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Dual Diffusion Model for Single-Cell Recording Data From the Superior Colliculus in a Brightness-Discrimination Task

¹Department of Psychology, Ohio State University, Columbus, Ohio; ²Department of Neurobiology and Physiology, Northwestern University, Evanston, Illinois; ³Neuroscience Research Institute, AIST, Tsukuba, Ibaraki, Japan; and ⁴Department of Psychology, University of Melbourne, Victoria, Australia

Submitted 13 April 2006; accepted in final form 15 November 2006

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Monkeys made saccades to one of two peripheral targets based on the brightness of a central stimulus. Task difficulty was manipulated by varying the ratio of stimulus black-and-white pixels. Correct response probability for two monkeys varied directly with difficulty. Deep layer SC neurons exhibited robust presaccadic activity the magnitude of which was unaffected by task difficulty when the stimulus specified a saccade toward a target within the neuron's response field. Activity after stimuli specifying saccades to targets outside the response field was affected by task difficulty, increasing as the task became more difficult. A quantitative model derived from studies of human decision-making was fit to the behavioral data. The model assumes that information from the stimulus drives two independent diffusion processes. Simulated paths from the model were compared with neuron activity, assuming that firing rate is linearly related to position in the accumulation process. The firing rate data show delayed availability of discriminative information for fast, intermediate, and slow decisions when activity is aligned on the stimulus and very small differences in discriminative information when aligned on the saccade. The model produces exactly these patterns of results. The accumulation process is highly variable, allowing the process both to make errors, as is the case for the behavioral performance, and also to account for the firing rate results. Thus the dual diffusion model provides a quantitative account for both the behavior in a simple decision-making task as well as the patterns of activity in competing populations of neurons.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Research in neural decision making is at the point of identifying the mechanisms that implement simple decisions (Glimcher and Sparks 1992; Gold and Shadlen 2000; Hanes and Schall 1996; Horwitz and Newsome 1999, 2001; Kim and Shadlen 1999; Krauzlis and Dill 2002; McPeek and Keller 2002; Ratcliff et al. 2003a; Roitman and Shadlen 2002; Romo et al. 2002; Sparks 1999). In parallel to this work, psychology has produced a set of models that describe simple rapid two-choice decision making (Busemeyer and Townsend 1992, 1993; Diederich 1997; LaBerge 1994; Laming 1968; Link 1975; Link and Heath 1975; Pike 1966, 1973; Ratcliff 1978, 1981, 1988; Ratcliff and Rouder 1998, 2000; Ratcliff and Smith 2004; Ratcliff et al. 1999; Roe et al. 2001; Smith 1995; Smith and Ratcliff 2004; Smith and Van Zandt 2000; Stone 1960; Townsend and Ashby 1983). We present a diffusion model that explicitly relates these two domains in the context of a single experiment. The model is applied to simple two-choice decisions and accounts for both behavioral data, namely, accuracy and correct and error response time (RT) distributions, and single-cell firing rate data for two competing populations of neurons. The model assumes that information from the stimulus drives two accumulators that are independent diffusion processes. The accumulation rates driving the two processes sum to a constant so that the more evidence there is for one response, the greater the accumulation rate in that accumulator and the smaller the accumulation rate in the other accumulator. But once these accumulation rates are set for a particular trial, the two diffusion processes proceed independently. The accumulation process is highly variable, and this allows the process to make errors. The model is fit to the behavioral data and simulated paths are generated from the model. The assumption linking the paths to the single-cell firing data are that firing rate is linearly related to position in the accumulation process.

In previous work, Ratcliff, Cherian, and Segraves (2003a) (see also Ratcliff 2001b) collected neural recordings from the superior colliculus (SC) in a task that required the monkeys to decide whether the separation between two dots was large or small. The response was made by a saccade to one of two target lights to the left and right of the stimulus dots. One target light was located in the middle of the receptive field of a SC build-up cell and the other at a mirror image position. Behavioral difficulty was varied by providing variable feedback in which large and small separations were rewarded for large and small responses with probability 0.98, but separations in the middle were rewarded with intermediate probabilities. Ratcliff and colleagues (2003a) then fit a diffusion model (Ratcliff 1978, 1988, 2002; Ratcliff and Rouder 1998; Ratcliff et al. 1999) to the behavioral data, namely accuracy, and RT distributions for correct and error responses. They then showed that simulated sample paths of the diffusion model, derived from fits of the model to behavioral data, were able to predict the build-up of discriminative information in the neural firing rates. Unlike the dual diffusion model presented here, however, the model did not predict firing rates for the populations of neurons corresponding to the two choices.

More specifically, the diffusion model used by Ratcliff and colleagues (2003a) assumes a noisy accumulation process in which a single accumulator integrates information toward either of two decision criteria. The decision process was simulated using the parameters obtained from fits to the behavioral data and position in the process was used as an analog of firing rate. The model was able to predict the growth of information that discriminated between the two decisions, i.e., the difference in firing rates between the target and competing neurons. Specifically, the firing rate data were aligned either on the stimulus onset or on the saccade, and the firing rate data were divided into three groups depending on the behavioral RT, i.e., firing rates for the fastest third, intermediate third, and slowest third of the responses (cf., Hanes and Schall 1996; Sato et al. 2001). Discriminative information was obtained by subtracting the firing rate for neurons corresponding to the response and neurons corresponding to the competitor. The analysis showed delayed onset of discriminative information among the groups corresponding to fast, intermediate, and slow responses when firing rates were aligned on the stimulus, but there was little difference in the growth of discriminative information for the three groups when aligned on the saccade.

