August 2011
Volume 11, Issue 9
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Article  |   August 2011
Rat performance on visual detection task modeled with divisive normalization and adaptive decision thresholds
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Journal of Vision August 2011, Vol.11, 1. doi:10.1167/11.9.1
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      Philip Meier, Pamela Reinagel; Rat performance on visual detection task modeled with divisive normalization and adaptive decision thresholds. Journal of Vision 2011;11(9):1. doi: 10.1167/11.9.1.

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Abstract

Performance on any perceptual task depends on both the perceptual capacity and the decision strategy of the subject. We provide a model to fit both aspects and apply it to data from rats performing a detection task. When rats must detect a faint visual target, the presence of other nearby stimuli (“flankers”) increases the difficulty of the task. In this study, we consider two specific factors. First, flankers could diminish the sensory response to the target via spatial contrast normalization in early visual processing. Second, rats may treat the sensory signal caused by the flankers as if it belonged to the target. We call this source confusion, which may be sensory, cognitive, or both. We account for contrast normalization and source confusion by fitting model parameters to the likelihood of the observed behavioral data. We test multiple combinations of target and flanker contrasts using a yes/no detection task. Contrast normalization was crucial to explain the rats' flanker-induced detection impairment. By adding a decision variable to the contrast normalization framework, our model provides a new tool to assess differences in visual or cognitive brain function between normal and abnormal rodents.

Introduction
We trained rats to detect a faint visual target, tested them on variety of contrast conditions, and provide a mathematical model for their behavioral performance. The model specifies how decreasing the target's contrast impairs performance, as well as how increasing the contrast of two flanking stimuli (“flankers”) impairs performance. 
During natural vision, multiple stimuli compete for a limited cognitive resource. The detection paradigm determines which stimulus is relevant for behavior. In our task, the central grating patch is always the target, and its presence or absence should determine the subject's action. A perfect subject would ignore the flankers and only be influenced by the contrast of the target. Yet flankers substantially influence rats. Thus, the behavioral impairment caused by the flankers is revealing about the encoding of the visual scene and the decision strategy of the rat. 
Successfully attending to the target alone involves suppressing the influence of the flankers. This is goal-oriented selection. Additionally, natural vision involves the automatic selection of statistically relevant features. It is likely that such stimulus-driven selection of features exploits spatial contrast normalization. It has been argued that attention and contrast normalization rely on common mechanisms (Reynolds & Heeger, 2009), such as lateral processing via inhibitory subnetworks. Therefore, we are interested in developing a unified model accounting for both. Many psychophysical methods emphasize the priority of finding the minimum contrast that is required to perform correctly on a fixed fraction of trials. Contrast thresholds are useful because they summarize a subject's aptitude with a single number and are presumably more robust to variability in subjects' strategies, such as idiosyncratic response biases. Often, determining the threshold involves measuring d′ that invokes signal detection theory to disambiguate the effects of bias and aptitude. Unfortunately, both d′ and contrast thresholds create a layer of assumptions and abstractions between a conceptual model that explains the data and the likelihood that the model gives rise to the data. Instead, we fit a mathematical model to the subjects' raw data of hits and false alarms. We treat hits and false alarms for every stimulus condition as independent binomial proportions. We then maximize the likelihood that a given model could give rise to the observed data. 
In order to explicitly model changes in bias caused by the flankers, we used a yes/no detection task. This differs from the more commonly used forced choice paradigm in which the subject chooses the stimulus at one of two spatial locations (2AFC) or one of two temporal intervals (2IFC). In these paradigms, both the target (T+) and non-target (T−) stimulus are provided on every trial. Presumably, subjects performing 2AFC and 2IFC make comparisons between two representations in short-term memory. We use a yes/no detection task that requires subjects to compare a single representation to an internal referent. As a result, we can independently measure a subject's responses to when the target is present (T+, hit rate) and when it is absent (T−, false alarm rate). We note that yes/no detection is similar but distinct from a go–no-go paradigm. Both are well suited for detection tasks. Our yes/no task has symmetric motor responses: a response port on the left or right side of the request port. On the other hand, go–no-go has an inherent asymmetry in withholding and engaging in an action. 
Using the framework of signal detection theory, we consider various models that differ in decision criteria and stimulus processing. Our strategy is to find a single class of models that quantitatively fits the results with a small number of parameters. To adequately constrain the fit, we collected behavioral data for four target contrasts and five flanker contrasts (20 randomly interleaved conditions). 
Below, we begin by presenting the raw data from a single subject and the standard measure of d′ and criterion for each stimulus condition independently. Then, we sequentially add features to the model, explaining why parameters were added. The best model incorporates divisive normalization and an adaptive decision criterion. Notably, subjects with very different responses can be fit by explicitly modeling the subjects' bias. Alternative inferior models are briefly considered. Finally, we consider the parameters of the model in order to explain why flankers impair detection in rats. 
Methods
Animal training
Data were collected from two male Long Evans rats (Harlan Laboratories). All experiments were conducted under the supervision and with the approval of the Institutional Animal Care and Use Committee at the University of California San Diego. All subjects that were trained on these stimuli were included in the study; no subjects were excluded. Training methods were the same as reported previously (Meier, Flister, & Reinagel, 2011). Each subject performed a 90-min session every weekday for a total of 95 sessions. During the testing phase, subjects performed about 30,000 valid trials. 
Task
Rats initiated each trial by licking a port in the center of the behavioral training chamber. The rats' request triggered a stimulus to appear in the middle of a CRT display for 200 ms. A target was present on 50% of the trials. Flankers were present on every trial, except for stimulus conditions where flanker contrast was set to zero. Subjects were required to select one of two response ports on the left- or right-hand side of the chamber to indicate that the target was either present or absent. The mapping between yes/no and left/right was opposite for the two subjects and constant for their entire life. For correct responses, rats were rewarded with a small amount of water, and for incorrect responses, rats had to wait 6–10 s before the next trial. 
A tone accompanied each trial request, providing the subject with confirmation that he had successfully initiated a trial and should proceed to ascertain and report target presence. During the time-out penalty after an incorrect response, a different tone and a flickering screen indicated to the rat that the system was non-responsive. 
Stimuli were presented in blocks of 150 trials. Within a block, the flanker contrast C F was constant and the target contrast was either zero or a fixed value C T. This study used four target contrasts and five flanker contrasts, for a total of twenty conditions. The stimulus with the target present is shown for all twenty conditions (Figure 1c). Each block was presented once in a random order before repeating the blocks in a new random order. Subjects completed roughly 3–4 blocks per day. 
Figure 1
 
Experimental design for yes/no 2AFC target detection. (a) A rat in the training chamber viewing a stimulus. The rat's task is to report the presence or absence of the central grating (the “target”). The rat is forced to choose either “yes” or “no” before proceeding to the next trial. The top and bottom gratings (“flankers”) contain no information about the correct response. In (a), the target is present in the stimulus, and thus, a response to the right port would be rewarded with a drop of water (a “hit”). A response to the left port would be punished with a time-out period and an aversive tone (a “miss”). For photographic purposes, the stimulus was left on indefinitely, but during the task, stimuli persisted for 200 ms. (b) The target was present on 50% of the trials (T+) and absent on other trials (T−). The rat was appropriately rewarded and punished for correct rejections and false alarms, just like hits and misses. (c) In the testing period, rats performed a block of 150 consecutive trials during which the target contrast and flanker contrast were held constant. Every 20 blocks visited each combination of target contrast (C T = [0.25, 0.5, 0.75, 1]) and flanker contrast (C F = [0, 0.25, 0.5, 0.75, 1]), in a random order. The probability of a target being absent (C T = 0) was always 50%, and thus, we refer to the C T of a block as the contrast that a target would be if it were present on a trial.
Figure 1
 
