Collinear facilitation is largely uncertainty reduction

Abstract

When flanked by collinear Gabor patches, detection thresholds for a target Gabor patch improve by up to a factor of 2. This result has been interpreted as evidence for collinear facilitation. However, facilitation has been observed only for targets near detection threshold, where observers seem uncertain about the location and other properties of the stimulus. So the effect of the flankers may be to reduce this uncertainty. If this is true, then other cues to target location should produce a similar improvement in thresholds. To test this hypothesis, we measured contrast detection thresholds for a Gabor target alone, and in the presence of either a faint circle surrounding the target location, or two high-energy flanking Gabor patches. We also used an adaptive procedure to measure the slope of the psychometric function to determine whether the slopes were considerably lower in the presence of location cues or flanking Gabors, as predicted by signal detection theory when uncertainty is reduced. As observed previously, the presence of collinear flankers improved detection thresholds by a factor of two. Yet, on average, the circle alone accounted for the most of the facilitation; for three of our five observers, it improved thresholds as much as the collinear flankers. Other cues that specified target location produced similar improvements in detection thresholds. The slopes of the psychometric functions were much shallower in the presence of these location cues or the collinear flankers compared to the target-alone condition. This change in the slopes indicates that the threshold improvement is largely due to a significant reduction in uncertainty.

Introduction

Starting with Polat and Sagi (1993), several studies have shown that the presence of collinear flankers improves the detection of a Gabor patch under central viewing conditions (e.g., Solomon, Watson, & Morgan, 1999; Zenger & Sagi, 1996). The facilitation effect is strongest (up to a factor of 2) when the flankers are closest to the target (without overlap) and collinear with it, but facilitation is also observed for larger separations and different flanker orientations/locations relative to the target (Polat & Sagi, 1994; Yu, Klein, & Levi, 2002). 

These results are often interpreted as evidence of long-range interactions in V1 and/or other low-level visual areas. Such interactions are usually assumed to be either a multiplicative (contrast gain) or an additive form of input modulation implemented via lateral cortical connections. It has been shown that such low-level input modulation could provide a basis for various image segmentation tasks, for example, contour integration or boundary detection (Li, 1998; Stemmler, Usher, & Niebur, 1995; Yen & Finkel, 1998). In support of this hypothesis, several neurophysiological studies have reported that lateral facilitation was a common effect in striate cortex. Notably, Kapadia, Ito, Gilbert, and Westheimer (1995) compared single-cell recordings in monkey V1 with the performance of human observers and found qualitative agreement between the two. In their study, 42% of complex cells responded more strongly to a bar presented in the suprathreshold (classical) receptive field if another (collinear) bar was presented in the surround. Other studies (e.g., Chen, Kasamatsu, Polat, & Norcia, 2001; Kapadia, Westheimer, & Gilbert, 2000; Nothdurft, Gallant, & Van Essen, 1999; Polat, Mizobe, Pettet, Kasamatsu, & Norcia, 1998; Sengpiel, Baddeley, Freeman, Harrad, & Blakemore, 1998; Toth, Rao, Kim, Somers, & Sur, 1996) have reported similar effects. 

Although the modulation of response due to stimuli beyond the classical receptive field is a well-established phenomenon, the interpretation of the facilitatory effect remains controversial. As Cavanaugh, Bair, and Movshon (2002a, 2002b) pointed out, stimuli outside the suprathreshold receptive field could still be in a cell's subthreshold summation zone. In this case, they do not elicit a direct neuronal response but can increase the response once another stimulus is presented within the receptive field. This form of collinear facilitation would be an artifact of an overly conservative definition of the receptive field, rather than a result of interactions across large cortical distances. In fact, when the excitatory receptive field was defined as the diameter of the smallest sine-grating patch that elicited at least 95% of the neuron's maximum response, the predominant effect of surround stimuli was inhibition rather than facilitation (Cavanaugh et al., 2002a, 2002b). 

To account for psychophysical facilitation, Solomon et al. (1999) proposed an explanation that essentially followed the same lines. They showed that facilitation can be explained by the traditional transducer model assuming that target summation zone is extended beyond the Gabor outlines, so that flankers overlap with the extended summation zone (i.e., a neuron's receptive field). However, the model produced a rather poor fit to the data. The amount of facilitation could not be fully explained, especially at larger target-flanker separations. 

