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From receptive profiles to a metric model of V1

  • Noemi MontobbioEmail author
  • Giovanna Citti
  • Alessandro Sarti
Article
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Abstract

In this work we show how to construct connectivity kernels induced by the receptive profiles of simple cells of the primary visual cortex (V1). These kernels are directly defined by the shape of such profiles: this provides a metric model for the functional architecture of V1, whose global geometry is determined by the reciprocal interactions between local elements. Our construction adapts to any bank of filters chosen to represent a set of receptive profiles, since it does not require any structure on the parameterization of the family. The connectivity kernel that we define carries a geometrical structure consistent with the well-known properties of long-range horizontal connections in V1, and it is compatible with the perceptual rules synthesized by the concept of association field. These characteristics are still present when the kernel is constructed from a bank of filters arising from an unsupervised learning algorithm.

Keywords

Visual cortex Connectivity kernel Functional architecture Metric space Association field Neurogeometry 

Notes

Acknowledgements

The authors have been supported by Horizon 2020 Project ref. 777822: GHAIA, PRIN 2015 “Variational and perturbative aspects of nonlinear differential problems”, and European Union’s Seventh Framework Programme ref. 607643: MAnET.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.

Supplementary material

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References

  1. Abbasi-Sureshjani, S., Favali, M., Citti, G., Sarti, A., ter Haar Romeny, B. M. (2016). Curvature integration in a 5D kernel for extracting vessel connections in retinal images. IEEE Transactions on Image Processing, 2018(27), 606–621.Google Scholar
  2. Angelucci, A., Levitt, J. B., Walton, E., Hup, J. M., Bullier, J., Lund, J.S. (2002). Circuits for local and global signal integration in primary visual cortex. The Journal of Neuroscience, 22, 8633–8646.Google Scholar
  3. Angelucci, A., & Bullier, J. (2003). Reaching beyond the classical receptive field of V1 neurons: Horizontal or feedback axons?. Journal of Physiology Paris, 97, 141–154.Google Scholar
  4. Anselmi, F., & Poggio, T. (2010). Representation learning in sensory cortex: a theory. CBMM memo n. 26.Google Scholar
  5. Antoine, J. -P., & Murenzi, R. (1996). Two-dimensional directional wavelets and the scale-angle representation. Signal Processing, 52(3), 241–272.Google Scholar
  6. Aronszajn, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc., 66, 937–404.Google Scholar
  7. August, J., & Zucker, S. W. (2000). The curve indicator random field: Curve organization via edge correlation. In Boyer, K., & Sarkar, S. (Eds.) Perceptual organization for artificial vision systems. Boston: Kluwer Academic.Google Scholar
  8. Barbieri, D., Citti, G., Sanguinetti, G., Sarti, A. (2014). An uncertainty principle underlying the functional architecture of V1. Journal of Physiology Paris, 106(5-6), 183–193.Google Scholar
  9. Barbieri, D., Cocci, G., Citti, G., Sarti, A. (2014). A cortical-inspired geometry for contour perception and motion integration. J. Math. Imaging Vis., 49(3), 511–529.Google Scholar
  10. Bekkers, E. J., Lafarge, M. W., Veta, M., Eppenhof, K. A. J., Pluim, J. P. W., Duits, R. (2018). Roto-translation covariant convolutional networks for medical image analysis. In Schnabel, J. A., Davatzikos, C., Alberola-López, C., Fichtinger, G., Frangi, A. F. (Eds.) Medical image computing and computer assisted intervention - MICCAI 2018 - 21st International Conference, 2018, Proceedings (pp. 440-448). (Lecture Notes in Computer Science; Vol. 11070).Google Scholar
