decision processes. The decision process in each paradigm that wins the race depends on different mechanisms. It has been shown that response latency is a stochastic process and has been proven to be an important measure of the cognitive control processes involved in response stopping in healthy and patient groups. Then, the inhibitory deficits in different brain diseases are reviewed, including schizophrenia and obsessive-compulsive disorder. Finally, new directions are suggested to improve the performance of models of response inhibition by drawing inspiration from successes of models in other domains.
This article is part of the themed issue ‘Movement suppression: brain mechanisms for stopping and stillness’.
Imagine you were playing a friendly game of soccer in the neighbourhood's field. At some point, the team's striker kicks the ball causing it to fly over the goal into the street. As you are about to chase the ball, you suddenly hear the honking of a car's horn. Not being able to stop in such a situation would have been detrimental to your life. How did you manage to stop? You were so ready to cross the street to get the ball, but then, in the blink of an eye, you were able to cancel this action and stop yourself. Such an inhibition may be the result of an external signal, like the honking of the car's horn, or simply happen because of your internal desire and decision to stop. This stopping behaviour is called response inhibition, which is the ability to suppress behaviours that are inappropriate, unsafe or no longer required. Response inhibition is the hallmark of successful cognitive and motor control. In everyday life, there are many examples of response inhibition, such as stopping yourself when you are about step over a snake during a hike, or stopping when an impatient car passes in a bike lane at a red light. Response inhibition deficits have also been linked to disorders such as attention-deficit/hyperactivity disorder, obsessive-compulsive disorder (OCD), schizophrenia and substance abuse disorders. Response inhibition and impulsivity deficits are discussed in more detail in Bari & Robbins , Chambers et al.  and Groman et al. . Investigations of response inhibition employ a variety of behavioural paradigms and it is commonly perceived that these paradigms share a mutual and closely related inhibitory mechanism. In the next section, I will review two behavioural paradigms used in eye movement research. Then, I will review the neural mechanisms of response inhibition and discuss various computational models while comparing and contrasting their performances in these two paradigms. Finally, I will highlight general research directions that may improve the performance of future models, uncover the mechanisms of stopping and suggest treatments for response inhibition deficits.
2. Behavioural tasks of response inhibition in eye movements in health and disease
A behavioural paradigm of response inhibition is the countermanding task (figure 1a), in which participants are required to make a saccade as quickly as possible to a GO signal (no-stop-signal trial). In few trials a STOP signal is presented and the response must be stopped (stop-signal trial). There are two possible behavioural outcomes in this paradigm: (i) participants fail to inhibit the saccade, producing a signal-respond trial with a signal-respond reaction time (RT); (ii) participants inhibit the saccade, producing a signal-inhibit or cancelled trial with a stop-signal reaction time (SSRT). Participants' ability to inhibit the response depends on the interval between the GO and STOP signal presentation, often referred as the stop-signal delay (SSD) (figure 1b). When the SSD is short, subjects often succeed at inhibiting; as SSD grows, subjects often fail to inhibit the response. When subjects fail to stop on stop-signal trials, their responses tend to be faster than on trials without a stop signal. The plot of the probability of inhibiting or the probability of responding as a function of SSD is called the inhibition function (figure 1c). The average SSRT in human subjects ranges between 125 and 145 ms and it is invariant with target luminance . In primates, SSRT is reported to be 95 ms [6,7]. In human patients suffering from schizophrenia and bipolar disorder [8,9] SSRT is significantly increased, which means that the patients required more time to inhibit a saccade than healthy controls.
Another behavioural paradigm of response inhibition is the antisaccade task, a choice RT task in which subjects are instructed to perform eye movements in the opposite direction from the location of a stimulus that appears in their right or left peripheral visual field while they are fixating on a central stimulus  (figure 2). During the antisaccade paradigm, two processes are taking place: (i) inhibition of an erroneous prosaccade toward the peripheral stimulus and (ii) generation of a volitional saccade to a position in the opposite direction (antisaccade) [12–14]. In any antisaccade task trial, any of the following three oculomotor behaviours are observed : (i) the subject makes an antisaccade or (ii) the subject makes an erroneous prosaccade (very rare) or (iii) the subject makes an erroneous prosaccade followed by a corrected antisaccade. Errors are considered movements made toward the peripheral stimulus instead of performing the antisaccade. The antisaccade performance involves the mean and standard deviation latency of each eye movement as well as the error rate . Healthy participants typically fail to suppress erroneous prosaccades toward the target on about 20–25% of trials, before correctly making a saccade toward a location in the opposite direction [13,16–19]. Unimodal skewed distributions of antisaccades, erroneous prosaccades and corrected antisaccades are observed. The mean latency and standard deviation of the antisaccade from a large cohort of healthy subjects are reported to be 270 ms and 56 ms, respectively . In the same group, the mean latency and standard deviation of the erroneous prosaccades are 208 ms and 46 ms, respectively, whereas the mean latency and standard deviation of the corrected antisaccades are 146 ms and 85 ms, respectively .
