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In vision science, cascades of Linear+Nonlinear transforms are verysuccessful in modeling a number of perceptual experiences [Carandini&Heeger12].However, the conventional literature is usually too focused on only describingthe input->output transform.  Instead, here we present the maths of such cascades beyond the forwardtransform, namely the Jacobians and the inverse. The fundamental reason forthis analytical treatment is that it offers useful insight into thepsychophysics, the physiology, and the function of the visual system. Forinstance, we show how the trends of the sensitivity (discrimination regions)and the adaptation of the receptive fields can be seen in the expression of theJacobian wrt the stimulus. This matrix also tells us which regions of thestimulus space are encoded more efficiently in multi-information terms. TheJacobian wrt the parameters shows which aspects of the model have bigger impactin the response, and hence bigger relevance. The analytic inverse impliesconditions for the response and the model to ensure decoding. From an appliedperspective, (a) the Jacobian wrt the stimulus is necessary in new experimentalmethods based on the synthesis of visual stimuli with interesting geometry, (b)the Jacobian matrices wrt the parameters are convenient to learn the model fromclassical experiments or alternative optimization goals, and (c) the inverse isa model-based alternative to blind machine-learning neural decoding that doesnot include meaningful biological information.  The theory is checked by building a derivable and invertible vision modelthat actually follows the modular program suggested by Carandini&Heeger. Tostress the generality of this modular setting we show examples where some ofthe canonical Divisive Normalization layers are substituted by equivalentlayers such as the Wilson-Cowan model at V1, or a tone-mapping model at theretina.

Derivatives and Inverse of Cascaded Linear+Nonlinear Neural Models