The diffusion model was able to fit these patterns of data reasonably well. The intuition behind this is that the model requires a great deal of variability in the accumulation process to produce both correct and error responses with a fixed drift rate driving the process, i.e., there has to be enough noise for the process to hit the correct boundary on some trials and the error boundary on other trials. Also when a process approaches a decision criterion, it is quite likely that it exits the process because variability is high. This means that processes that remain in the decision process for a long time have an average position near the starting point. Thus there is delayed availability of discriminative information when the process is aligned on stimulus onset because processes remain on average near the starting point. Also there is little difference in the growth of discriminative information when the process is aligned on the decision.

Despite the ability of the model to fit the growth of discriminative information, it is unable to fit the separate firing rate functions for the neurons corresponding to the two decisions. Although the diffusion model might be seen as a model of the difference between two processes separately accumulating evidence, no analysis that would connect the single diffusion process to two racing processes was presented.

Our aim in this article is to present a model closely related to the diffusion model, a dual diffusion model, in which evidence is accumulated in two separate accumulators. Other research has found that this model mimics the diffusion model over a number of sets of experimental data (Ratcliff and Smith 2004; and data from Ratcliff et al. 2004c), but comprehensive comparisons over a wider range of parameter values than those obtained in fits to existing data sets have not been carried out. The aim is to fit the behavioral data and then use the parameters to generate simulated paths of the processes in the two accumulators. The positions of the processes in the two diffusion processes are assumed to be the analog of firing rates. The nearer the position to a decision criterion, the higher the firing rate.

In a purely behavioral approach to the results we report, any shortcomings of the model would be supplemented with additional hypotheses to allow the model to better fit the data. But because we have both behavioral and neural data, so long as the model captures many of the phenomena in the data, misses between the model and data can focus attention on what might be key aspects of the data that require additional mechanisms to those already represented in the model. Such additions will provide a more accurate description of behavior and its neural substrates.

Preliminary reports of these findings have appeared in abstract form (Hasegawa et al. 2003, 2004).

	METHODS

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Animals and surgery

Two female adult rhesus monkeys (Macaca mulatta) were used for these experiments. Northwestern University's Animal Care and Use Committee approved all procedures for training, surgery, and experiments performed. Each monkey received preoperative training followed by an aseptic surgery to implant a subconjunctival wire search coil, a plastic cilux recording cylinder aimed at the SC, and a titanium receptacle to allow the head to be held stationary during behavioral and neuronal recordings. All of these methods have been described in detail elsewhere (Dias and Segraves 1999; Helminski and Segraves 2003). Surgical anesthesia was induced with the short-acting barbituate thiopental (5–7 mg/kg iv) and maintained using isoflurane (1.0–2.5%) inhaled through an endotracheal tube.

Behavioral task

The behavioral task was designed to satisfy a number of criteria. First, we wanted to use a different task than that used by Ratcliff and colleagues (2003a) to determine whether the empirical results obtained in that study replicated across stimuli and tasks and to allow us to argue that the decision process is not dependent of the specifics of the stimulus and task. Second, we wanted to use a task that had been successfully used in the human literature, both experimentally and as the basis for model fitting. Third, we wanted a task in which the stimulus did not vary on the same dimension as was used to make the response. The task we chose was a brightness-discrimination task. In the majority of earlier studies of decision-making in monkeys, the stimulus and response often have a strong spatial correspondence. For example, in a stimulus-detection task, a single light is presented with the required response an eye movement to it (Hanes and Schall 1996). The spatial location of the light is the same location as the target of the saccade. In a visual-search task, several dots are arranged in a circle, and when one of them changes color, the required response is a saccade to the location of the dot changing color (Bichot et al. 2001; Schall et al. 1995; Thompson et al. 1996). In other experiments, some aspect of the stimulus information physically indicates the response to be made. For example, in the random dot motion task, stimulus dots move in the direction of the saccade target (Horwitz and Newsome 2001; Newsome et al. 1989; Roitman and Shadlen 2002; Shadlen and Newsome 2001). In the task used by Ratcliff and colleagues (2003a), the separation of the dots was the stimulus dimension that determined which response was to be rewarded, but the vertical separation did not point in the direction of the response target. In the brightness-discrimination task, the brightness of the patch does not contain any spatial content.

The experiments in which the response is correlated with the stimulus suggests views of processing in which visual information flows through the system with little transformation. Thus if cells are driven by multiple factors, for example, if they are visually responsive while also increasing their firing rate during the guidance of motor activity, then based on the decision made about the stimulus, if the cell's receptive field contains the target, it may be difficult to identify the factor responsible for increasing firing rates. This means that both sources are potentially telling the system the same thing, and it is hard to separate decision processes from stimulus information (but see Basso and Wurtz 1998). Although such cells do not usually fire unless the stimuli are relevant for planning saccades to a target inside the response field of the neuron, when the stimuli are relevant, it might be possible for some leakage to occur. With our paradigms, because visual information is uncorrelated with the response and visual stimulus information, no part of the differential increase in activity of a population of cells corresponding to one of the two decisions could be attributed to such leakage. This means that these paradigms are capable of providing information about decision processes that are decoupled from the stimulus information.

