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In perceptual discrimination tasks, a subject’s response time is determined both by sensory and motor processes. Measuring the time consumed by the perceptual evaluation step alone is thus complicated by factors such as motor preparation, task difficulty and speed-accuracy tradeoffs. Here we present a task design that minimizes these confounds and allows us to track a subject’s perceptual performance with unprecedented temporal resolution. We find that monkeys can make accurate color discriminations in less than 30 ms. Furthermore, our simple task design provides a novel tool for elucidating how neuronal activity relates to sensory versus motor processing, as demonstrated with neural data from cortical oculomotor neurons. In these cells, perceptual information acts by accelerating and decelerating the ongoing motor plans associated with correct and incorrect choices, as predicted by a race-to-threshold model, and the time course of these neural events parallels the time course of the subject's choice accuracy.
Perceptual decision-making capacity has been intensely studied in psychophysical and neurophysiological experiments as a function of signal quality, strength, and subjective value1–10. However, there is a fundamental question that has been more difficult to address: how long does it take to make a perceptual judgement? This issue is relevant to many real-life situations in which a choice must be made very quickly. Here is a common example. A driver sees a traffic light and must rapidly decide whether to step on the brake or the accelerator. The ensuing action on the pedal will take at least a few hundred milliseconds to be initiated11,12. How much of this reaction time (RT) is dedicated to the perceptual analysis of the visual scene, i.e. to determining whether the light is red or green?
The basic measurement seems deceptively simple. In well-trained macaque monkeys, which are the subjects of our study, a fixational eye movement (saccade) to a highly salient, unambiguous target takes about 150–200 ms to execute13. This must be the time that the motor apparatus needs to produce an unambiguous response, so the difference between this number and the RT in a two-alternative forced-choice task should be equal to the time that is necessary for making an additional sensory discrimination. Although this subtractive approach is a classic one11,12,14–17, it has three shortcomings. (1) Forced-choice paradigms typically require cognitive resources other than just sensory analysis, such as working memory and a rule that links sensory stimuli to appropriate motor responses, so the two situations may not be directly comparable. (2) In tasks involving a sensory discrimination, what the experimenter measures is typically the point in time at which the motor action is initiated, but this is not necessarily the same as the point in time at which the sensory processing ends, which is what is needed. (3) When a change in a task parameter causes a change in RT, it is usually accompanied by a concurrent change in performance, and vice versa. Hence, a given change in RT may be caused either by an actual change in sensory processing speed or by an internal tradeoff between speed and accuracy to which the experimenter has no access.
Another approach is to manipulate the duration of stimulus presentation. Studies based on this idea1,18–20 have shown that sensory information gradually accumulates over time in order to influence a behavioral choice, and that visual discrimination improves with stimulus duration. However, the accuracy with which sensory processing speed can be inferred with this method is limited, particularly for highly discriminable stimuli, which may be processed very rapidly. The main problem, besides the often necessary use of a mask18,19 that introduces further complexities and unknown sources of variance21,22, is again that changes in stimulus duration produce both variations in performance and large, correlated variations in RT (i.e., point 3 above).
The compelled-response paradigm described below largely avoids these problems and can be easily adapted to a variety of experimental conditions. Its crucial element is that the sensory cues to be discriminated (two colored spots in this experiment) are revealed after the go signal that instructs the subject to respond. Thus, the percentage of correct choices varies with the time gap between the go and the cues, whereas motor execution and mean RT remain approximately constant. This approach results in an accurate time course for perceptual performance, which can be temporally correlated with neuronal activity and from which sensory processing speed can be directly measured.
The compelled-saccade (CS) task is schematized in Figure 1. A monkey subject first fixates on a central spot, and the color of this spot, red or green, indicates the color of the target. Then two yellow spots — potential targets — appear in the periphery. Next, the disappearance of the fixation spot is the go signal that tells the animal to initiate a saccade, although at this point the identities of the target and distracter are still unknown; these are revealed later, after a time gap that varies between 50 and 250 ms in duration, at the point marked “Cue”. At the cue, one yellow spot turns red and the other green. Finally, a saccade is executed, and if the response is correct, a drop of liquid is given as a reward. The RT is the time between the onset of the go signal and the onset of the saccade, and the key parameter is the time gap, which varies pseudo-randomly across trials.