  11. Ben-Shahar, O., Huggins, P., Izo, T., Zucker, S. W. (2003). Cortical connections and early visual function: intra- and inter-columnar processing. J. Physiol. Paris., 97(2-3), 191–208.Google Scholar
  12. Ben-Shahar, O., & Zucker, S. (2004). Geometrical computations explain projection patterns of long-range horizontal connections in visual cortex. Neural Computation, 16(3), 445–476.Google Scholar
  13. Bonfiglioli, A., Lanconelli, E., Uguzzoni, F. (2007). Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Math. Berlin: Springer.Google Scholar
  14. Boscain, U., Chertovskih, R., Gauthier, J. P., Remizov, A. (2014). Hypoelliptic diffusion and human vision: a semi-discrete new twist. SIAM Journal on Imaging Sciences, 7(2), 669–695.Google Scholar
  15. Bosking, W., Zhang, Y., Schoenfield, B., Fitzpatrick, D. (1997). Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. Journal of Neuroscience, 17(6), 2112–2127.Google Scholar
  16. Bressloff, P. C., & Cowan, J. D. (2003). The functional geometry of local and long-range connections in a model of V1. J. Physiol. Paris, 97(2-3), 221–236.Google Scholar
  17. Bressloff, P. C., Cowan, J. D., Golubitsky, M., Thomas, P. J., Wiener, M. C. (2002). What Geometric Visual Hallucinations Tell Us about the Visual Cortex. Neural Computation, 14, 473–491.Google Scholar
  18. Citti, G., & Sarti, A. (2006). A Cortical Based Model of Perceptual Completion in the Roto-Translation Space. Journal of Mathematical Imaging and Vision archive, 24(3), 307–326.Google Scholar
  19. Cohen, T., & Welling, M. (2016). Group equivariant convolutional networks. Int. Conf. on Machine Learning, 2990–2999.Google Scholar
  20. Cocci, G., Barbieri, D., Sarti, A. (2012). Spatiotemporal receptive fields of cells in V1 are optimally shaped for stimulus velocity estimation. Journal of the Optical Society of America. A, vol. 29, no. 1.Google Scholar
  21. Coifman, R. R., & Lafon, S. (2006). Diffusion maps. Applied and Computational Harmonic Analysis, 21, 5–30.Google Scholar
  22. Daugman, J. G. (1985). Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. Journal of the Optical Society of America, A2, 1160–1169.Google Scholar
  23. Deng, C. -X., Li, S., Fu, Z. -X. (2010). The reproducing kernel Hilbert space based on wavelet transform, Proceedings of the 2010 international conference on wavelet analysis and pattern recognition. Qingdao, 370–374.Google Scholar
  24. Dobbins, A., Zucker, S., Cynader, M. (1987). Endstopped neurons in the visual cortex as a substrate for calculating curvature. Nature, 329(6138), 438–441.Google Scholar
  25. Duits, R. (2005). Perceptual organization in image analysis: A mathematical approach based on scale orientation and curvature. Phd thesis: Eindhoven University of Technology.Google Scholar
  26. Duits, R., Felsberg, M., Granlund, G., ter Haar Romeny, B. M. (2007). Image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the euclidean motion group. International Journal of Computer Vision, 72(1), 79–102.Google Scholar
  27. Duits, R., & Franken, E. M. (2010a). Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores, Part I: Linear Left-Invariant Diffusion Equations on SE(2). Quarterly of Applied Mathematics, 68, 293–331.Google Scholar
  28. Duits, R., & Franken, E. M. (2010b). Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores, Part II: Nonlinear left-invariant diffusions on invertible orientation scores. Quarterly of Applied Mathematics, 68, 255–292.Google Scholar
  29. Duits, R., Führ, H., Janssen, B., Bruurmijn, M., Florack, L., van Assen, H. (2013). Evolution Equations on Gabor Transforms and their Applications. Applied and Computational Harmonic Analysis, 35(3), 483–526.Google Scholar