In this study, each monkey was trained to make a conditional saccade based on a brightness discrimination (Fig. 1 A). After the monkey had fixated a small central spot (yellow spot; 0.25° diam) for 500–1,000 ms, two peripheral targets appeared and remained on for the duration of the trial. The targets were located opposite to one another with one in the left hemifield, the other in the right hemifield. After an additional 800 ms of fixation, a brightness stimulus was presented at the central fixation point. The brightness stimulus consisted of a 2 x 2° square of white and black pixels. The monkey was free to make a saccade to one of the targets as soon as the stimulus appeared and was allowed 500 ms after stimulus appearance to complete the saccade. For the brightness-discrimination task, saccades to the target in the left hemifield were linked to discrimination of a bright stimulus and saccades to the target in the right hemifield linked to discrimination of a dark stimulus. A correct response was rewarded with a drop of water. Trials with saccades made to either response target light were scored as valid and included in the data used for this report, regardless of whether the saccade was rewarded or not. Trials in which the monkey failed to make a saccade, made a saccade that did not terminate at either response target, or made a saccade in <120 ms after stimulus appearance were rejected.

View larger version (26K): [in this window] [in a new window]   FIG. 1. Oculomotor bridghtness-discrimination task. A: representative trials of the task for leftward/bright and rightward/dark responses. Each trial was initiated by fixation of a central spot (yellow dot) for 500–1,000 ms. Next, 2 peripheral targets (green dots) appeared, 1 in each hemifield. After continued fixation of 800 ms, the brightness stimulus (square) was presented at the fixation point. Beginning with the appearance of the stimulus, the monkey was required to make a saccade, within 500 ms, to one of the peripheral targets, based on a discrimination of the brightness stimulus. For "bright" stimuli, saccades to the leftward target were rewarded, and for "dark" stimuli, saccades to the rightward target were reinforced. B: brightness stimuli. Task difficulty was manipulated by varying the percentage of white and black pixels. Numbers beneath the stimuli reflect the percentage of white pixels in the stimulus. C: pair of peripheral target locations were adjusted to place one target in a neuron's response field (RF, shaded). The remaining target was positioned at a symmetric location in the opposite hemifield.

During training, we presented nine pairs of target locations that were in three orientations (45° spacing) at three eccentricities (5, 10, and 15° away from the fixation point) in separate blocks (Fig. 1C). We used the REX system (Hays et al. 1982) running on a PC computer for behavioral control and eye-position monitoring. Visual stimuli were generated by a second PC controlled by the REX machine and rear-projected onto a tangent screen in front of the monkey by a CRT video projector (Sony VPH-D50, 75-Hz noninterlaced vertical scan rate, 1,024 x 768 resolution).

Recording

The location of the SC was confirmed by stereotaxic coordinates, the response properties of isolated neurons, and the characteristics of its topographically organized visual/motor map. We recorded from neurons in the deep layers of the SC. In this report, we define deep layers as all collicular layers located below the superficial layers (superficial gray and stratum opticum), including the intermediate and deep gray layers. Assurance that the neurons included in this study were confined to the deep layers of the SC is based on the fit of our electrode penetrations to the highly reproducible map of the SC, the ability to evoke saccades from our recording sites with current intensities of <50 µA, and the match of recorded activity to established cell activity types in these layers (Cynader and Berman 1972; Mays and Sparks 1980; Munoz and Wurtz 1995; Robinson 1972). The recording of single- and multiunit activity was done with tungsten microelectrodes (A-M Systems) introduced through stainless steel guide tubes that pierced the dura, using a Crist grid system (Crist et al. 1988). A 16-channel Plexon system was dedicated to on-line spike discrimination and the generation of pulses marking action potentials, which were stored by the REX system. The Plexon system could isolate two neuron waveforms from each electrode. We normally used one or two electrodes for recording from a maximum of four neurons. A gap saccade task was used to aid in the classification of neurons based on established criteria (Munoz and Wurtz 1995). The gap task began with a variable period of fixation; after the disappearance of the fixation light, a gap period of 400 ms was inserted before the appearance of the peripheral target light. When the peripheral target appeared, the monkey was required to make a saccade to it within 500 ms and was rewarded after the completion of the correct movement. The gap task was also used to map the response field (RF), within which the maximal response was obtained. The neurons included in this study were all located in the saccade-related region of the SC and were chosen, in part, based on their exclusion of any responsiveness to a central visual stimulus. After the RF was defined by checking the on-line histograms in the REX system, we switched to the brightness-discrimination task, in which we presented one target in the RF (preferred location) and the other at a nonpreferred location set to the same amplitude and rotated 180° from the preferred location (Fig. 1C). Every difficulty condition for both bright and dark stimuli was tested >30 times.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Behavioral results

For all 64 experimental sessions (37 in monkey 11 and 27 in monkey 12), the monkeys showed a smooth change in behavioral response as a function of stimulus brightness (Fig. 2, Table 1). The monkeys performed the task at a near perfect level when the stimulus was very bright (98% of white pixels) or very dark (2% of white pixels). On the other hand, their performance was in the range of 60–65% accuracy when the stimulus was only slightly bright (55%) or slightly dark (45%). The intermediate stimuli (65 or 35%) were associated with performance at a level of 80–90% accuracy. These results convinced us that a monkey's level of behavioral accuracy was dependent on the brightness of the central stimulus. The mean RT (saccadic latency referenced to appearance of the brightness stimulus), however, was fairly stable across the various difficulty conditions in both monkeys (206 ± 33 ms, n = 16,551 correct trials for monkey 11; 239 ± 61 ms, n = 8,209 correct trials for monkey 12), suggesting that the monkeys might have put more emphasis on a rapid response than on accuracy. Although one-way ANOVA for effect of brightness on saccadic latency was significant for both monkeys (P < 0.001), the range of this effect was small (monkey 11: 25 ms; monkey 12: 19 ms).