In contrast with healthy participants, patients with frontal lobe lesions [20,21] and patients suffering from schizophrenia  make more antisaccade errors and their antisaccade latencies are more variable within and across subjects [22–25]. An increase in correct antisaccade mean latency in schizophrenia patients was recently reported by Damilou et al. . Impaired antisaccade task performance has also been reported in patients with recent onset schizophrenia and first-episode schizophrenia [23,27–30,31], chronic schizophrenia [22,32–35] and remitted schizophrenia [33,34]. Aberrant antisaccade performance has also been reported by first degree unaffected biological relatives of schizophrenia patients [36–38]. The antisaccade performance deficit in schizophrenia patients is reported to be due to: (i) a deficit in top–down inhibition control of the erroneous response [13,25,33,34,39], (ii) a deficit in response generation of the antisaccade [13,25,33,34,39] or (iii) an emergent property of competing noisy decision accumulating processes (the erroneous prosaccade and the antisaccade) [40,41].
Various psychopharmacological manipulations in healthy and schizophrenia suffering participants, including lorazepam [42,43], risperidone [44,45], nicotine [46–49], amphetamine  and modafinil , reported changes in their antisaccade performance. In schizophrenia patients, risperidone, a serotonergic and dopaminergic antagonist, has been observed to improve error rates in some schizophrenia patients [44,45]. Nicotine administration in schizophrenia patients improves their antisaccade performance [48,49,51].
The experimental literature on the antisaccade performance of OCD patients is more variable and contradicting. Initial studies reported increased error rates in OCD patients compared with healthy controls, but no difference in their latencies of antisaccades . Other studies reported higher antisaccade latencies in OCD patients compared with healthy controls, while their error rate did not differ significantly [53,54]. Another study observed no differences in error rates and latencies of antisaccades between OCD patients and healthy subjects . The most recent study reported an increase in error rates and in latency of corrected antisaccades . The antisaccade performance deficit in OCD is reported to be due to a common dysfunctional network of brain structures including the (pre)frontal and posterior parietal cortices and superior colliculus (SC). In this network, there is a reported deficit in erroneous response inhibition control [13,25,33,34,39,56].
3. Neural control mechanisms of response inhibition
Before discussing various computational models of response inhibition in health and disease, I will briefly review here two of the cortical areas directly involved in saccade eye movements in macaque monkeys, namely the frontal eye fields (FEF) and SC. An excellent review of response inhibition in humans can be found in Verbruggen & Logan . Although the primate brain circuits involved in eye movement preparation and execution are far more complex, involving many more brain areas (for reviews see Munoz & Everling  and Schall & Boucher ), I will focus on these two brain areas because they are directly linked to response inhibition in the countermanding and antisaccade tasks.
FEF is an area in the dorsolateral prefrontal cortex located in the rostral bank of the arcuate sulcus in macaque monkeys. FEF neurons respond to visual stimuli, while other neurons are involved in the production of saccadic eye movements [6,59–61]. The FEF movement-related neurons have been shown to discharge before and during saccades [6,62,63] and innervate other movement-related neurons in SC and brainstem [60,61,64]. Other FEF and SC neurons have been shown to discharge during fixation, but suppress their activities during saccades. These fixation cells are critical for controlling saccade initiation. Electrical stimulation of SC fixation cells can interrupt saccades in monkeys  and their deactivation results in excessive saccade initiation . Two experimental studies have described the activities of movement-related and fixation neurons in FEF and SC in macaque monkeys performing a saccade stop signal task [7,67]. Once the GO signal was given (target appeared), then movement-related neuronal activity in both areas grew towards a threshold. In trials where no STOP signal was presented (non-cancelled trial), these activities continued to grow (figure 5a left). The activities of corresponding fixation neurons were decreased (figure 5a right). If a STOP signal was presented (cancelled trial), then the movement-related activities were inhibited (figure 5a left) and the fixation cells generated rapid bursts (figure 5a right). This activation reciprocity of movement-related and fixation cells naturally reinforces the idea that they share a mutually inhibitory relationship.
