Optimal coding through divisive normalization models of V1 neurons

Current models of the primary visual cortex (V 1) include a linear filtering stage followed by a gain control mechanism that explains some of the nonlinear behaviour of neurons. This nonlinear stage consists of a divisive normalization in which each linear response is squared and then divided by a weighted sum of squared linear responses in a certain neighbourhood plus a constant. Simoncelli and Schwartz (1999 Adv. Neural Inform. Process. Syst. 11 153-9) have suggested that divisive normalization reduces the statistical dependence between neuron responses when the weights are adapted to the statistics of natural images, which is consistent with the efficient coding hypothesis. Nevertheless, there are still important open issues, such as, for example, how to obtain the values for the parameters that minimize statistical dependence? Does divisive normalization give a total independence between responses? In this paper, we present the general mathematical formulation of the first of these two questions. We arrive at an expression which permits us to compute, numerically, the parameters of a quasi-optimal solution adapted to an input set of natural images. This quasi-optimal solution is based on a Gaussian model of the conditional statistics of the coefficients resulting from projecting natural images onto an orthogonal linear basis. Our results show, in general, lower values of mutual information, that is, responses are more independent than those provided by previous approximations.