Skip to main content

The symplectic structure of the primary visual cortex


We propose to model the functional architecture of the primary visual cortex V1 as a principal fiber bundle where the two-dimensional retinal plane is the base manifold and the secondary variables of orientation and scale constitute the vertical fibers over each point as a rotation–dilation group. The total space is endowed with a natural symplectic structure neurally implemented by long range horizontal connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures, such as the medial axis of a shape.

This is a preview of subscription content, access via your institution.


  • Bar O, Sompolinsky H and Ben-Yishai R (1995). Theory of orientation tuning in visual cortex. PNAS 92: 3844–3848

    Article  Google Scholar 

  • Ben S (2003). Geometrical computations explain projection patterns of long-range horizontal connections in visual cortex. Neural Comp 16(3): 445–476

    Google Scholar 

  • Ben S and Zucker S (2004). Geometrical computations explain projection patterns of long-range horizontal connections in visual cortex. Neural Comput 16: 445–476

    Article  Google Scholar 

  • Blum H (1973). Biological shape and visual science. J Theor Biol 38: 205–287

    Article  PubMed  CAS  Google Scholar 

  • Bosking W, Zhang Y, Schoenfield B and Fitzpatrick D (1997). Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. J Neurosci 17(6): 2112–2127

    PubMed  CAS  Google Scholar 

  • Bressloff PC, Cowan JD, Golubitsky M, Thomas PJ and Wiener M (2001). Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. Phil Trans Roy Soc B 40: 299–330

    Article  Google Scholar 

  • Bressloff PC and Cowan JD (2003). Functional geometry of local and horizontal connections in a model of V1. J Physiol (Paris) 97: 221–236

    Article  Google Scholar 

  • Carandini M and Ringach DL (1997). Predictions of a recurrent model of orientation selectivity. Vision Res 37: 3061–3071

    Article  PubMed  CAS  Google Scholar 

  • Citti G and Sarti A (2006). A cortical based model of perceptual completion in the roto-translation space. J Math Imaging Vision 24(3): 307–326

    Article  Google Scholar 

  • Das A and Gilbert CD (1995). Long range horizontal connections and their role in cortical reorganization revealed by optical recording of cat primary visual cortex. Nature 375: 780–784

    Article  PubMed  CAS  Google Scholar 

  • De Angelis GC, Ozhawa I and Freeman RD (1995). Receptive-field dynamics in the central visual pathways. Trends Neurosci 18(10): 451–458

    Article  Google Scholar 

  • Field DJ, Hayes A and Hess RF (1993). Contour integration by the human visual system: evidence for a local association Field. Vision Res 33: 173–193

    Article  PubMed  CAS  Google Scholar 

  • Fitzpatrick D (1996). The functional organization of local circuits in visual cortex: insights from the study of tree shrew striate cortex. Cerebral Cortex 6: 329–341

    Article  PubMed  CAS  Google Scholar 

  • Hubel DH (1988) Eye, brain and vision. Scientific American Library

  • Issa NP, Trepel C and Stryker MP (2000). Spatial frequency maps in cat visual cortex. J Neurosci 22(22): 8504–8514

    Google Scholar 

  • Kimia BB (2003) On the role of medial geometry in human vision. In: Petitot J, Lorenceau J (eds) Neurogeometry and visual perception. J Physiol Paris 97(2–3):155–190

  • Lund J, Fitzpatrick D and Humphrey AL (1985). The striate visual cortex of the tree shrew. In: Jones, EG and Peters, A (eds) Cerebral cortex, pp 157–205. Plenum, New York

    Google Scholar 

  • Marr D (1982). Vision. Freeman, San Francisco

    Google Scholar 

  • Mitchinson G and Crick F (1982). Long axons within the striate cortex: their distribution, orientation and patterns of connections. PNAS 79: 3661–3665

    Article  Google Scholar 

  • Miller KD, Kayser A and Priebe NJ (2001). Contrast-dependent nonlinearities arise locally in a model of contrast-invariant orientation tuning. J Neurophysiol 85: 2130–2149

    PubMed  Google Scholar 

  • Parent P and Zucker SW (1989). Trace interference, curvature consistency and curve detection. IEEE Trans Pattern Anal Machine Intell 11: 823–839

    Article  Google Scholar 

  • Petitot J (1989). Modèles morphodynamiques pour la grammaire cognitive et la sémiotique modale. RSSI Can Semiotic Assoc 9(1-2-3): 17–51

    Google Scholar 

  • Petitot J, Tondut Y (1999) Vers une Neurogeometrie. Fibrations corticales, structures de contact et contours subjectifs modaux, Mathematiques, Informatique et Sciences Humaines, vol 145. EHESS, CAMS, Paris, pp 5–101

  • Petitot J (2003) The neurogeometry of pinwheels as a sub-Riemannian contact structure. In: Petitot J, Lorenceau J (eds) Neurogeometry and visual perception. J Physiol Paris 97(2–3):265–309

  • Rockland KS and Lund JS (1982). Widespread periodic intrinsic connections in the tree shrew visual cortex. Science 215: 532–534

    Article  Google Scholar 

  • Rockland KS and Lund JS (1993). Intrinsic laminar lattice connections in the primate visual cortex. Journal Comp Neurol 216(3): 303–318

    Article  Google Scholar 

  • Shelley M, Wielaard DJ, McLaughlin D and Shapley R (2000). A neuronal network model of macaque primary visual cortex (V1): orientation selectivity and dynamics in the input layer 4C α. PNAS 97: 8087–8092

    Article  PubMed  Google Scholar 

  • Yen SC and Finkel LH (1998). Extraction of perceptually salient contours by striate cortical networks. Vision Res 38(5): 719–741

    Article  PubMed  CAS  Google Scholar 

  • Swindale NV (2004). How different feature spaces may be represented in cortical maps. Network 15: 217–242

    Article  PubMed  CAS  Google Scholar 

  • Swindale NV (2000). How many maps are there in visual cortex. Cerebral Cortex 7: 633–643

    Article  Google Scholar 

  • Ts’o D, Gilbert CD and Wiesel TN (1986). Relationships between horizontal interactions and functional architecture in cat striate cortex as revealed by cross-correlation analysis. J Neurosc 6(4): 1160–1170

    CAS  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Alessandro Sarti.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sarti, A., Citti, G. & Petitot, J. The symplectic structure of the primary visual cortex. Biol Cybern 98, 33–48 (2008).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Symplectic Form
  • Symplectic Structure
  • Contact Structure
  • Primary Visual Cortex
  • Math Image Vision