View larger version (30K): [in this window] [in a new window]   FIG. 2. Mean accuracy and response time values for each monkey in the brightness judgment experiment. x, data; - model's best predictions. Bars represent ±2 SE.

We recorded 137 neurons with presaccadic activity in the SC of the two monkeys (82 neurons from monkey 11; 55 neurons from monkey 12) while they performed the oculomotor brightness-discrimination task. The analyses included data from all cells identified as build-up cells by exhibiting build-up activity in the gap-saccade task. Build-up activity was monitored during the final 200 ms of the gap period, before the appearance of the target light, in a gap saccade task, and compared for significant increases (Wilcoxon sign-rank test, P < 0.05) above fixation period activity measured during the final 200 ms before the disappearance of the fixation light and beginning of the gap period. Figure 3 illustrates how the activity of a right SC neuron varied according to task difficulty. In the brightness-discrimination task, the cell exhibited strong activity after the appearance of "bright" stimuli instructing the monkey to make a saccade to the leftward target. The level of this activity for bright stimuli was similar for each of the three levels of difficulty (Fig. 3, A, C, and E). In contrast, the neuron's activity generated after the appearance of "dark" stimuli instructing rightward saccades appeared to be modulated by task difficulty. The peak and mean frequencies of this activity increased as the task became more difficult (Fig. 3, B, D, and F). Thus a manipulation of task difficulty, induced by changing the brightness level of a central stimulus, altered the level of presaccadic activity for a collicular neuron with a peripheral activity field.

View larger version (53K): [in this window] [in a new window]   FIG. 3. Activity of a superior colliculus (SC) neuron during the oculomotor brightness-discrimination task. Trials are synchronized to the appearance of the brightness stimulus (left, vertical line through each histogram) or the saccade onset (right). For rasters aligned to stimulus appearance, the red symbol in each trial indicates the beginning of the saccade. A, C, and E: trials where the monkey correctly identified a bright stimulus and made a leftward saccade into the neuron's response field. B, D, and F: trials where the monkey identified a dark stimulus and made a rightward saccade away from the neuron's response field. A and B: easy condition. C and D: middle condition. E and F: hard condition. The spike density histogram below each raster was generated by first convolving single trial neuronal activity using a Gaussian kernel with a sigma of 10 ms (Richmond et al. 1987), then averaging the spike density profile across trials. The eye position is shown as blue (horizontal) and green (vertical) traces under each histogram.

View larger version (22K): [in this window] [in a new window]   FIG. 4. Two examples of the effect of task difficulty on neuronal activity. A: averages (±SE) of the activity of the neuron in Fig. 3 during the stimulus aligned epoch (left; 50 ms interval extending from 150 to 200 ms after the appearance of the brightness stimulus) and during the presaccadic epoch (right; 50 ms interval extending from 50 to 0 ms before the start of the saccade) are plotted against the different levels of task difficulty. and , brighter and darker stimuli, respectively. B: another set of averages of activity for a neuron the response field of which was in the right hemifield. For this neuron, darker stimuli were linked to its preferred direction.

View larger version (16K): [in this window] [in a new window]   FIG. 5. Effect of task difficulty on activity across the population of neurons. A pair-wise comparison (easy vs. hard) of average discharge rate in the 50-ms interval immediately before the saccade onset is shown. Each symbol represents activity for a single neuron. For the preferred direction (left), activity was similar for easy and hard conditions. For the nonpreferred direction (right) activity associated with the hard condition tended to be greater than it was for the easy condition.

Dual diffusion model

Our model assumes that the decision process involves two racing diffusion processes as shown in Fig. 6. Figure 6 also shows the equation for the evolution of evidence as a function of time in the model along with a list of the parameters of the model. Evidence corresponding to the two decisions is accumulated in two separate accumulators each with a separate decision criterion (c_a and c_b). The accumulation process is very noisy, and noise is assumed to be normally distributed as in the Wiener diffusion process (e.g., Ratcliff 1978). Because we are using evidence in the accumulation process to mimic firing rates, it is assumed that evidence cannot go below zero. If an update to evidence in the process would have taken it below zero, it is reset to zero. The rate of accumulation of evidence varies as a function of difficulty in the task, but it is assumed that the sum of the rates for the two accumulators is constant across conditions (i.e., they are negatively correlated), this means, if one has a high drift rate, the other has a low drift rate. This assumption means that the total rate of information accumulation is the same across conditions. This assumption leads to one parameter for the sum of drift rates and one parameter for the drift rate for one accumulator for each condition; the other drift rate is the difference between the sum and the drift rate for the other accumulator.