Experimental design for yes/no 2AFC target detection. (a) A rat in the training chamber viewing a stimulus. The rat's task is to report the presence or absence of the central grating (the “target”). The rat is forced to choose either “yes” or “no” before proceeding to the next trial. The top and bottom gratings (“flankers”) contain no information about the correct response. In (a), the target is present in the stimulus, and thus, a response to the right port would be rewarded with a drop of water (a “hit”). A response to the left port would be punished with a time-out period and an aversive tone (a “miss”). For photographic purposes, the stimulus was left on indefinitely, but during the task, stimuli persisted for 200 ms. (b) The target was present on 50% of the trials (T+) and absent on other trials (T−). The rat was appropriately rewarded and punished for correct rejections and false alarms, just like hits and misses. (c) In the testing period, rats performed a block of 150 consecutive trials during which the target contrast and flanker contrast were held constant. Every 20 blocks visited each combination of target contrast (C T = [0.25, 0.5, 0.75, 1]) and flanker contrast (C F = [0, 0.25, 0.5, 0.75, 1]), in a random order. The probability of a target being absent (C T = 0) was always 50%, and thus, we refer to the C T of a block as the contrast that a target would be if it were present on a trial.
Correction trials were used during initial training. After making a mistake, a correction trial began with 50% probability. During a correction trial, a computer algorithm would present a stimulus in which the correct response was the opposite of the subject's last response. This prevented the rat from perseverating on a single port at chance reward rates. The correction trial persisted until the subject made a correct response. Correction trials were not present during the testing phase. Data were not analyzed from trials immediately following errors, allowing for the possibility that the rats were biased to switch sides after an error. This also served to avoid short-term effects of contrast normalization to the flickering visual signal presented during the time-out period (Gaudry & Reinagel, 2007). 
We increased rewards for consecutive correct answers. The first correct response after an error yielded an 80-ms valve opening (approximately 16 μl). The 2nd to 4th consecutive correct responses earned 100-, 150-, and 250-ms rewards. Consecutive responses thereafter earned 250-ms rewards; the first incorrect response reset this schedule to the beginning value of 80 ms. Using ramped rewards discourages guessing strategies. Without a ramp, a rat performing at 75% correct only earns 50% more rewards than guessing randomly. With the ramp imposed, the same performance yields more than twice the rewards expected by chance guessing. 
Stimuli
The target stimulus was an oriented square-wave grating with a spatial frequency of 0.22 cyc/deg, presented in a Gaussian mask. The standard deviation of the Gaussian mask was 6.8°. Subjectively, this left about 3 visible periods in the grating. Previous experiments (N = 2 subjects, not shown) confirmed that detection is contrast limited between contrasts of 0.25 and 1.0 at this spatial frequency, even for larger grating patches. The pixel pattern of each flanker was the same as the target. The two flanking stimuli were on the axis defined by the stimulus orientation, at a distance of 3λ, where λ (=4.5°) is the spatial scale determined by the spatial frequency of the grating. Flankers did not spatially overlap the target and subjectively appeared separated from the target. In order to be consistent with their previous training stimuli, Subject 1 viewed orientations tilted 15°, and Subject 2 viewed gratings tilted 22.5°. We do not think that this small difference particularly mattered. Target and flanker stimuli were always collinear. When viewing the screen from the request port, a rat's eye is roughly 10 cm from the monitor, 10 cm below its center. At this position, the center of the target grating is roughly 14 cm away. No effort was made to invert perspective of the non-tangent display screen, and thus, the orientation and spatial frequency of the three stimulus patches varied slightly due to perspective. 
To present visual stimuli, we used PsychToolbox (Kleiner, Brainard, & Pelli, 2007) to control standard OpenGL capable graphics cards (Nvidia GeForce 7600) via Matlab (Mathworks, Natick, MA). Stimuli were presented on a CRT monitor (NEC FE992, 19, 100 Hz, 1024 × 768 resolution) with linearized luminance output. A linearization table was created for the monitor by fitting a power law with gain and offset (y = b * x γ + m) to photodiode measurements (Thorlabs, PDA55) of a rectangular patch in the center of the screen. The minimum, mean, and maximum luminance were set to 4, 42, and 80 cd/m2, respectively (Colorvision, spyder2express). 
During initial training, rats viewed stimuli with geometric configurations such that the target orientation, flanker orientation, and angular position were randomly and independently chosen to be either clockwise or counterclockwise by a fixed amount. During testing on the task reported in this paper, contrast was varied and geometric configurations were held constant to minimize the number of independent stimuli tested. Target orientation, flanker orientation, and angular position remained constant at the same counterclockwise angle. Thus, all stimuli had a collinear configuration. 
Model fitting
Models were created with a small number of parameters (4–7) that determined the noise distribution, the signal distribution, and the decision criterion for each stimulus condition. Optimal parameters were selected by maximizing the likelihood that the model could have produced the data. If the experimental data are independent observations of a hit rate and a false alarm rate (Ogilvie & Creelman, 1968), then the likelihood is the product of binomial processes. For this experiment with 20 stimulus conditions, the likelihood can be calculated as the product of 40 binomial processes: 
p ( d a t a | θ ) = i = 1 20 ( H i + M i H i ) h i H i ( 1 h i ) M i i = 1 20 ( F i + C i F i ) f i F i ( 1 f i ) C i ,
(1)
where H, M, F, and C are integer counts of the number of hits, misses, false alarms, and correct rejections, and h and f are the rates predicted by the model. All counts and rates are indexed by the stimulus condition i, of which there were twenty in this study. Minimizing the negative log likelihood is numerically more stable than maximizing the likelihood. Additionally, we will isolate the model-independent factors into a constant a that is defined purely by the data and need not be recalculated in the optimization procedure. The log likelihood is then basically a sum of products: 
L L = a + i = 1 20 ( H i log ( h i ) + M i log ( 1 h i ) + F i log ( f i ) + C i log ( 1 f i ) ) .
(2)
 
Each model was fit by choosing the best of 10–40 random starts of the Nelder–Mead simplex method (fminsearch in MATLAB) with a cost function proportional to the negative log likelihood. As some models have more parameters than others, we provide the Bayesian information criterion (BIC) as a measure of goodness of fit (Schwarz, 1978): 
B I C = L L + k 2 log ( n ) .
(3)
 
The BIC is equal to twice the negative log likelihood plus a penalty for the number of parameters k in the model. Each parameter is penalized by log(n), where n is the number of independent observations, equal to the number of trials performed by the rat. For our data set (n ∼ 30,000), each additional parameter incurred a penalty of ∼10 units of a natural logarithm (nats). If a model produced a perfect fit to our data and had no parameters, the BIC would equal the negative likelihood, which is about 137 nats. Our best model has a fit around 238 nats, as compared to inferior explanatory models that have fits over 500 nats. 
The variability of the fitting procedure is characterized by refitting the model 50 times to data that are sampled from the posterior of the model (Goris, Zaenen, & Wagemans, 2008). Assuming the best fit model is a true model of the data, this provides the variability in the parameters expected from limited data and the fitting procedure. 
In the Results section, models will be increased in complexity by adding more parameters. The final model contains seven parameters that were fit; they are summarized in Table 1 at the end of the results. Some parameters were constrained to a limited range by passing the fit value through a sigmoid: α, λ ∈ [0 1]; or an exponential function: k T, k F, γ, c 50 ∈ [0 ∞]. The bias parameter is defined as a log ratio and, thus, was not constrained, b
R
Table 1
 
Summary of the final model. There are seven free parameters that are fit from 40 data points.
Table 1
 
Summary of the final model. There are seven free parameters that are fit from 40 data points.
Stage 1: Effective contrast
γ Acceleration of the contrast response non-linearity
c 50 Contrast at 50% response saturation
λ Space constant of normalization pool
→These determine effective contrasts of target and flanker, C T′ and C F′ (Equations 8 and 9).

Stage 2: Decision variable
k T Weight of target contrast C T′ input to decision variable
k F Weight of flanker contrast C F′ input to decision variable
→These determine the means of the signal and noise distributions, μ S and μ N (Equations 10 and 11).

Stage 3: Decision threshold
b Response bias of subject→with μ S and μ N, determines ζ bias
α Relative weight of global vs. local threshold
→These determine the 20 condition-specific decision thresholds, ζ (Equation 13).
Results
Target contrast improves detection
We begin by considering the detection of a faint target when no flankers are present. If a target is so faint that it provides no information (is not visible), the rat must behave the same whether or not the target was there. In this case, the hit rate and the false alarm rate would be equal (diagonal line in Figure 2a). The rat is above chance performance for even the weakest contrast stimulus presented in this study. We find that increasing the target's contrast increases the rat's hit rate and decreases the rat's false alarm rate (Figure 2a). 
Figure 2
 
Higher target contrasts increase detection performance and bias rats to say yes. (a) The probability that a rat responds “yes” given that the target was absent (false alarm rate) or present (hit rate). If the data fall on the diagonal line, the animal's response contains no information about the target. Perfect performance is in the upper left. The four data points indicate performance from blocks where the flanker was absent and the target varied in contrast. The horizontal and vertical components of each plus symbol indicate the 95% confidence interval of a binomial proportion, for the false alarms and hit proportions, respectively. (b) A representation of the medium-contrast target condition, assuming that a rat uses a single decision variable that has a Gaussian distribution with equal variance for the noise (T−, black) and signal (T+, blue). The common equal-variance Gaussian assumption will fail to fit our later models, but we present it here to explain the framework of signal detection theory. If a decision variable is greater than threshold criterion (gray line), then the model produces a “yes” response. The means of the distributions are separated by 1.3σ. Thus, the measure d′ = 1.3. (c) For a higher contrast (C T = 1.0), the distributions overlap less, and the target is easier to detect (d′ = 2.2). Notice that the threshold criterion is greater (gray line shifted right), resulting in fewer false alarms. (d) Higher contrast targets have a larger d′. Each data point represents an estimate from a single block of trials. The vertical bars cover ±1 SD. The variability of data is larger than expected from limited sampling; other factors beyond the stimulus also affect performance. (e) The average bias criterion is greater than zero, indicating that this rat favored no responses on all of these conditions. The subject's mild bias for no responses was reduced as the target contrast increased.
Figure 2
 
Higher target contrasts increase detection performance and bias rats to say yes. (a) The probability that a rat responds “yes” given that the target was absent (false alarm rate) or present (hit rate). If the data fall on the diagonal line, the animal's response contains no information about the target. Perfect performance is in the upper left. The four data points indicate performance from blocks where the flanker was absent and the target varied in contrast. The horizontal and vertical components of each plus symbol indicate the 95% confidence interval of a binomial proportion, for the false alarms and hit proportions, respectively. (b) A representation of the medium-contrast target condition, assuming that a rat uses a single decision variable that has a Gaussian distribution with equal variance for the noise (T−, black) and signal (T+, blue). The common equal-variance Gaussian assumption will fail to fit our later models, but we present it here to explain the framework of signal detection theory. If a decision variable is greater than threshold criterion (gray line), then the model produces a “yes” response. The means of the distributions are separated by 1.3σ. Thus, the measure d′ = 1.3. (c) For a higher contrast (C T = 1.0), the distributions overlap less, and the target is easier to detect (d′ = 2.2). Notice that the threshold criterion is greater (gray line shifted right), resulting in fewer false alarms. (d) Higher contrast targets have a larger d′. Each data point represents an estimate from a single block of trials. The vertical bars cover ±1 SD. The variability of data is larger than expected from limited sampling; other factors beyond the stimulus also affect performance. (e) The average bias criterion is greater than zero, indicating that this rat favored no responses on all of these conditions. The subject's mild bias for no responses was reduced as the target contrast increased.
Even when the target is not present, the visual system still produces a noise response and the organism must internally process it to make a decision. On a given trial, an internal decision variable that summarizes the sensory evidence could take on a range of possible values. We adopt the framework of signal detection theory to mathematically model this decision variable (Green & Swets, 1966). We represent its possible values with a Gaussian distribution (see Discussion section for alternatives). The rat shown in Figure 1a was tested on four target contrasts (Figure 2a). We illustrate hypothetical underlying signal and noise distributions for a high-contrast condition (Figure 2b, C T = 1) and a medium-contrast condition (Figure 2c, C T = 0.5). 
We refer to the distribution caused by a blank screen (T−) as the noise (black curves, Figures 2b and 2c) and the response to a target (T+) as the signal (blue curves, Figures 2b and 2c). The signal distribution increases with the target's contrast, which makes sense because there is increased sensory evidence that it was present. The metric d′ measures the separation between the noise and signal, and it increases with contrast (Figure 2d). Notice that the noise distribution is the same regardless of the condition; it is always caused by a blank screen. All models in this paper assume that the sensory representation of a blank screen is not different by virtue of its being randomly interleaved with targets of higher contrast. We observe that the probability that a rat chooses yes when the target is absent (the false alarm rate) is dependent on the contrast of the target in that condition (Agresti–Caffo test between false alarms of C T = 0.25 and C T = 1.0, p < 10−4 for both subjects). How could the false alarm rate change if the stimulus is the same? Rats can adjust their decision threshold. For 150 consecutive trials, the rat views trials of the same condition. For blocks with low-contrast targets, the subject reduces his threshold for making a yes decision (Figure 2b; gray line is shifted left compared to Figure 2c). 
In Figure 2e, we present the bias as measured on each individual stimulus. This common measure of bias criterion is equal to the average of the normal transform of hit rate and false alarm rate: −[z(h) + z(f)] / 2. This value is zero when the subject chooses “yes” and “no” with equal probability; zero corresponds to a threshold at the midpoint of two equal-variance Gaussians. The positive criterion bias (Figure 2e) indicates that this rat slightly favors no responses. This bias is reduced when the target is higher contrast. Note that the “decision threshold” increases with high contrast (Figure 2c; gray line moves right), while the “criterion bias” decreases (gray bar is closer to the intersection of the distributions). We clarify this point because some explanations of decision theory conflate these terms by translating the x-axis of the decision variable so that the terms are equal. We avoid this translation, because it removes information about the means of the distributions. In all graphical depictions, we will preserve the raw means of the distributions because in some models the absolute decision thresholds interact between different stimulus conditions. 
Flanker contrast impairs detection
The presence of flankers makes the detection task difficult for the rats (Meier et al., 2011). Flankers decrease performance both by decreasing the hit rate and by increasing the false alarm rate (Figure 3a). To facilitate comparison, a condition with no flanker contrast is replotted from Figure 2 (blue symbol in Figure 3a, all of Figure 3b). Despite the high-contrast target in all conditions, the separation between signal and noise is poor when then flanker contrast is high (Figure 3c). There are reasons why the noise distribution might change in the presence of flankers. In this model, however, we apply the approximation that the noise distribution is constrained to be a normal Gaussian with zero mean and standard deviation of one. 
Figure 3
 
