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Edited by: Misha Tsodyks, Weizmann Institute of Science, Israel

Reviewed by: Maurizio Mattia, Istituto Superiore di Sanità, Italy; Herve Rouault, Janelia Research Campus, USA

*Correspondence: Tobias Teichert, Department of Psychiatry, University of Pittsburgh, 200 Lothrop Street, Pittsburgh, PA 15261, USA

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Inhibitory control is an important component of executive function that allows organisms to abort emerging behavioral plans or ongoing actions on the fly as new sensory information becomes available. Current models treat inhibitory control as a race between a Go- and a Stop process that may be mediated by partially distinct neural substrates, i.e., the direct and the hyper-direct pathway of the basal ganglia. The fact that finishing times of the Stop process (Stop-Signal Reaction Time, SSRT) cannot be observed directly has precluded a precise comparison of the functional properties that govern the initiation (GoRT) and inhibition (SSRT) of a motor response. To solve this problem, we modified an existing inhibitory paradigm and developed a non-parametric framework to measure the trial-by-trial variability of SSRT. A series of simulations verified that the non-parametric approach is on par with a parametric approach and yields accurate estimates of the entire SSRT distribution from as few as ~750 trials. Our results show that in identical settings, the distribution of SSRT is very similar to the distribution of GoRT albeit somewhat shorter, wider and significantly less right-skewed. The ability to measure the precise shapes of SSRT distributions opens new avenues for research into the functional properties of the hyper-direct pathway that is believed to mediate inhibitory control.

Inhibitory control is an important component of executive function that allows us to stop a planned or ongoing thought and action on the fly as new information becomes available (Logan,

The success of the model depends on its ability to estimate the duration of the Go and Stop process. The Stop-signal reaction time (

In contrast to the finishing times of the Go process, the finishing times of the Stop process cannot be observed directly and need to be inferred through the absence of a response. Based on the assumption of independence between the Go and Stop-process, the horse-race model enables the estimation of the mean and variance of the SSRT distribution, independent of its precise shape (Logan and Cowan,

In 1990, two labs presented a theoretical method to derive

Here we present an alternative approach to this problem by using a different type of inhibitory task that conveys more information about SSRT on each trial. We based our task on the complex movement inhibition task by Logan (

The current study focuses the novel theoretical framework and analysis technique that we developed to extract information about the speed of inhibitory control from the SeqIn task or similar tasks that require subjects to stop an ongoing motor sequence with discrete behavioral output such as typing. In particular, the study highlights the feasibility of a novel deconvolution method to extract non-parametric estimates of entire SSRT distributions. The study also provides a detailed description of the subjects' behavior in the SeqIn task to understand and rule out potential confounds.

All participants provided written signed informed consent after explanation of study procedures. Experiments and study protocol were approved by the Institutional Review Boards of Columbia University and New York State Psychiatric Institute.

We developed a novel

In the SeqIn task subjects placed the fingers of their left and right hands on a keyboard as they would for typing. Each finger was assigned one particular key (left pinky: “

A trial was defined as a single go-period followed by a single stop-period. One run consisted of a series of 25 go and 25 stop periods that were presented in immediate succession with no time between them (Figure

_{i}_{1}). The total number of observed button presses is defined as _{N}_{N}_{N}_{N+1}_{N}_{N}

The current study aimed to explore the possibilities of the SeqIn task in the best possible circumstances. Hence, particular care was taken to recruit subjects that had already performed other experiments and were known to be reliable psychophysical subjects. All subjects performed at least 3 runs of 25 trials of the SeqIn task on a day prior to the start of the main experiment.

The experiments were performed on MacBook Pro Laptop computers. The task was programmed and executed with Matlab2009a using routines from Psychtoolbox-3 (Kleiner et al.,

The participants were 6 experienced human psychophysics subjects (2 female) of age 22–42, including the first author. All participants provided written informed consent after explanation of study procedures.