View larger version (20K): [in this window] [in a new window]   FIG. 6. Parameters: 1. Two criteria, ca and cb 2. Leak 3. Ter (mean nondecision component), with range st 4. Sum of accumulation rates, vsum 5. Accumulation rate for each condition (va + vb = vsum) 6. Range of starting points (negatively correlated: sum = sx) 7. Minimum base starting point (x1): starting point xa = x1 + sx – xb) Note: The SD in Gaussian noise, , is scaled to 1 The equation for the update to evidence dxi for evidence xI in accumulator i, is Illustration of the dual diffusion model and the parameters of the model. There are 2 accumulators with decision criteria ca and cb, accumulation rates va and vb (va + vb = vsum), starting points xa and xb (a random number x is selected from a uniform distribution with range sx and lower limit zero, and xa = x + xI and xb = sx – x + xI, where xI is a minimum baseline starting point), and decay rate (–x). The equation for the update of evidence in accumulator i (dxi) at the bottom of the page has time steps of size h, set to 1 ms in the fits of the model to the behavioral data. Variability in processing within a trial (Gaussian noise) is normally distributed with SD = 1. Across trials, the nondecision components are uniformly distributed with mean Ter and range st.

To produce slow errors, the diffusion model (Ratcliff et al. 1999) assumes that drift rate is variable from trial to trial. Initial fits to the behavioral data from this study that included variability in drift rates produced estimates of variability that were small and so variability in drift was set to zero for subsequent fits.

As noted earlier, there is a great deal of variability within trials due to intrinsic noise in the accumulation process. This variability is necessary to produce error responses even when drift rates are high. Figure 7 illustrates variability in processing by presenting four simulated paths of the two diffusion processes. In two cases, the process represented by the black line wins, whereas in the other two cases, the process represented by the green line wins. The negative correlation in starting points can be seen in the initial starting levels: when one is high, the other is low. To model firing rates after a saccade is made, we assume when one process terminates, evidence decays back to the starting level.

View larger version (29K): [in this window] [in a new window]   FIG. 7. Examples paths of the accumulation of evidence for the same accumulation rate for each diffusion process. The 2 horizontal lines at the top of each panel represent the decision criteria (black for the black accumulator and green for the green accumulator).

The nondecision components of processing are combined into a single parameter T_er which is assumed to be variable from trial to trial. Ratcliff and Tuerlinckx (2002) assumed a uniform distribution of variability with range s_t. A uniform distribution was assumed because it was simple and because convolution with a decision time distribution with larger SD produces a distribution with shape about the same as that of the decision time distribution. This means that the assumption about the shape of the distribution of nondecision components does not affect predictions of the model so long as the SD is three or more times smaller than that of the decision component.

One key feature of this model that is addressed later is that the stimulus information that drives the decision process is stationary. It assumes that prior to the decision process, there is no evidence accumulation. Then during the decision process, the mean and variance of the stimulus information driving the decision process both remain constant during the time taken to make a decision. For this experiment, it is natural to assume that the stimulus provides a constant perceptual representation that drives the decision process. However, in other work in which a stimulus is presented briefly and masked, we have conceptualized perceptual processing as producing a short-term memory representation of the relevant characteristics of the stimulus that then provides constant drift rate to the decision process (Ratcliff 2002; Ratcliff and Rouder 2000; Smith et al. 2004). Thus during the decision process, all the parameters of the process are constant over time. Then when one of the processes reaches a decision criterion, the decision is made (response output processes are initiated), and evidence decays back to the initial level. The view that perceptual processing produces a representation in memory that drives the decision process means that in our conceptualization, the stimulus is not simply directly tied to the decision process. This also suggests that issues of how stimuli are represented and how the representation drives the decision process need to be understood. Later, we will discuss relaxing some of these stationarity assumptions to better account for misses between predictions of the model and firing rate data.

A model very similar to this model is the leaky competing accumulator model of Usher and McClelland (2001) that has two accumulators with Gaussian distributed noise (i.e., 2 racing diffusion processes) and with decay in evidence. However, each accumulator is assumed to inhibit the other accumulator by an amount proportional to the evidence in the accumulator. We also fit this model, but decay and inhibition were modest in size, and the fits were qualitatively and quantitatively similar to those presented here. We take up this model in the general discussion. Also, versions of the dual diffusion model have been considered by Smith (2000) and by Ratcliff and Smith (2004).

Behavioral data fits

To generate predictions for fitting the model to data, the dual diffusion process is simulated with step size 1 ms (see Brown et al. 2006). To produce a set of accuracy and RT predictions for one condition and one set of parameter values, 20,000 simulations of the process are performed. This is repeated for each of the conditions in the experiment (for 6 different drift rates) to produce a set of predictions to be used to match the results for the six conditions in the experiment.

We fit the model to the behavioral data using the ² method presented in Ratcliff and Tuerlinckx (2002). In this method, a ² statistic is computed from both predictions and data to represent goodness of fit of the model to data. An iterative nonlinear minimization routine (SIMPLEX) (Nelder and Mead 1965) is used to minimize ² by adjusting parameter values. The fitting routine begins with an initial set of parameter values and a range for each of the parameter values (in our case, 10% of the parameter value). The algorithm produces n + 1 sets of parameter values based on the mean and ranges in the parameter values and then computes n + 1 ² values. The largest value of ² is selected, parameter values are adjusted, and the ² value is recomputed. The largest value of ² is again selected (which may or may not be a different one) and parameter values again adjusted until the process creeps toward a minimum ² value.