The idea behind the compelled-response paradigm is to separate the perceptual decision-making and motor-planning stages of the task by always instructing the subject when to respond. Because the motor response is triggered first, at the go signal, mean RTs should stay approximately constant, and the perceptual information should influence a saccadic choice process that is already ongoing. The expected task performance is best understood at two extremes. At long or infinite gaps the sensory information never becomes available; the subject must guess which spot is the target because both of them are yellow. In contrast, at zero or short gaps the sensory information is revealed early; the subject is able to identify target and distracter and look at the appropriate spot (red or green). Thus, performance is expected to change systematically as a function of gap, but motor execution is not.
We examined the behavior of two macaque monkeys trained to perform the CS task. First we analyzed their eye movements (Fig. 2) and found that, for a given pair of targets, the velocity profiles generated in short- and long-gap trials were statistically identical. That is, for each monkey, the mean peak velocity was the same across gaps (Wilcoxon, p > 0.25 for both monkeys; see Fig. 2), and so was the mean width at half height (Wilcoxon, p > 0.29 for both monkeys; see Fig. 2). Thus, the gap had no discernible effect on saccadic execution.
In contrast, perceptual performance was indeed strongly dependent on the gap, as revealed by the psychometric curves (Fig. 3a). The percentage of correct choices was much higher at short gaps, when the cue was revealed early and subjects could discriminate the red and green spots, than at long gaps, when the cue was revealed late and subjects had to guess the target’s location. Importantly, however, mean RTs changed little over this range of gaps (Fig. 3b), indicating that the monkeys did not try to wait for the cue (see Supplementary Information). So, as anticipated, variations in performance in the CS task were largely dissociated from variations in motor execution.
Crucially, the behavioral data just discussed (Fig. 3a,b) can be sorted differently to directly reveal the perceptual processing speed of each subject — but before discussing that result, it is useful to consider a simple heuristic model that reproduces much of the observed behavior (Fig. 4). As in previous models11,19, 23–31, a saccadic choice is conceived as a race toward a threshold. In our case there are two variables, xL and xR, which represent the activities of two neuronal populations that trigger saccades to different locations. A movement to the left spot is produced if xL reaches threshold first, and vice versa, the movement is to the right when xR wins the race. Our model, however, uniquely combines random and deterministic choices.
In each trial, the go signal starts the race (after an afferent delay); this means that a positive build-up rate is drawn for one of the variables and a lower or negative rate is drawn for the other. These initial rates are assigned randomly because at this point the two spots are still yellow. Thus, if xL or xR reaches threshold during this pre-cue stage, the outcome of the race is a coin toss (Fig. 4d,e). However, once the cue information becomes available (after a total delay equal to gap + afferent delay), the build-up rates change depending on the target and distracter locations. For instance, Figure 4a shows a trial in which initially xL approached threshold faster than xR, but then, because the target was on the right, xL decelerated and xR accelerated, eventually winning the race. One of the parameters in the model (τ in Eqn. 3, Methods) is proportional to the speed with which the perceptual process occurs and thus determines how fast xL and xR are accelerated, i.e., how fast the cue information can alter the ongoing motor plans once the cue is revealed. In each trial, the model produces a RT and a motor outcome, left or right (for details, see Methods).
Importantly, in this model the time during which the color information is available for guiding the choice in each trial, which we call the effective processing time (ePT), is clearly defined. It is
where the constant TND is the total non-decision time (afferent + efferent delay). Note that this expression requires gap and RT values specific for each trial, not mean values. In the five examples of Figure 4, each ePT period is indicated by a horizontal bar. Races that end after the cue information becomes available have positive ePTs and an accuracy that increases with increasing ePT (Fig. 4a–c); in contrast, races that end before the cue information becomes available have negative ePTs and random outcomes (Fig. 4d,e), thus leading to 50% accuracy.