The horse-race model treats response inhibition as a race between a Go- and a Stop process. If the Go process finishes first, the response is executed. If the Stop process finishes first, the response is successfully inhibited. We have expanded this framework to include multiple Go processes to account for the multiple button presses in the SeqIn task. Figure

Let _{i} i_{i} relative to the time of the stop signal_{i}_{1} relative to the go signal. Let _{N}_{i}

For simplicity we assume that all Δ_{i}

Let _{N}_{N}_{N}_{N}

Based on Equation (3) we can estimate the unobservable distribution of _{N}_{i}_{M}_{i}_{1}. To give subjects time to establish a steady pressing routine, we use an even stricter criterion and ensure that _{1}. For each value of

The second approach is based on the density of inter-button press intervals _{Δ}. To that aim we first assume that Δ_{N+1} is known and equal to a fixed value _{i}, S_{N}_{N}_{N+1} is equal to

We can solve and differentiate Equation (4) numerically to obtain an estimate of the density of

We confirmed that both methods yield numerically highly similar estimates of the distribution of

After estimating the density of _{N}_{N}_{N}

In addition to the model-free non-parametric deconvolution algorithm outlined above, it is also possible to use a parametric approach. Here we convolved an exponentially modified Gaussian distribution (ex-Gauss) with the distribution of X, and then adjusted the parameters of the ex-Gauss such that the result matched the observed times of the last button press, _{N}_{1}, or

To verify the deconvolution approach, we used the same method to recover the known _{1}). To that aim, we first convolved the GoRT distribution with the distribution of X. This was done empirically: for each trial

All computations were performed with the statistical software-package R (R Development Core Team, _{N}_{N}_{N}

In addition to the deconvolution approach described above, it is possible to estimate the mean SSRT from the data using a simpler method that does not rely on the deconvolution method. This simpler method does not provide an estimate of the entire distribution, but provides reliable estimates of mean SSRT from as little as 75 trials. The method relies on the definition _{N}_{N}_{N}_{N}_{N}_{N}

The deconvolution algorithm provides estimates of SSRT and GoRT density. To visualize the shape of the distributions independent of inter-subject differences in mean and variance we subtracted out the mean and normalized the standard deviation to one. The normalization was performed on the labels of the time-bins: Let

For visualization purposes we estimated the average normalized density over all subjects. To that aim we averaged the individual normalized

The averaging process described above was aimed at removing differences in mean and standard deviation to highlight potential differences in shape. We took another step to re-introduce the information regarding mean and standard deviation while maintaining the information about the shape of the distributions. To that aim we first re-scaled the default time-axis with the square root of the mean of the individual variances. Then we then added the mean of the individual means to the rescaled time-axis. The entire transformation allowed us to give a precise estimate of the average shape as well as their width and position.

We developed a novel theoretical framework and analysis technique to facilitate the parameter-free estimation of

Before moving on to the calculation of the entire SSRT distribution, we describe some basic properties of the button press patterns in the SeqIn task. In particular, we addressed three main points: (1) What is the rate of button presses that the subjects achieved? (2) Is there any evidence that the subjects used stereotyped motor patterns? (3) Are parts of the sequence ballistic or can the motor sequence be interrupted with equal probability at any point in time? (4) Are there any systematic changes of SSRT or GoRT over time? All of these analyses are important to put the results of the following deconvolution analysis into perspective. The first and third points allow us to quantify the amount of information that can be gained from a single SeqIn trial. The second point helps us gauge the cognitive effort involved in maintaining the motor sequences: it has been suggested that the generation of truly random sequences may require significant cognitive effort. Hence, the use of a stereotyped and presumably automated response pattern supports the idea that the subjects were free to focus on starting and stopping demands of the task without being distracted by the task of maintaining the motor sequence. The fourth point will help us understand the potential contribution of systematic changes in SSRT on the results of the deconvolution. Note that for these analyses we estimate mean SSRT using a more robust method that does not depend on the deconvolution approach (see Methods).

Figure

_{i}^{*}.

The systematic variation of mean IRI with button press number indicates that most subjects used stereotyped mini-sequences. We followed up on this assumption by calculating the cross-correlation of the mean IRIs for the first 40 button presses. If subjects use stereotyped mini-sequences, we would expect cyclic modulation of inter-button-response intervals. Given that a sequence would likely consist of all 8 fingers from both hands (subjects were not allowed to use their thumbs), we would expect the inter-button press intervals to be auto-correlated with a lag of 8 button-presses. This is indeed what we observed in most of the subjects (Figure

The results depicted in Figure

^{*}.