The parameter spaces of stochastic models of this kind have not been explored theoretically. For example, Usher and McClelland (2001) used a simulated annealing fitting method to produce fits to the data from a few experiments. This was done to address two main concerns, namely, that the model fitting process might fall into local minima or into regions where changes in parameter values do not change chi-square values. To allow our fitting program to deal with such problems, we ran the simplex minimization six times in succession using the parameters of the previous fit as the starting values with the original 10% ranges in the values. This was an attempt to functionally mimic Usher and McClelland's method.

There was one parameter that was adjusted to attempt to produce better fits to the neural firing data, and that was the baseline level of activity in the accumulators. This was adjusted in a series of exploratory fits so that the baseline positions in the accumulators roughly match baseline activity in the firing rate functions so that the relative size of the baseline to peak positions matched those for the firing rate functions.

Figure 2 presents the fits of the model to accuracy and mean RT. Tables 2 and 3 present the parameter values of the model for the fits. The top two panels of Fig. 2 present the probability of the bright response as a function of the proportion of white pixels in the stimulus display. The model fits the accuracy values well except for the two dark stimuli for monkey 11 and the extreme stimuli (2 and 98% white pixels) for monkey 12 (maximum discrepancy of 4–5%). The bottom four panels of Fig. 2 show mean RT as a function of the proportion of white or black pixels in the display. The error bars represent ±2 SE. There are significant misses to several of the conditions. Two of most note are correct responses when the proportion of black or white pixels is 0.98, i.e., the easiest conditions. The data from these conditions are quite unlike those we see for human subjects. Invariably, the conditions with the highest accuracy are also the fastest. Here we find the most accurate conditions are a little slower than less accurate conditions. We do not understand why this might have occurred except to point our that these two conditions were trained first, for several sessions, before the conditions with the lower proportions of black or white pixels were introduced later in training. There are also significant misses in the error RTs in the extreme conditions (0.02 white or black pixels). Because there are very few of these errors, a few responses might have been produced by guesses when attention had wandered, and these would have been counted as errors that would produce longer RTs.

Figure 8 includes quantile probability functions (Ratcliff 2001a; Ratcliff and Smith 2004) that show quantiles of the RT distributions plotted against the probability that the response is made to the stimulus. In the quantile probability functions shown here, data from "bright" and "dark" responses are plotted separately. Data for each stimulus are used to produce RT quantiles, and these are plotted vertically on the y axis and their location on the x axis is the probability of the bright or dark response. This method of presenting data was developed by Ratcliff (2001a) to allow response probability and the shapes of the RT distributions for correct and error responses to be all displayed in one plot and allow visualization of how the different dependent variables co-vary. The gray ellipses around some of the points provide a representative sample of 95% confidence intervals around the data (see Ratcliff et al. 2003a).

View larger version (46K): [in this window] [in a new window]   FIG. 8. Quantile probability functions for each monkey (top 4 panels). The diffusion model (fitted values are , connected by —) were fit to the behavioral data (x). The 5 crosses in each column represent the 0.1, 0.3, 0.5 (median), 0.7, and 0.9 quantile response times for a particular condition. Their location on the x axis represents the probability that the response is made to the stimulus. The top 2 panels represent "bright" responses for stimuli with 2, 35, 45, 55, 65, and 98% white pixels moving from left to right across the figure. There are very few responses for the extreme error quantiles. To illustrate how the quantiles are related to RT distribution shape, the bottom 2 panels show RT distributions from 2 single conditions in the quantile probability functions above (see text for detailed description). The gray ellipses in the top four panels represent 95% confidence intervals on the quantile RTs and accuracy values. The SE is computed from the means across RTs and accuracy values across trials.

Apart from the misses noted earlier, the general shapes of the RT distributions are captured by the model. In particular, both the empirical and theoretical RT distributions are skewed for all the conditions as can be seen by the wider separation of the 0.7 and 0.9 quantiles relative to the 0.1 and 0.3 quantiles.

In analyzing the data, we found that the monkeys had good days and bad days, i.e., mean RT could differ by 30–50 ms between days. We produced two groups of data, one with faster days, the other with slower days. We fit the model to these two groups for the two monkeys and found that in each case, the only parameters that varied systematically were boundary setting, for the slower days it was 6% larger than for the faster days, and the sum of drift rates, which was 16% smaller for slow days than fast days. These two parameters do not trade off against each other because changes in boundary position produce changes in the leading edge of the distribution while changes in the sum of drift rates affects skew with only a very small effect on leading edge. Thus the model based interpretation is that on bad days, the monkeys get poorer information from the stimulus than on good days and adopt more conservative decision criteria than on good days.