  30. Duits, R., Boscain, U., Rossi, F., Sachkov, Y. (2014). Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2). Journal of Mathematical Imaging and Vision, 49(2), 384–417.Google Scholar
  31. Elder, J. H., & Goldberg, R. M. (2002). Ecological statistics of Gestalt laws for the perceptual organization of contours. Journal of Vision, 2(4), 5,324–353.Google Scholar
  32. Favali, M., Citti, G., Sarti, A. (2017). Local and Global Gestalt Laws: A Neurally Based Spectral Approach. Neural Computation, 29(2), 394–422.Google Scholar
  33. Federer, H. (1969). Geometric measure theory. Berlin: Springer-Verlag.Google Scholar
  34. Field, D. J., Hayes, A., Hess, R. F. (1993). Contour integration by the human visual system: evidence for a local association field. Vision Res, 33, 173–193.Google Scholar
  35. Geisler, W. S., Perry, J. S., Super, B. J., Gallogly, D. P. (2001). Edge co-occurrence in natural images predicts contour grouping performance. Vision Research, 41, 711–724.Google Scholar
  36. Gilbert, C. D., Das, A., Ito, M., Kapadia, M., Westheimer, G. (1996). Spatial integration and cortical dynamics. Proceedings of the National Academy of Sciences USA, 93, 615–622.Google Scholar
  37. Gilbert, C. D., & Wu, L. (2013). Top-down influences on visual processing. Nature Reviews Neuroscience, 14, 350–363.Google Scholar
  38. Grossberg, S., & Mingolla, E. (1985). Neural dynamics of perceptual grouping: Textures, boundaries, and emergent segmentations. Perception & Psychophysics, 38(2), 141–171.Google Scholar
  39. Hansen, T., & Neumann, H. (2008). A recurrent model of contour integration in primary visual cortex. J. of Vision, 8(8), 1–25.Google Scholar
  40. Hausdorff, F. (1918). Dimension und ausseres Mass̈. Mathematische Annalen, 79(1-2), 157–179.Google Scholar
  41. Hubel, D. H., & Wiesel, T. N. (1962). Receptive fields, binocular interaction and functional architecture in the cat visual cortex. J. Physiol. (London), 160, 106–154.Google Scholar
  42. Jones, J. P., & 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, 1233–1258.Google Scholar
  43. Kapadia, M. K., Westheimer, G., Gilbert, C. D. (1999). Dynamics of spatial summation in primary visual cortex of alert monkeys. Proc Natl Acad Sci USA, 96, 12073–12078.Google Scholar
  44. Karas, P., & Svoboda, D. (2013). Algorithms for efficient computation of convolution, in Design and Architectures for Digital Signal Processing, InTech.Google Scholar
  45. Kruger, N. (1998). Collinearity and parallelism are statistically significant second order relations of complex cell responses. Neural Processing Letters, 8, 117–129.Google Scholar
  46. Lawlor, M., & Zucker, S. W. (2013). Third-order edge statistics: contour continuation, curvature, cortical connections. In Burges, C. J. C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K. Q. (Eds.) Advances in neural information processing systems 26, (Vol. 26 pp. 1763–1771). Red Hook: Curran Associates, Inc.Google Scholar
  47. LeCun, Y., Bengio, Y., Hinton, G. (2015). Deep learning. Nature, 521, 436–444.Google Scholar
  48. Lee, H., Battle, A., Raina, R., Ng, A. Y. (2007). Efficient sparse coding algorithms. In Proceedings of the 19th annual conference on neural information processing systems (pp. 801–808). Cambridge: MIT Press.Google Scholar
  49. Lee, T. S. (1996). Image Representation Using 2D Gabor Wavelets. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 10.Google Scholar
  50. Liang, M., & Hu, X. (2015). Recurrent convolutional neural network for object recognition, CVPR.Google Scholar
  51. Martinez, L. M., & Alonso, J. -M. (2003). Complex receptive fields in primary visual cortex. The Neuroscientist, 9(5), 317–331.Google Scholar
