Discriminating natural image statistics from neuronal population codes

Abstract

The power law provides an efficient description of amplitude spectra of natural scenes. Psychophysical studies have shown that the forms of the amplitude spectra are clearly related to human visual performance, indicating that the statistical parameters in natural scenes are represented in the nervous system. However, the underlying neuronal computation that accounts for the perception of the natural image statistics has not been thoroughly studied. We propose a theoretical framework for neuronal encoding and decoding of the image statistics, hypothesizing the elicited population activities of spatial-frequency selective neurons observed in the early visual cortex. The model predicts that frequency-tuned neurons have asymmetric tuning curves as functions of the amplitude spectra falloffs. To investigate the ability of this neural population to encode the statistical parameters of the input images, we analyze the Fisher information of the stochastic population code, relating it to the psychophysically measured human ability to discriminate natural image statistics. The nature of discrimination thresholds suggested by the computational model is consistent with experimental data from previous studies. Of particular interest, a reported qualitative disparity between performance in fovea and parafovea can be explained based on the distributional difference over preferred frequencies of neurons in the current model. The threshold shows a peak at a small falloff parameter when the neuronal preferred spatial frequencies are narrowly distributed, whereas the threshold peak vanishes for a neural population with a more broadly distributed frequency preference. These results demonstrate that the distributional property of neuronal stimulus preference can play a crucial role in linking microscopic neurophysiological phenomena and macroscopic human behaviors.

Figure 1. Modeling of visual stimuli and neuronal responses.

(a) Models of amplitude spectra of images, where the amplitude spectrum of each image satisfies . Each curve represents a spectrum with a particular falloff parameter from −1 to 2. Contrast-determining parameter was set to 0.05 for each image. The thick curve represents the spectrum of an image with . (b) Spatial frequency tuning of model neurons within a hypercolumn. Seven example neurons with most-preferred frequencies are spaced evenly from 0.01 to 10 cycles/deg (colored from red to indigo). The tuning curves are modeled with log Gaussian functions. (c) Demonstration of population activity evoked by the images satisfying the power law with different values. Dot colors are matched to those in panel b. Each curve links the unit activities evoked by a common stimulus. The thick flat line represents the responses to the image with . (d) Hypothetical response curves as functions of , derived from activity profiles shown in panel c.

Figure 2. Empirically and theoretically obtained thresholds for discriminating falloff parameters .

(Left panel) Data extracted from Hansen and Hess . (Right panel) The discrimination thresholds predicted by the current models of multiresolutional population codes. Brown and red curves show the thresholds for fovea and parafovea, respectively.

Figure 3. Derivations of Fisher information carried by neural populations.

(a–c) Fovea. Preferred spatial frequencies of the units varied from 0.01 to 10 cycles/deg. (a) Hypothetical -tuning curves (identical to Fig. 1d). Inset illustrates the spatial frequency tuning curves and the distribution of the preferred frequencies of the model neurons (same as shown in Fig. 1b). The figure shows seven example units with preferred frequencies spaced evenly from 0.01 to 10 cycles/deg (colored from red to indigo). (b) The derivative functions of the -tuning differentiated by the falloff parameter (i.e., ). (c) The local Fisher information of the individual units (thin colored curves) and their average (i.e., information per unit; thick black curve). Colors of curves in panels b and c are matched to those in panel a. (d–f) Same as (a–c), but computed for the parafovea, where the units' preferred spatial frequencies varied from 0.01 to 1 cycles/deg. Seven example units, whose preferred frequencies are spaced evenly from 0.01 to 1 cycle/deg (colored from red to indigo).

Figure 4. Estimating neuronal distributions so that they fit the model prediction with the data.

Data are the same as in Fig. 2. (a,b) Model predictability (a) without and (b) with gain control within hypercolumn. In both models, we fitted the data by varying the numbers of neurons as the fitting parameters. (c) Data fitting by model that takes into account gain control within hypercolumn. (d) Estimated neuronal distributions using the model with gain control. The arrows above the histograms indicate the mean preferred spatial frequency of neurons within foveal (brown) or parafoveal (red) hypercolumns. We set the model parameters and in Eq. (7) as and .

Figure 5. Hypothetical response curves as functions of with model that takes into account gain control within hypercolumn.

The seven representative neurons in the foveal condition, in which the neuronal distribution is estimated as shown in Fig. 4d (brown histogram). The most-preferred frequencies of sampled neurons are spaced evenly from 0.01 to 10 cycles/deg (colored from red to indigo).

Figure 6. The unit contribution to the total Fisher information carried by the whole population, ranked according to the proportions of contribution.

For each unit, we calculated the average of unit-wise Fisher information within , and then analyzed the averaged contributions for 50 neurons having different preferred spatial frequencies between , which are the same as those used in the model fitting in Fig. 5. When compared to the case for no neuronal interaction (gray line), the models with response gain control within the hypercolumn suggest more broad distributions of information both in fovea (brown) and parafovea (red). Note the slightly different result between fovea and parafovea when considering gain control among neurons, because the distribution of cell number affects the -tuning curves of the individual units.