G11 Contact Geometry of the Visual Cortex

Columnar structure and fiber bundles

The primary visual cortex V1 carries another kind of geometric structure besides the ones met so far. Hubel and Wiesel found a columnar structure in V1: neurons sensitive to orientation record a pair of data , where

• is a position on the retina, and
• is a line in the plane (an orientation).

So the data have a local product structure: writing for the retinal surface and for the set of lines through a point of the plane, there is a projection The slide illustrates this with the classical block picture of a piece of cortex: thin slabs (orientation columns) cut by alternating ocular dominance columns labeled L and R.

Fiber bundles

V1 as a circle bundle

The model of V1 is a bundle with base , the retinal surface, and fiber , the set of lines in the plane. Topologically is a circle, so locally V1 is a product : a circle bundle over a 2-dimensional surface. The slide shows circle bundles over a torus and over a sphere, drawn as small circles attached at points of the surface.

We will see that this leads to geometric models of V1 based on contact geometry.

Contact geometry on 3-manifolds

Plane fields and contact structures

A real 1-form on determines at each point a linear map Its kernel is either a plane or all of . If for all , then is a plane field. All plane fields are locally of the form for some 1-form .

Darboux’s theorem

Locally, all contact structures look like the standard one.

Example: the contact structure on

The slide shows two computer pictures: the orbits of the Hopf vector field, which are linked great circles, and the Hopf fibration of seen after stereographic projection to , as nested tori filled by linked circles.

Contact structures and complex manifolds

Let be a complex manifold with , with boundary of real dimension , and with complex structure on . Take a function near the boundary with (on a collar neighborhood of the boundary). The complex tangencies of form the plane field (on ), and this is a contact structure if and only if is a non-degenerate 2-form on the planes .

Contact geometry and symplectic geometry

Darboux’s theorem for symplectic forms: locally , that is in suitable coordinates (like a cotangent bundle).

• Fillability by a complex manifold is a special case: is symplectic.
• Not all contact structures are fillable by symplectic structures: if a contact structure is symplectically fillable, then it is tight (defined below).
• Note: one can always extend to a symplectic form on the cylinder , namely , but then is not .

Tight and overtwisted contact structures

The generic singularities of a characteristic foliation are of the two kinds shown below: an elliptic point, out of which all nearby leaves emanate (here curving as they leave), and a hyperbolic point, a saddle where two leaves come in, two go out, and the others pass by like hyperbolas.

Some facts about contact structures and 3-manifolds (Eliashberg, Gromov, Etnyre, Honda, Bennequin, and others):

• All 3-manifolds admit contact structures.
• Some 3-manifolds do not admit any tight contact structure (though most of them do).
• If a contact structure is symplectically fillable, then it is tight.
• A contact plane field has an Euler class , and for tight structures it satisfies a genus bound.

Legendrian and transverse knots

Every knot in a contact manifold can be continuously approximated by a Legendrian knot. In the standard contact structure on with , a Legendrian knot is recorded by its front projection to the -plane: along the knot , so the missing coordinate is the slope of the front, and fronts have cusps where an ordinary knot diagram would have vertical tangencies. Fronts look like these (a Legendrian unknot with two crossings, and a Legendrian trefoil):

Invariants of Legendrian knots are used to study contact manifolds (see the Bennequin invariants, and others).

For the standard contact structure, written here (after the change of coordinates ) as , the projections of transverse knots to the -plane cannot have segments like these:

The reason is that along a transverse knot. A vertical tangency would have and , which violates this; in the second case (the crossing) is bounded by the slope in the -plane, which decides which strand passes in front.

Any transverse knot in the standard contact structure is transversely isotopic to a closed braid.

