Skip to main content
SpringerLink
Log in

Nonlinearity in Simple and Complex Cells in Early Biological Visual Systems

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

The nonlinear model put forward in Mahmoodi (J Math Imaging Vis 54(2):138–161, 2016) for early visual systems is investigated in detail in this paper to explain some nonlinear behaviours of complex and some simple cells. Nonlinear cells are modelled as systems with linear–nonlinear structures where the linear sub-unit is constructed by the layers proposed in Mahmoodi (2016) and nonlinear sub-units are the results of an axon (modelled as a transmission line) carrying a series of spikes. In this paper, the nonlinear sub-systems of complex cells are investigated in more detail to show the mechanism by which nonlinear neurons work. Here the nonlinear systems modelling nonlinear sub-units of complex cells are represented by their first- and second-order responses. Our analytical as well as numerical results show good agreements with biological recordings reported in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Adelson, E.H., Bergen, J.R.: Spatiotemporal energy models for the perception of motion. J. Opt. Soc. Am. Assoc. 2, 284–299 (1985)

    Article  Google Scholar 

  2. Azzopardi, G., Petkov, N.: A CORF computational model of a simple cell that relies on LGN input outperforms the Gabor function model. Biol. Cybern. 106, 177–189 (2012)

    Article  Google Scholar 

  3. Azzopardi, G., Rodriquez-Sanchez, A., Piater, J., Petkov, N.: A push–pull CORF model of a simple cell with antiphase inhibition improves SNR and contour detection. PLoS ONE 9(7), 1–13 (2014)

    Article  Google Scholar 

  4. Carandini, M., Fester, D.: Membrane potential and firing rate in cat primary visual cortex. J. Neurosci. 20(1), 470–484 (2000)

    Google Scholar 

  5. DeAngelis, G.C., Anzai, A.: A modern view of the classical receptive field: linear and non-linear spatiotemporal processing by V1 neurons. In: Chalupa, L.M., Werner, J.S. (eds.) The Visual Neuroscience, vol. 1, pp. 704–719. MIT Press, Cambridge (2004)

    Google Scholar 

  6. DeAngelis, G.C., Ohzawa, I., Freeman, R.D.: Spatiotemporal organization of simple cell receptive fields in the cat’s striate cortex. II. Linearity of temporal and spatial summation. J. Neurophysiol. 69(4), 1118–1135 (1993)

    Google Scholar 

  7. Duits, R., Florack, L., de Graaf, J., ter Haar, Romeny B.: On the axioms of scale space theory. J. Math. Imaging Vis. 22, 267–298 (2004)

    Article  MathSciNet  Google Scholar 

  8. Emerson, R.C., Citron, M.C., Vaughn, W.J., Klein, S.A.: Nonlinear directionally selective sub-units in complex cells of cat striate cortex. J. Neurophysiol. 58(1), 33–65 (1987)

    Google Scholar 

  9. Florack, L., ter Haar Romney, B., Koenderink, J.J., Viergever, M.: Scale and the differential structure of images. Image Vis. Comput. 10(6), 376–388 (1992)

    Article  Google Scholar 

  10. Florack, L.: Image Structure, Series in Mathematical Imaging and Vision. Springer, Dordrecht (1997)

    Book  Google Scholar 

  11. Koenderink, J.J.: Scale-time. Biol. Cybern. 58, 159–162 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Korenberg, M.J.: Identification of biological cascades of linear and static nonlinear systems. Proc. Midwest Symp. Circuit Theory 18(2), 1–9 (1973)

    Google Scholar 

  13. Korenberg, M.J., Hunter, L.W.: The identification of nonlinear biological systems: LNL cascade models. Biol. Cybern. 55, 125–134 (1986)

    MathSciNet  MATH  Google Scholar 

  14. Lindeberg, T., Fagerstrom, D.: Scale-space with causal time direction. In: Proceedings of 4th European Conference on Computer Vision, vol. 1064, pp. 229–240. Springer, Berlin (1996)

  15. Lindeberg, T.: Generalized Gaussian scale-space axiomatic comprising linear scale-space, affine scale-space and spatio-temporal scale-space. J. Math. Imaging Vis. 40, 36–81 (2011)

  16. Lindeberg, T.: A computational theory of visual receptive fields. Biol. Cybern. 107, 589–635 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mahmoodi, S.: Linear neural circuitry model for visual receptive fields. J. Math. Imaging Vis. 54(2), 138–161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mahmoodi, S., Saba, N.: Nonlinear model for complex neurons in biological visual systems. In: \(9^{th}\) International Conference on Bio-Inspired Systems and Signal Processing, Rome, 6 pages (2016)

  19. Spekreijse, H.: Rectification in the goldfish retina: analysis by sinsoidal and auxiliary stimulation. Vis. Res. 9, 1461–1472 (1969)

    Article  Google Scholar 

  20. ter Haar Romeny, B., Florack, L., Nielsen, M.: Scale-time kernels and models. In: Scale-Space and Morphology, Proceedings of Scale-Space’01, Vancouver, Canada, LNCS, Springer, Berlin (2001)

  21. ter Haar, Romeny B.: Front-End Vision and Multi-Scale Image Analysis. Springer, Dordrecht (2003)

    Google Scholar 

  22. Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations. Dover, New York (1959)

    MATH  Google Scholar 

  23. Weickert, J., Ishikawa, S., Imiya, A.: On the history of Gaussian scale-space axiomatic, Chap. 4. In: Sporring, J., Nielsen, M., Florack, L.M.J., Johansen, P. (eds.) Gaussian Scale Space Theory, vol. 8. Computational Imaging and Vision Series, vol. 8, pp. 45–59. Springer, Dordrecht, Netherlands (1997)

  24. Weickert, J., Ishikawa, S., Imiya, A.: Linear scale space has first been proposed in Japan. J. Math. Imaging Vis. 10, 237–252 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wiener, N.: Nonlinear Problems in Random Theory. MIT Press, Cambridge, Massachusetts (1958)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Electronic and Computer Science, Southampton University, Southampton, Hampshire, UK

    Sasan Mahmoodi

Authors
  1. Sasan Mahmoodi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sasan Mahmoodi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mahmoodi, S. Nonlinearity in Simple and Complex Cells in Early Biological Visual Systems. J Math Imaging Vis 58, 179–188 (2017). https://doi.org/10.1007/s10851-016-0698-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-016-0698-9

Keywords

Search

Navigation