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.

*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.

*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.

*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.

*μ*l). 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.

*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 3

*λ*, 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.

*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/m

^{2}, respectively (Colorvision, spyder2express).

*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:

*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):

*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.

*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*∈

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). |

*C*

_{T}= 1) and a medium-contrast condition (Figure 2c,

*C*

_{T}= 0.5).

*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).

*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.

*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.

*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.

*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.

*i*, we obtain

*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.

*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.

*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.

*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:

*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*

_{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 2

*λC*

_{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).

*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).

*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).

*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.

*k*

_{F}?

*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.

*σ*

_{RF}

*λ*?

*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).

*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.

*d*′ in Figures 4b and 4c). However, this did not raise performance above the level of the target-alone condition.

*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.

*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.

*d*′.

*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.