For each monkey, we fitted the model parameters to the experimental data (Fig. 3a–d; see Methods). We computed an ePT for each trial performed by the monkeys by applying Equation 1 using the TND value from the model. The resulting ePT distributions are shown in Figure 3c. Note that, for ePT ≤ 0, the ePT distributions for correct and incorrect trials are practically identical, indicating that, as in the model, performance is at chance level in this range. In contrast, as ePT increases above 0 ms, a larger proportion of the trials tends to be correct.
The ePT is the amount of time during which the cue is effectively visible in each trial; it is the fundamental variable that determines perceptual performance in the CS task. This is seen most clearly by plotting the percentage of correct responses as a function of ePT. The resulting curve (Fig. 3d) may be referred to as a `tachometric curve’, because it reveals the speed of the perceptual process embedded in the task. For instance, for monkeys S and G, an accuracy of 75% correct was achieved with ePTs of 26 ± 2 ms (± s.e., from bootstrap) and 42 ± 2 ms, respectively. Importantly, the parameter TND is not crucial to this analysis; it simply sets the origin of the x-axis in Figure 3c,d without affecting the shapes of the curves (see Supplementary Information). The key quantity is the difference between the RT and the gap in each trial, which we call the raw processing time (rPT).
For each monkey, we found model parameter values that best fitted the data in Figure 3a–d (Methods). Importantly, only the proportions of errors and the overall variability of the RTs at each gap were used to fit the model. We then generated model RT distributions for correct and incorrect responses at each gap, with the idea that any salient features in the shapes of the simulated distributions would thus constitute specific predictions that we could analyze in order to validate the model.
RT distributions obtained from the experiments and from computer simulations at various gaps are shown side by side in Figure 3e. The shapes of these distributions indeed change dramatically from short to long gaps, and the progression predicted by the model agrees extremely well with that observed in the monkey data. The key is that the model mixes random choices (of 50% accuracy) with deterministic choices (of nearly 100% accuracy) in the right proportions for each gap. These correspond to the monkeys’ guesses and informed discriminations, respectively. Further quantification of the match between model and experimental results is described in the Supplementary Information online.
We interpret the slope of the tachometric curve (Fig. 3d) as a direct indicator of the speed of the perceptual process that guides the subject's choices in the CS task. If this is correct, then manipulations that alter the motor responses but not the perceptual difficulty of the task should lead to little or no change in slope. To test this, we varied the task conditions so that the monkeys would develop a motor bias, a tendency to make more saccades to one side than to the other32. We gave the monkeys a large reward (0.3 ml) following correct saccades to one side and a small reward (0.1 ml) following correct saccades to the other (Fig. 5; see Methods). As a consequence, on average the animals chose the high-reward side about 76% of the time (75% for monkey S, 78% for monkey G; Fig. 5a). This result agrees well with the 3:1 ratio expected from Herrnstein’s matching law4,33,34, but does not capture the full complexity of the effects generated by the asymmetric reward, which depended strongly on the availability of the sensory information32.
We analyzed the choices that the monkeys made to the high- and low-reward sides and found three prominent effects: (1) the bias was stronger at long than at short gaps (Fig. 5a), (2) responses in the high-reward direction were much more prone to errors than in the low-reward direction (Fig. 5b), and (3) responses in the high-reward direction were also initiated sooner (Fig. 5c). Results are shown for monkey S only (Fig. 5) but were similar with monkey G (Supplementary Information). Crucially, in spite of these striking differences in motor execution, the slopes of the tachometric curves were not significantly different across reward conditions (monkey S, p = 0.18; monkey G, p = 0.53, permutation test). This, we believe, is because the perceptual discrimination itself did not change.