Higher flanker contrasts decrease detection performance and bias rats to say yes. (a) A scatter plot of the false alarms and hits for five stimulus conditions, varying in flanker contrast. Symbols are 95% confidence interval of a binomial proportion, as in Figure 2a. All stimuli have a target contrast of 1. The blue symbol indicates zero flanker contrast; increasing flanker contrast is indicated by redness. (b) The discriminability of the target when there is no flanker present. (c) The discriminability when the flanker contrast is increased. The decision criteria are represented by a vertical gray line and it fit to exactly match the data with no error. Note that we plot the means of the noise distribution at zero. In subsequent models, the means of the signal and noise distributions may both be non-zero. Notably, the flanker contrast can make the discriminability go down by increasing noise distribution, even if the signal distribution were unchanged. (d) The discriminability decreases with flanker contrast. (e) The subjects bias criterion shifts with flanker contrast. For consistency with the past literature, we present the criterion here as (z(cr) + z(fa)) / 2. The change in the parameter reflects the rat's increasing bias to say yes as flanker contrast increases, as also evidenced by the raw data in (a).
Figure 3
 
Higher flanker contrasts decrease detection performance and bias rats to say yes. (a) A scatter plot of the false alarms and hits for five stimulus conditions, varying in flanker contrast. Symbols are 95% confidence interval of a binomial proportion, as in Figure 2a. All stimuli have a target contrast of 1. The blue symbol indicates zero flanker contrast; increasing flanker contrast is indicated by redness. (b) The discriminability of the target when there is no flanker present. (c) The discriminability when the flanker contrast is increased. The decision criteria are represented by a vertical gray line and it fit to exactly match the data with no error. Note that we plot the means of the noise distribution at zero. In subsequent models, the means of the signal and noise distributions may both be non-zero. Notably, the flanker contrast can make the discriminability go down by increasing noise distribution, even if the signal distribution were unchanged. (d) The discriminability decreases with flanker contrast. (e) The subjects bias criterion shifts with flanker contrast. For consistency with the past literature, we present the criterion here as (z(cr) + z(fa)) / 2. The change in the parameter reflects the rat's increasing bias to say yes as flanker contrast increases, as also evidenced by the raw data in (a).
As flanker contrast increases, detection sensitivity d′ decreases (Figure 3d). Flankers bias the rat to report yes more often (Figure 3e). Measuring each stimulus condition separately, the bias criterion becomes more negative with flanker contrast. The impairment and the bias are consistent with theory that rats are confused about the source of the perceived contrast, causing accidental responses to the flankers alone. The reduction in sensitivity is also consistent with the theory that spatial contrast normalization decreases the effective strength of the target, rendering it hard to detect. However, the impact of spatial contrast normalization is more difficult to intuit. To isolate these components, we fit a model to data from a range of conditions that independently vary target contrast and flanker contrast. 
Humans that detect targets sometimes improve their sensitivity when the oriented target is presented between collinear flankers, compared to without flankers (Chen & Tyler, 2008; Polat & Sagi, 2007). This increase in performance seems to be dependent on the target being low contrast. Here, we ask, will low-contrast targets also be easier for rats to detect when they are flanked by collinear visual features? The hit rates and false alarms are presented for twenty conditions, including the previously described target-only conditions (blue, Figure 2) and variable flanker contrast data with a high-contrast target (red, Figure 3). The additional observations for the remaining conditions (gray, Figure 1c) follow the same trends (Figure 4a): increasing flanker contrast always impoverishes performance (Figures 4b and 4c), and it always biases the rats to report yes more often (Figures 4d and 4e). Thus, regardless of the target contrast, collinear flankers do not facilitate detection in rats. 
Figure 4
 
The impairment and bias caused by flankers are present for all target contrasts. (a) The false alarm rate vs. the hit rate for twenty conditions including all combinations of four target contrasts (C T = [0.25, 0.5, 0.75. 1]) and five flanker contrasts (C F = [0, 0.25, 0.5, 0.75, 1]). Error bars are binomial confidence intervals. (b) Increasing flanker contrast impairs performance. Lines connect stimuli with constant target contrast. Error bars are ±1 SEM from n = 14–20 measurements. (c) Same as (b) but for Subject 2. (d) Increasing flanker contrast increases the probability to say yes, which is a decrease in bias criterion. Lines connect stimuli with constant target contrast. (e) Same as (d) but for Subject 2.
Figure 4
 
The impairment and bias caused by flankers are present for all target contrasts. (a) The false alarm rate vs. the hit rate for twenty conditions including all combinations of four target contrasts (C T = [0.25, 0.5, 0.75. 1]) and five flanker contrasts (C F = [0, 0.25, 0.5, 0.75, 1]). Error bars are binomial confidence intervals. (b) Increasing flanker contrast impairs performance. Lines connect stimuli with constant target contrast. Error bars are ±1 SEM from n = 14–20 measurements. (c) Same as (b) but for Subject 2. (d) Increasing flanker contrast increases the probability to say yes, which is a decrease in bias criterion. Lines connect stimuli with constant target contrast. (e) Same as (d) but for Subject 2.
At higher target contrasts, each increment in flanker contrast mildly reduces the hit rate and substantially increases the false alarm rate. Interestingly, for low-contrast targets, the hit rate increases with the flanker contrast. However, the rise in false alarms is still greater than the rise in hit rate, reflecting an overall loss of sensitivity (Figures 4b and 4c). 
Fitting a model to the data
It is common for perceptual models to quantify the relationship between contrast and performance by accounting for the d′ measurements and ignoring the decision criteria. This is sensible when the criteria vary between subjects and a given subject has a fixed criterion for all stimuli. Our data would be poorly fit by such a model because the criteria shift between the stimulus conditions. That is, for a given stimulus condition, the model would not be able to predict the observed hit rate and false alarm rate. 
Our approach is to find a model with a small number of parameters to fit the raw data on all conditions. For each subject, there are twenty stimulus conditions, each with a hit rate and false alarm rate, totaling 40 independent estimates of a binomial proportion per subject. In Figure 4, these values were transformed into 20 estimates of d′ and estimates of 20 criteria. This is guaranteed to “fit” the data exactly and represents no savings in parameters. To find a compact representation of the data, we begin with a very simple model of four parameters and incrementally increase its complexity, explaining the parameters as we include them. The goal of the model is to predict the signal (S) and noise (N) distributions and the decision threshold (ζ) for each stimulus condition. This level of description requires at least three parameters per stimulus condition, and so it is important to compute these parameters from a simpler set of rules. 
In our models, we assume that the internal noise is stimulus independent and Gaussian; thus, the standard deviation of each distribution is fixed at 1. The means of the distributions are calculated from a linear fit to the effective contrast, which employs a power law to characterize an accelerating non-linearity. Thus, for each stimulus condition i, we obtain 
μ S [ i ] = k T ( C T [ i ] ) γ + 2 k F ( C F [ i ] ) γ ,
(4)
 
μ N [ i ] = 2 k F ( C F [ i ] ) γ .
(5)
 
The parameters k T and k F are coefficients that determine the contribution of the target contrast and the flanker contrast to the decision variable. The coefficient for the flanker k F is multiplied by two because there are two flankers. A subject ideally suited for this task would have a large k T and a k F of zero. Consequently, when relying on the state of the decision variable, the subject would have no confusion about whether the source of the signal was the target contrast or the flanker contrast. The ratio of k T/k F indicates how well the subject is attending to the target. Large values correspond to good selectivity of spatial attention. Values near 1 indicate that the rat is confused about the source of the contrast. If the ratio is 1, then all contrasts are treated equally by the rat, and it is not preferentially attending to the target location. Notice that the means of the signal distribution (μ S) and noise distribution (μ N) in fact use the same stimulus-generating equation. The noise distribution is defined as the case where the target is not present (C T = 0), in which case Equation 4 reduces to Equation 5
What remains is a specification of the decision threshold. To begin, we consider the case where the model can only select a single decision threshold to account for the subject's performance on all stimulus conditions. The prediction of the hit rate and false alarms for all 40 observations is summarized by drawing iso-contrast curves (Figure 5a). The intersections of the lines correspond to the model's prediction of the data. The model fit captures the gross topology: Target contrast increases performance and flanker contrast biases. However, the model fails in two respects. First, flanker contrast does not impair the model's performance: Shifting the signal and noise distributions by a constant amount will not change d′. Second, the model predicts that changing the target contrast will not change the false alarm rate. This is graphically apparent by the presence of vertical lines in the iso-contrast curves (Figure 5a). Since rats do change their false alarm rate, even when the flanker is absent (Figure 2a), we know that the class of models with fixed threshold will be a poor approximation. 
Figure 5
 