The analysis in Figure

We then tested if mean SSRT and mean GoRT changed systematically over the course of the experiment. To that aim we divided the data from each subject into 10 equally sized bins (deciles) and calculated mean GoRT and SSRT for each bin (Figure

We further tested if mean GoRT and mean SSRT remain stable over the course of each recording session. To that aim we calculated GoRT and SSRT for the first, second and third block of each recording day separately. Figure

After establishing the general properties of the button-press patterns in the task, we then turned to the deconvolution algorithm to recover the entire SSRT distribution from the times of the last button press (see Figure

A series of simulations was conducted to estimate the accuracy of the deconvolution algorithm to recover the true distribution of the SSRT. For each simulated trial, a _{1} was defined as GoRT and the subsequent _{i}_{1} and the randomly drawn inter-response intervals. The duration of the go-period was arbitrarily set to 1200 ms. _{N}

The simulations show that the deconvolution algorithm accurately recovers the different shapes of the distributions, in this case a left-, non- and right-skewed shape. The accuracy was quantified in two ways. First, we extracted 95% confidence intervals (mean ± 1.96^{*}standard deviation) of the estimated mean, standard deviation and skew. Figure

_{N}_{N}_{N}

Note that the confidence intervals of the parameters of the deconvolved distributions are only a small fraction larger than the confidence intervals of the parameters of the simulated sample. This indicates that a large fraction of the variance is due to variability of the simulated sample around its true parameters rather than errors introduced by the deconvolution algorithm. After subtracting the recovered parameters from those of the underlying simulated sample (rather than the ones of the theoretical distribution) the confidence intervals are approximately half as wide (Figure

_{Gauss}_{ex-Gauss} = 0.77), standard deviation (ρ_{Gauss}_{ex-Gauss}_{Gauss}_{ex-Gauss}

We further quantified the accuracy of the deconvolution algorithm by comparing it to the accuracy of a parametric approach based on the fit of an ex-Gauss distribution. Figure

Finally we compared the accuracy of the two approaches using the maximum difference between the recovered and the smoothed empirical distribution function (Kolmogorv-Smirnov Statistic, Figure

To verify the deconvolution approach on real data, we empirically convolved the known GoRT distribution with our estimate of

Using the same approach we deconvolved _{N}

We then directly compared the first three moments of the recovered

^{*}:^{**}:

Reaction time | Raw | 208 ± 14 | 29 ± 7 | 1.5 ± 0.5 |

Binned | 208 ± 14 | 29 ± 8 | 1.7 ± 0.7 | |

Smoothed | 208 ± 14 | 34 ± 7 | 1.2 ± 0.7 | |

Recovered reaction time | Deconvolution | 208 ±14 | 33 ± 7 | 1.1 ± 0.6 |

Ex-Gauss Fit | 208 ± 14 | 27 ± 7 | 1.2 ± 0.4 | |

Recovered SSRT | Deconvolution | 199 ± 23 | 38 ± 7 | 0.9 ± 0.7 |

Ex-Gauss Fit | 199 ± 23 | 31 ± 6 | 0.6 ± 0.6 |

Setting

The overall picture conveyed by both methods is that of strikingly similar

Response inhibition is a critical aspect of motor and cognitive control, and is thought to involve prefrontal cortex and basal ganglia; specifically, the hyperdirect cortico-striatal pathway. Using a small sample of young healthy control subjects trained on the task, the current study showcases the feasibility of the non-parametric method to estimate entire

This reduction in the number of required trials derives from the specific design of the SeqIn task. Rather than using a single discrete motor act, it uses a quasi-continuous motor sequence. Hence, our approach is related to an SSRT-paradigm developed by Morein-Zamir and colleagues (the _{N}_{N+1}). Because on average, button presses occur once every ~30 ms, each trial narrows down SSRT to a window of ~30 ms. This is substantially more information than available from individual stop-signal trials of the standard SSRT task: SSRTs are either longer (failed inhibition) or shorter (successful inhibition) than a particular value (determined by SSD and mean RT), and hence the information content is to a first approximation binary.

A second reason for the substantial reduction in the number of trials is that in the SeqIn task every trial is a stop-signal trial that directly contributes to the estimation of

The third reason that leads to the reduction of the number of trials is that the deconvolution approach applies temporal smoothing to the distributions of the last button presses _{N}

Here we used the SeqIn task to recover entire

While both

At this point we want to provide a brief comparison of our non-parametrically recovered SSRT distributions with the parametrically recovered ones from the study of Matzke et al. (

Any estimate of SSRT or GoRT distributions implicitly assumes that the variable in question is stationary during the data-acquisition period. Hence, we tested if there is any indication of systematic changes in GoRT and SSRT over the course of the experiment. Our data indicated that GoRT and SSRT stayed constant over the course of the experiment. This suggests that the subjects had enough training and were performing at ceiling levels during the entire experiment. However, we also tested whether performance stayed constant within each behavioral session consisting of 3 blocks of 25 trials. We observed a significant increase of SSRT over the course of each behavioral session. GoRTs in contrast, remained stable. This systematic SSRT slow-down affects the comparison between GoRT and SSRT distributions that is based on all of the data, including blocks where SSRTs have already slowed down. In particular, it may have led to an overestimation of mean SSRT and the width of the SSRT distribution. In addition it may have led to an underestimation of the skew of SSRT.