A similar inhibitory relationship exists between fixation and saccade neurons in macaque monkeys performing the antisaccade task [68,69]. Fixation cells are tonically active when subjects are fixating and they pause their activities when a saccade is executed. Saccade neurons, on the other hand, discharge when a saccade is initiated, but remain silent during fixation. Two populations of saccade neurons have been recorded in SC: build-up and burst cells [66,70,71].
4. Computational models of response inhibition in health and disease
(a) Modelling the countermanding task in health
The oldest and most successful model of the countermanding task is the ‘independent race model’ (figure 3a; ). The model is intuitively simple consisting of two independent accumulators, a GO process and a STOP process, which race each other until some predefined threshold. If the GO process, initiated at GO stimulus onset, reaches the threshold first, then the subject saccades towards the peripheral stimulus and the trial ends; if the STOP process, initiated at the STOP stimulus onset, reaches the threshold first, then the subject stops and the trial ends. The STOP process wins the race if SSRT + SSD is less than the GO-RT. Then, the response is inhibited and a signal-inhibit trial takes place. If the SSRT + SSD is greater than the GO-RT, then the GO process wins, the response is executed and a signal-respond trial occurs. Because the GO-RT and SSRT are assumed, in the model, to be independent random variables, then the outcome of the race is random. Increases in SSD favour the GO process and so the STOP process wins less often. Thus, the probability of inhibiting the response decreases and the probability of responding increases. Such a relationship produces the inhibition function (figure 4b, left). The ‘independent race model’ also explains why the responses in the stop-signal trials are faster than the no-stop-signal ones and how they change in relation to SSD. When SSD is short, only the fastest GO-RTs are faster than SSRT + SSD, so signal-respond RT is very short, reflecting the lower tail of the GO-RT distribution. As SSD increases, more GO-RTs are fast enough to win the race, so signal-respond RT increases. If SSD is long enough, all GO-RTs will win the race and signal-respond RT will approach no-stop-signal RT. In other words, the failed-stop reaction times (RTs) are faster than the no-stop-signal RTs because only the fastest GO responses escape inhibition. The model has been successfully able to quantitatively simulate the inhibition function as a function of SSD and the SSRT distributions across age, patient groups and experimental conditions (figure 4b, right). However, the model has failed to show any commitment to the underlying neural processes. Furthermore, recent experimental evidence has ruled against the independent race of the GO and STOP processes .
A model similar to the Logan's & Cowan's  ‘independent race model’ is the countermanding LATER model of Hanes & Carpenter  (figure 3b). The countermanding LATER model consists of two independent LATER units, a GO unit and a STOP unit, racing each other until a threshold. The rate of rise of each LATER unit activity takes values from two normal distributions with different means and standard deviations. In the model as in the experiment, the GO unit is activated first, followed by activation of the STOP unit, which in turn inhibits the GO unit. If the GO unit reaches the threshold before the STOP unit, then a GO response is generated. If the STOP unit reaches the threshold before the GO unit, then no response is generated, because the STOP unit stops the activity of the GO unit. If the value of the slope of the GO unit activity is higher than the slope value of the STOP activity, then a response is generated. If the slope value of the STOP unit activity is greater than the slope value of the GO unit activity, then a response is inhibited. The model is able to simulate accurately the RT distribution in a reciprobit scale and inhibition function as a function of SSD (figure 4c). However, it also failed to provide any insights into the neural mechanisms of stopping.
A few years ago, Boucher et al.  introduced the ‘interactive race model’ (figure 3c) as an alternative to the previous race models in order to overcome their limitations, quantitatively simulate the behaviour of the countermanding task and specify the neural underpinnings of stopping. They proposed that lateral inhibition between competing neural accumulators is the mechanism for stopping. This model assumption was in line with the experimentally observed neural activity modulation from recurrently inhibited neurons in SC and FEF, two brain areas directly involved in saccade eye movements (discussed in §3; [7,67]). As in the previous models, the interactive model consisted of two accumulators, the GO process and the STOP process. The GO process was modelled as a stochastic accumulator that integrated neural activity over time until it reached a threshold, after which a response was made. Similarly, the STOP process was modelled as a stochastic accumulator that stopped the response by inhibiting the GO activation and preventing it from reaching the threshold. The STOP and GO processes were mutually inhibitory, but their strengths were asymmetric with the STOP process inhibiting the GO process more than the GO process inhibits the STOP process. Because in the model the GO and STOP processes proceed independently for most of their duration, the stop unit strongly inhibited the go unit only very late in their processing. This produces finishing times that appear independent, as in the independent model. The model has been successful at accurately simulating both behavioural (figure 4d) and neurophysiological data including the activation profiles of gaze-responding neurons (movement and fixation cells; figure 5b) in the signal-inhibit and no-stop-signal trials.