Various decision criteria. (a) The best fit model (k T, k F, γ, α = 0) assuming that the subject only chooses a single fixed decision criterion for all stimuli (α = 0, see Equation 7). The blue vertical line indicates the results of a pure change in target contrast. Other black lines indicate the influence of target contrast when the flanker contrast is higher. The red curve represents a change in the flanker contrast for a target contrast of 1. Other black curves represent pure changes in flanker contrast if the target contrast is lower. If the data perfectly fit the model, the intersections of the lines would match the observed data. Blue crosses from Figure 1a would fall on the blue line, and red crosses from Figure 2b would fall on the red line. Gray crosses indicate all possible combinations of the target and flanker contrasts (see Figure 1c) and should be located at the intersection of the black lines. The gray contour indicates d′ for each model's best fit to the high-contrast condition; the curve spans all possible decision criterion thresholds. The signal and noise distributions of the model are displayed for three representative stimuli: (b) a low-contrast target alone, (c) a high-contrast target alone, and (d) a high-contrast target with a high-contrast flanker. (e) The best fit model (k T, k F, γ, α = 1) if the subject chooses the optimal decision threshold for each stimulus condition (α = 1). Notice that allowing for the optimal choice for any symmetric distribution results in data that fall on a single diagonal line with a slope of −1 that extends from pure chance behavior to perfect performance. The dots along the line correspond to the model's prediction of performance for the four target contrasts. Each dot represents five overlapping conditions because the model predicts no effect of flankers. However, the data from the rat do not fall on a line; they are spread over a plane. This model is clearly wrong. The poor fit is reflected in the substantial rise in the Bayesian information criterion (BIC). The decision criteria are presented in (f)–(h) using the same stimulus conditions as before. Note that the entire model is refit, and so the signal and noise distributions may vary slightly as well. (i) The best fit model (k T, k F, γ, α) assuming that the subject's decision criterion is the weighted average between a single criterion (a = 0) and the optimal criterion for that stimulus condition (α = 1). The best relative weight (α) is fit to the model. The value of α = 0.79 indicates that the decision criterion is close to the optimal but ∼20% influenced by a global criterion that is modeled as the average of criterion across all conditions.
Figure 5
 
Various decision criteria. (a) The best fit model (k T, k F, γ, α = 0) assuming that the subject only chooses a single fixed decision criterion for all stimuli (α = 0, see Equation 7). The blue vertical line indicates the results of a pure change in target contrast. Other black lines indicate the influence of target contrast when the flanker contrast is higher. The red curve represents a change in the flanker contrast for a target contrast of 1. Other black curves represent pure changes in flanker contrast if the target contrast is lower. If the data perfectly fit the model, the intersections of the lines would match the observed data. Blue crosses from Figure 1a would fall on the blue line, and red crosses from Figure 2b would fall on the red line. Gray crosses indicate all possible combinations of the target and flanker contrasts (see Figure 1c) and should be located at the intersection of the black lines. The gray contour indicates d′ for each model's best fit to the high-contrast condition; the curve spans all possible decision criterion thresholds. The signal and noise distributions of the model are displayed for three representative stimuli: (b) a low-contrast target alone, (c) a high-contrast target alone, and (d) a high-contrast target with a high-contrast flanker. (e) The best fit model (k T, k F, γ, α = 1) if the subject chooses the optimal decision threshold for each stimulus condition (α = 1). Notice that allowing for the optimal choice for any symmetric distribution results in data that fall on a single diagonal line with a slope of −1 that extends from pure chance behavior to perfect performance. The dots along the line correspond to the model's prediction of performance for the four target contrasts. Each dot represents five overlapping conditions because the model predicts no effect of flankers. However, the data from the rat do not fall on a line; they are spread over a plane. This model is clearly wrong. The poor fit is reflected in the substantial rise in the Bayesian information criterion (BIC). The decision criteria are presented in (f)–(h) using the same stimulus conditions as before. Note that the entire model is refit, and so the signal and noise distributions may vary slightly as well. (i) The best fit model (k T, k F, γ, α) assuming that the subject's decision criterion is the weighted average between a single criterion (a = 0) and the optimal criterion for that stimulus condition (α = 1). The best relative weight (α) is fit to the model. The value of α = 0.79 indicates that the decision criterion is close to the optimal but ∼20% influenced by a global criterion that is modeled as the average of criterion across all conditions.
It is natural to wonder if a dynamic threshold will predict the data if the rat always chooses the threshold that maximizes its rewards in the current block of trials. This model involves even one less parameter, because the decision threshold is determined by the distributions (ζ = ζ opt, Equation 6). The results of this model are degenerate: all the outputs lie on a single line (Figure 5e). Later in this paper, we consider the addition of non-linear terms and a global bias. Even these parameters will not change the one-dimensional organization of the iso-contrast curves; at most, these parameters serve to curve the line. In no case does the topology of the model prediction span a plane as we see in the raw data. Thus, the hypothesis that rats know the signal and noise distributions and choose the optimal threshold for each condition also fails to account for the observed data. 
What decision thresholds could explain the data in this framework? It seems possible that rats strive for the optimal threshold but do not confidently know what it is. If this were the case, the rat's choice of threshold could be explained by the combination of his prior for the optimal threshold with some weak evidence of the current optimal threshold. We speculate that this model could be formulated as the weighted combination of two thresholds: ζ prior and ζ opt. We allow ζ opt to be defined by the signal and noise distributions and take ζ prior as the expected value of ζ opt averaged over all conditions. The relative weight between the two hypotheses is determined by the parameter α that ranges from 0 to 1 and is fit empirically from the data. Thus, for each stimulus condition i, we obtain a threshold ζ[i] as a weighted sum of the optimum threshold and a global prior: 
ζ o p t [ i ] = μ S [ i ] + μ N [ i ] 2 ,
(6)
 
ζ [ i ] = α ζ o p t [ i ] + ( 1 α ) ζ p r i o r .
(7)
 
When the model includes a weighted hypothesis, it fits the data better (Figure 5i). The Bayesian information criterion (BIC) is substantially reduced (see Methods section for the interpretation of BIC). Notably, the model is improved because it can produce diagonal lines for the curves of constant flanker contrast, even when flanker contrast is zero (Figure 5i, blue line). However, as with the constant threshold model (Figure 5a), the influence of a flanker still does not impoverish performance as measured by d′. The reason is that the only influence of flanker contrast is linearly additive to the mean of the decision variable. Next, we incorporate the non-linearity of spatial contrast normalization, which can be described by a Naka–Rushton equation. This will cause flankers to also influence the decision variable indirectly by reducing the effective contrast of the target. Rather than using a simple power law, we use the following equations to calculate the effective contrast of the target (C T′) and the flanker (C F′), for each stimulus condition i: 
C T [ i ] = ( C T [ i ] ) γ C 50 γ + ( C T [ i ] + 2 λ C F [ i ] ) γ ,
(8)
 
C F [ i ] = ( C F [ i ] ) γ C 50 γ + ( C F [ i ] + λ C T [ i ] + λ 2 C F [ i ] ) γ .
(9)
 
The two terms in the denominator include the semi-saturation point (c 50) and the normalization pool. In other models, the normalization pool is sometimes taken to be the sum of the activity in a local neighborhood (Foley, 1994; Heeger, 1992; Tolhurst & Heeger, 1997). Here, we estimate this activity by taking the weighted sum of the target and flanker contrasts. The parameter λ represents the decrease in contribution to the normalization pool at the spatial separation between target and flanker that was used in the experiments. In Equation 8, the normalization pool includes the term 2λC F because there are two flankers, and each one contributes λC F. In Equation 9, the flankers' effective contrast is only influenced by one adjacent stimulus, the target, which is captured by the term λC T. Contrast normalization is a locally weighted phenomena, and so the other flanker, which is farther away, should not have the same influence. Because the flanker is twice as far away, we square the coefficient; this operation assumes an exponential decay of influence. Because the value of λ is always less than 1, the squared term will always have less of an effect. In pilot tests, we removed the influence of one flanker on the other, and the models were not qualitatively different. We kept the term because the simplicity of treating all contrast features equally (all stimuli in the display had the potential to contribute to the normalization pool) seemed more important than the parsimony of simply removing a term (insisting that the far flanker could not contribute to the normalization pool). 
When applying the divisive normalization, the calculation of the means of the signal and noise distributions remains the same. This is the same as Equations 4 and 5 except that the effective contrasts are now calculated with Equations 8 and 9: 
μ S [ i ] = k T C T [ i ] + 2 k F C F [ i ] ,
(10)
 
μ N [ i ] = 2 k F C F [ i ] .
(11)
 
The best model fit with divisive normalization (k T, k F, γ, α, c 50, λ) has a spatial falloff of 0.33. This means that each flanker contributes about a third as much as the target does to the target's normalization pool. The effective contrast of the target is reduced by the presence of the flanker contrast (Figure 6a). A weaker contribution from the surrounding flankers would have less of an effect on the target's contrast (Figure 6b). From this experiment, it is unclear what the shape of the spatial profile is for divisive normalization. If we assume that the contribution from spatial neighbors falls off with a Gaussian profile, we can calculate the size of the normalization pool (σ DN) that corresponds to the best fit model (Figure 6c). The contribution is the spatial integral of the product of the contrast with the normalization pool, A F = ∫∫C F(x, y) * DN(x, y)δxδy. The flanker's contribution to this pool was represented in units normalized to the strength of the target's contribution, which is equal to the parameter that we fit (λ = A F/A T). The best fit model has a normalization pool (σ DN) about 4 times as large as the size of the target stimulus (Figure 6c, red circle is the optimal fit). The corresponding model fits the data quite well, especially the reduction of performance caused by increasing contrast (Figure 6e). Notably, the curvature of the iso-target contrast curves is correct. If the parameters are held constant, except for halving the influence of the flankers on the normalization pool, then the curvature of the iso-contrast contours change and fail to fit the data (Figure 6f). 
Figure 6
 