It is important to note that, unlike neuronal facilitation, psychophysical facilitation has been observed only for targets near detection threshold (Chen & Tyler, 2002; Williams & Hess, 1998). In these conditions, the target is barely visible and, when shown on its own, is hard to localize without a large degree of uncertainty. This uncertainty requires an observer to attend to a larger spatial area, which likely increases the pool of neurons engaged in the detection task. Elevated detection thresholds are predicted by signal detection theory for this case (Pelli, 1985). From this perspective, the high-contrast Gabor patches presented simultaneously with the target provide excellent cues to the target's exact location (including Vernier cues) that would explain the improved performance. 

If facilitation is produced by a reduction in uncertainty, then other cues to target location should produce the same effect. Faint, low-contrast cues, such as thin lines or circles, could be particularly revealing because they should not stimulate lateral interactions or pedestal effects in any significant way. We can also look for effects of uncertainty in a different form. According to signal detection theory (Green & Swets, l966), the uncertainty reduction provided by surround cues would make the slope of the psychometric function shallower. To the contrary, facilitation based on the input modulation mechanism should modify the thresholds but not the psychometric slopes. 

In this study, we tested two possible sources of collinear facilitation: (i) input modulation at the level of individual neurons as implemented by contrast gain control, for example, and (ii) modulation of the number of the relevant neuronal detectors as determined by the degree of uncertainty about the target's location. We compared the change in detection thresholds and slopes of psychometric functions produced by a conventional pair of collinear Gabor flankers to the effects of low-energy location cues (lines and circles). Our results strongly suggest that uncertainty plays a major role in the observed facilitation. 

Methods

Stimuli

The target was a 3-cpd cosine phase Gabor (σ = λ/2) in which ∼1.5 periods of the sinusoidal pattern were visible (Figure 1). The Gabor could be slanted 45 deg. either right or left from the vertical. There were two types of location cues: (i) a dark circle centered on the target and (ii) four dark lines surrounding the target. The cues were 1 pixel (1.2 arcmin) wide and their contrast was 15%. The circle was 2.5λ (0.83 deg) in diameter, the lines were 2λ (0.67 deg) long. The lines had the same orientation as the target Gabor and were positioned 2.5λ away from the target's center in a square fashion, as shown in Figure 1. 

Figure 1

 

Diagram of the stimuli. The Gabor target alone is shown on the left. The top row illustrates the circle cue alone and in conjunction with two surrounding Gabors in “sides” and “ends” configurations. Analogously, the bottom row illustrates the four-line cue.

To compare the effect of the location cues with the effect of the conventional surround stimuli, we used a pattern composed of two 30% contrast Gabor patches otherwise identical to the target. The two Gabors were positioned either at the ends of the target, or at the sides of the target, 2.5λ away from the target's center in both cases. The location cues were displayed on their own and in conjunction with the two Gabor patterns. Thus, for each cue we tested four configurations: (a) target alone; (b) target and the cue; (c) target, cue, and two parallel Gabors; and (d) target, cue, and two collinear Gabors. The four configurations are shown in Figure 1. In Experiment 1, all four conditions were randomly interleaved within each experimental session. In Experiment 2, only one condition was presented within each session. The two target orientations were randomly interleaved within each session in both experiments. 

Stimuli were displayed on a gray background (42 cd/m²) and viewed through Wheatstone stereoscope on a pair of linearized Sony Trinitron G220 monitors. Viewing distance was 65 cm. The stereoscope was used for a series of studies on the spatial properties of surround effects, including disparity, but for this study, the binocular images were presented at zero disparity for all components. The video signal was rendered with nominal 8-bit precision, but an additional factor of 4 increase in precision was attained using an ordered 2 × 2 block pixel dithering (analogous to the newspaper halftone technique). The resulting 2 × 2 pixel size was 2.4 arcmin, whereas the dithering artifacts (0.8% contrast modulation at 22 cpd) were approximately 30 times below the detection threshold. The effective luminance resolution of the screen at the background level (after gamma-correction) was confirmed to be 0.2% (9 bits) by counting the number of gray levels in the stimulus screenshots. 

