**Estimation of perceptual variables is imprecise and prone to errors. Although the properties of these perceptual errors are well characterized, the physiological basis for these errors is unknown. One previously proposed explanation for these errors is the trial-by-trial variability of the responses of sensory neurons that encode the percept. In order to test this hypothesis, we developed a mathematical formalism that allows us to find the statistical characteristics of the physiological system responsible for perceptual errors, as well as the time scale over which the visual information is integrated. Crucially, these characteristics can be estimated solely from a behavioral experiment performed here. We demonstrate that the physiological basis of perceptual error has a constant level of noise (i.e., independent of stimulus intensity and duration). By comparing these results to previous physiological measurements, we show that perceptual errors cannot be due to the variability during the encoding stage. We also find that the time window over which perceptual evidence is integrated lasts no more than ∼230 ms. Finally, we discuss sources of error that may be consistent with our behavioral measurements.**

*τ*(Figure 1A); (b) The spike-count tuning curve,

*R*(

*θ*,

*τ*), is monotonic with respect to both

*θ*and

*τ*, where

*θ*is the perceptual variable; note that this tuning curve could describe either single neurons or a combined variable due to a population of neurons; (c) The variability in the spike count (

*σ*) can be approximated by

*σ*=

*β*·

*R*(

*θ*,

*τ*)

*, where*

^{ρ}*β*and

*ρ*are empirically determined constants (Dean, 1981). For example, a value of

*ρ*= 0.5 would represent a Poisson-like relationship between firing rate and noise, whereas a value of

*ρ*= 0 would represent a constant noise level, independent of the firing rate.

*τ*is reduced, the spike count is reduced, the distributions are altered (Figure 1B), and the overlap region (and thus JND) increases. The increase in the overlap depends both on the change in the mean spike count and on the variance of the spike count. Therefore, for different statistical models of noise, the change in overlap will be quantifiably different. In the example shown in Figure 1, we observe that decreasing

*τ*has a more deleterious effect on discrimination ability in the

*ρ*= 0 case than in the

*ρ*= 0.5 case (Figure 1C). The relationship between

*τ*and the

*JND*can be mathematically formalized with the following derivations.

*τ*

_{sat}) such that for observation times

*τ*>

*τ*

_{sat}, performance at sensory discrimination will not improve. This assumption is motivated by experimental data, both ours and others (G. E. Legge, 1978; Watson, 1979), which exhibit performance saturation as a robust phenomenon. However, were our data sets not to include time points beyond

*τ*

_{sat}, our current theory would still be able to determine ρ.

_{θ}is the standard deviation of the stimulus estimation,

*σ*is the standard deviation of the encoding variable

_{E}*R(θ, τ), σ*is the standard deviation of the decoding variable, and

_{D}*R*′ is the derivative of

*R*with respect to

*θ*. Note that

*R(θ, τ)*is a tuning curve in terms of spike count, and depends both on the encoded parameters

*θ*, and on the duration of the stimulus presentation

*τ. R*can be either a single neuron-tuning curve or a population coding based tuning curve. When

*σ*= 0, Equation 1 is exactly the inverse of the Fisher information for many forms of encoding statistics (e.g., Gaussian, Poisson); for others, it is a close approximation (Paradiso, 1988; Seung & Sompolinsky, 1993). Due to the Cramer-Rao lower bound, this sets the lower limit of the accuracy of stimulus estimation, and for many distributions the optimal decoder is efficient and can attain the lower bound (Wijsman, 1973). Therefore, any significant behavioral errors in excess of this lower bound are decoding errors.

_{D}*R*(

*θ*,

*τ*). However, for simplicity the function can be decomposed into

*R*(

*θ*) =

*τ*·

*r*(

*θ*), where

*τ*is the duration of the stimulus, and

*r*(

*θ*) is the firing rate function. Here we complete the derivation with the assumption that the firing rate is static, but we show in Appendix 01 that the following conclusions hold for dynamic firing. The need to know the exact details of

*r*(

*θ*) can be eliminated if we make a ratio of

*σ*at two values of

_{θ}*τ*. For simplicity sake, we use

*τ*

_{sat}(as described above) as one of these values, and

*τ*

_{1}as the other, where

*τ*

_{1}<

*τ*

_{sat}. This gives us

*σ*≪

_{T}*σ*) our analysis results in predictions that are testable at the behavioral level. It is these predictions that are central to the experiment performed here.