Over the years, model variations of the popular ‘interactive race model’ have been introduced [73–75,78–80]. Salinas' & Stanford's  ‘cancellable rise-to-threshold model’ is built on the foundation of the compelled saccade task , which attempts to separate the contribution of perceptual and motor processing in a countermanding task. As in the interactive race model, programming a saccade involves a process building activity to a threshold. The model assumes that once a stimulus is presented then some perceptual mechanism detects it with a particular speed and reliability. Once the signal is detected, then the movement plan decelerates and the movement is stopped. The model successfully replicates the experimentally observed countermanding behaviour, but not the dynamics of neural recordings. However, it has been ruled as profoundly similar to the ‘interactive race model’ and hence no additional new information is provided by this model (see Bissett  for a critique of the ‘cancellable rise-to-threshold model’). Wong-Lin et al.  advanced an extension (figure 3d) of the ‘interactive race model’ by including an additional pre-target input to account for the experimentally reported high firing rates of fixation neurons . They demonstrate that this simulated pre-target fixation neuronal activity reproduces countermanding behaviour (figure 4e) that maximizes reward rate as a function of SSD, fraction of stop-signal trials, intertrial interval, duration of timeout and relative reward value. Further, it simulated accurately the neural traces of movement and fixation cells in no-stop and cancelled trials (figure 5c). Lo et al.  introduced an attractor-based model of neural spiking dynamics (figure 3e). The neural network model was multi-modular consisting of a pre-movement module that controlled inhibition of movements and a control module that provided a top–down inhibitory control over the pre-movement module. The model replicated the probability of responding and RTs on correct and error trials (figure 4f) as well as the patterns of activity observed in SC and FEF (figure 5d; [7,67]). The model made new predictions and suggested new experiments. Particularly, their model predicted that the neural activities of movement and fixation neurons are negatively correlated earlier in non-cancelled trials than in cancelled trials. It demonstrated that inhibitory control in the countermanding task is due to modulation of the strength of the top–down drive to movement module. Weaker simulated top–down control resulted in shorter RTs, more errors, but unchanged SSRTs. Despite its successes, the model left out many of the crucial neurophysiological details of critical importance to gaze control brain areas such as supplementary frontal eye fields (SEF) and anterior cingulate cortex (ACC). Furthermore, the model failed to separate the stimulus-driven input and the top–down input to fixation neurons in order to assess their effects on inhibitory control .
Recently, Logan et al.  introduced new variations of the original interactive race model , namely the ‘blocked-input 1.0 race model’ (figure 3f), the ‘blocked-input 2.0 race model’ (figure 3g) and the ‘boosted-fix 1.0 race model’ (figure 3h). The ‘blocked-input 1.0 race model’ assumed that the STOP unit did not directly inhibit the GO activation, but instead it activated a top–down process that turned off the GO activation and sets the mean neural GO activity to zero once it reached the threshold. If the input is blocked early enough, then the GO neural activation did not reach the threshold and the response was inhibited. If the GO activation reached the threshold before its input was blocked, then inhibition failed and the GO response was executed. The ‘blocked-input 1.0 race model’ fitted the behavioural data (figure 4g) as well as the original interactive model indicating that direct inhibition from the STOP process on the GO process is not needed to account for the countermanding behaviour. The ‘blocked-input 1.0 race model’ produced better fits to the monkey neurophysiological data than the original interactive model (figure 5e and fig. 6 and 7 in ) by predicting growth rates, decay rates and cancel times that fell within the 95% confidence intervals of the observed monkey data. The ‘blocked-input 2.0 race model’ was the same as the original interactive model with mutual inhibition between the GO and STOP processes except that it also includes a third independent top–down process that also inhibits the GO signal. Version 2.0 of the ‘blocked-input race model’ fitted the behavioural (figure 4h) and neurophysiological data (figure 5f and fig. 11–15 in Logan et al. ) as well as version 1.0, but better than the original interactive model. The ‘boosted-fix 1.0 race model’ assumed that a similar top–down process that this time boosted the STOP activation instead of blocking the GO activation. It fitted the data (figures 4i and 5g) as well as the ‘blocked-input 2.0 race model’. Further neurophysiological experimentation will show whether such a hypothesized top–down signal that blocks the GO activation and boosts the STOP activation really exists and what are the mechanisms that allow it to switch modes of operation.
