In this work we study the communication efficiency of a psychophysically-tuned cascade of Wilson-Cowan and Divisive Normalization layers that simulate the retina-V1 pathway. This is the first analysis of Wilson-Cowan networks in terms of multivariate total correlation. The parameters of the cortical model have been derived through the relation between the steady state of the Wilson-Cowan model and the Divisive Normalization model.

The communication efficiency has been analyzed in two ways: First, we provide an analytical expression for the reduction of the total correlation among the responses of a V1-like population after the application of the Wilson-Cowan interaction. Second, we empirically study the efficiency with visual stimuli and statistical tools that were not available before: (1) we use a recent, radiometrically calibrated, set of natural scenes, and (2) we use a recent technique to estimate the multivariate total correlation in bits from sets of visual responses which only involves univariate operations, thus giving better estimates of the redundancy.

The theoretical and the empirical results show that although this cascade of layers was not optimized for statistical independence in any way, the redundancy between the responses gets substantially reduced along the neural pathway. Specifically, we show that (1) the efficiency of a Wilson-Cowan network is similar to its equivalent Divisive Normalization model, (2) while initial layers (Von-Kries adaptation and Weber-like brightness) contribute to univariate equalization, the bigger contributions to the reduction in total correlation come from the computation of nonlinear local contrast and the application of local oriented filters, and (3) psychophysically-tuned models are more efficient (reduce more total correlation) in the more populated regions of the luminance-contrast plane. These results are an alternative confirmation of the Efficient Coding Hypothesis for the Wilson-Cowan systems. And from an applied perspective, they suggest that

Subjective image quality databases are a major source of raw data on how the visual system works in naturalistic environments. These databases describe the sensitivity of many observers to a wide range of distortions of different nature and intensity seen on top of a variety of natural images. Data of this kind seems to open a number of

possibilities for the vision scientist to check the models in realistic scenarios. However, while these natural databases are great benchmarks for models developed in some other way (e.g., by using the well-controlled *artificial stimuli *of traditional psychophysics), they should be carefully used when trying to fit vision models. Given the high dimensionality of the image space, it is very likely that some basic phenomena are under-represented in the database. Therefore, a model fitted on these large-scale natural databases will not reproduce these under-represented basic phenomena that could otherwise be easily illustrated with well selected artificial stimuli. In this work we study a specific example of the above statement. A standard corticalmodel using wavelets and divisive normalization tuned to reproduce subjective opinion on a large image quality dataset fails to reproduce basic cross-masking. Here we outline a solution for this problem by using artificial stimuli and by proposing a modification that makes the model easier to tune. Then, we show that the modified model is still competitive in the large-scale database. Our simulations with these artificial stimuli show that when using steerable wavelets, the conventional unit norm Gaussian kernels in divisive normalization should be multiplied by high-pass filters to reproduce basic trends in masking. Basic visual phenomena may be misrepresented in large natural image datasets but this can be solved with model-interpretable stimuli. This is an additional argument* in praise of artifice* in line with Rust and Movshon (2005).

In vision science, cascades of Linear+Nonlinear transforms are very successful in modeling a number of perceptual experiences [1]. However, the conventional literature is usually too focused on only describing the forward input-output transform. Instead, in this work we present the mathematics of such cascades beyond the forward transform, namely the Jacobian matrices and the inverse. These analytic results are important for three reasons: (a) they are strictly necessary in new experimental methods based on the synthesis of visual stimuli with interesting geometrical properties, (b) they are convenient to learn the model from classical experiments or alternative goal optimization, and (c) they are a promising model-based alternative to blind machine-learning methods for neural decoding. Moreover, the statistical properties of the neural model are more intuitive by using this kind of vector formulation. The theory is checked by building and testing a vision model that actually follows the modular program suggested in [1]. Our derivable and invertible model consists of a cascade of modules that account for brightness, contrast, energy masking, and wavelet masking. To stress the generality of this modular setting we show examples where some of the canonical Divisive Normalization modules are substituted by equivalent modules such as the Wilson-Cowan interaction model [2, 3] (at the V1 cortex) or a tone-mapping model [4] (at the retina). In the Discussion we address three illustrative applications. First, we show how the Jacobian (w.r.t. the input) plays a major role in setting the model by allowing novel psychophysics based on the geometry of the neural representation (as in [5]). Second, we show how the Jacobian (w.r.t. the parameters) can be used to find the model that better reproduces classical psychophysics of image distortion. In fact, thanks to the presented derivatives, this cascade of isomorphic canonical modules has been psychophysically tuned to work together for the first time. Third, we show how the analytic inverse may improve regression-based visual brain decoding.

The Wilson-Cowan equations were originally proposed to describe the low-level dynamics of neural populations (Wilson&Cowan 1972). These equations have been extensively used in modelling the oscillations of cortical activity (Cowan et al. 2016). However, due to their low-level nature, very few works have attempted connections to higher level psychophysics (Herzog et al. 2003, Hermens et al. 2005) and, to the best of our knowledge, they have not been used to predict contrast response curves or subjective image quality. Interestingly (Bertalmío&Cowan 2009) showed that Wilson-Cowan models may lead to (high level) color constancy. Moreover, these models may have positive statistical effects similarly to Divisive Normalization, which is the canonical choice to understand contrast response (Watson&Solomon 1997, Carandini&Heeger 2012): while Divisive Normalization reduces redundancy due to predictive coding (Malo&Laparra 2010), Wilson-Cowan leads to local histogram equalization (Bertalmío 2014), another route to

increase channel capacity.

Here we show that the functional (statistical) similarities between Wilson-Cowan and Divisive Normalization actually hold and may be extended to contrast perception. Specifically, first we fitted the Wilson-Cowan model using a procedure reported for Divisive Normalization: following (Watson&Malo 2002, Laparra&Malo 2010), we maximized the correlation with human opinion in quality assessment. Secondly, we used the resulting model to predict the visibility of textured patterns on top of backgrounds of different frequencies and contrasts as in classical masking experiments. Finally, we checked the redundancy reduction of Wilson-Cowan and Divisive Normalization in the same way (as in Malo&Laparra 2010). Results show that (1) Wilson-Cowan is as good as Divisive Normalization in reproducing image distortion psychophysics, (2) Wilson-Cowan dynamics induces saturating responses that attenuate with the contrast of the background, particularly when the background resembles the test; and (3) mutual information between V1-like responses after the Wilson-Cowan interaction decreases similarly as in Divisive Normalization.

1 aBertalmío, Marcelo1 aCyriac, Praveen1 aBatard, Thomas1 aMartinez-García, Marina1 aMalo, Jesús uhttp://ip4ec.upf.edu/ContrastPerception