Derivatives and Inverse of Cascaded Linear+Nonlinear Neural Models

In vision science, cascades of Linear+Nonlinear transforms are very successful in modeling a number of perceptual experiences [Carandini&Heeger12]. However, the conventional literature is usually too focused on only describing the input->output transform. Instead, here we present the maths of such cascades beyond the forward transform, namely the Jacobians and the inverse. The fundamental reason for this analytical treatment is that it offers useful insight into the psychophysics, the physiology, and the function of the visual system. For instance, we show how the trends of the sensitivity (discrimination regions) and the adaptation of the receptive fields can be seen in the expression of the Jacobian wrt the stimulus. This matrix also tells us which regions of the stimulus space are encoded more efficiently in multi-information terms. The Jacobian wrt the parameters shows which aspects of the model have bigger impact in the response, and hence bigger relevance. The analytic inverse implies conditions for the response and the model to ensure decoding. From an applied perspective, (a) the Jacobian wrt the stimulus is necessary in new experimental methods based on the synthesis of visual stimuli with interesting geometry, (b) the Jacobian matrices wrt the parameters are convenient to learn the model from classical experiments or alternative optimization goals, and (c) the inverse is a model-based alternative to blind machine-learning neural decoding that does not include meaningful biological information. The theory is checked by building a derivable and invertible vision model that actually follows the modular program suggested by Carandini&Heeger. To stress the generality of this modular setting we show examples where some of the canonical Divisive Normalization layers are substituted by equivalent layers such as the Wilson-Cowan model at V1, or a tone-mapping model at the retina.