The SSRT slowdown was an incidental finding outside of the main focus of the study that was aimed at exploring technical feasibility of the deconvolution algorithm. It is not our intent to draw conclusions about SSRT slowdown form the small sample of subjects. Nevertheless, the finding was intriguing enough to warrant some speculation about its potential origin. In particular we want to rule out two trivial explanations. (1) The observed SSRT slowdown can not be explained by a reduction of attention or arousal. If so, we would also expect a corresponding reduction of GoRT. (2) SSRT slowdown cannot be explained by a tradeoff in the balance between going and stopping: first, the minor reduction of GoRT does not seem to be consistent with the substantially larger increase of SSRT. Second, the SeqIn task is not a dual task (as the countermanding task) where subjects need to prioritize either one or the other of the tasks.

We want to end the discussion by addressing certain limitations and anticipating potential criticisms. (1) The analysis and interpretation of our data depends on the assumption of independence between the Go and Stop process. Our study did not allow us to explicitly test this assumption. However, the assumption of independence is central not only to our paradigm, but also to all other independent race-models. Studies using countermanding tasks have (a) indicated that violations of independence are moderate, and (b) indicated that SSRT measured in countermanding tasks are reasonably robust against violations of the assumption of independence. Future studies will be necessary to test if the same is true for the SeqIn task.

(2) The overwhelming majority of studies of inhibitory control use the countermanding paradigm in which the to-be-inhibited response is the result of a binary decision process. This type of task has been extremely useful to study response inhibition in healthy controls, and response inhibition deficits in a number of neuro-psychiatric conditions. However, the concept of a Stop process and SSRT, have been formulated independent of this particular paradigm. In fact, other paradigms have been developed in the past, and were shown to correlate with SSRT measured in the countermanding paradigm. Hence, while terms like “SSRT” and “Stop process” have been intimately linked with the countermanding paradigm, their use in the current context is very much within the original definition that does not specify that the to-be-inhibited action must be the result of a binary decision process. Nevertheless, we want to caution that the SeqIn task should not be used as a substitute for established countermanding paradigms such as the Stop-It, SST or Vink task until it has formally been shown to measure the same construct. However, based on the similarity of our task with the complex motion task by Logan and the continuous tracking task by Morein-Zamir both of which are believed to measure the same construct we are confident that a validation study will confirm that the SeqIn task measures the same construct.

(3) As pointed out above, it is not yet 100% clear if the SeqIn task measures the same variant of inhibitory control as countermanding tasks. On the flipside of this argument, it is not clear if countermanding paradigms measure the same construct of inhibitory control that is involved in stopping ongoing motor sequences. Situations in which an ongoing motor sequence needs to be inhibited are prevalent in real life and constitute an important area of study. For example, a quarterback may need to abort a particular play immediately after the snap, right before the ball leaves his hand, or at any point during the execution of the complex motor sequence that takes place between the snap and the pass. In fact, even the standard example of inhibitory control—a baseball player aborting a swing at a ball outside the strike-zone—arguably shares more similarity with the SeqIn than the countermanding task. Similarly, many situations of inhibitory control of relevance in neuropsychiatric conditions require the interruption of an ongoing motor sequence, such as the interruption of perseverative hand washing in obsessive compulsive disorder. In the most likely scenario, the two types of inhibitory control involve identical neural mechanisms and insight from one type of task will be relevant to both types of scenario. However, it is also theoretically possible that different neural mechanisms are involved in the two tasks and that insights from standard SSRT tasks do not extrapolate to the inhibition of ongoing motor sequences. In this case the SeqIn task will be one of only very few tasks to measure inhibitory control of ongoing motor sequences.

(4) Based on the fact that SSRT and GoRT have similar means (~200 ms), it has been argued by some that the motor sequence in the SeqIn task may be stopped without the need to engage inhibitory control. Rather than inhibiting the ongoing motor sequence it might be sufficient to just refrain from issuing additional motor commands, or issue a new motor command (“

This work was supported by National Institutes of Health Grant MH059244 (V.P.F.) and German Research Foundation Grant TE819/1-1 (T.T.).

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