Spatial contrast normalization improves the model fit. Spatial contrast normalization characterizes the non-linear contrast response by dividing the target's actual contrast by a normalization term to yield the effective contrast (C T′). The full equation involves a semi-saturation constant and a power law (see Equation 8). The appropriate non-linearity is also applied to the flanker contrast (Equation 9) but is not shown here. (a) The effective vs. actual contrast of the target is colored blue for the condition where no flanker is present. The effective target contrast is reduced as the flanker contrast is increased (indicated by increasing redness). The curves display the non-linearity for the best fit model (k T, k F, γ, α, c 50, λ) in which each flanker contributes 39% as much as the target to the normalization pool (λ = 0.39). (b) A smaller suboptimal parameter setting (λ = 0.2) is displayed to facilitate intuitions. The flankers contribute less to the normalization pool and do not reduce the effective contrast of the target as much. (c) The relationship between the contribution of the flankers (λ) to the normalization pool and the spatial extent of the normalization pool, assuming a Gaussian profile. Here, λ is calculated as the ratio of the contrast contribution from a single flanker (A F) to the contribution from the target (A T), see Methods section. The size of divisive normalization pool (σ DN) is plotted in units scaled to the size of the stimulus (σ stim). The optimal fit of the model had a flanker contribution (λ = 0.33, red line) that corresponds to a normalization pool about four times the radius of the stimulus. The suboptimal setting (yellow line) is also displayed. (d) A schematic representation of a stimulus where the intensity represents the contrast from the Gaussian-masked gratings. The three gray contours represent the 2 SD boundary of the contrast for the target and flanker patches. The red contour indicates the 2 SD boundary for the optimum spatial region fit by the model. The yellow contour indicates the suboptimal region that is too small. (e) The best fit for the model. (f) A suboptimal model with all parameters the same expect for λ. The red contour is curved in such a way that faint flankers do not sufficiently impair detection.
Figure 6
 
Spatial contrast normalization improves the model fit. Spatial contrast normalization characterizes the non-linear contrast response by dividing the target's actual contrast by a normalization term to yield the effective contrast (C T′). The full equation involves a semi-saturation constant and a power law (see Equation 8). The appropriate non-linearity is also applied to the flanker contrast (Equation 9) but is not shown here. (a) The effective vs. actual contrast of the target is colored blue for the condition where no flanker is present. The effective target contrast is reduced as the flanker contrast is increased (indicated by increasing redness). The curves display the non-linearity for the best fit model (k T, k F, γ, α, c 50, λ) in which each flanker contributes 39% as much as the target to the normalization pool (λ = 0.39). (b) A smaller suboptimal parameter setting (λ = 0.2) is displayed to facilitate intuitions. The flankers contribute less to the normalization pool and do not reduce the effective contrast of the target as much. (c) The relationship between the contribution of the flankers (λ) to the normalization pool and the spatial extent of the normalization pool, assuming a Gaussian profile. Here, λ is calculated as the ratio of the contrast contribution from a single flanker (A F) to the contribution from the target (A T), see Methods section. The size of divisive normalization pool (σ DN) is plotted in units scaled to the size of the stimulus (σ stim). The optimal fit of the model had a flanker contribution (λ = 0.33, red line) that corresponds to a normalization pool about four times the radius of the stimulus. The suboptimal setting (yellow line) is also displayed. (d) A schematic representation of a stimulus where the intensity represents the contrast from the Gaussian-masked gratings. The three gray contours represent the 2 SD boundary of the contrast for the target and flanker patches. The red contour indicates the 2 SD boundary for the optimum spatial region fit by the model. The yellow contour indicates the suboptimal region that is too small. (e) The best fit for the model. (f) A suboptimal model with all parameters the same expect for λ. The red contour is curved in such a way that faint flankers do not sufficiently impair detection.
The model fits surprisingly well, considering that there is no parameter that explicitly accounts for the rat's bias. We consider the bias to be a stimulus-independent reflection of the rat's preference for one port over the other. A bias could be caused if the rats had a prior expectation that a target would be present or absent. However, the target probability was 50% across their entire training and testing experience, and so we do not think this played a role. Alternatively, a bias could be caused if the rat expected a larger reward or smaller penalty on one side than the other. In our experiment, penalties and rewards were symmetric; nonetheless, rats still displayed some bias. 
We note that animals using the same apparatus with the same reward schedule displayed different biases. Thus, we suspect that subjects' idiosyncratic strategies may affect their perceived utility of a left or right response. It is useful to have a free parameter to fit this bias. There is only a single parameter for the rat's overall bias and it is applied to all the stimulus conditions. The bias is determined by finding a threshold in the log-likelihood ratio of the noise and signal distributions that best fits all of the observed data. 
For equal-variance Gaussian distributions, the bias (b) affects the threshold by translating it proportional to the distance between the signal and noise distributions. 
ζ b i a s [ i ] = ζ o p t [ i ] + b μ S [ i ] μ N [ i ] .
(12)
 
The effect of the adaptation is appropriately updated from Equation 7 to use the rat's biased threshold: 
ζ = α ζ b i a s + ( 1 α ) ζ p r i o r .
(13)
 
Including a bias term improves the fit of data (k T, k F, γ, α, c 50, λ, b) from Subject 1, reducing the BIC from 257 to 234 (Figures 7a and 7b). The bias is only a slight preference to respond yes more often. On the other hand, the fit to the data from Subject 2 is very poor without the bias term, and the model improves substantially from BIC of 2351 to 246 (Figures 7c and 7d). The bias parameter captures the rats' tendency to favor no responses. In fact, without the bias term, the previous divisive normalization model completely failed to account for the data from Subject 2 (Figure 7c). 
Figure 7
 
Bias allows the model to generalize to different subjects. (a) The best model for Subject 1 when the utility for a correctly rejected trial is equal to the utility for a hit. In this panel, the log of the utility ratio is zero, b = log(util(cr) / util(hit)) = 0. b is a single bias term that affects all 40 stimulus conditions. (b) Subject 1 displays a modest improvement when a bias parameter is added, because he reports “yes” slightly more often. The decrease in the Bayesian information criterion is ∼23 nats, despite the penalty of ∼5 nats per parameter. (c) Subject 2 favors “no” responses, which is fit poorly by a model without a bias term. (d) The addition of bias term enables the model to be fit quite well.
Figure 7
 
Bias allows the model to generalize to different subjects. (a) The best model for Subject 1 when the utility for a correctly rejected trial is equal to the utility for a hit. In this panel, the log of the utility ratio is zero, b = log(util(cr) / util(hit)) = 0. b is a single bias term that affects all 40 stimulus conditions. (b) Subject 1 displays a modest improvement when a bias parameter is added, because he reports “yes” slightly more often. The decrease in the Bayesian information criterion is ∼23 nats, despite the penalty of ∼5 nats per parameter. (c) Subject 2 favors “no” responses, which is fit poorly by a model without a bias term. (d) The addition of bias term enables the model to be fit quite well.
Discussion
We measured rats' ability to detect a dim target at a known location and onset. Additionally, we quantified the rats' performance as a function of the contrast of the target and the contrast of nearby flanking stimuli. Using an automated training and testing procedure, we observed 30,000 binary choices from each of two subjects. The experimental paradigm was a yes/no forced choice, which enables us to explicitly observe changes in the rats' bias per stimulus condition. We find that the most parsimonious description of rats' detection abilities requires three components: a decision criterion that is influenced by the ensemble of stimuli the rat views, spatial contrast normalization, and a global bias acting as a threshold of the log-likelihood ratio. 
Having accounted for the rats overall bias, the two sources of stimulus-specific bias are a linear additive term that we use to model source confusion (k F) and a non-linear divisive term that we use to model the flanker's contribution (λ) to spatial contrast normalization. Collectively, the models provide a quantitative description for how the flankers bias and impair the subject's performance. According to our models, flankers impair detection in two ways: Flanker contrast biases rats to report the target is present (source confusion), and flanker contrast reduces the effective contrast of the target (contrast normalization). Rats also adapt their decision criterion to the stimulus distribution. 
What is the meaning of the parameter kF?
The target contrast had more impact on the rat's decision than the flanker contrast. This is summarized at the ratio of k T to k F, which was larger than 1 in the best model for each of the two subjects (Figure 8). If these values were equal, it would indicate that the rat was not selective as to the source of the contrast that influenced the decision variable to increase and, thus, caused more yes responses. The fact that k F was lower indicates a selectivity, but it is mild. Thus, a substantial amount of the time, the rat is confusing the contrast from the flankers “as if” it came from the target. 
Figure 8
 
Confidence intervals for each of the parameters. (a) The parameter values for Subject 1. (b) Subject 2. Parameter distributions represent the variability due to limited data and the estimation procedure, assuming the best fit model was a true model of the data. For each parameter estimate, 30,000 trials of data were sampled from the posterior of the model, matching the total amount of real data observer for each subject. The fitting procedure was applied to this synthetic data. This process was repeated 50 times. The height of each box indicates the 1st and 3rd quartiles and the slash in the middle is the median. If parameter values were distributed Gaussian, the whiskers would extend 2.7σ, covering 99.3% of the values. However, the data are not Gaussian, and the actual coverage shown here is less. The seven parameter symbols are displayed on the bottom, along with the median value. Notably, the target's contribution is larger than the flanker's contribution.
Figure 8
 