It is possible to directly compare performance on human and monkey data on this brightness-discrimination task. In human data, accuracy is higher than for the monkey data. The human experiment (Ratcliff 2002; Ratcliff et al. 2003b) uses a presentation-time manipulation and a brightness manipulation as well as a speed-accuracy instruction manipulation. Presentation time does not have a large effect on performance, but as for the monkey data presented here, brightness has a large effect. The experiment here has the most difficult conditions with 0.45 and 0.55 white pixels, but in the human experiments, the conditions closest to these are 0.425, 0.475, 0.525, and 0.575 white pixels. Accuracy for the average of the 0.45 and 0.55 conditions for the monkey data are 0.65 and accuracy for the human data in the speed instruction condition is 0.75, and for the accuracy instruction condition, 0.80. We have fit the dual diffusion model to the human data, and the decision criteria are set to much larger values than for the fits to the monkey data (about double). If we increase the monkey decision criteria to the human speed instruction value, keeping all other parameters constant, the predicted accuracy value increased from 0.65 to 0.77, i.e., quite close to the human accuracy value.

Although the fits shown in Figs. 2 and 8 are not as good as those obtained for behavioral data for human subjects, they do capture most of the main features of the data. We now use them to produce predicted paths for the build up in evidence for the two accumulators.

Decision process paths and neural firing rates

To relate the dual diffusion model to the neural firing rate data, we make the simplest assumption: collicular build-up activity for the target and nontarget receptive fields correspond to the evidence in the two accumulators. The nearer the process is to the criterion, the higher the firing rate. The firing rate data presented are aggregates over trials for a single neuron and over sessions for different neurons. Thus the firing rate data are assumed to represent the behavior of the population of cells with build-up activity that are proposed to implement the decision.

To generate predicted average evidence in each accumulator (the analog of firing rates), we used the parameters of the model in Tables 2 and 3 to generate 2,000 simulated processes. These processes provide the amount of evidence in each accumulator as a function of time. The only parameter that was adjusted in fitting the behavioral data to accommodate the firing rate data is the initial level of evidence. This was adjusted manually in the process of fitting the behavioral data to produce initial levels of evidence that match the initial firing rates so that the relative differences between peak activity and baselines are similar to the firing rate data. Once one of the processes reaches a decision criterion, both processes are assumed to exponentially decay (with noise turned off) to the resting level with decay rates selected by eye to match the data (with exponential time constants 45 and 20 ms for monkeys 11 and 12, respectively). The other alignments that allow the model to predict the neural firing rate data involve aligning the functions on the time axis. Because the behavioral model combines encoding and response output (and other nondecision processes) into one parameter, this has to be divided up into a component prior to the decision process and a component after. This was done by eye, and for monkey 11, the estimate of the component of the nondecision component occurring after the presaccadic peak activity is 14 ms and for monkey 12, it is 10 ms. The other adjustment is vertical scaling from evidence to firing rate.

Firing rate data for RT tasks are typically plotted two ways (e.g., Hanes and Schall 1996; Roitman and Shadlen 2002), aligned on stimulus onset and aligned on the saccade (response initiation). There are small differences in mean RT across conditions and there are small differences in the initial onset of firing rates of neurons corresponding to the two decisions for both ways of plotting the firing rate data. In Ratcliff, Cherian, and Segraves (2003a), peak firing rates were about the same height across conditions in neurons corresponding to the decision made, but the peak firing rate in the neurons corresponding to the competing alternative becomes larger as the condition becomes more difficult [e.g., compare Fig. 8 of Ratcliff and colleagues (2003a) to Fig. 3 of this report].

Hanes and Schall (1996) grouped firing rate functions depending on the RT of the trial. This allowed the rate of growth of firing rates to be examined as a function of speed of the response. The patterns of results they obtained allowed more detailed qualitative tests of models of the growth of firing rate with time. Ratcliff, Cherian, and Segraves (2003a) found that when firing rates were aligned on the stimulus, firing rates for the fastest responses rose early in neurons corresponding to both the response made and the competing neurons. The initial rise was slower for slower responses. But the difference in the firing rates for neurons corresponding to the decision made and the competitor (the difference corresponding to discriminative information) was delayed for slower responses by as much as 80 ms (see also Sato and Schall 2001). For firing rates aligned on the saccade, there were differences in onset of firing rate functions prior to the saccade as large as 50 ms. But the growth of discriminative information (the difference in firing rates) was almost identical for firing rates corresponding to faster versus slower trials.

For data from this brightness-discrimination task, for firing rates aligned on stimulus onset, the functions for the fastest, middle, and slowest terciles rise at close to the same point in time but at different rates for both the target and competing neuron. Firing rates for the difference between the target and competitor represent the growth in discriminative information (both populations of neurons become active after stimulus onset, but information that discriminates between them can become available later), and this shows delays in onset for the slowest tercile relative to the middle tercile and the middle tercile relative to the fastest tercile. For data aligned on the saccade, the functions for the three terciles are similar. Only for the slowest tercile does the rise in activity begin a little earlier relative to the saccade and rise at a slower rate. The rise in discriminative information shows almost no difference between terciles.

This pattern of data suggests that on average, a successful model has to accumulate relatively little information prior to 75–100 ms before the saccade initiation followed by a rapid build up, and this has to occur whether the stimulus to response duration is short or long. In the brightness-discrimination data presented in the following text, stimulus onset can be between 300 and 180 ms before the saccade for the first and third tercile, but the difference in the rise in discriminative information may only differ in onset by 20 ms.