These results shed some light on how reward information and sensory evidence are combined when two motor responses are rewarded differently. They suggest that the low-reward side is chosen only if there is little uncertainty about the target’s position, i.e., when there is enough time to discriminate red from green accurately; otherwise, the high-reward side is the default. Two observations support this explanation. First, consider the ePT distributions obtained for correct and error trials. For high-reward choices the results are indicative of a large fraction of guesses (Fig. 5d), whereas for low-reward choices the results are indicative of a majority of correct perceptual judgements (Fig. 5e). Second, consider the fraction of choices made to the low-reward side as a function of ePT (Fig. 5g). The curve shows that the monkey’s preference varied extremely sharply as a function of ePT (compare to the weak dependence as a function of gap shown in Fig. 5a). Therefore, the amount of time during which a monkey is exposed to the sensory cue — that is, the ePT — is an excellent predictor of his choice. This effect was also seen in the model (Fig. 5g), which, with minor modifications (see Methods), was able to reproduce quite accurately all the major differences between high- and low-reward choices (Fig. 5, “Model”).
In our race model, xL and xR are conceived as the firing rates of two populations of neurons that generate movements to the left and right targets but are also influenced by perceptual information8,19,28,35–39. If this representation is correct, oculomotor neurons in cortical areas that participate in the saccadic choice process should qualitatively behave like xL and xR; in particular, they should exhibit similar dependencies on processing time. To investigate this, we recorded neuronal activity from oculomotor neurons in the frontal eye field (FEF) while monkeys S and G performed the CS task. We report results from a population of 30 neurons, 18 from monkey S and 12 from monkey G, that had stereotypical motor properties24,40 and little or no sensory-evoked activity (Methods; Supplementary Information). Such “movement” cells were selected because they correspond most closely to the model variables xL and xR.
In a first analysis, we subdivided all recorded trials into short- and long-rPT groups (Fig. 6a) and compared the neuronal responses across groups. Recall that in short-rPT trials choices are not guided by sensory information, whereas in long-rPT trials they are. According to the model, the fundamental action of the cue information is to accelerate the developing motor plan for the target side and decelerate the plan for the distracter side, producing curved trajectories (Fig. 4a–c). Here we mean acceleration in the literal, physical sense, because the cue information changes the second derivatives of the decision variables xL and xR (Eqn. 3). By averaging over many simulated trials, this effect becomes manifest in two ways: (1) the oculomotor activity leading to saccades away from the cells’ movement field (MF) should be stronger with long rPTs than with short rPTs, but should also decline markedly before movement onset (Fig. 6b, green traces), and (2) the activity leading to saccades into the MF should rise to threshold more steeply with long than with short rPTs (Fig. 6b, red traces). Are these patterns found in the neural data?
The responses of a single neuron during the CS task are shown in Figure 6c,d. Activity is aligned on saccade onset, with trials sorted according to saccade direction (Fig. 6c, into the MF; Fig. 6d, away from the MF) and processing time (left side, short rPTs; right side, long rPTs). This unit’s behavior was generally consistent with the model, but its responses were quite variable, as expected. However, the population average showed good agreement with the model’s predictions: before saccades away from the MF (Fig. 6e, green traces), the firing rate reaches a higher level and then declines more sharply at long rPTs than at short rPTs (see ref. 31); and before saccades into the MF (Fig. 6e, red traces), the firing rate trajectory is steeper at long rPTs than at short ones. This difference in curvature is somewhat subtle, but was very highly significant (p = 0.0006, permutation test).
In the CS task, performance changes sharply with the amount of exposure to the sensory cue. We thus hypothesized that, as a function of rPT, the neural activity associated with the oculomotor choice should change with a time course similar to that observed behaviorally. To test this, we measured the mean convexity of the firing rate trajectories as a function of rPT. We chose convexity because this quantity is closely related to acceleration, and its calculation is simple and robust to noise (Fig. 7a). Trajectories that decelerate have negative convexity and curve downward, whereas trajectories that accelerate have positive convexity and curve upward.