Confidence intervals for each of the parameters. (a) The parameter values for Subject 1. (b) Subject 2. Parameter distributions represent the variability due to limited data and the estimation procedure, assuming the best fit model was a true model of the data. For each parameter estimate, 30,000 trials of data were sampled from the posterior of the model, matching the total amount of real data observer for each subject. The fitting procedure was applied to this synthetic data. This process was repeated 50 times. The height of each box indicates the 1st and 3rd quartiles and the slash in the middle is the median. If parameter values were distributed Gaussian, the whiskers would extend 2.7σ, covering 99.3% of the values. However, the data are not Gaussian, and the actual coverage shown here is less. The seven parameter symbols are displayed on the bottom, along with the median value. Notably, the target's contribution is larger than the flanker's contribution.
Why do rats respond to flanker contrast as if it were target contrast? One simple possibility is that the size of the rats' receptive fields is so large that the relevant neurons span both the target and the flanker. Thus, the target's signal is contaminated—correlated with the flanker contrast—at the earliest stages of encoding. However, target stimuli were at least 13° and receptive fields of the rat ganglion cells are about 5° wide (Anishchenko et al., 2010). This estimate involves a schematic model eye (Hughes, 1979; Hughes & Wassle, 1979) and a conservative approximation that the receptive field falls within 3 standard deviation of a Gaussian fit: 3σ RF
* 100 μ m σ R F * 1 59 μ m
= 5°. Some of the cell classes may vary in size but are roughly 4.5–6° wide. The point spread function caused by the optics of the rat's eye may add some blur to the retinal image and increase the receptive field size (Artal, Herreros de Tejada, Munoz Tedo, & Green, 1998), but this increase in width would be well less than a degree. Thus, retinal receptive fields measured as linear transfer functions do not require correlation between the target and flankers. If contrast signals are pooled due to large receptive fields, this occurs in later stages of processing. 
An alternative possibility is that the rat does not know which neurons to monitor to make a decision, despite the consistent location of the target on the screen. This would be true if the rat's gaze varied from trial to trial and the decision process did not have access to the information about the gaze. Thus, any contrast signal in a visual area like V1 has multiple possible sources: It could have come from the target or the flanker. Given such a correlated noisy encoding, attributing target evidence to flanker contrast may be the result of optimally decoding a correlated signal. In other words, if a learning algorithm assigned weights to V1 neurons at a slow time scale spanning hundreds to thousands of trials, it should choose high weights for neurons that most likely to respond to the target and weak but non-zero weights for the neighbors. The width of these weights would reflect the variability of the stimulus encoding including behavioral variability. 
Finally, it is possible that lateral interactions in the visual system cause the presence of the flankers to “fill in”(Chen & Tyler, 2008; Polat & Sagi, 2007). This is a conceptually interesting possibility, but with this data set, we cannot distinguish it from the other possible lateral interactions in the visual system or the uncertainty about the target's location (Wu & Chen, 2010). The strength of our model is that we explicitly represent the influence of source confusion and, thereby, isolate it as distinct from the effects of contrast normalization. 
What is the meaning of the parameter λ?
For both subjects, the best fit of the flanker's contribution (λ) to spatial contrast normalization was about 66% of the target's contribution. Specifically, each flanker contributed about a third as much as the target. The Naka–Rushton equation has effectively modeled many saturating components of perceptual systems. Extending the normalization to include a spatial pool of local neural responses has proven to be an effective model for the non-linearities of circuit-level neural processing in the early visual system (Carandini, Heeger, & Movshon, 1997; Geisler & Albrecht, 1992; Heeger, 1992). It also provides compact descriptions in behavioral data (Boynton, Demb, Glover, & Heeger, 1999; Chen & Tyler, 2008; Foley, 1994). This pooling may be selective to a specific visual channel at a given orientation or spatial frequency (Parkes, Lund, Angelucci, Solomon, & Morgan, 2001; Watson & Solomon, 1997), though this is not known for rats. Rats have orientation-tuned neurons in V1 (Girman, Sauve, & Lund, 1999) but lack smoothly varying orientation maps (Ohki, Chung, Ch'ng, Kara, & Reid, 2005). 
The model we present is a mathematical description that does not implicate a specific part of the brain. It is interesting that the same model can describe the activity of neurons in the primary visual cortex. Consequently, it is possible that neurons responsible for the interaction between the target and the flanker reside in the primary visual cortex. Future experimental studies could confirm this hypothesis. However, it is also possible that the behavior we observe, which is consistent with normalization, is the result of computations in earlier visual areas, subsequent visual areas, or even part of a non-visual decision-making stage. 
It has been argued that the control of attention is mediated by engaging the same mechanisms that are used in contrast normalization (Grossberg & Raizada, 2000; Reynolds & Heeger, 2009). 
Assuming that contrast and attention share the same normalization pool, the behavioral model developed and fit to our data could also account for attention. The influence of attention could be modeled as an additional gain control preceding the normalization. This could competitively outweigh the flanker's contribution (λ) to the normalization pool. We note that this is different from the coefficient that we fit after the normalization (k T), which takes on the more cognitive role of weighting evidence. This experiment did not include an independent manipulation of the subject's spatial attention, and so we cannot isolate the influence of attention from the influence of contrast. 
Theory
We have considered two ways in which the presence of flanking gratings could impair target detection performance: contrast normalization and source confusion. 
How might contrast normalization impair? Flankers with a common onset should drive synchronous activity in the neurons surrounding the ones that encode the target. This is likely to induce contrast normalization: the divisive suppression of the spikes that encode the target (Carandini et al., 1997; DeAngelis, Robson, Ohzawa, & Freeman, 1992; Heeger, 1992). Thus, contrast normalization should impair detection because it weakens the signal encoding the target. Because suppression is divisive, it would have little or no effect when the target is absent. How might source confusion impair? One possibility is that visual neurons lack the spatial resolution to separate flanker from target contrast. In this case, source confusion can arise from an optimal decision strategy, given that visual detection of the signal involves some contamination from nearby regions of visual space. Alternatively, source confusion could arise at a cognitive level, for example, poor spatial resolution in the rats' representation of what screen location qualifies as a valid target. Among models we tested, the best fit to our data was achieved by models with both contrast normalization and source confusion. 
Interestingly, the presence of flanking gratings never facilitated the rats' ability to detect the target. This is contrary to the observation that flanking gratings can sometimes facilitate human detection performance. Such facilitation may be caused by decreasing the uncertainty about the target's location in space and time. This would enable the decision process to attend to a smaller number of sensory neurons, thus avoiding the false positives caused by noise from irrelevant sensors. Alternately, flanking stimuli could provide a pedestal of excitation that would enable weak signals to cross a threshold for activation, similar to stochastic resonance (Goris et al., 2008; Wiesenfeld & Moss, 1995). Both theories suggest that facilitation may occur for low target contrasts. However, facilitation was not found for rats at any combination of target and flanker contrasts. It is possible that there was very mild facilitation with a flanker contrast of 0.75 and a target contrast of 0.25 (see small bump in d′ in Figures 4b and 4c). However, this did not raise performance above the level of the target-alone condition. 
Future recommendations
The model presented here represents the groundwork for a core model of visual detection in the presence of distracters. We recommend four ways in which a future model could be improved: by adding priors for the model parameters, by considering alternate decision rules, by analyzing trial by trial fluctuations in behavior, and by using non-Gaussian models of internal noise. 
In this paper, the model provided was fit to the likelihood, assuming a flat prior for all parameters. A Bayesian approach that provides a prior for the parameters would result in more stable fits. Providing priors, even just for incidental parameters like γ and c 50, can decrease the variance in estimates of parameters of interest, such as λ. Including priors may be especially useful if taking multiple measurements over time or measuring a larger population of subjects. 
We provided a decision rule as the linear combination of two thresholds because it is the simplest mathematical form that fit data well. Intuitively, it has some appeal because the weighted combination of the two may correspond to the limited evidence the subject has for the current optimal threshold. The model we provide does not have a temporal component of the decision process and cannot explain why the rats' bias is correlated with their reaction time on a trial-to-trial basis. We note that a simple diffusion model of decision making does not fit the reaction time data because accuracy does not decrease with reaction time (data not shown). 
Another possible improvement for future models is to consider the trial to trial changes in behavior. The current model treated each stimulus condition as a constant binomial proportion for all trials with the same stimulus. However, we know that rats change their bias slightly based on the context of their previous response and whether it was correct or incorrect. Specifically, rats are more likely to choose the opposite response immediately after an error. This is similar to the well-known tendencies of human subjects to alternate and perseverate responses (Cho et al., 2002). It is possible that the rats increased their alternation strategy due to our use of correction trials in initial training (see Methods section). The consequence of this is that animals may appear to be slightly less biased on hard stimuli where they make a large number of errors. In our data set, we avoided the variability due to post-error switching by only analyzing trials in which the previous trial was correct. Future models could explicitly model the switching tendencies. A simple example is to add or subtract a fixed switching bias (b s) to the bias parameter (b) after errors, +b s after false alarms and −b s after misses. This shift in bias is equivalent to context-specific adjustment to the prior odds. 
Finally, it is possible that future models may describe decision variables with probability functions that are more interpretable with physiological parameters. For example, a Gaussian distribution is capable of having negative values; this does not make sense for firing rates, which must be positive. Additionally, it seems that noise increases with signal strength in neural representations (Shadlen & Newsome, 1998), especially for low-contrast signals that are not saturated. We performed preliminary tests on three families of distributions that are bounded at zero and, thus, change their shape as the decision variable increases: the exponential distribution, the gamma distribution, and the lognormal distribution. When comparing the fits with the BIC, the gamma distribution performs slightly better, but the normal distribution performs quite well, especially when the variance of the noise distribution scales with the signal such that the variance is twice the mean (data not shown). In the long run, other probability models may add explanatory power and provide a better fit to the data in terms of a lower BIC. However, for this data set, we found that the improvements from non-Gaussian probability functions were modest. Thus, for the purpose of presenting this model, we focused on the Gaussian distribution that has a clear relationship to d′. 
Applications
The behavioral paradigm presented in this paper may prove useful for high-throughput screening of perceptual and cognitive defects in rodents. For example, schizophrenic patients exhibit abnormal visual processing to high-contrast visual stimuli (Brenner et al., 2009; Chapman & McGhie, 1962) and atypical performance in the visual tasks with flankers (Gooding, Braun, & Studer, 2006; Keri, Kelemen, & Benedek, 2009), presumably due to abnormal function of GABA-ergic interneurons (Bullock, Cardon, Bustillo, Roberts, & Perrone-Bizzozero, 2008; Cruz, Weaver, Lovallo, Melchitzky, & Lewis, 2009; Lewis, Hashimoto, & Volk, 2005). A task and model like ours can be used to separately measure the influence of spatial contrast normalization and cognitive confusion in animal models of schizophrenia. Other disease states implicated with abnormal lateral processing in inhibitory networks may also have unique behavioral signatures that could be discovered and quantified by estimating the strength of the contrast normalization in basic detection tasks. 
Applying an automated method to train animals (Meier et al., 2011), we provide here a framework to fit models to the likelihood of the raw data and examples of key parameters required to fit the bias and deficits caused by flanking stimuli. Within the model provided, the contribution of the flankers to non-linear divisive normalization (λ) is separated from the linear contribution of the flanker contrast (k F) to the decision variable. Measuring both of these parameters, as well as nuisance parameters like the animal's overall bias, provides an assay that is sensitive to changes in the spatial contrast normalization. 
Acknowledgments
Philip Meier designed the experiment, collected the data, and built the model. Philip Meier and Pam Reinagel interpreted the results and wrote the manuscript. 
We thank Erik Flister for his continued support of the software for the rat training apparatus. We thank Sarah Meder and Danielle Dickson for running daily behavioral sessions. We thank Thomas Albright for feedback on a draft of the paper. 
This work was supported by a Hellman Foundation Fellowship, an Innovative Research Grant from the Kavli Institute for Brain and Mind, a Scholar Award from the J. S. McDonnell Foundation, and NIH R01 Grant EY016856. Philip Meier was supported by NSF IGERT Grant DGE-0333451 to GW Cottrell/VR de Sa and by an NSF Graduate Research Fellowship. 
Commercial relationships: none. 
Corresponding author: Philip Meier. 
Email: pmeier@ucsd.edu. 
Address: 9500 Gilman Drive, Campus Box 0357, La Jolla, CA 92093-0357, USA. 
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Figure 1
 