_{R}*σ*=

*β*·

*R*(

*θ*,

*τ*)

*into Equation 2. Once simplified, this results in the following equation:*

^{ρ}*σ*can be considered the equivalent of the traditional JND used in psychometric experiments. Here the JND measured will depend on the presentation times (

_{θ}*τ*). With this in mind, we now define the variable LJR, which is the logarithm of the above equation:

*is the JND for a time window of length*

_{τ}*τ*, and JND

_{sat}is the JND for

*τ*≥

*τ*

_{sat}. For

*τ > τ*

_{sat}, the LJR = 0. Equation 4 is limited to the condition τ ≥ τ

_{sat}. By plotting

*LJR*against log(

*τ*), we arrive at a line whose slope and intercept can be used to calculate

*ρ*and

*τ*

_{sat}. This function is plotted in Figure 1D for two specific cases:

*ρ*= 0 (constant noise) and

*ρ*= 0.5 (Poisson-like noise). From this formulation, we have designed a contrast discrimination experiment with a range of integration times (

*τ*) that can determine both

*τ*

_{sat}and

*ρ*for contrast perception.

*, we found the stimulus value that would correspond to 84% (1σ) on the determined psychometric function. JND*

_{τ}_{sat}(necessary for solving Equation 4) was found by averaging the JND

*for the 300- and 600-ms conditions. These time values are presumed to be above*

_{τ}*τ*

_{sat}because JND

_{300}and JND

_{600}are statistically indistinguishable from one another when combined across all subjects (unpaired

*t*test,

*p*= 0.28). JND

_{sat}was calculated individually for each subject.

_{sat}permits us to calculate the

*LJR*(see Equation 4 above) for each subject and condition (Figure 3). For each subject, we used an iterative least square estimation method (as implemented with the “nlinfit” function within MATLAB) to do a nonlinear fit of the data to Equation 4 and extract the parameters

*τ*

_{sat}and

*ρ*. Several distinct fitting methods were used and yielded similar results.

*ρ*= −0.20 ± 0.31. This value of

*ρ*is significantly different (

*ρ*≅ 0.00014) from the Poisson-like value of

*ρ*= 0.58 found through an electrophysiology experiment (Dean, 1981). However, it is statistically indistinguishable from a constant noise condition (

*ρ*≅ 0). We also find that

*τ*

_{sat}= 232 ± 83 (Figure 3D and Table 1). Note that this value of

*τ*

_{sat}is consistent with the estimation of JND

_{sat}done at the end of the Methods section, which showed

*τ*

_{sat}should be less than 300 ms. Similar results are also derived and presented in both Table 1 and Figure 3 under “Combined data” by combining all the data points across subject, time, and reference conditions, and fitting Equation 4.

*τ*

_{sat}values. Largely, we find that no such effect exists, with the exception of our lowest contrast level (see Figure 4). At extremely low contrast intensities, subjects see a significant reduction in both

*ρ*and

*τ*

_{sat}. We interpret this as a possible ramification of the dipper function, the well described phenomenon of nonlinear thresholds at low contrast values (Bradley & Ohzawa, 1986; Legge & Foley, 1980; Tolhurst & Barfield, 1978), although this theory would take additional data collection to confirm.

*σ*). Using some general assumptions, including that timing errors must be proportional to the length of time being estimated, we have calculated analytically and confirmed using simulations how timing errors affect the LJR (For details, see “The impact of errors in estimating temporal intervals” in Appendix 01). We find that, for any system in which the stimulus estimation errors arise primarily from errors in temporal estimation, the predicted slope of the LJR graph (e.g., Figure 1D) would be 0 which corresponds to

_{θ}*ρ*= 1. This result is obviously inconsistent with our data. Further, the fact that variance is additive leaves no way for a combination of Poisson and timing related noise to result in a

*ρ*≅ 0, suggesting that timing-related errors make an insubstantial contribution to perceptual errors (

*σ*).