Procedure

We used a 2-temporal interval forced choice procedure (2IFC). Each trial consisted of two consecutive stimulus presentations only one of which contained the target. The two stimulus intervals were indicated by a pair of dark peripheral disks appearing in the top of the screen. The location cues or Gabor masks were presented in both intervals. The task was to indicate with a button press in which 2IFC interval the target was shown. Stimulus duration in each of the two intervals was 150 ms; the interstimulus interval was 500 ms. The stimuli were displayed in the center of the screen; viewing was binocular. A fixation pattern comprised of two low-contrast concentric circles and a pair of nonius lines was displayed at the target location in the beginning of each trial and also in the interstimulus interval, but it disappeared 150 ms before the stimulus onset to avoid overlay and forward masking. Thus, unlike location cues, which were present at all times during the stimulus interval, the fixation mark was never displayed at the same time as the target. 

We used the adaptive staircase algorithm of Kontsevich and Tyler (1999) to estimate thresholds α and steepness β of psychometric functions. The 2IFC psychometric function of the target contrast c was fitted by  

$P_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}}\left(c\right)=N\left(\frac{d^{\prime }}{\sqrt{2}}\right),$

(1)

where  

$N\left(\mu \right)=\frac{1}{\sqrt{2\pi }}\underset{-\infty }{\overset{\mu }{\int }}{e}^{-{\tau }^2/2}d\tau$

(2)

is the cumulative function for the Gaussian distribution, and  

$d^{\prime }={\left(\frac{c}{\alpha }\right)}^{\beta }$

(3)

is the signal detectability. 

Threshold contrast α corresponds to 76% correct responses. Parameter β describes the steepness (slope) of the psychometric function. For β = 1, P_correct(c) gives the signal detection theory prediction for a single linear detector with normally distributed additive noise characterized by standard deviation α. Larger values of β would signify either a nonlinear relationship between the stimulus contrast and the detector input or alternatively indicate multiple detectors and inherent uncertainty about which detector receives the signal (see 1). 

In Experiment 1 where only detection thresholds were measured, 4 blocks of 52 trials per condition were done by each observer. In Experiment 2 where both thresholds and psychometric slopes were measured, 3 blocks of 300 trials were accumulated for each condition. Results for two target orientations were averaged. Estimates of uncertainty for threshold and slope values were obtained from variation in between the blocks. 

Observers

Six observers with normal or corrected visual acuity were tested. Two of the observers were naive to the purpose of the study; all except EF were experienced psychophysical observers. Observers were trained for a short time (2–5 min) to get acquainted with the stimuli and the task. 

Results

Experiment 1

In this experiment, we tested whether the addition of thin low-energy (i.e., low-contrast) location cues would result in facilitation of target detection. As described in Methods, we tried two kinds of cues: a circle and a set of four collinear lines. The four-line cue had the same energy as the circle cue, but it differed in two important aspects: (i) it did not enclose the target completely, and (ii) it provided an orientation cue along with the location cue. Because in each trial the target could have one of two possible orientations, the orientation cue could improve the performance. We tested the effect of the low-energy cues alone and in conjunction with the usual pattern of two flanking Gabors positioned either at the ends or at the sides of the target Gabor. It can be easily appreciated from Figure 1 that the two Gabors were significantly higher energy than the location cues. To make it more quantitative, we estimated energy as the squared RMS contrast confined to a 2.5λ diameter disk around each Gabor patch location. This calculation gives ∼1:16 ratio between the circle cue presented on its own and in conjunction with the two flanking Gabors. 

The measured detection thresholds are shown in Figure 2. Individual observer's data are marked with different colors and average performance is shown in black. To make the comparison between observers easier, thresholds for each observer were normalized by the target-alone threshold: 3.3% (PV), 2.0% (YP), 3.4% (SPM), 2.4% (LM), and 2.8% (EF). On average, the circle cue in conjunction with two Gabors at the ends of the target produced the strongest reduction of thresholds: 46%. The circle cue alone produced a 28% reduction. For the four-line cue, the average reductions were 40% and 27% with and without Gabors, respectively. Considering the standard errors of our data, there was no significant difference between the effects of the two types of location cues. Thus, the location cues alone accounted for approximately 65% of collinear facilitation from the flanking Gabors positioned at the ends of the target. This proportion was even higher (85%) in the case when the two Gabors were positioned on the sides. Note that for two observers (LM and YP) the effect of the location cues alone was large enough to account for most of collinear facilitation. 

Figure 2

 

The effect of target surround on contrast detection thresholds. Thresholds were normalized by the results for target-alone condition (leftmost group of bars in each graph).

Experiment 2

In this experiment, we measured both thresholds and slopes of the psychometric functions. Because a reliable estimate of the psychometric slope requires roughly threefold increase in the number of trials, only the circle cue was used. Also, unlike Experiment 1, no location cue (only flanking Gabors) was presented in configurations c and d. 