The visual cortex as a contact bundle

Hoffman’s model

W.C. Hoffman, The visual cortex is a contact bundle (1989):

• Hubel–Wiesel microcolumns in the columnar structure of the V1 cortex exhibit both directional and areal response. Hoffman models these directional-areal response fields as contact plane directions.
• “Orientation response” refers to the directionally sensitive response field of a single cortical neuron.
• Microelectrode penetration measurements (Hubel, Wiesel) of the directional and areal response of neurons in the cat visual cortex show contact planes.

The first figure reproduced from Hoffman is Hubel and Wiesel’s drawing of electrode penetrations: a cross-section of the cat cortex (areas 17, 18, 19), several electrode tracks (numbered I–VI), and along each track the receptive fields of the cells recorded at successive sites, drawn as small rectangles marking their position and preferred orientation. The second (Hoffman’s Fig. 8) is a typical orientation response field (ORF) in the visual cortex: the neuronal firing rate response to shining a rectangular slit of light on the receptive field of a neuron whose “orientation” (i.e., directional) response is maximal in the vertical direction. The slit is shown at a sequence of orientations next to the recorded spike trains, which are densest for the vertical slit.

Relation to the shape of Gabor filters

The following pictures are from Sarti, Citti and Petitot, Functional geometry of the horizontal connectivity in the primary visual cortex (2009). Their Fig. 2 shows an in vivo registered odd receptive field (from De Angelis et al., 1995) next to a schematic representation of it as a Gabor filter: one dark lobe and one bright lobe side by side, with faint side lobes. Across the lobes, an odd Gabor filter is a sine wave under a Gaussian envelope:

Their Fig. 3 shows the odd part of Gabor filters with different orientations, and schemata of odd simple cells arranged in a hypercolumn of orientations: a vertical stack of disks, each split into a dark and a light half along a line whose angle increases up the stack. The orientation-tuned cells over one retinal point thus fill out the fiber of the bundle. (The slides then show the picture of the standard contact structure on again.)

Visual pathways and connections on fiber bundles

The next slide shows the visual pathways from the retina to the visual cortex: from the left and right eyes through the optic chiasma and the lateral geniculate nucleus to the visual cortex, with the corpus callosum connecting the two hemispheres.

• Paths (visual contours) are lifted along the visual pathways from the retina to the visual cortex.
• Patterns of “constancies” are detected (shape, size, motion, color, etc.), then higher forms (areas 18 and 19 of the human visual cortex).
• The path lifting property (from retina to cortex) is, geometrically, path lifting from the base to the total space of the fibration with fiber :
• To lift a path along the projection of a fibration, one needs to choose a horizontal direction at each point in the total space of the fibration (there is always a well-defined vertical direction). A connection determines the choice of a horizontal direction.

In the figure (after Nakahara), is the vertical subspace at a point of the total space , tangent to the fiber, and is a horizontal subspace, a complement to it; projects to the point of the base. The next figure, from Nakahara, Geometry, Topology, and Physics, shows the trivialization defined by a local section: a section over gives an identification of with under which corresponds to , and a tangent vector at is carried to .

Hoffman’s picture of path lifting to the visual cortex shows a visual contour drawn on the retinal surface, the cortical microcolumns standing over points of as fibers, and small rectangles along the fibers for the orientation response fields; the projection takes each fiber down to its point of .

Connection 1-form and contact planes

View a connection as a splitting of the exact sequence of tangent spaces of the fibration: a choice of horizontal direction at each point. It is achieved by a 1-form (scalar valued because the fiber of the circle bundle is ), while the vertical direction is .

Geometric model: the orientation response fields (ORFs) are the contact planes determined by the connection 1-form that performs the path lifting from the retina to the visual cortex.

Horizontal connectivity in the primary visual cortex

The model of Sarti, Citti and Petitot works on the product , where has coordinates and has coordinate .

The figure shows the contact planes at the points of one fiber, together with the orthogonal vector : as goes from to the plane turns around the fiber.