Direct comparison of the performance of computational models in the countermanding task can be found in table 1.
(b) Modelling the antisaccade task in health and disease
The first ever attempt to simulate the antisaccade task and uncover its neural mechanisms of stopping was made by Cutsuridis et al.  when they introduced a neural nonlinear accumulator model with competition via lateral inhibition between its components. The model was a one-dimensional layer neuronal arrangement of the intermediate layer of SC with three different types of cells, namely the fixation, build-up and burst neurons (figure 6a). In the model, the fixation cells were activated by the fixation stimulus, and the build-up and burst cells by two inputs, a reactive input that represented the error prosaccade motor plan and originated from the lateral intraparietal (LIP) area  and a planned input that represented the antisaccade motor plan and originated from FEF . The inputs were linear ramping processes until a maximum value after which they were either abruptly brought or smoothly decayed to zero. The slope values of these linearly ramping processes took values from Gaussian distributions with different means and standard deviations for each input. Both inputs were integrated by spatially distant build-up and burst cells, which competed with one another via lateral inhibition . Each simulation trial started with the model fixation cells firing maximally for as much time as the subjects were fixating to the central stimulus and build-up and burst cells being silent. As soon as the peripheral stimulus appeared and the subjects had to make an antisaccade, then the model fixation activity started to decay to zero and the build-up cell activity started to rise. In the model, the build-up cells had the role of accumulator cells that integrated evidence until some user-pre-set threshold. Once the threshold was crossed, then the cue was given to the model burst cells to fire maximally, but for a short period of time (phasic activation). In the model, the burst activity represented the final motor command given to the eyes to move. Occasionally in some simulated trials both error prosaccade and antisaccade build-up cells crossed the threshold, which then cued their corresponding burst cells to fire. Behaviourally that meant that the virtual subject made an error prosaccade first and corrected with an antisaccade. The model was able to produce all three oculomotor behaviours of the antisaccade task, namely the error prosaccade, the antisaccade and the corrected antisaccade. The model was also able to simulate accurately the antisaccade performance (latency distributions of the error prosaccades and antisaccades as well as the error rates) of 10 virtual groups of participants with only five free parameters. The model predicted that competition via lateral inhibition and not a third inhibitory signal accounts for the antisaccade performance of a large cohort of healthy participants. Despite its successes, the model suffered from shortcomings. The simulated discharged rates of the fixation and build-up cells were unrealistically high (roughly 600 Hz) [66,70,71]. For lower discharge rates, the model can still accurately simulate the behavioural and neurophysiological properties of the antisaccade task, but for different parameter values (V. Cutsuridis 2007, unpublished observations). Furthermore, the model failed to uncover the ionic and synaptic mechanisms that may produce the range of values of input slopes needed to produce the latency distributions of the error prosaccades and antisaccades.
In a subsequent modelling study, Cutsuridis et al.  embarked to uncover these cellular mechanisms. They introduced a multi-modular neural network model consisting of two cortical modules (FEF and LIP) that drove the SC module to decide the winner motor command to move the eyes. Each cortical module was a network of Hodgkin–Huxley type excitatory and inhibitory neurons connected together. The SC module was the same as in the Cutsuridis et al.  study. No connectivity was assumed between the cortical modules although it has been experimentally observed . Symmetric and asymmetric connection types were tried. Background noise and synaptic noise were also included in the cortical model neurons and in their connections to simulate homogeneous and heterogeneous neuronal firings. The population activity from each cortical network was extracted and a line was fitted to its ramping activity to estimate its slope. Variations in all model ionic and synaptic conductances were attempted to uncover which current(s) and what range of their conductance values reproduced the full range of slope values of the planned and reactive inputs to the SC model needed to reproduce the latency distributions and error rates of the virtual groups of participants in the Cutsuridis et al.  study. The model predicted that only conductance variations of the persistent Na+, NMDA and AMPA currents could produce the necessary slope variability in the cortical decision signals to reproduce the latency distributions and response probabilities of the virtual subjects.