Figure 3 shows the shapes of the psychometric curves for two observers obtained via the adaptive procedure used for this study (see Methods). The data were fitted using the Pelli (1985) uncertainty model (see 1). As expected from the previous experiment, the target-alone condition produced the highest detection thresholds. Importantly, it was also characterized by the steepest psychometric curves. Thresholds for the other two conditions shown here (circle, and two collinear Gabors positioned at the ends of the target) were much lower, and the slopes were shallower. 

Figure 3

 

Psychometric functions for two observers. Symbols show proportion-correct responses for three conditions: target with no cue (circles), target with a circle cue (squares), and target and two collinear Gabors at the target ends (diamonds). Curves show fits calculated using Pelli's uncertainty model (see 1).

To present the relationship between thresholds and slopes in a more illuminating way, we plotted the observed thresholds α and steepness parameters β (for definition, see Methods) as a correlation plot in Figure 4. For all five observers, the steepness increased as a function of threshold. As in Experiment 1, the two collinear Gabors positioned at the ends of the target produced the strongest facilitation compared to the target-alone condition, but they also produced the largest drop in steepness. Note that for three observers (YP, LM, and AMW), the facilitatory effect of the Gabor pattern was just slightly larger than the effect of the circle cue. The experimental results were fitted with the predictions of the Pelli (1985) uncertainty model. The model is based on signal detection theory and on the assumption of probability summation over multiple noisy detector outputs (see 1). Because the noise level and internal scaling of the contrast signal were unknown, the uncertainty model fits were carried out with one free parameter representing the unknown signal-to-noise factor for each observer. In all cases, the uncertainty model gave a good fit to the observed threshold–steepness correlation. 

Figure 4

 

Psychometric steepness β versus threshold α. Results for the five tested conditions are marked by different symbols. Five observers are marked by different colors. Curves show a one-parameter uncertainty model fit (Pelli, 1985).

Discussion

The results of this study confirm previous findings of collinear facilitation in a contrast detection task. In agreement with Polat and Sagi (1994), we found that collinear Gabors positioned 2.5λ away from the target Gabor lowered detection threshold by almost a factor of two, whereas the same Gabors positioned at the sides of the target produced a smaller degree of facilitation. However, we discovered that a comparable degree of facilitation can be induced by a faint circle, or a set of four faint lines surrounding the target. Such low-energy patterns can cue an observer to the target location or to both its location and orientation, respectively. We also showed that the improved performance is accompanied by a significant decrease in the slope of the psychometric functions both for flanking high-energy Gabors and low-energy cues. 

These two findings along with the fact that substantial collinear facilitation is never observed above detection thresholds, that is, where the target is plainly visible, argue very strongly that collinear facilitation is largely a result of uncertainty reduction brought about by the flanking stimuli. Experiment 1 indicates that it is mainly the uncertainty about the target location that matters. Because we observed no significant difference in facilitation between circle and line cues, we conclude that the effect of two possible target orientations was too small to be detected. We also tested the possible effect of orientation uncertainty on the two observers that showed the least facilitation due to the circle cue alone (SPM and PV). They repeated Experiment 2 with target at one orientation at all times using the circle as the location cue. The results are shown in Figure 4 by shaded squares. Although thresholds decreased slightly for both observers, the effect of eliminating uncertainty about target orientation was close to the experimental noise. 

It is important to recognize how big a change in the number of detectors is required to affect performance. The uncertainty model predicts that a rather large change to this number is required. For example, for the fits shown in Figure 3, the estimated number of detectors changed by two orders of magnitude between cued and noncued conditions (from 3 to 4 to 200 for YP, and from 10 to 3000 for AMW). Because the effect of increasing the possible target orientations from one to two in our experiments corresponds to a mere doubling of the number of relevant detectors, it might have been insufficient to alter performance noticeably. 

One might want to estimate what spatial area is an observer effectively monitoring when the target is presented without any location cues. Suppose that the monitored receptive fields are arranged in a hexagonal lattice that has its center to center spacing equal to half that of the target Gabor wavelength, that is, the minimal spatial sampling required to represent the carrier grating according to the Nyquist theorem. A simple calculation shows that observers could be monitoring up to 100 detectors if they were attending to an area of 0.8 deg. Of course this assumes uncertainty in location only. If we consider uncertainty in orientation and in phase, this could increase the number of detectors in that same area by an order of magnitude. 