The 1-form relates local tangent vectors (in the lift of the retinal image) and forms integral curves, either along the contact planes (Legendrian) or transverse to them. This is the mechanism responsible for creating regular and illusory contours. (The slide shows integral curves running along the contact planes, drawn over a fan of contact planes.)

The scale variable

Add a scale variable : think of the visual-field information recorded in the lift to the visual cortex as a smeared distribution with Gaussian parameter (Gabor frames), instead of a delta function. When one recovers the geometric picture described above, with integral curves.

The geometric space is now , with coordinates . For a 2-form on , scale and set This is symplectic:

It is not a symplectic filling: blows up as , so there is no at the boundary. The form , by contrast, does restrict to on the contact planes at (though it is not closed: ).

In terms of a frame, where is the 1-form dual to the vector field , and the corresponding vector fields are (So and .) For small the contact planes are predominant; for large , the -planes. The slide shows the integral curves in the -planes (a fan of curves leaving a small plane in all the directions of that plane) and in the -planes.

Geometry of Gabor filters. Sarti–Citti–Petitot’s Fig. 7 shows an in vivo registered even receptive field (from Niell and Stryker, 2008) and a schematic representation of it as a Gabor filter, with positive sign in white and negative in black: a bright central lobe between two dark flanks. Across the lobes, an even Gabor filter is a cosine wave under a Gaussian envelope:

With both rotation and dilation, their Fig. 8 shows the two-dimensional fiber of even simple cells, obtained via rotation and dilation of the mother filter: a grid of even filters, rotated across each row and shrinking down each column.

Projection in the retinal plane. Their Fig. 12: the projection of the integral curves of the symplectic structure on the retinal plane (left) reveals the pattern of co-axial and trans-axial connections found by neurophysiological experiments (right, from Yen and Finkel (1998)). The projected curves form a bow-tie fan of arcs through a point, tangent to one direction, crossed by a straight segment in the orthogonal direction; the experimental pattern of connections shows the same arrangement.

Pinwheel structure in the visual cortex

The next pictures are from Petitot. The first shows the V1 cortex of the tree shrew Tupaia: optical imaging of the responses to stimuli at four orientations, and the resulting orientation map with different orientations coded by colors, with zooms on a regular point (the colors vary smoothly) and on a singular point (all colors meet: a pinwheel). The second draws the iso-orientation (isochromatic) lines of the V1 cortex: curves along which the preferred orientation is constant, which run into the pinwheel points.

Given a section of the fibration (an orientation map, choosing an orientation at each retinal point) one gets a surface . The idea is to read the iso-orientation curves through the characteristic (canonical) foliation of this surface, and the pinwheels in as overtwisted disks for this foliation: near a pinwheel the iso-orientation lines all run into one singular point, like the radial model foliation of an overtwisted disk.

Petitot’s picture of networks of pinwheels marks the pinwheel points on the orientation map and joins neighboring ones into a network.

• Projected down to by , one has a network of pinwheels on , via the identification of and given by .
• The fiber over each pinwheel point is .
• One can view these fibers as the (real) blowup of at the pinwheel points.

An image by Charles Staats shows the real blowup of at a point as a twisted, helicoid-like sheet over the plane, with the exceptional fiber standing over the blown-up point. Petitot’s picture of pinwheels in the base and fibers shows a square grid of pinwheels in the base and, in perspective, the stack of lines (the fiber) over each pinwheel point.

Observed relation between pinwheel structure and ocular dominance domains. In Petitot’s picture, the boundaries of the ocular dominance domains are drawn over the iso-orientation lines: the pinwheels’ iso-orientation lines cut the boundaries of the ocular dominance domains transversely and nearly orthogonally.

Contact invariants and knots in the visual cortex

There are interesting invariants of contact manifolds, such as contact homology, a Morse or Morse–Bott theory for the functional on loops in with .

• In the neuroscience setting we are especially interested in transverse and Legendrian knots that occur via the path lifting property from the retinal surface to the contact bundle .
• We want some nice invariants of contact geometry built from transverse and Legendrian knots.