  52. Mitchison, G., & Crick, F. (1982). Long axons within the striate cortex: their distribution, orientation, and patterns of connection. Proceedings of National Academy of Sciences USA, 79, 3661–3665.Google Scholar
  53. Montobbio, N., Sarti, A., Citti, G. (2017). A metric model for the functional architecture of the visual cortex (submitted).Google Scholar
  54. Montgomery, R. (2002). A tour of subriemannian geometries, their geodesics and applications, Mathematical surveys and monographs, Vol. 91 American mathematical society, Providence, RI.Google Scholar
  55. Mumford, D. (1993). Elastica and computer vision. In Bajaj, C (Ed.) Algebraic geometry and its applications (pp. 507–518). Berlin: Springer-Verlag.Google Scholar
  56. Neumann, H., & Mingolla, E. (2001). Computational neural models of spatial integration in perceptual grouping. In Shipley, T.F., & Kellman, P. J. (Eds.) Advances in psychology, 130, From fragments to objects: Segmentation and grouping in vision, 353– 400.Google Scholar
  57. Olshausen, B. A., & Field, D. J. (1996). Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381, 607–609.Google Scholar
  58. Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Lerer, A.A. (2017). Automatic differentiation in pytorch, in NIPS-W.Google Scholar
  59. Petitot, J. (2008). Neurogeométrie de la vision - Modèles mathématiques et physiques des architectures fonctionnelleś, Éditions de l’École Polytechnique.Google Scholar
  60. Petitot, J., & Tondut, Y. (1999). Vers une neuro-geométrie. Fibrations corticales, structures de contact et contours subjectifs modaux́, Mathématiques, Informatique et Sciences Humaines, vol. 145, 5–101. EHESS, Paris.Google Scholar
  61. Sanguinetti, G., Citti, G., Sarti, A. (2010). A model of natural image edge co-occurrence in the rototranslation group. Journal of Vision 10(14).Google Scholar
  62. Sarti, A., & Citti, G. (2015). The constitution of visual perceptual units in the functional architecture of V1. Journal of Computational Neuroscience, 38(2), 285–300.Google Scholar
  63. Sarti, A., Citti, G., Petitot, J. (2008). The symplectic structure of the visual cortex. Biological Cybernetics, 98(1), 33–48.Google Scholar
  64. Sifre, L., & Mallat, S. (2013). Rotation, scaling and deformation invariant scattering for texture discrimination, CVPR, IEEE 1233–1240.Google Scholar
  65. Sigman, M., Cecchi, G. A., Gilbert, C. D., Magnasco, M. O. (2001). On a common circle: Natural scenes and Gestalt rules. Proceedings of the National Academy of Sciences, 98(4), 1935–1940.Google Scholar
  66. Spoerer, C. J., McClure, P., Kriegeskorte, N. (2017). Recurrent convolutional neural networks: a better model of biological object recognition. Frontiers in psychology, 8, 1551.Google Scholar
  67. Sturm, K. -T. (1995). On the geometry defined by Dirichlet forms. In Bolthausen, E. et al. (Eds.) Seminar on stochastic analysis, random fields and applications (pp. 231–242). Boston: Birkhäuser.Google Scholar
  68. Sturm, K. -T. (1998). Diffusion processes and heat kernels on metric spaces. Annals of Probability, 26(1), 1–55.Google Scholar
  69. Vedaldi, A., & Lenc, K. (2015). MatConvNet - convolutional neural networks for MATLAB, Proc. of the ACM Int. Conf. on Multimedia.Google Scholar
  70. Worrall, D. E., Garbin, S. J., Turmukhambetov, D., Brostow, G. J. (2017). Harmonic networks: Deep translation and rotation equivariance. CVPR 5028–5037.Google Scholar
  71. Yen, S. C., & Finkel, L. H. (1998). Extraction of perceptually salient contours by striate cortical networks. Vision Res, 38(5), 719–741.Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversità di BolognaBolognaItaly
  2. 2.CAMS Center of Mathematical Analysis (CNRS - EHESS)ParisFrance

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