?url_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rft.title=On+the+analysis+and+interpretation+of+inhomogeneous+quadratic+forms+as+receptive+fields&rft.creator=Berkes%2C+Pietro&rft.creator=Wiskott%2C+Laurenz&rft.subject=Neural+Modelling&rft.subject=Computational+Neuroscience&rft.subject=Neural+Nets&rft.description=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%2C+making+it+particularly+suitable+for+application+to+second-order+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%2C+i.e.+the+stimuli+that+maximize+and+minimize+the+considered+quadratic+form%2C+respectively%2C+given+a+fixed+energy+constraint.+The+analysis+of+the+optimal+stimuli+is+completed+by+considering+their+invariances%2C+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%2C+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+two-layer+neural+network%2C+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.%0A&rft.date=2005-02&rft.type=Preprint&rft.type=NonPeerReviewed&rft.format=application%2Fpdf&rft.identifier=http%3A%2F%2Fcogprints.org%2F4081%2F1%2Fquadratic.pdf&rft.format=application%2Fpostscript&rft.identifier=http%3A%2F%2Fcogprints.org%2F4081%2F2%2Fquadratic.ps&rft.identifier=++Berkes%2C+Pietro+and+Wiskott%2C+Laurenz++(2005)+On+the+analysis+and+interpretation+of+inhomogeneous+quadratic+forms+as+receptive+fields.++%5BPreprint%5D+++++&rft.relation=http%3A%2F%2Fcogprints.org%2F4081%2F