Heegaard Floer invariants of Legendrian and transverse knots

These come from Lisca, Ozsváth, Stipsicz and Szabó, Heegaard Floer invariants of Legendrian knots in contact three-manifolds (2009).

• Datum: null-homologous Legendrian and transverse knots in a closed contact 3-manifold .
• Note: the null-homologous condition is meaningful in the context of neuroscience applications, since one wants closed curves to be (lifted) contours of two-dimensional surfaces.

A very sketchy overview:

• Given a null-homologous knot in a 3-manifold , there is an associated homology theory (Heegaard Floer theory) (in different versions), where is a -structure, such that isotopic knots have the same Floer homology.
• Given a null-homologous Legendrian knot , there is a construction of a class (with determined by ), and the invariant is the equivalence class under the -module structure of : the isomorphism class of the pair under -module isomorphisms.
• There is a global invariant of contact manifolds (also a class in a Floer homology , up to a flip of orientation), the Ozsváth–Szabó invariant: if then all ; if then for some large .

• If is strongly symplectically fillable, then for all null-homologous Legendrian knots .
• If is overtwisted, then for any null-homologous Legendrian knot there is a with .
• For a transverse knot in a contact 3-manifold there is an “approximation by a Legendrian knot” , and one sets .

These provide invariants to study null-homologous transverse and Legendrian knots in contact manifolds. An example of the properties of such knots that they detect: is loose if its complement contains an overtwisted disk, and non-loose if is overtwisted but the complement of is tight (this is about the position of the overtwisted disks with respect to the knot). If is loose then .

Gabor frames from contact geometry

This part follows Liontou and Marcolli, Gabor frames from contact geometry in models of the primary visual cortex (2023).

• There is an interesting interplay in the V1 cortex between the geometry describing the connectivity in terms of a contact structure and the signal analysis carried out by Gabor filters associated to the orientation-sensitive receptor fields.
• What is the mathematical structure that gives rise to signal analysis from contact geometry?
• The two seem to live in different places: Gabor analysis is a linear structure, contact geometry a nonlinear manifold structure.

Circle bundles of contact elements

• Classification (R. Lutz) of Legendrian circle bundles: either the unit cosphere bundle , or a -fold cyclic cover of it for , with the genus.
• For the V1 model, has genus zero and .
• The contact form on is determined by the tautological Liouville form on .

The tautological Liouville form

This 1-form determines the canonical symplectic form of the cotangent bundle . The restriction of the Liouville 1-form to the unit spheres in induces a contact 1-form on the cosphere bundle .

Complex structure and the twisted tautological form

The surface , the Riemann sphere, has a complex structure (also responsible for the conformal retinotopic mapping). With complex coordinate on , the induced complex structure on is .

So there are two contact structures and on , where is the restriction of , with Reeb field .

• The contact planes of these two contact structures intersect along the fiber direction, spanned by , so the fibers are Legendrian circles with respect to both.
• The Reeb field of is Legendrian for , and the Reeb field is Legendrian for .
• The complex structure fixes and exchanges and .

The bundle of signal planes

Let be the real 2-plane bundle over the contact 3-manifold given by the pullback of the tangent bundle of the surface to along the projection of the unit cosphere bundle.

• At a point , with on and on the circle fiber over , the fiber is the same as the fiber .
• Meaning: is a curved surface, is the collection of its local linear approximations , and the bundle places such linear approximations at each location in .
• Think of this as a retinotopy designed to transfer linear signal-analysis filters in the linear spaces (for signals on ) to the 3-dimensional columnar structure (with its information on orientation sensitivity ) of the V1 cortex.
• A signal on the retina is lifted to signals on the linear spaces , and then to a signal on .