Although we used the uncertainty model to fit our data and to explain the observed correlation between psychometric thresholds and slopes, it is possible that not all of the model assumptions are strictly valid for real observers. Thus, the model assumes statistical independence of detectors, which might not be the case, particularly for central vision, where receptive fields are known to be heavily overlaid (Dow, Snyder, Vautin, & Bauer, 1981; Van Essen, Newsome, & Maunsell, 1984). Because of that, we used the uncertainty model not as a quantitative tool, but rather as a qualitative descriptor of the observed behavior in this study. 

We also do not claim that there is no low-level neural input to collinear facilitation. In particular, our results show that although uncertainty reduction explains most of the effect, it does not explain all of it. Two observers (PV, and to a lesser degree SPM) showed much less facilitation from low-energy cues than from the collinear Gabors. However, the remaining facilitation can be easily accommodated by the transducer model of Solomon et al. (1999), which seems to do a good job when the separation between target and flankers is small (<3λ). 

We have assumed that the location cues used in our study had a sufficiently low energy to prevent them from contributing to low-level neuronal interactions either via long-range lateral connections or via the direct stimulation of the properly defined “extended” receptive field as a pedestal mask (see Introduction). Yet, it is possible that the low-energy surround stimuli can significantly modify a cell's response if there is a strong output nonlinearity. The observed change in psychometric slopes argues against this interpretation. It indicates instead that the surround stimuli act as location cues (e.g., via attention allocation). If this were true, then the spacing between the cue and the target would not be crucial, provided that the target was located at the cue's center of symmetry, where its position can be easily estimated. To verify this, Experiment 2 was repeated with a 10λ diameter circle as the location cue. Other circle parameters were the same as before. The results for three observers (YP, PV, and LM) are shown in Figure 4 with open squares. The 10λ circle produced as much facilitation as the 2.5λ circle, which supports the uncertainty explanation. This is reminiscent of the studies of Kovacs, Feher, and Julesz (1998) and Kovacs and Julesz (1994), which showed that contrast sensitivity was enhanced when a Gabor probe was positioned at symmetrical points (medial axis) of the enclosing boundary, even for large distances between the boundary and the target. They interpreted this as evidence for “skeletal” representation of objects, but our results suggest that uncertainty reduction would be a more natural explanation for the effect. 

Because the changes in psychometric slopes are an important part of our argument, it is worth mentioning that other studies have found the same effect. Levi, Klein, and Hariharan (2002) showed that collinear Gabors positioned at the ends of the target reduced the averaged psychometric slopes (expressed as Weibull exponents) from 1.7 to 1.1, whereas parallel Gabors positioned on the sides reduced them to 1.4. This is in good agreement with our data (see Figure 4), which also shows that noncollinear flankers on average induce a smaller change in psychometric slopes (and thresholds) than collinear flankers. Levi et al. also suggested a possible role of uncertainty reduction in collinear facilitation. The same explanation was mentioned earlier by Williams and Hess (1998). More recently, Giorgi, Soong, Woods, and Peli (2004) and also Shani and Sagi (2005) found that it is important to consider uncertainty effects when looking for facilitation in the parafovea. Shani and Sagi noted that psychometric slopes increase with thresholds, just as shown in Figure 4. 

Thus, an uncertainty explanation for collinear facilitation has been considered in the previous studies but it has not gained complete support. Primarily, it has been criticized as being an unsatisfactory explanation for the effects of different flanker configurations. In particular, it was claimed that opposite-phase Gabors produce no facilitation (Williams & Hess, 1998), which seems to be inconsistent with the uncertainty explanation. A later study by Solomon et al. (1999) showed that a weak opposite-phase facilitation exists, albeit at larger target-flanker separations. Note, however, that opposite-phase flankers overlapping with the target summation zone will suppress target detection, as predicted by the Solomon et al. transducer model. Our results for four-line cue show that opposite-phase stimuli (dark lines) can produce a strong facilitation when they have low energy and thus are unlikely to cause any significant degree of overlap suppression, for example, pedestal masking. 

