Cogprints

On the analysis and interpretation of inhomogeneous quadratic forms as receptive fields

Berkes, Pietro and Wiskott, Laurenz (2005) On the analysis and interpretation of inhomogeneous quadratic forms as receptive fields. [Preprint]

Full text available as:

[img]
Preview
 PDF
794Kb
[img]
Preview
 Postscript
2238Kb

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.

Item Type:Preprint
Keywords:Quadratic forms, receptive fields, nonlinear analysis, visualization
Subjects:Neuroscience > Neural Modelling
Neuroscience > Computational Neuroscience
Computer Science > Neural Nets
ID Code:4081
Deposited By: Berkes, Pietro
Deposited On:08 Feb 2005
Last Modified:11 Mar 2011 08:55

References in Article

Select the SEEK icon to attempt to find the referenced article. If it does not appear to be in cogprints you will be forwarded to the paracite service. Poorly formated references will probably not work.

Adelson, E. H. and Bergen, J. R. (1985). Spatiotemporal energy models for the perception of motion. Journal Optical Society of America A, 2(2):284-299.

Baddeley, R., Abbott, L., Booth, M., Sengpiel, F., Freeman, T., Wakeman, E., and Rolls, E. (1997). Responses of neurons in primary and inferior temporal visual cortices to natural scenes. Proc R Soc Lond B Biol Sci., 264(1389):1775-83.

Bartsch, H. and Obermayer, K. (2003). Second-order statistics of natural images. Neurocomputing, 52-54:467-472.

Berkes, P. and Wiskott, L. (2002). Applying slow feature analysis to image sequences yields a rich repertoire of complex cell properties. In Dorronsoro, J. R., editor, Artificial Neural Networks - ICANN 2002 Proceedings, Lecture Notes in Computer Science, pages 81-86. Springer.

Berkes, P. and Wiskott, L. (2003). Slow feature analysis yields a rich repertoire of complex-cell properties. Cognitive Sciences EPrint Archive (CogPrint) 2804, http://cogprints.ecs.soton.ac.uk/archive/00002804/.

Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press.

Dayan, P. and Abbott, L. F. (2001). Theoretical neuroscience: computational and mathematical modeling of neural systems. The MIT Press.

Fortin, C. (2000). A survey of the trust region subproblem within a semidefinite framework. Master's thesis, University of Waterloo.

Hashimoto, W. (2003). Quadratic forms in natural images. Network: Computation in Neural Systems, 14(4):765-788.

Hubel, D. and Wiesel, T. (1962). Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. Journal of Physiology, 160:106-154.

Hyvaerinen, A. and Hoyer, P. (2000). Emergence of phase and shift invariant features by decomposition of natural images into independent feature subspaces. Neural Computation, 12(7):1705-1720.

Hyvaerinen, A. and Hoyer, P. (2001). A two-layer sparse coding model learns simple and complex cell receptive fields and topography from natural images. Vision Research, 41(18):2413-2423.

Jones, J. P. and Palmer, L. A. (1987). An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex. Journal of Neurophysiology, 58(6):1233-1257.

Jutten, C. and Karhunen, J. (2003). Advances in nonlinear blind source separation. Proc. of the 4th Int. Symp. on Independent Component Analysis and Blind Signal Separation (ICA2003), pages 245-256.

Koerding, K., Kayser, C., Einhaeuser, W., and Koenig, P. (2004). How are complex cell properties adapted to the statistics of natural scenes? Journal of Neurophysiology, 91(1):206 212.

MacKay, D. (1985). The significance of feature sensitivity. In Rose, D. and Dobson, V., editors, Models of the visual cortex, pages 47-53. John Wiley & Sons Ltd.

Pollen, D. and Ronner, S. (1981). Phase relationship between adjacent simple cells in the visual cortex. Science, 212:1409-1411.

Rust, N. C., Schwartz, O., Moxshon, J. A., and Simoncelli, E. (2004). Spike-triggered characterization of excitatory and suppressive stimulus dimensions in monkey V1. Neurocomputing, 58 60:793-799.

Schoelkopf, B., Smola, A., and Mueller, K.-R. (1998). Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10(5):1299-1319.

Stork, D. and Levinson, J. (1982). Receptive fields and the optimal stimulus. Science, 216:204-205.

Touryan, J., Lau, B., and Dan, Y. (2002). Isolation of relevant visual features from random stimuli for cortical complex cells. Journal of Neuroscience, 22(24):10811-10818.

Walter, W. (1995). Analysis 2. Springer Verlag.

Wiskott, L. and Sejnowski, T. (2002). Slow feature analysis: Unsupervised learning of invariances. Neural Computation, 14(4):715-770.

Metadata

Repository Staff Only: item control page

Cogprints is powered by EPrints 3 which is developed by the School of Electronics and Computer Science at the University of Southampton. More information and software credits.