The exponential map and the injectivity radius

On a Riemannian manifold one can locally map the flat linear tangent space to a curved coordinate neighborhood of the point through the exponential map. (The slide’s picture shows a curved surface with its tangent plane at a point : a vector in the tangent plane is sent to the point of the surface.)

In general is not a diffeomorphism: geodesics on can close up, and injectivity fails. This is measured by the injectivity radius.

The pathway of signals: faithfulness

• A signal is a real-valued function on the retinal surface .
• It is lifted to a collection of functions on the linear tangent spaces by pulling back through the exponential map, .
• This gives a single function with , hence by pullback a function with .
• The injectivity radius function of the exponential map describes a region where the signal on the bundle faithfully represents the retinal signal .

Lattices and Gabor frames

Consider the dual vector bundle (the bundle of linear functionals on ) The symmetry of the Reeb fields under the complex structure twist gives dual local bases and on a dense open .

(Its real and imaginary parts, and , are the even and odd Gabor profiles pictured earlier.)

The injectivity radius and the scaling function

• Empirical cutoff scale: a finite (though large) number of receptor profiles is involved in signal analysis in V1, hence only filters with contained in some ball of radius .
• Geometric cutoff scale: the injectivity radius, where moving signals between the curved and the linear stops being faithful.
• The scaling factor compares these two cutoffs: It is the intrinsic scale of the Gabor system construction for signal analysis in V1.

Symplectization and contactization

There are two constructions going back and forth between contact and symplectic manifolds:

1. a contact manifold gives a symplectic manifold (symplectization);
2. a symplectic manifold gives a contact manifold (contactization).

The variables and in the symplectic form account for the orientation dependence (through ) and for scale. The slide illustrates this with Gabor filters of several orientations and sizes, drawn as light and dark elongated ellipses.

• The symplectization of a contact manifold is an exact symplectic manifold, , so there is always a contactization of the symplectization.
• When is not exact this is sometimes still possible, via “prequantization bundles”.

The 5-dimensional contact manifold corresponds to a model in which an additional pair of dual variables is introduced, describing the phase and velocity of spatial wave propagation: orientation dependence in the 3-manifold , and additional direction/phase sensitivity in the 5-manifold . This model was proposed by Baspinar, Sarti and Citti, A sub-Riemannian model of the visual cortex with frequency and phase (2020).

References

• Jean Petitot, Neurogéométrie de la Vision, Les Éditions de l’École Polytechnique, 2008.
• John B. Etnyre, Introductory Lectures on Contact Geometry, arXiv:math/0111118.
• W.C. Hoffman, The visual cortex is a contact bundle, Applied Mathematics and Computation 32 (1989) 137–167.
• O. Ben-Shahar, S. Zucker, Geometrical Computations Explain Projection Patterns of Long-Range Horizontal Connections in Visual Cortex, Neural Computation 16, 3 (2004) 445–476.
• Alessandro Sarti, Giovanna Citti, Jean Petitot, Functional geometry of the horizontal connectivity in the primary visual cortex, Journal of Physiology - Paris 103 (2009) 3–45.
• V. Liontou, M. Marcolli, Gabor frames from contact geometry in models of the primary visual cortex, Math. Neuro. and Appl. 3 (2023), article no. 2, 1–28.
• M. Nakahara, Geometry, Topology, and Physics, CRC Press, 2003.
• S.C. Yen, L.H. Finkel, Extraction of perceptually salient contours by striate cortical networks, Vision Res. 38 (1998) no. 5, 719–741.
• Paolo Lisca, Peter Ozsváth, András I. Stipsicz, Zoltán Szabó, Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. 11 (2009) 1307–1363.
• E. Baspinar, A. Sarti, G. Citti, A sub-Riemannian model of the visual cortex with frequency and phase, The Journal of Mathematical Neuroscience 10 (2020) Article 11, 31 pp.
• De Angelis et al. (1995) and Niell and Stryker (2008), cited in figure captions from Sarti–Citti–Petitot for the in vivo receptive fields.