_{θ}*τ*

_{sat}was found to range from 50 to 1000 ms, depending on the spatial frequency of the stimuli. For the spatial frequency closest to ours (0.75 cycles/° of vision),

*τ*

_{sat}was found to be approximately 100 ms. Whereas the results are somewhat different, there are significant distinction between that experiment and our own: It used a staircase procedure (Wetherill & Levitt, 1965) to estimate the parameters of the psychometric curve rather than Bayesian adaption (Kontsevich & Tyler, 1999), it had a smaller number of subjects, and it used a contrast detection task rather than a discrimination task. This last fact may have contributed significantly to the differences observed. In Legge (1978), stimuli were determined to be present or absent compared to a blank screen. To compare, our study displayed similar gratings, and subjects chose the one with the higher contrast. This means our study captures decision behavior over a wider array of comparisons, which have established nonlinear effects—e.g., the dipper function (Bradley & Ohzawa, 1986; Legge & Foley, 1980; Tolhurst & Barfield, 1978). Additionally, we used a substantially higher contrast mask (∼1 vs. 0.2) than Legge (1978). Within his experiment, increasing the magnitude of the contrast mask appeared to increase

*τ*

_{sat}; however, it is difficult to generalize this relationship due to limited data within both experiments.

*ρ*value is significantly different (

*p*≅ 0.00014) from the previously determined physiological value of

*ρ*= 0.58 for single contrast encoding neurons in V1 (Dean, 1981).

*ρ*from the most comparable portion of the data set obtained by Legge (1978, i.e., spatial frequency of 0.75), and found it to be approximately 0.25. Although this differs from our estimate, due to the difference in the actual experiments we cannot determine the origin of this difference and cannot even determine if it is statistically different from our result.

*ρ*that we find in this experiment, as well as that just estimated from previous data (Legge, 1978), is very different than the value one expects from single neurons that encode contrast (Churchland et al., 2010; Dean, 1981). One possible explanation of this discrepancy is that, since the stimulus is encoded by an ensemble of correlated neurons (and not by a single neuron), some forms of averaging over the population may give rise to a constant noise distribution across the entire ensemble. There is some empirical support for this notion (Chen, Geisler, & Seidemann, 2006). However, such an interpretation is very puzzling if one assumes that perceptual noise (

*σ*) were to indeed arise from the encoding neurons. Although a combination of correlated Poisson-like neurons could have statistics that differ significantly from Poisson statistics, there seems to be no way of combining such noisy signals to obtain a constant noise source. This would require a situation where the neural signal accumulates over time while noise does not, which seems to be a logical impossibility. Therefore, it is highly unlikely that the behavioral errors in contrast perception arise from the variability of the encoding sensory neurons in V1.

_{θ}*ρ*= 0). This suboptimal decoding process would overwhelm any noise from spike count or timing variability (Johnson, 1980). Such noise added during decoding would be independent of the stimulus duration or contrast. This interpretation is consistent with observations that perceptual errors are far greater than expected from averaging over many sensory neurons (Britten et al., 1993; Tolhurst, Movshon, & Dean, 1983). However, this alternative seems inconsistent with experiments that report significant choice probabilities in sensory cortical neurons (Britten et al., 1996). One possible explanation is that significant choice probabilities arise from sensory neurons that get top-down feedback from the decoding neurons. In such a case, the behavioral variability does not arise from the variability of the sensory neurons; rather, the decoding process affects the variability of the sensory neurons. There exists some evidence for this (Cumming & Nienborg, 2016). Further experiments are needed to resolve this apparent contradiction between our results and previous results regarding choice probabilities of sensory neurons.

*n*biases the response to trial

*n*+ 1, often in such a way that cannot be detected by averaging across the entire data set (as we do in Appendix 01 section “Analyzing for bias and order effects”). It is possible that this could leave a noise signature in the behavior, although extricating this bias is difficult. Methods to isolate this bias (Neri, 2010) are not compatible with adaptive tests, and further experiments would need to be pursued to determine the extent that this effects our calculated

*ρ*.