Besides varying the phase of the flankers, Solomon and Morgan (2000) also varied the number of flanking Gabors between two and eight. The eight Gabors were positioned in a wreath formation around the target (for both the same- and opposite-phase flankers). For low flanker contrast (<30%), the wreath decreased detection thresholds to the same degree as the two collinear Gabors. But, at higher contrasts, the extra flankers significantly reduced the facilitatory effect of the collinear Gabors. While the low-contrast result is consistent with flankers reducing uncertainty, the high-contrast result seems to contradict the uncertainty explanation. But, in fact, Solomon and Morgan (2000) showed that the transducer model fits the data well in the high-contrast range. Thus, the inhibitory input of extra flankers is predicted as a result of pedestal masking effects. 

Polat and Sagi (1993) and later Chen and Tyler (2002) showed that flankers with an orientation orthogonal to the target had a much smaller effect on target detection. But how well does such a flanker configuration serve as a location cue? Originally, Keeble and Hess (1998) reported that collinear carrier orientation had no effect on the alignment of Gabor patches. More recently, Popple, Polat, and Bonneh (2001) studied the alignment precision of a 50% contrast Gabor target in the presence of two flanking Gabors. Their results demonstrated that the precision is tightly tuned to the flanker orientation relative to the target: the precision peaked for a collinear orientation but dropped by a factor of three for the orientation differences of 45 deg or more. Also, see Popple and Levi (2002) response to Keeble and Hess. Thus, orthogonal Gabors appear to be much less precise indicators of the target position, which may explain why facilitation is smaller in this case. 

In summary, our study shows that contrast detection thresholds for a Gabor target are strongly reduced in the presence of low-energy location cues, such as faint circles, or lines. The effect is strong enough to account for most of the facilitation usually produced by collinear Gabors. The slopes of psychometric functions change along with thresholds in a manner consistent with the predictions of the uncertainty model. Our results strongly suggest that collinear facilitation is largely due to uncertainty reduction. 

Appendix

The uncertainty model assumes that the observer monitors multiple independent detectors M in each interval. We also assume that a physical value of contrast c is linearly translated into a response kc plus noise, where k refers to the gain of each detector, that is, the proportionality constant between contrast and number of spikes. The gain parameter k is related to the psychometric threshold (i.e., the horizontal position of the psychometric curve) whereas the uncertainty parameter M is related to the steepness of the curve. All M detectors have the same gain k and each detector produces an independent noisy response from a Gaussian distribution. The observer finds the largest of these responses in each interval and then chooses the interval with the larger response. Errors arise when the interval without the increment produces a larger response and the probability of error increases with the number of detectors that the observer monitors. This formulation is based on Pelli's (1985) uncertainty model. For our two-interval forced choice task, the probability of a correct response is given by the probability that the interval with the signal produces the larger response. Alternately, the probability of a correct response is one minus probability that the blank interval produces the larger response: 

$\begin{array}{cc}P(\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t})& =P(\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{>}\mathrm{b}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{k})=1-P(\mathrm{b}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{>}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l})\\ & =1-M\underset{-\infty }{\overset{\infty }{\int }}f\left(x\right)N\left(x{)}^{2M-2}N\right(x-kc)dx\hfill \end{array},$

(A1)

where c is the contrast of the signal, f(x) is Gaussian probability density function, and N(x) is the cumulative function for the Gaussian distribution. 

The blank interval produces the largest response when one of the M detectors in this interval has a value x and all the other detectors have a value less than x. These other detectors include a total of 2M − 2 detectors that see no contrast (the remaining M − 1 from the blank interval, and M − 1 from the signal interval), and the one detector that sees the signal contrast c. The probability that a detector that does not see a signal produces a response x is given by the Gaussian density f(x). The probability that a detector that sees signal produces a response x is f(x − kc), where k is the gain parameter and c is the signal contrast. The probability that all the other detectors will produce a contrast less than x is the product of all their cumulative distributions, that is, N(x)^{2M − 2} N(x − kc), where the first term is due to the 2M − 2 detectors that see no contrast and the second term is from the one detector in the signal interval that sees the signal. The expression is multiplied by M because any one of the M detectors in the nonincrement interval can produce the largest response. The integral calculates the probability of an incorrect response over all values of x. 

Acknowledgments

This research was supported by National Institutes of Health Grants EY-06644 (SPM), NASA grant NAG 9-1461 and NSF grant 0347051 (PV), and a Ruth L. Kirschstein NRSA Fellowship (YP). 

Commercial relationships: none. 

Corresponding author: Yury Petrov. 

Email: [e-mail]. 

Address: Smith-Kettlewell Eye Research Institute, 2318 Fillmore Street, San Francisco, CA 94115.