Experimental design for yes/no 2AFC target detection. (a) A rat in the training chamber viewing a stimulus. The rat's task is to report the presence or absence of the central grating (the “target”). The rat is forced to choose either “yes” or “no” before proceeding to the next trial. The top and bottom gratings (“flankers”) contain no information about the correct response. In (a), the target is present in the stimulus, and thus, a response to the right port would be rewarded with a drop of water (a “hit”). A response to the left port would be punished with a time-out period and an aversive tone (a “miss”). For photographic purposes, the stimulus was left on indefinitely, but during the task, stimuli persisted for 200 ms. (b) The target was present on 50% of the trials (T+) and absent on other trials (T−). The rat was appropriately rewarded and punished for correct rejections and false alarms, just like hits and misses. (c) In the testing period, rats performed a block of 150 consecutive trials during which the target contrast and flanker contrast were held constant. Every 20 blocks visited each combination of target contrast (C T = [0.25, 0.5, 0.75, 1]) and flanker contrast (C F = [0, 0.25, 0.5, 0.75, 1]), in a random order. The probability of a target being absent (C T = 0) was always 50%, and thus, we refer to the C T of a block as the contrast that a target would be if it were present on a trial.
Figure 1
 
Experimental design for yes/no 2AFC target detection. (a) A rat in the training chamber viewing a stimulus. The rat's task is to report the presence or absence of the central grating (the “target”). The rat is forced to choose either “yes” or “no” before proceeding to the next trial. The top and bottom gratings (“flankers”) contain no information about the correct response. In (a), the target is present in the stimulus, and thus, a response to the right port would be rewarded with a drop of water (a “hit”). A response to the left port would be punished with a time-out period and an aversive tone (a “miss”). For photographic purposes, the stimulus was left on indefinitely, but during the task, stimuli persisted for 200 ms. (b) The target was present on 50% of the trials (T+) and absent on other trials (T−). The rat was appropriately rewarded and punished for correct rejections and false alarms, just like hits and misses. (c) In the testing period, rats performed a block of 150 consecutive trials during which the target contrast and flanker contrast were held constant. Every 20 blocks visited each combination of target contrast (C T = [0.25, 0.5, 0.75, 1]) and flanker contrast (C F = [0, 0.25, 0.5, 0.75, 1]), in a random order. The probability of a target being absent (C T = 0) was always 50%, and thus, we refer to the C T of a block as the contrast that a target would be if it were present on a trial.
Figure 2
 
Higher target contrasts increase detection performance and bias rats to say yes. (a) The probability that a rat responds “yes” given that the target was absent (false alarm rate) or present (hit rate). If the data fall on the diagonal line, the animal's response contains no information about the target. Perfect performance is in the upper left. The four data points indicate performance from blocks where the flanker was absent and the target varied in contrast. The horizontal and vertical components of each plus symbol indicate the 95% confidence interval of a binomial proportion, for the false alarms and hit proportions, respectively. (b) A representation of the medium-contrast target condition, assuming that a rat uses a single decision variable that has a Gaussian distribution with equal variance for the noise (T−, black) and signal (T+, blue). The common equal-variance Gaussian assumption will fail to fit our later models, but we present it here to explain the framework of signal detection theory. If a decision variable is greater than threshold criterion (gray line), then the model produces a “yes” response. The means of the distributions are separated by 1.3σ. Thus, the measure d′ = 1.3. (c) For a higher contrast (C T = 1.0), the distributions overlap less, and the target is easier to detect (d′ = 2.2). Notice that the threshold criterion is greater (gray line shifted right), resulting in fewer false alarms. (d) Higher contrast targets have a larger d′. Each data point represents an estimate from a single block of trials. The vertical bars cover ±1 SD. The variability of data is larger than expected from limited sampling; other factors beyond the stimulus also affect performance. (e) The average bias criterion is greater than zero, indicating that this rat favored no responses on all of these conditions. The subject's mild bias for no responses was reduced as the target contrast increased.
Figure 2
 
Higher target contrasts increase detection performance and bias rats to say yes. (a) The probability that a rat responds “yes” given that the target was absent (false alarm rate) or present (hit rate). If the data fall on the diagonal line, the animal's response contains no information about the target. Perfect performance is in the upper left. The four data points indicate performance from blocks where the flanker was absent and the target varied in contrast. The horizontal and vertical components of each plus symbol indicate the 95% confidence interval of a binomial proportion, for the false alarms and hit proportions, respectively. (b) A representation of the medium-contrast target condition, assuming that a rat uses a single decision variable that has a Gaussian distribution with equal variance for the noise (T−, black) and signal (T+, blue). The common equal-variance Gaussian assumption will fail to fit our later models, but we present it here to explain the framework of signal detection theory. If a decision variable is greater than threshold criterion (gray line), then the model produces a “yes” response. The means of the distributions are separated by 1.3σ. Thus, the measure d′ = 1.3. (c) For a higher contrast (C T = 1.0), the distributions overlap less, and the target is easier to detect (d′ = 2.2). Notice that the threshold criterion is greater (gray line shifted right), resulting in fewer false alarms. (d) Higher contrast targets have a larger d′. Each data point represents an estimate from a single block of trials. The vertical bars cover ±1 SD. The variability of data is larger than expected from limited sampling; other factors beyond the stimulus also affect performance. (e) The average bias criterion is greater than zero, indicating that this rat favored no responses on all of these conditions. The subject's mild bias for no responses was reduced as the target contrast increased.
Figure 3
 
Higher flanker contrasts decrease detection performance and bias rats to say yes. (a) A scatter plot of the false alarms and hits for five stimulus conditions, varying in flanker contrast. Symbols are 95% confidence interval of a binomial proportion, as in Figure 2a. All stimuli have a target contrast of 1. The blue symbol indicates zero flanker contrast; increasing flanker contrast is indicated by redness. (b) The discriminability of the target when there is no flanker present. (c) The discriminability when the flanker contrast is increased. The decision criteria are represented by a vertical gray line and it fit to exactly match the data with no error. Note that we plot the means of the noise distribution at zero. In subsequent models, the means of the signal and noise distributions may both be non-zero. Notably, the flanker contrast can make the discriminability go down by increasing noise distribution, even if the signal distribution were unchanged. (d) The discriminability decreases with flanker contrast. (e) The subjects bias criterion shifts with flanker contrast. For consistency with the past literature, we present the criterion here as (z(cr) + z(fa)) / 2. The change in the parameter reflects the rat's increasing bias to say yes as flanker contrast increases, as also evidenced by the raw data in (a).
Figure 3
 
Higher flanker contrasts decrease detection performance and bias rats to say yes. (a) A scatter plot of the false alarms and hits for five stimulus conditions, varying in flanker contrast. Symbols are 95% confidence interval of a binomial proportion, as in Figure 2a. All stimuli have a target contrast of 1. The blue symbol indicates zero flanker contrast; increasing flanker contrast is indicated by redness. (b) The discriminability of the target when there is no flanker present. (c) The discriminability when the flanker contrast is increased. The decision criteria are represented by a vertical gray line and it fit to exactly match the data with no error. Note that we plot the means of the noise distribution at zero. In subsequent models, the means of the signal and noise distributions may both be non-zero. Notably, the flanker contrast can make the discriminability go down by increasing noise distribution, even if the signal distribution were unchanged. (d) The discriminability decreases with flanker contrast. (e) The subjects bias criterion shifts with flanker contrast. For consistency with the past literature, we present the criterion here as (z(cr) + z(fa)) / 2. The change in the parameter reflects the rat's increasing bias to say yes as flanker contrast increases, as also evidenced by the raw data in (a).
Figure 4
 
The impairment and bias caused by flankers are present for all target contrasts. (a) The false alarm rate vs. the hit rate for twenty conditions including all combinations of four target contrasts (C T = [0.25, 0.5, 0.75. 1]) and five flanker contrasts (C F = [0, 0.25, 0.5, 0.75, 1]). Error bars are binomial confidence intervals. (b) Increasing flanker contrast impairs performance. Lines connect stimuli with constant target contrast. Error bars are ±1 SEM from n = 14–20 measurements. (c) Same as (b) but for Subject 2. (d) Increasing flanker contrast increases the probability to say yes, which is a decrease in bias criterion. Lines connect stimuli with constant target contrast. (e) Same as (d) but for Subject 2.
Figure 4
 
The impairment and bias caused by flankers are present for all target contrasts. (a) The false alarm rate vs. the hit rate for twenty conditions including all combinations of four target contrasts (C T = [0.25, 0.5, 0.75. 1]) and five flanker contrasts (C F = [0, 0.25, 0.5, 0.75, 1]). Error bars are binomial confidence intervals. (b) Increasing flanker contrast impairs performance. Lines connect stimuli with constant target contrast. Error bars are ±1 SEM from n = 14–20 measurements. (c) Same as (b) but for Subject 2. (d) Increasing flanker contrast increases the probability to say yes, which is a decrease in bias criterion. Lines connect stimuli with constant target contrast. (e) Same as (d) but for Subject 2.
Figure 5
 
Various decision criteria. (a) The best fit model (k T, k F, γ, α = 0) assuming that the subject only chooses a single fixed decision criterion for all stimuli (α = 0, see Equation 7). The blue vertical line indicates the results of a pure change in target contrast. Other black lines indicate the influence of target contrast when the flanker contrast is higher. The red curve represents a change in the flanker contrast for a target contrast of 1. Other black curves represent pure changes in flanker contrast if the target contrast is lower. If the data perfectly fit the model, the intersections of the lines would match the observed data. Blue crosses from Figure 1a would fall on the blue line, and red crosses from Figure 2b would fall on the red line. Gray crosses indicate all possible combinations of the target and flanker contrasts (see Figure 1c) and should be located at the intersection of the black lines. The gray contour indicates d′ for each model's best fit to the high-contrast condition; the curve spans all possible decision criterion thresholds. The signal and noise distributions of the model are displayed for three representative stimuli: (b) a low-contrast target alone, (c) a high-contrast target alone, and (d) a high-contrast target with a high-contrast flanker. (e) The best fit model (k T, k F, γ, α = 1) if the subject chooses the optimal decision threshold for each stimulus condition (α = 1). Notice that allowing for the optimal choice for any symmetric distribution results in data that fall on a single diagonal line with a slope of −1 that extends from pure chance behavior to perfect performance. The dots along the line correspond to the model's prediction of performance for the four target contrasts. Each dot represents five overlapping conditions because the model predicts no effect of flankers. However, the data from the rat do not fall on a line; they are spread over a plane. This model is clearly wrong. The poor fit is reflected in the substantial rise in the Bayesian information criterion (BIC). The decision criteria are presented in (f)–(h) using the same stimulus conditions as before. Note that the entire model is refit, and so the signal and noise distributions may vary slightly as well. (i) The best fit model (k T, k F, γ, α) assuming that the subject's decision criterion is the weighted average between a single criterion (a = 0) and the optimal criterion for that stimulus condition (α = 1). The best relative weight (α) is fit to the model. The value of α = 0.79 indicates that the decision criterion is close to the optimal but ∼20% influenced by a global criterion that is modeled as the average of criterion across all conditions.
Figure 5
 
