Research Project (2002-2004)
Analysis of inhomogeneous quadratic forms as receptive fields
Pietro Berkes and Laurenz Wiskott
In theoretical as well as experimental studies receptive fields are
often formalized as linear or (inhomogeneous) quadratic forms (functions).
While the interpretation and visualization of a linear receptive field is
trivial and straightforward, it gets more complicated for quadratic forms.
In context of our project on the selforganization of receptive fields with
SFA we have developed different methods of visualizing the functional
properties of a quadratic form. Two of them are particularly useful and
generally applicable, namely those yielding the optimal stimuli and the
Given a quadratic Form y(x) = 1/2 xT
H x + fT x + c
is the input image written as a vector and y is the output activity of the
unit, while H is a square matrix, f is a vector, and c is a
scalar. If y(x) is supposed to represent a receptive field, the
first question one might ask is "What is the optimal stimulus that drives
this unit best?". To make this question meaningful, one has to constrain
x somehow, because otherwise one could generate arbitrarily high
output activities by simply making x large enough (if x
points in a direction where the quadratic term has a positive value). If
we constrain the norm of x to a certain value that is typical for
the input stimuli, the optimal excitatory stimulus with that norm can be
calculated. For the quadratic forms resulting from applying SFA to image
sequences we get optimal stimuli that look like Gabor wavelets (Fig. 1),
which is in nice agreement with physiological results for complex cells in
primary visual cortex. One can similarly calculate the optimal inhibitory
stimuli, which in our case also look like Gabor wavelets (not shown).
Figure 1: Optimal stimuli of quadratic forms that were
selforganized by applying SFA to quasi-natural image sequences. Six out of
hundred units ordered by slowness (slowest first) are shown.
The optimal stimulus only gives partial information about the quadratic
form. The next question one might ask is "How can I vary the optimal
stimulus such that the output decreases as little as possible?", which is
the question of invariances. Again the norm of x must be fixed to
the set value. A single unit usually has several (almost) invariant
directions. Interestingly these directions can often be interpreted quite
nicely. Animation 1 shows the first five invariances of unit # 14.
Animation 1: First five invariances of unit # 14. They can be
interpreted as phase, rotation, shift, curvature, and size invariance.
We have also considered other ways of visualizing quadratic forms, such
as Volterra-kernels and eigenvectors of matrix H and discuss the pro
See also the project page "On the
analysis and interpretation of inhomogeneous quadratic forms as receptive
fields" by Pietro Berkes, which includes Matlab code for the
methods we have developed.
Black colored reference are the principal
ones. Gray colored references are listed for the
sake of completeness only. They contain little additional
information. .ps-files are optimized for printing; .pdf-files are
optimized for viewing at the computer.
Berkes, P. and Wiskott, L. (1. February 2007).
Analysis and interpretation of quadratic models of receptive fields.
Nature Protocols 2(2):400-407.
Berkes, P. and Wiskott, L. (August 2006).
On the analysis and interpretation of inhomogeneous quadratic forms as receptive fields.
Neural Computation, 18(8):1868-1895.
(bibtex, abstract.html, paper.pdf)
Berkes, P. and Wiskott, L. (17. March 2005).
Analysis of inhomogeneous quadratic forms for physiological and theoretical studies.
Proc. Computational and Systems Neuroscience, COSYNE'05, Salk Lake City, Utah, March 17-20, (abstract).
Berkes, P. and Wiskott, L. (8. February 2005).
On the analysis and interpretation of inhomogeneous quadratic forms as
Cognitive Sciences EPrint Archive
(CogPrints) 4081, http://cogprints.org/4081/.
setup February 7, 2005; updated August 16, 2006
Laurenz Wiskott, http://www.neuroinformatik.ruhr-uni-bochum.de/PEOPLE/wiskott/