Another notable modelling attempt of the antisaccade paradigm was Noorani's & Carpenter's  three-unit model (figure 6b). The model consisted of three LATER units racing to threshold: an ANTI unit, a PRO unit and a STOP unit. An important model feature was that the ANTI unit was identical (μ and σ) to the PRO unit. In the model, the STOP unit prevented the PRO unit from reaching threshold, thus allowing the ANTI unit to reach a different threshold a little later. The authors hypothesized that the threshold level of the PRO unit was higher than the ANTI unit's threshold, reflecting the advice given by the experimenters to every subject to avoid errors. How often the STOP unit cancelled the PRO unit depended on its rate of accumulation (μ) and its variance (σ2). The model's performance was contrasted against the performance of five healthy subjects performing the antisaccade task. The model captured most of the response repertoire observed in the antisaccade task, namely the antisaccades and the error prosaccades, their corresponding latency distributions and the error response rate. Despite the model's successes, it had several shortcomings. The model failed to produce ‘the error prosaccade followed by the corrected antisaccade’ behaviour. Moreover, the model postulated the existence of a third inhibitory signal (the STOP signal), which occasionally stopped the error prosaccade response and indirectly allowed just the antisaccade response to be expressed. Recent experimental evidence has challenged the existence of such a third signal .
To address some of these shortcomings, Noorani & Carpenter  extended their previous model  by including a RESTART mechanism (figure 6c). In this case, when the PRO unit reached the threshold first, it restarted the ANTI unit allowing it to reach the threshold and generate the antisaccade response. Their new model successfully reproduced the ‘error prosaccade followed by the corrected antisaccade’ behaviour, but failed now to reproduce the just error prosaccades. This shortcoming was inherent in their model. The authors postulated that if the STOP signal did not prevent the error prosaccade response, then the PRO unit will always restart the ANTI unit . This meant the error prosaccades followed by corrected antisaccades will always be produced. If the STOP unit did prevent the PRO unit, then the ANTI unit would not re-start (the corrected antisaccade will not be produced), and an antisaccade response would be generated . In either scenario, just an error prosaccade response cannot be generated. The authors claimed in their studies that participants never made any just error prosaccades (private communication of the author with Roger Carpenter, 2015). However, psychophysical studies of a large group of 2006 participants performing the antisaccade task  reported that subjects do make the just error prosaccades, but their response frequency is low. Another limitation of their new model was their consideration that the simulated latency of the corrected antisaccade is the result of the linear sum of latencies of the error prosaccade and the antisaccade minus the latency of the STOP activity. This shortcoming was inherent in the model, because its units are considered linear encoders of the input information.
Recently, Cutsuridis extended his neural nonlinear accumulator with competition model of antisaccade performance  to the realms of schizophrenia and OCD [40,94]. In the new model (figure 6d), variations in the integration constants of build-up cell activities in the model SC and not in the slopes of the ramping phases of the cortical inputs produced the error rates and latency distributions of the error prosaccades, antisaccades and corrected antisaccades of healthy controls , schizophrenia and OCD  suffering subjects. The model showed that the poor antisaccade performance in schizophrenia is due to a more noisy accumulation of information, but the schizophrenia patients are as confident (threshold level is unchanged) as their healthy counterparts . By contrast, in OCD, the accumulation of information is also noisy, but the OCD subjects are less confident (threshold level is changed) than the healthy participants . In both disorders the model predicted that the antisaccade performance is not due to a deficit in the top–down inhibitory control of the erroneous response as many speculated, but instead it is due to a local inhibitory mechanism in the form of a competitive race to a threshold between the build-up cell representations of the erroneous prosaccade and antisaccade in SC. In favour of this competitive race to a threshold between competing prosaccade and antisaccade signals is the Massen  study, which selectively manipulated the exogenous and endogenous processes in the antisaccade task (e.g. slowing down or speeding up one of these or both processes) and observed the effects of this manipulation on error rate. Massen  observed that if a manipulation slowed the generation of antisaccades, while having no effect on prosaccade generation, then the error rate was increased. If, however, manipulation influenced both pro- and antisaccade generation to the same degree, then the error rate remained unchanged. Massen  argued that antisaccade performance is explained in terms of a competition between two parallel programmes for saccade execution: if the volitional antisaccade is programmed fast enough (e.g. reaches some threshold for activation), then it will win the competition, and the reflexive prosaccade will be cancelled. Alternatively, if the prosaccade is programmed fast enough or the computation for the antisaccade is too slow, an erroneous prosaccade will be made first followed by the correct antisaccade. This account, in line with the Cutsuridis et al. [40,86,87,94] computational studies, favoured the concept of an active inhibitory mechanism in the form of competition between competing decision signals as being critical to antisaccade performance.
