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In this paper we introduce some mathematical and numerical tools to analyze and interpret inhomogeneous quadratic forms. The resulting characterization is in some aspects similar to that given by experimental studies of cortical cells, making it particularly suitable for application to secondorder approximations and theoretical models of physiological receptive fields. We first discuss two ways of analyzing a quadratic form by visualizing the coefficients of its quadratic and linear term directly and by considering the eigenvectors of its quadratic term. We then present an algorithm to compute the optimal excitatory and inhibitory stimuli, i.e. the stimuli that maximize and minimize the considered quadratic form, respectively, given a fixed energy constraint. The analysis of the optimal stimuli is completed by considering their invariances, which are the transformations to which the quadratic form is most insensitive. We introduce a test to determine which of these are statistically significant. Next we propose a way to measure the relative contribution of the quadratic and linear term to the total output of the quadratic form. Furthermore, we derive simpler versions of the above techniques in the special case of a quadratic form without linear term and discuss the analysis of such functions in previous theoretical and experimental studies. In the final part of the paper we show that for each quadratic form it is possible to build an equivalent twolayer neural network, which is compatible with (but more general than) related networks used in some recent papers and with the energy model of complex cells. We show that the neural network is unique only up to an arbitrary orthogonal transformation of the excitatory and inhibitory subunits in the first layer.

Berkes
Pietro

Wiskott
Laurenz
Quadratic forms, receptive fields, nonlinear analysis, visualization
FALSE
FALSE
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 neuromod
 compneurosci
 compscineuralnets
On the analysis and interpretation of inhomogeneous quadratic forms as receptive fields
published
200502
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