Research Project (2001-2003)

Slow feature analysis yields a rich repertoire of complex cell properties

Pietro Berkes and Laurenz Wiskott

In the project Slow feature Analysis: Unsupervised learning of invariances I have shown that the slowness principle can be used to learn translation and other invariances for whole objects in a hierarchical network model. However, the retina was only one-dimensional and the stimuli were very artificial. In this project we wanted to move to more natural stimuli, but for simplicity we first considered only one layer rather than a whole hierarchical network. Thus we applied slow feature analysis (SFA) to quasi-natural image sequences (derived from static natural images) and obtained units that were (inhomogeneous) quadratic forms (functions). We compared these with complex cells in primary visual cortex of primates and found that they have many properties in common.

Obtaining the results was fairly straightforward and a matter of weeks, really. Analyzing the units (quadratic forms) we had gotten and comparing them with physiological results was the hard part. For the analysis we have partly used traditional methods known from experiments and partly have developed new methods, see the project Analysis of inhomogeneous quadratic forms as receptive fields. The units we have obtained

Thus, our units are remarkably similar to complex cells in primary visual cortex. What we don't get is much of a localization of the receptive fields. They usually cover the whole image patch we are simulating.

tuning curves of complex cells and SFA units (66 kB)

Figure 1: Orientation tuning curves of complex cells (left, black lines) and SFA units (right, blue lines) in comparison. The response of the cells/units is plotted in radial direction as a function of the orientation of a grating shown to the cell/unit. Drifting gratings were used and responses 180° apart come from the same grating orientation but different drifting direction. Thus, the cell/unit at the bottom left is direction selective while the others are not.

SFA units and
complex cells with end- and side-inhibition (38 kB)

Figure 2: As one moves a grating into a receptive field of a complex cell (left) the response typically increases. There are some cells where the response starts to decrease again at some point (middle). Such cells are called end- or side-inhibited, depending on the orientation of the grating relative to the direction of movement. Some of our SFA units show exactly that same bahavior (right). The level to which the response dropes varies greatly within each class of cells/units so that the differences seen here are not significant.

Interestingly, the results that we get do not depend on the higher-order statistics of the images. We get virtually identical results if we start with colored noise images. What matters are the transformations that we apply to the static images to generate the image sequences. The most important one is translation.

See also the project page "Slowness as a computational principle for the visual cortex" by Pietro Berkes, which includes Matlab code for performing the simulations of this project. Python source code for SFA and several other learning algorithms written by Pietro Berkes and Tiziano Zito is available at

Relevant Publications:

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.

  1. Wiskott, L., Franzius, M., Berkes, P., and Sprekeler, H. (15. September 2007).
    Is slowness a learning principle of the visual system?
    Proc. 39th Annual European Brain and Behaviour Society, EBBS, Triest, Italy, September 15-19, eds. Alessandro Treves et al., special issue of Neural Plasticity, Article ID 23250, pp. 14-15 (abstract).
    (bibtex, abstract.html)

  2. Wiskott, L., Sprekeler, H., and Berkes, P. (29. March 2007).
    Towards an analytical derivation of complex cell receptive field properties.
    Proc. 7th Meeting of the German Neuroscience Society - 31st Göttingen Neurobiology Conference, Göttingen, March 29 - April 1, S12-2 (abstract).
    (bibtex, abstract.html)

  3. Wiskott, L. (1. February 2006).
    Is slowness a learning principle of visual cortex?
    Proc. Japan-Germany Symposium on Computational Neuroscience, Wako, Saitama, Japan, February 1-4, publ. RIKEN Brain Science Institute, p. 25, (abstract).
    (bibtex, abstract.html)

  4. Berkes, P. and Wiskott, L. (20. July 2005).
    Slow feature analysis yields a rich repertoire of complex cell properties.
    Journal of Vision, 5(6):579-602,, doi:10.1167/5.6.9.
    (bibtex, abstract.html, paper, paper.pdf, paper.pdf)

  5. Berkes, P. and Wiskott, L. (28. May 2004).
    Slow feature analysis yields a rich repertoire of complex-cell properties.
    Proc. Early Cognitive Vision Workshop, Isle Of Skye, Scotland, May 28-June 1.
    (bibtex, abstract.html)

  6. Wiskott, L. and Berkes, P. (2003).
    Is slowness a learning principle of the visual cortex?
    Proc. Jahrestagung der Deutschen Zoologischen Gesellschaft 2003, Berlin, June 9-13, special issue of Zoology, 106(4):373-382.
    (bibtex, abstract.html)

  7. Berkes, P. and Wiskott, L. (12. June 2003).
    Slow feature analysis yields a rich repertoire of complex-cell properties.
    Proc. 29th Göttingen Neurobiology Conference, Göttingen, June 12-15 (abstract).
    (bibtex, abstract.html)

  8. Berkes, P. and Wiskott, L. (4. March 2003).
    Slow feature analysis yields a rich repertoire of complex-cell properties.
    Cognitive Sciences EPrint Archive (CogPrints) 2804,
    (bibtex, abstract.html, paper, paper.pdf,, results.shtml)

  9. Berkes, P. and Wiskott, L. (27. August 2002).
    Applying slow feature analysis to image sequences yields a rich repertoire of complex cell properties.
    Proc. Int'l Conf. on Artificial Neural Networks, ICANN'02, Madrid, August 27-30, ed. José R. Dorronsoro, in series Lecture Notes in Computer Science, publ. Springer-Verlag, pp. 81-86.
    (bibtex, abstract.html, paper.pdf, results.shtml)

  10. Wiskott, L. and Berkes, P. (18. April 2002).
    Is slowness a principle for the emergence of complex cells in primary visual cortex?
    Proc. Berlin Neuroscience Forum 2002, Bad Liebenwalde, April 18-20, ed. Helmut Kettenmann, publ. Max-Delbrück-Centrum für Molekulare Medizin (MDC), Berlin, p. 43, (abstract).
    (bibtex, abstract.html)

Related Projects:

setup April 24, 2002; updated April 3, 2008
Laurenz Wiskott,