Direct comparison of the antisaccade performance of computational models in the antisaccade task can be found in table 2.
5. Conclusions and future directions
I have provided here an extensive review (see also ) and comparison of computational attempts of response inhibition in two behavioural paradigms of eye movement research, the countermanding task and the antisaccade task. The models embodied a hypothesis and/or a theory of mechanisms of underlying performance in both paradigms. Each model's main computational elements were highlighted and a critical analysis of each model's strengths and weaknesses was also provided. All models assumed a race of decision processes accumulating evidence until a threshold. In the countermanding paradigm, two schools of thought were reviewed: the ‘independent race’ type models [4,76] and the ‘interactive’ type models [5,73,78,83,84]. The ‘interactive’ type models appear to more accurately fit the experimental data (behavioural and neurophysiological). In the antisaccade task, two other schools of thought were reviewed: the ‘independent race’ between three separate decision (the ERROR, the ANTI and the STOP) processes type models [88,89], and the ‘competitive nonlinear accumulating race between two decision (the ERROR and the ANTI) processes' type models [40,41,86,87,94]. The latter models have been applied to schizophrenia and OCD realms with considerable success [40,94]. The competitive type models display a better antisaccade performance.
I suggest below general research directions that were omitted from the current models in order to improve their performance in response inhibition and decipher the neural mechanisms of stopping. Future models should include sufficient details of all known neuroanatomy and neurophysiology of brain structures involved in response inhibition in ocular motor research. Aside from FEF and SC, other brain areas involved in response inhibition include SEF, ACC, posterior parietal cortex, basal ganglia and brainstem [14,58] with each one serving a particular set of functions. As I discussed, neurons in FEF respond in visual stimuli, while others are involved in the production of saccadic eye movements [6,59–61]. Neurons in SEF signal the production of errors, the anticipation and delivery of reinforcement and the presence of response conflict [58,97,98]. Neurons in AAC signal the production of errors, the anticipation and delivery of reinforcement and performance monitoring [99–105]. Several notable large-scale neural models of cortical and subcortical dynamics in eye movements have been introduced in the past [106–109]; see also in Girard & Berthoz  for a review of such modelling attempts. Brown et al.  introduced a neural model of cortico (FEF), basal ganglia and SC dynamics where the indirect pathway of the basal ganglia played a key role in the stopping process when a delay intervened before a saccade. This stopping process operated by increasing the activity in the subthalamic nucleus that produced additional activation of the substantia nigra pars reticulata, thus increasing the inhibition on the saccade production circuit. Incorporation of stochastic elements in the large-scale models of saccade generation capable of accounting for the range of saccade latencies in the stop-signal tasks such as the countermanding and antisaccade ones would without a doubt uncover the potential sources of variability observed in the behavioural ocular motor responses.
Future modelling attempts should build on previous modelling attempts of the cellular mechanisms of stochastic accumulating evidence to a threshold  to examine the effects of neurotransmitters such as GABA, and neuromodulators such as dopamine and acetylcholine in inhibitory deficit disorders such as schizophrenia, OCD, attention-deficit disorder, bipolar disorder and others. The differential effects of D1 versus D2 receptor activity modulation on the ramping population activities of cortical networks have recently uncovered that reduced levels of dopamine (hypo-dopamine) and not increased ones (hyper-dopamine) account for observed variability in error prosaccade and antisaccade RTs of a large sample of participants in the antisaccade task .
Finally, future models should simulate the effects of medications (e.g. buprenorphine, naltrexone, topiramate, etc.) on impulse control disorders. They should explain how increasing/decreasing dopamine and GABA levels relate to impulse control in OCD.
I declare I have no competing interests.
I received no funding for this study.
One contribution of 17 to a theme issue ‘Movement suppression: brain mechanisms for stopping and stillness’.
- Accepted December 5, 2016.
- © 2017 The Author(s)
Published by the Royal Society. All rights reserved.