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*The Annals of Statistics**y*axis from

*percent correct*to

*percent perceived as higher*. See Figure A1 for a description and graphical representation of the differences between these two methods. Both methods are used somewhat interchangeably within the psychophysical literature, which can cause significant confusion. The following equation represents the function that our algorithm optimizes for:

*θ*is the magnitude of the stimulus. The two variables

*β*(the slope) and

*T*(the threshold) are the variables that the adaptive algorithm changes to better fit the psychometric function of the test subject. Prior probability distributions were constructed for

*β*and

*T*. An analysis of our data (shown in “Analyzing for bias and order effects”) suggested little to no bias in user responses (i.e., the user was equally likely to be correct for test stimuli both above and below the reference stimuli), so the prior for the threshold was chosen to be very narrow. Since our primary goal was to determine the slope of the curve, a very broad, nearly flat distribution was used. A copy of the code used, as well as the raw data from subjects, is available upon request.

*β*) and threshold (

*T*) of the psychometric function (Equation A1). For clarity, we will call the true values of these variables

*β*

_{true}and

*T*

_{true}. At the start of each experimental condition, the adaptive algorithm is given a prior probability distribution for

*β*and

*T*, which we will denote as

*β*

_{prior}and

*T*

_{prior}. These distributions can be interpreted as the adaptive algorithms initial “belief” (and confidence about that belief) about what

*β*

_{true}and

*T*

_{true}are (see Figure 3.1B for a graphical representation). Over the course of the experiment, the algorithm updates the prior distributions based on the responses of the subject. Essentially, the algorithm starts with a (potentially wrong) belief, and converges onto the correct answer. Here, we developed a Monte Carlo simulation to test for this convergence in a variety of conditions.

*β*

_{true}) and threshold (

*T*

_{true}) we controlled. Each trial, the simulation was presented with a numerical value (

*θ*), which represented the contrast intensity that would have been shown to a real subject.

*θ*was then inputted into the psychometric function (

*ψ*(

*θ*)), which returned the probability that the computer would respond “Yes” to the prompt “Was the second stimulus greater than the first?” A random number was generated, and if it was less than the returned value, the simulation recorded a “Yes”; otherwise it recorded “No.”

*β*

_{true}and

*T*

_{true}), as well as different prior probabilities for the adaptive algorithm (

*β*

_{prior}and

*T*

_{prior}). In Figure A2, we demonstrate several simulations on the convergence rate of

*β*. Three different simulated subjects are presented, with a wide (yet plausible) range of

*β*

_{true}. Additionally, we present two different sets of

*β*

_{prior}—one where

*β*

_{prior}is 40% less than

*β*

_{true}, and one where it is 40% more. In all cases, we show that it converges on to the true answer within 45 trials, and often before. This result extends to all realistic conditions.

*t*test, we failed to reject the null hypothesis (i.e., that the data is not biased in a statistically significant way) in the normal condition (

*p*= 0.93), the reverse condition (

*p*= 0.19), or a combination of the two conditions (

*p*= 0.37). This failure to reject suggests that there is no bias in response.

*t*test to test between the normal and reverse conditions, and failed to reject the null hypothesis (i.e., that the two conditions are statistically indistinguishable) at a

*p*= 0.26. This lends credence to the hypothesis that there is no order effect.

*t*test, no condition reaches statistical significance when corrected for multiple comparisons. When left uncorrected, one condition (150 ms × 0.125 Michelson Contrast) exceeds a traditional significance of

*α*= 0.5; this is to be expected via random chance. Another test that can be performed is comparing all the JND values directly rather than averaging across subjects first. To do this, we used a paired-sample

*t*test, which makes a direct comparison for every Subject × Time × Reference condition in the normal and reversed presentation condition. Once again, we fail to reject the null hypothesis (

*p*= 0.40), further reinforcing the notion that there is no order effect.

*r*(

*θ*), was a static function (i.e., the firing rate does not change over time). However, this is not a biologically realistic assumption; firing rates often have temporal dynamics. In this section, we demonstrate why this does not matter for our model.