We now show that the model fits the patterns of activity recorded in the brightness-discrimination experiment. After presenting the correspondence between the model predictions and neural firing rate data, we focus on the main misses between theory and data and how the model could be augmented to deal with these misses. As noted earlier, the model is stationary, that is, it assumes that the instant the decision process begins, the mean and variance of the stimulus information are both constant until the process hits a decision criterion at which point the decision process is over, that is, none of the parameters of the decision process change during the decision process. Augmentation focuses on how we might allow processes to be nonstationary which might better match what is known about the neurophysiology of this system.

Firing rates aligned on the stimulus

Figure 9 shows firing rate functions aligned on the stimulus for target and nontarget for correct responses (dark responses) for the 98% black pixel condition and 55% black pixel. The plots are for terciles of the data, that is, firing rate functions for the fastest third of the behavioral responses, the intermediate third, and the slowest third. For the activity for the target, the functions grow from 100 ms after stimulus onset to peaks at 180 and 200 ms for the first tercile to 220 and 270 ms for the longest tercile for monkeys 11 and 12, respectively. The predicted functions show about the correct relative heights and they decay at about the same rate as the data. But the functions miss in the initial rise, especially for monkey 12. For the activity for the nontarget, the peak activity levels for monkey 11 match, but for monkey 12, the peak is a little too high for the 55% black pixel condition and in all cases, the firing rates fall more quickly than the predictions. The results for the 65% black pixel condition fall between these shown here.

View larger version (34K): [in this window] [in a new window]   FIG. 9. Neural firing rates averaged over cells and aligned on the stimulus. The firing rates are divided into thirds as a function the behavioral response (fastest third, middle third, and slowest third). Left: easy conditions, dark responses to 98% black pixels; right: difficult conditions, dark responses to 55% black pixels. First and 3rd rows: firing rates for cells the receptive field of which contained the target for the correct response when a correct response is made (target cell). Second and 4th rows: firing rates for cells the receptive field of which contained the target for the incorrect response when a correct response is made (nontarget cell). The solid lines are the data and the green lines are predictions from the dual diffusion model. The points marked "a" show what appears to be a suppression prior to the rise in activity leading to a decision. The vertical lines represent the median saccade initiation times for the 3 conditions.

Firing rates aligned on the saccade

Figure 10 shows the same firing rate data but aligned on the saccade. The predicted initial rises and peaks for correct responses to the targets match the data reasonably well, and the decay matches well. There are some misses in the initial rises on the order of tens of milliseconds, but overall the match between the empirical and theoretical shapes are reasonably good. The firing rate functions for the nontargets show a large and systematic miss. For the data, the competing functions produce bimodal functions (that appear in all the conditions) that rise to a peak and then dip 20 ms prior to the saccade (dips marked by b in Fig. 10) and then rise again at the saccade to produce a second peak. The model can only produce a single peaked function. This dip also occurs in the data of Ratcliff, Cherian, and Segraves (2003a) (Fig. 10).

View larger version (34K): [in this window] [in a new window]   FIG. 10. Neural firing rates averaged over cells and aligned on the saccade. Zero time is the median point at which the saccade begins and the vertical line to the right of 0 is the median point at which the saccade terminates on the choice target. Vertical lines to the left of 0 represent onset of the brightness stimulus. For monkey 12, stimulus onset for the slowest group of responses occurred before the earliest time represented by the x axis of these plots. The points "b" represent the points at which activity in the competing neurons dips just prior to the saccade initiation (see the text).

View larger version (35K): [in this window] [in a new window]   FIG. 11. Quality of the fit for firing rates aligned on stimulus and saccade. A plot of the middle tercile of the firing rate functions shown in Figs. 9 and 10 for the 55% black pixel condition for alignment on the stimulus and saccade for the target and competitor cells. One hundred bootstrap samples were generate by randomly sampling with replacement from the data. Thus the data from a cell might be included 0 times, once, twice, or so on. The functions were sorted into order from highest to lowest based on the average firing rate for 50 ms around the peak. The 5th, middle, and 95th functions are plotted (···, —, and - - -, respectively). These represent 90% confidence intervals around the firing rates that would be obtained if different random samples of cells were selected.

Difference in firing rates

Figure 12 shows the difference in firing rates and the differences in the amounts of evidence from the theoretical predictions for the functions shown in Figs. 9 and 10. These functions show the growth of discriminative information for fast, intermediate, and slow responses and as a function of two levels of stimulus intensity. The results show for functions aligned on the stimulus, discriminative information is delayed for slow versus fast responses both in the data and in the model predictions. The mismatch between the model and data was much less than in the individual functions (Fig. 9); this suggests that the misses in the initial growth were not the result of growth of information that discriminates between the two choices in the decision process. For functions aligned on the saccade, the growth of discriminative information began to rise 50–80 ms before the saccade and showed little difference for fast, intermediate, or slow responses. The predictions and the data match quite well (better than the individual functions in Fig. 10) except for a miss of 10–20 ms in some initial rises. Some of the tails fall to asymptotes that are lower than the starting level for the reasons discussed in the preceding text.

View larger version (36K): [in this window] [in a new window]   FIG. 12. Firing rates for the growth of discriminative information. The functions are obtained by subtracting the target cell firing rates and competitor cell firing rates shown in Figs. 9 and 10.

View larger version (20K): [in this window] [in a new window]   FIG. 13. Empirical and predicted peak firing rates as a f