We first analyzed the convexity associated with the model variables xL and xR, and found that the responses in and out of the MF behave in opposite ways as functions of rPT: the former have an approximately constant convexity at first, which then increases steadily (Fig. 7b), whereas for the latter, the convexity also stays constant initially but then decreases toward strongly negative values (Fig. 7c). This simply restates the result in Figure 6b and quantifies it with a fine temporal grain. That is, the firing rate trajectories should curve upward before correct saccades into the MF, whereas they should turn downward before correct saccades away from the MF. Crucially, however, both effects must start at about the same time as the subject’s rise in performance (Fig. 7d), and their magnitude must depend on the amount of exposure to the cue.
This prediction was verified. We plotted the mean convexity of the neuronal responses as a function of rPT for trials into (Fig. 7e) and away from the MF (Fig. 7f). The results from the 30 FEF neurons were somewhat noisy, but the time courses clearly matched those expected from the model. In particular, the two transition points at which convexity starts changing steadily (arrows in Figs. 7e,f) occurred very close to each other and to the transition point of the tachometric curve, as predicted (transition points were determined by fitting the data with analytical functions that contain such transitions; see Methods). Furthermore, after the transitions, the slopes of the convexity plots had a very high significance (p = 0.002 and p = 0.0004 for trials into and away from the MF, respectively, permutation test). Therefore, during the task the cue information accelerates the motor plan for the target and decelerates the plan for the distracter — the main premise of the race model — and the onset of these dynamical changes coincides with the onset of the subject’s rise in performance above chance.
The compelled-response paradigm presented here is able, to a large extent, to decouple the perceptual-evaluation and motor-execution steps of a behavioral choice. Part of its success depends on simultaneously controlling the timing of the sensory information (cue) and of the response trigger (go), as suggested by earlier studies in motor control41,42, but its minimalistic design goes much further: it achieves fine temporal resolution in a perceptual discrimination process without additional stimuli or masks, and monkeys readily learn the task. The resulting tachometric curve is an excellent diagnostic of perceptual ability because performance varies much more rapidly with ePT (Fig. 3d) than with RT or with gap (Fig. 3a). This means that, by computing the tachometric curve, a major source of variability unrelated to sensory processing is filtered out. Importantly, however, this is achieved with a very simple calculation (Eqn. 1).
Our results indicate that, given highly discriminable stimuli, color information needs to be processed for just 25–50 ms to have a sizeable impact on behavioral outcome. But note that our model does not attribute this processing time to a specific color-sensitive area, such as the retina, or area V4, and that it also remains agnostic regarding the biophysical origin of this time scale, i.e., whether it results from slow sensory transduction or from internal noise in more central circuits25,26,36,38, for instance. What the model does show, however, is that such a time scale for sensory processing is consistent with all the behavioral data in the task — e.g., RT distributions, choice preferences under bias, etc. — and with the oculomotor activity observed in FEF.
Our results provide converging evidence that, within a choice process that takes a few hundred milliseconds, the sensory evaluation step may consume much less than 100 ms, as suggested by previous experiments43–46. In particular, our findings are close to a measurement of ~25 ms made in humans46, and which likely reflects the minimum time necessary for color identification only, without the computational overhead needed to compare two stimuli and communicate the result of the discrimination (red or green) to the correct motor output (left or right) on the fly. Our results are also in agreement with an earlier study of FEF visuo-movement cells43, which showed that an ideal observer can reliably differentiate the activity evoked by a target from that evoked by a distracter within ~25 ms of the onset of neural discrimination, on average. Importantly, however, the advantage of the present approach is that it does not require any assumptions about the relationship between neuronal and behavioral performance (e.g., identities and numbers of participating neurons, ideal-observer criterion, etc.), but rather is based on a direct behavioral manifestation of the subject’s perceptual discrimination process — the tachometric curve — which explicitly depicts how the perceptual decision unfolds in time.
More generally, the techniques described here may be used to understand how reward contingencies and perceptual knowledge are combined, as shown with the motor bias experiment, and to reveal how perceptual and motor processes interact, as illustrated with the FEF data. Furthermore, the present task design is not limited to color discrimination; it can be extended to investigate processing speed under many other experimental conditions. For instance, the task can be easily adapted to the auditory modality and to visual features other than color, such as object shape or motion direction. These applications will be explored in future studies.