*R*(

*θ*), is separable, and can be written as a multiplication of two functions:

*g*(

*τ*) that accounts for the spike count for a given window

*τ*, and

*f*(

*θ*) that modulates the spike count for different stimulus intensities. Further, we can say that for presentation times over approximately 75 ms, we can assume that the firing rate is falling [See Heller et al. (1995) or figure 8 in Albrecht et al. (2002) for evidence.] This means that

*g*(

*τ*) is less than linear, and can be approximated with the following equation:

*R*(

*θ*) function into Equation 1 from the main text, we arrive at

*σ*(

_{θ}*τ*

_{1}) and

*σ*(

_{θ}*τ*

_{sat}), we arrive at

*LJR*against log(

*τ*

_{1}), the slope (

*m*) of this line will be equivalent to the coefficient of the above equation:

*m*= −1.11 ± 0.09. From Equation A9,

*η*would have to equal 2.64 for our original hypothesis of

*ρ*= 0.58 to be true. As stated when we introduced the

*η*variable, a biologically reasonable approximation for

*η*would be 0 <

*η*< 1. This implies that, even assuming a dynamically varying firing rate, we would still estimate

*ρ*< 0. Thus, for physiologically plausible firing rate dynamics, the behavioral statistics are inconsistent with the premise of spike count variability as the source of behavioral noise.

*S*) as a proxy for spike rate (

*r*), since we are comparing stimuli that have the same duration. However, this approximation hides an assumption that the time window is perfectly estimated, which may not be the case. Certainly at the behavioral level, there is substantial literature on errors in temporal estimation (Buhusi & Meck, 2005; Church, 2003; Gibbon et al., 1984). In this section, we analyze the impact of these temporal estimation errors on stimulus magnitude estimation.

*r*=

*S*/

*τ*, where

*τ*is the presentation time of the stimulus. Assume that there are two sources of noise: noise in the spike count (Δ

*S*), and noise in the estimation of the temporal interval (Δ

*τ*). Therefore, on each trial

*S*= 〈

*S*〉 + Δ

*S*and

*τ*= 〈

*τ*〉 + Δ

*τ*, where 〈 〉 represents the expectation value. For simplicity, these calculations assume that Δ

*τ*≪

*τ*and Δ

*S*≪

*S*. We will mostly consider here the case where Δ

*S*≪ Δ

*τ*. Therefore, we obtain

*r*to obtain its variance. First, taking the square of Equation A10:

*τ*or Δ

*S*must vanish. Additionally, if the distribution of

*τ*is independent of the distribution of

*S*, terms with Δ

*τ*· Δ

*S*as coefficients must also disappear:

*S*is small, and define

*σ*=

_{T}*α*〈

*τ*〉, we obtain

*τ*, and that the spike count depends on the parameter through a tuning curve

*f*(

*θ*) such that 〈

*S*(

*θ*,

*τ*)〉 =

*τ*·

*f*(

*θ*). Using this, we get

*τ*. We will now use a modification of Equation A4 (which was used in estimating the error of the decoded variable θ) where we use the deduced rate rather than the spike count:

*S*(

*θ*,

*τ*) =

*f*(

*θ*) ·

*g*(

*τ*), where

*g*(

*τ*) is not the density, but rather the cumulative number of spikes between 0 and time

*τ*. Similar to the linear case, we can retrieve

*f*(

*θ*) by dividing the spike count with

*g*(

*τ*). Note that in the constant firing rate case

*g*(

*τ*) =

*τ*, so this formulation is equivalent to the constant firing case. Let's define this new variable

*q*=

*S*(

*τ*,

*θ*)/

*g*(

*τ*), which is an attempt to retrieve

*f*(

*θ*). Here, we will only examine the simple case where Δ

*S*is very small and can be ignored.

*τ*

^{2}〉 =

*α*

^{2}·

*τ*

^{2}, and using

*S*(

*θ*,

*τ*) =

*f*(

*θ*) ·

*g*(

*τ*) we get that

*g*(

*τ*) =

*τ*we get that this ratio is one, just like the constant firing rate case (

^{η}*η*= 0).

*τ*≪

*τ*

_{sat}does not hold, our conclusions hold for a wide range of Δ

*τ*values. This include values of Δτ up to and including

*τ*

_{sat}.

*ρ*≅ 1. Given that noise from different systems would combine in an additive way, there is no way it could combine (solely) with Poisson neural noise (

*ρ*≅ 0.5) to give us our findings (

*ρ*≅ 0).