Prior Expectations in Visual Speed Perception Predict Encoding Characteristics of Neurons in Area MT

Abstract

Bayesian inference provides an elegant theoretical framework for understanding the characteristic biases and discrimination thresholds in visual speed perception. However, the framework is difficult to validate because of its flexibility and the fact that suitable constraints on the structure of the sensory uncertainty have been missing. Here, we demonstrate that a Bayesian observer model constrained by efficient coding not only well explains human visual speed perception but also provides an accurate quantitative account of the tuning characteristics of neurons known for representing visual speed. Specifically, we found that the population coding accuracy for visual speed in area MT ("neural prior") is precisely predicted by the power-law, slow-speed prior extracted from fitting the Bayesian observer model to psychophysical data ("behavioral prior") to the point that the two priors are indistinguishable in a cross-validation model comparison. Our results demonstrate a quantitative validation of the Bayesian observer model constrained by efficient coding at both the behavioral and neural levels.SIGNIFICANCE STATEMENT Statistical regularities of the environment play an important role in shaping both neural representations and perceptual behavior. Most previous work addressed these two aspects independently. Here we present a quantitative validation of a theoretical framework that makes joint predictions for neural coding and behavior, based on the assumption that neural representations of sensory information are efficient but also optimally used in generating a percept. Specifically, we demonstrate that the neural tuning characteristics for visual speed in brain area MT are precisely predicted by the statistical prior expectations extracted from psychophysical data. As such, our results provide a normative link between perceptual behavior and the neural representation of sensory information in the brain.

Keywords: Bayesian model; Weber's law; efficient coding; neural representation; speed prior.

Figure 1.

Bayesian observer model constrained by efficient coding. A, We model speed perception as an efficient encoding, Bayesian decoding process (Wei and Stocker, 2012, 2015). Stimulus speed v is encoded in a noisy and resource-limited sensory measurement, m, with an encoding accuracy that is determined by the stimulus prior p(v) via the efficient coding constraint (Eq. 1). Ultimately, a percept is formed through a Bayesian decoding process that combines the likelihood p(m|v) and prior p(v) to compute the posterior p(v|m), and then selects the optimal estimate $\widehat{v}$ according to a loss function. Encoding and decoding are linked and jointly determined by the prior distribution over speed. B, Efficient coding determines the accuracy of the neural representation of visual speed (i.e., the tuning characteristics of neurons in area MT). C, Embedding the Bayesian observer within a decision process provides a model to predict psychophysical behavior in a 2AFC speed discrimination task.

Figure 2.

Extracting the behavioral prior. A, We jointly fit the Bayesian observer model to psychophysical data across all contrast and reference speed conditions (72 conditions total). Shown are a few conditions for exemplary Subject 1. Circle sizes are proportional to the number of trials at that test speed. The dashed curves are Weibull fits to each condition, and the solid blue curves represent the model prediction. See the Data availability section for instructions to create a full display of psychometric curves and model fits of all 72 conditions for individual subjects. B, Log-likelihood values of the best-fitting model for each subject using four different prior parameterizations including a power-law function, Gamma distribution, piece-wise log-linear function, and Gaussian distribution, respectively. Values are normalized to the range set by a coin flip model (lower bound) and Weibull fits to individual psychometric curves (upper bound). C, The relative BIC values for the different parameterizations as well as the original, less constrained Bayesian observer model by Stocker and Simoncelli (2006). Values are normalized to the range set by the efficient Bayesian observer model with power-law parameterization and the coin flip model (lower is better). For details, see Materials and Methods.

Figure 3.

Contrast-dependent noise. Fit parameter values are shown as a function of stimulus contrast plotted for every subject. Bold lines represent fits with a parametric description of the contrast response function of cortical neurons $h(c)={\left[r_{\max }{c}^q/\left(c^q+c_{50}^q\right)+r_{\text{base}}\right]}^{-1/2}$ (Albrecht and Hamilton, 1982; Sclar et al., 1990; Heuer and Britten, 2002).

Figure 4.

Effect of prior parameterization. Shown are the best fit prior density functions for each subject using four different parameterizations including a power-law function, a Gamma distribution, a piece-wise log-linear function, and a Gaussian distribution, respectively. Note that the Gaussian provides a relative poor fit of the data (see also Fig. 2B).

Figure 5.

Predicted contrast-dependent bias and discrimination threshold. A, Ratios of the test relative to reference speed at the PSE extracted from individual Weibull fits to the data (black) and our model (blue). Shading levels correspond to different contrast levels (0.05, 0.1, 0.2, 0.4, 0.8) of the test stimulus (darker means higher contrast). The reference stimulus has a contrast of 0.075 in the left column, and 0.5 in the right column. B, Speed discrimination thresholds, defined as the difference in stimulus speed at the 50% and 75% points of the psychometric curve, at two different contrast levels (0.075, 0.5), extracted from individual Weibull fits to the data (black) and our model (blue). Error bars indicate the 95% confidence interval across 500 bootstrap runs.

Figure 6.

Predicted Weber fraction. Shown are the predicted Weber fractions Δv/v based on the reverse-engineered behavioral priors of the four individuals and the average subject, compared with previously reported psychophysical measurements (McKee et al., 1986; De Bruyn and Orban, 1988). We can analytically show that the modified power-law prior with an exponent c₀ = –1 will predict both the constant Weber fraction at higher speed and its deviation at slow speeds (see Materials and Methods). Note that since the Weber fraction is predicted up to a factor, the predictions are scaled to the level of the data. De Bruyn and Orban (1988) also found deviations from Weber's law at extremely high speeds (256 deg/s), which is not depicted here.

Figure 7.

Extracting the neural prior. According to the efficient coding constraint (Eq. 1), the population FI should directly reflect the prior distribution of visual speed. A, Single-trial mean firing rates as a function of stimulus speed v_test (dots) shown for two example MT neurons from the dataset (Nover et al., 2005), together with their fit log-normal tuning curves (black, left y-axis) and corresponding FI (red, right y-axis) assuming a Poisson noise model. B, Histogram of the goodness of the log-normal tuning curve fit across all neurons measured by R². C, Individual FI of 25 example neurons (gray, left y-axis) and the population FI (red, right y-axis), calculated as the sum of the FI over all 480 neurons in the dataset. D, The normalized square root of population FI assuming independent Poisson noise (red), adjusted for the variance by estimating the Fano factor explicitly (orange), and the linear Fisher Information (dashed black). E, The normalized square root of population FI assuming independent Poisson noise (red) and the linear population FI based on a speed tuning-dependent, limited-range noise correlation model (Abbott and Dayan, 1999) for three levels of correlation strength as illustrated by the histogram of pairwise correlation coefficients on the right. For details, see Materials and Methods.

Figure 8.

Comparing neural and behavioral prior. A, Reverse-engineered behavioral priors of every subject and their average (dark blue) superimposed by the neural prior (red). Slope values are computed from a linear fit of the curves in log–log coordinates. B, The cross-validated log-likelihood of the model using the subject's best-fitting behavioral prior (green/blue) or the fixed neural prior (red), and the log-likelihood of the original model (Stocker and Simoncelli, 2006; gray). The log-likelihood value is normalized to the range defined by a “coin-flip” model (lower bound) and a Weibull fit to each psychometric curve (upper bound). Error bars represent ±SD across 100 validation runs according to a fivefold cross-validation procedure. For details, see Materials and Methods.