All protocols complied with NIH guidelines, USDA regulations, and policies set forth by Wake Forest University School of Medicine. Before training, under general anesthesia, an MRI-compatible titanium post was attached to each monkey’s skull. The post served to stabilize their heads during subsequent experimental sessions. In monkey S, eye movements were monitored with an eye coil that provided an analog eye-position signal at a rate of 500 Hz. For monkey G, we used an EyeLink1000 infrared tracking system (SR Research) with a sampling rate of 1000 Hz. Color stimuli were delivered through an array of light-emitting diodes placed at a viewing distance of 145 cm from the monkey’s chair. Pairs of saccade targets were separated by 10–20° of visual angle. RT was measured as the amount of time from the go signal until the velocity of the saccade reached a cutoff value of 50°/s. To discourage the monkeys from waiting for the cue, they were allowed to initiate a response up to 600 ms after the go signal, and a trial was aborted if the response took longer. In the motor bias sessions, large and small rewards were 0.3 and 0.1 ml in volume, respectively. The high-reward side was kept constant for a block of 150–250 trials and was then switched. In all conditions, standard and biased, red and green targets were presented with equal probability at all target locations.
In both monkeys, stereotactic coordinates and MRI images were used to place a recording cylinder (20 mm dia.) over the arcuate sulcus of frontal cortex. Putative FEF neurons were recorded using tungsten microelectrodes (FHC). The majority of neurons reported on here (23/30) were constrained to recording sites in which we verified that saccade-like eye movements could be evoked with microstimulation at low current threshold (< 50 µA). Once a single unit was isolated, a single-target delayed-saccade task was used to render an online estimate of preferred response direction and eccentricity prior to testing with the CS task. To explore the predictions of the present model, we explicitly chose for analysis a subset of neurons having strong saccade-related bursts but little or no transient activation linked to stimulus onset (see Supplementary Information).
The race-to-threshold model has ten free parameters that varied for each monkey. The free parameters were optimized to minimize the root-mean-square error between the simulated and the experimental data for each animal. In the standard CS task, the error function had six terms, one for each of the curves in Figure 3a–d plus one for the standard deviations included in Figure 3b. All simulations were run in Matlab (The Mathworks). A Supplementary Movie showing ten simulated trials is available online, and a Matlab script that replicates the model results in Figure 3 is available from the authors upon request.
In the race model, the competing variables xR and xL represent developing motor plans. They start at 0 and the race is over when one of them reaches a threshold of 1000 (arbitrary units). When xL (xR) wins, the oculomotor response is considered to be to the left (right). In each simulated trial there are two integration stages, a random and a deterministic one. In the first stage of each race, before the cue information is available, the two build-up rates, rR and rL, are drawn from a bivariate Gaussian distribution with mean rG (same for both variables), standard deviation σG (same for both variables), and correlation coefficient ρ. Thus, xR and xL change according to
where the rates are constant. Because ρ is negative, one variable tends to approach threshold faster than the other, but this assignment is random in each trial. This integration process lasts a time equal to the gap plus ΔT, where ΔT is drawn from a Gaussian distribution with zero mean and s.d. of 10 ms. The term ΔT represents variability in the afferent delay (or whatever processes occur before the integration starts). The second stage of the race begins gap + ΔT ms after the start of the first stage, at which point the target and distracter locations become known to the circuit. At this transition, two things happen. First, the integration is halted for a time TI, which is drawn from a Gaussian distribution with mean μI and s.d. σI, but such that any negative TI values are discarded. This interruption time is not essential to the model but accounts for a small dip near zero that is often seen in the ePT distributions (most obvious in Fig. 3c, Monkey G). During TI, none of the variables change. Second, integration is resumed after the interruption, but the build-up rates start changing towards asymptotic values rT for target (large, positive value) and rD for distracter (negative value). For instance, if the target is on the right, the rate for xR begins to change toward rT and the rate for xL begins to change toward rD. That is, during this second stage, xL and xR still follow the equations above, but also
where r0R and r0L are the rate values drawn originally, during the first stage, and τ is the time required to reach the final values rT and rD. We stress that these equations (Eqns. 3) apply only after the cue has been revealed. Once rR and rL are equal to rT and rD, respectively, they do not change any more. Of course, if the target is on the left, then rL tends to rT and rR tends to rD instead. The integration process (Eqns. 2) continues until xL or xR reaches threshold. In most trials, the build-up rates rR and rL are correctly assigned according to the target and distracter locations, as described. However, there is a small probability pe that this assignment is incorrect (i.e., reversed). This accounts for trials in which, even though the ePT is long, the response is wrong (e.g., the curve for monkey G in Fig. 3d does not reach 100%). The duration of the second stage is equal to the ePT. The reaction time in each trial is computed as RT = gap + ePT + TND, where TND is the total non-decision time. Note that although it is easier to think of separate afferent and efferent delays, as schematized in Figure 4, the model is sensitive only to the total non-decision time TND (see Supplementary Information). Parameter values are listed in Table 1.