Various decision criteria. (a) The best fit model (k T, k F, γ, α = 0) assuming that the subject only chooses a single fixed decision criterion for all stimuli (α = 0, see Equation 7). The blue vertical line indicates the results of a pure change in target contrast. Other black lines indicate the influence of target contrast when the flanker contrast is higher. The red curve represents a change in the flanker contrast for a target contrast of 1. Other black curves represent pure changes in flanker contrast if the target contrast is lower. If the data perfectly fit the model, the intersections of the lines would match the observed data. Blue crosses from Figure 1a would fall on the blue line, and red crosses from Figure 2b would fall on the red line. Gray crosses indicate all possible combinations of the target and flanker contrasts (see Figure 1c) and should be located at the intersection of the black lines. The gray contour indicates d′ for each model's best fit to the high-contrast condition; the curve spans all possible decision criterion thresholds. The signal and noise distributions of the model are displayed for three representative stimuli: (b) a low-contrast target alone, (c) a high-contrast target alone, and (d) a high-contrast target with a high-contrast flanker. (e) The best fit model (k T, k F, γ, α = 1) if the subject chooses the optimal decision threshold for each stimulus condition (α = 1). Notice that allowing for the optimal choice for any symmetric distribution results in data that fall on a single diagonal line with a slope of −1 that extends from pure chance behavior to perfect performance. The dots along the line correspond to the model's prediction of performance for the four target contrasts. Each dot represents five overlapping conditions because the model predicts no effect of flankers. However, the data from the rat do not fall on a line; they are spread over a plane. This model is clearly wrong. The poor fit is reflected in the substantial rise in the Bayesian information criterion (BIC). The decision criteria are presented in (f)–(h) using the same stimulus conditions as before. Note that the entire model is refit, and so the signal and noise distributions may vary slightly as well. (i) The best fit model (k T, k F, γ, α) assuming that the subject's decision criterion is the weighted average between a single criterion (a = 0) and the optimal criterion for that stimulus condition (α = 1). The best relative weight (α) is fit to the model. The value of α = 0.79 indicates that the decision criterion is close to the optimal but ∼20% influenced by a global criterion that is modeled as the average of criterion across all conditions.
Figure 6
 
Spatial contrast normalization improves the model fit. Spatial contrast normalization characterizes the non-linear contrast response by dividing the target's actual contrast by a normalization term to yield the effective contrast (C T′). The full equation involves a semi-saturation constant and a power law (see Equation 8). The appropriate non-linearity is also applied to the flanker contrast (Equation 9) but is not shown here. (a) The effective vs. actual contrast of the target is colored blue for the condition where no flanker is present. The effective target contrast is reduced as the flanker contrast is increased (indicated by increasing redness). The curves display the non-linearity for the best fit model (k T, k F, γ, α, c 50, λ) in which each flanker contributes 39% as much as the target to the normalization pool (λ = 0.39). (b) A smaller suboptimal parameter setting (λ = 0.2) is displayed to facilitate intuitions. The flankers contribute less to the normalization pool and do not reduce the effective contrast of the target as much. (c) The relationship between the contribution of the flankers (λ) to the normalization pool and the spatial extent of the normalization pool, assuming a Gaussian profile. Here, λ is calculated as the ratio of the contrast contribution from a single flanker (A F) to the contribution from the target (A T), see Methods section. The size of divisive normalization pool (σ DN) is plotted in units scaled to the size of the stimulus (σ stim). The optimal fit of the model had a flanker contribution (λ = 0.33, red line) that corresponds to a normalization pool about four times the radius of the stimulus. The suboptimal setting (yellow line) is also displayed. (d) A schematic representation of a stimulus where the intensity represents the contrast from the Gaussian-masked gratings. The three gray contours represent the 2 SD boundary of the contrast for the target and flanker patches. The red contour indicates the 2 SD boundary for the optimum spatial region fit by the model. The yellow contour indicates the suboptimal region that is too small. (e) The best fit for the model. (f) A suboptimal model with all parameters the same expect for λ. The red contour is curved in such a way that faint flankers do not sufficiently impair detection.
Figure 6
 
Spatial contrast normalization improves the model fit. Spatial contrast normalization characterizes the non-linear contrast response by dividing the target's actual contrast by a normalization term to yield the effective contrast (C T′). The full equation involves a semi-saturation constant and a power law (see Equation 8). The appropriate non-linearity is also applied to the flanker contrast (Equation 9) but is not shown here. (a) The effective vs. actual contrast of the target is colored blue for the condition where no flanker is present. The effective target contrast is reduced as the flanker contrast is increased (indicated by increasing redness). The curves display the non-linearity for the best fit model (k T, k F, γ, α, c 50, λ) in which each flanker contributes 39% as much as the target to the normalization pool (λ = 0.39). (b) A smaller suboptimal parameter setting (λ = 0.2) is displayed to facilitate intuitions. The flankers contribute less to the normalization pool and do not reduce the effective contrast of the target as much. (c) The relationship between the contribution of the flankers (λ) to the normalization pool and the spatial extent of the normalization pool, assuming a Gaussian profile. Here, λ is calculated as the ratio of the contrast contribution from a single flanker (A F) to the contribution from the target (A T), see Methods section. The size of divisive normalization pool (σ DN) is plotted in units scaled to the size of the stimulus (σ stim). The optimal fit of the model had a flanker contribution (λ = 0.33, red line) that corresponds to a normalization pool about four times the radius of the stimulus. The suboptimal setting (yellow line) is also displayed. (d) A schematic representation of a stimulus where the intensity represents the contrast from the Gaussian-masked gratings. The three gray contours represent the 2 SD boundary of the contrast for the target and flanker patches. The red contour indicates the 2 SD boundary for the optimum spatial region fit by the model. The yellow contour indicates the suboptimal region that is too small. (e) The best fit for the model. (f) A suboptimal model with all parameters the same expect for λ. The red contour is curved in such a way that faint flankers do not sufficiently impair detection.
Figure 7
 
Bias allows the model to generalize to different subjects. (a) The best model for Subject 1 when the utility for a correctly rejected trial is equal to the utility for a hit. In this panel, the log of the utility ratio is zero, b = log(util(cr) / util(hit)) = 0. b is a single bias term that affects all 40 stimulus conditions. (b) Subject 1 displays a modest improvement when a bias parameter is added, because he reports “yes” slightly more often. The decrease in the Bayesian information criterion is ∼23 nats, despite the penalty of ∼5 nats per parameter. (c) Subject 2 favors “no” responses, which is fit poorly by a model without a bias term. (d) The addition of bias term enables the model to be fit quite well.
Figure 7
 
Bias allows the model to generalize to different subjects. (a) The best model for Subject 1 when the utility for a correctly rejected trial is equal to the utility for a hit. In this panel, the log of the utility ratio is zero, b = log(util(cr) / util(hit)) = 0. b is a single bias term that affects all 40 stimulus conditions. (b) Subject 1 displays a modest improvement when a bias parameter is added, because he reports “yes” slightly more often. The decrease in the Bayesian information criterion is ∼23 nats, despite the penalty of ∼5 nats per parameter. (c) Subject 2 favors “no” responses, which is fit poorly by a model without a bias term. (d) The addition of bias term enables the model to be fit quite well.
Figure 8
 
Confidence intervals for each of the parameters. (a) The parameter values for Subject 1. (b) Subject 2. Parameter distributions represent the variability due to limited data and the estimation procedure, assuming the best fit model was a true model of the data. For each parameter estimate, 30,000 trials of data were sampled from the posterior of the model, matching the total amount of real data observer for each subject. The fitting procedure was applied to this synthetic data. This process was repeated 50 times. The height of each box indicates the 1st and 3rd quartiles and the slash in the middle is the median. If parameter values were distributed Gaussian, the whiskers would extend 2.7σ, covering 99.3% of the values. However, the data are not Gaussian, and the actual coverage shown here is less. The seven parameter symbols are displayed on the bottom, along with the median value. Notably, the target's contribution is larger than the flanker's contribution.
Figure 8
 
Confidence intervals for each of the parameters. (a) The parameter values for Subject 1. (b) Subject 2. Parameter distributions represent the variability due to limited data and the estimation procedure, assuming the best fit model was a true model of the data. For each parameter estimate, 30,000 trials of data were sampled from the posterior of the model, matching the total amount of real data observer for each subject. The fitting procedure was applied to this synthetic data. This process was repeated 50 times. The height of each box indicates the 1st and 3rd quartiles and the slash in the middle is the median. If parameter values were distributed Gaussian, the whiskers would extend 2.7σ, covering 99.3% of the values. However, the data are not Gaussian, and the actual coverage shown here is less. The seven parameter symbols are displayed on the bottom, along with the median value. Notably, the target's contribution is larger than the flanker's contribution.
Table 1
 
Summary of the final model. There are seven free parameters that are fit from 40 data points.
Table 1
 
Summary of the final model. There are seven free parameters that are fit from 40 data points.
Stage 1: Effective contrast
γ Acceleration of the contrast response non-linearity
c 50 Contrast at 50% response saturation
λ Space constant of normalization pool
→These determine effective contrasts of target and flanker, C T′ and C F′ (Equations 8 and 9).

Stage 2: Decision variable
k T Weight of target contrast C T′ input to decision variable
k F Weight of flanker contrast C F′ input to decision variable
→These determine the means of the signal and noise distributions, μ S and μ N (Equations 10 and 11).

Stage 3: Decision threshold
b Response bias of subject→with μ S and μ N, determines ζ bias
α Relative weight of global vs. local threshold
→These determine the 20 condition-specific decision thresholds, ζ (Equation 13).
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