ON THE ANALYSIS AND INTERPRETATION OF
INHOMOGENEOUS QUADRATIC FORMS AS RECEPTIVE FIELDS
by Pietro Berkes and Laurenz Wiskott
Abstract:
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
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,
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 two-layer 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.
Available from:
http://cogprints.org/4081/
Reference:
Berkes, P. and Wiskott, L. (2005).
On the analysis and interpretation of inhomogeneous quadratic forms as
receptive fields.
Cognitive Sciences EPrint Archive (CogPrints) 4081,
http://cogprints.org/4081/ (<add date of your document download here>).
-- Dr. Laurenz Wiskott, Institute for Theoretical Biology, Berlin http://itb.biologie.hu-berlin.de/~wiskott/ l.wiskott@biologie.hu-berlin.de
|
Home Login Meetings Courses Belgium Maillist Credits Servers Links Archive
Please send comments and suggestions to
Mike Wijnants. |
|
Page last updated on © BBF 1998 all rights reserved |