In simulations of the motor-bias experiment, the race model is the same but the parameters that determine xR and xL are no longer identical (the only exception is τ). For instance, in the first stage of each race, the two build-up rates, rR and rL, are now drawn from a bivariate Gaussian distribution with means rG + ΔrG (for rR) and rG − ΔrG (for rL), standard deviations σG + ΔσG (for rR) and σG − ΔσG (for rL), and correlation coefficient ρ. The bias terms ΔrG and ΔσG thus make the left and right sides asymmetric. With such manipulation, the model still generates random choices before the cue information is revealed, but it tilts the odds in favor of one side (Table 1).
We obtained several curves with a particular characteristic in common: they remain approximately flat for a certain range of x-axis values and then start increasing or decreasing steadily (Fig. 7b–f). We refer to the point at which the slope becomes non-zero in those curves as the transition point xtrans. This is an important quantity in our analysis, because according to the model (Fig. 7b,c), it should be nearly the same for the tachometric curve (Fig. 7d) and for the two convexity curves derived from the neuronal data (Fig. 7e,f). We estimated xtrans as follows. For a given curve, we fitted the data to an analytic function composed of two line segments, a flat segment and a non-flat segment, that intersect at xtrans. We refer to this as a threshold-linear function. Each threshold-linear function is characterized by three parameters: the height of the flat segment along the y-axis, the slope of the non-flat segment, and the transition point xtrans. Optimal parameter values were found with standard algebraic methods used to fit a single straight line to a data set (i.e., least-squares minimization). The Supplementary Information online illustrates the resulting fits.
The error in xtrans was estimated using the jackknife method, which is based on the idea of recalculating a statistic multiple times, each time deleting a different contributing sample47,48. In our case, we recalculated xtrans 30 times, on each pass omitting the data contributed by one of the recorded neurons and repeating the fit.
Finally, each linear-threshold fit produced a slope for the increasing or decreasing part of a curve. To calculate its significance we used a permutation test49. For each curve, we shuffled the values on the y axis and computed the slope of the line that best fitted the shuffled data between xtrans and the maximum rPT value used in the original fit. This randomization was repeated 5000 times. Then we computed the fraction of times that the slopes of the randomized data equaled or exceeded the slope of the original, non-shuffled data. This fraction was taken as the probability of having observed the original slope value just by chance, when there was actually no correlation between the quantities on the x and y axes.
Research was supported by the National Institutes of Health/National Eye Institute grant R01 EY12389 to TRS.
Author Contributions: TRS conceived the task, supervised all experiments and data analyses, and co-wrote the manuscript; SS contributed to the collection, analysis and modeling of the behavioral data; DPM contributed to the design of the experiments and to the collection of behavioral data; MGC contributed to the collection of behavioral data and to the collection and analysis of neural data; ES developed the race model, contributed to the experimental design and data analysis, and co-wrote the manuscript.