It’s always nice, intellectually, when two apparently unrelated areas collide.

I had an experience of this sort recently with an area of mathematics — one very familiar to me — and an ostensibly completely distinct area of science.

On the one hand, contact geometry — a field of pure mathematics, pure geometry.

And on the other hand, the brain and its functioning. More particularly, the visual cortex, and how it processes incoming signals from the eyes.

Now, contact geometry has lots of applications: arguably it goes back to Huygens’ work on optics. It is closely related to thermodynamics. It is the odd-dimensional sibling of symplectic geometry, which is related to classical mechanics and almost every part of physics.

But applications to neurophysiology? Now that’s new.

Well, it’s only new to me. It’s been in the scientific literature for some time. It goes back at least to a paper from 1989:

- William C. Hoffman, “The Visual Cortex is a Contact Bundle”, Applied Mathematics and Computation 32:137-167 (1989). (Also available here.)

And the discussion below is largely based on this article:

###### What’s the connection?

Contact geometry is the study of contact structures. And a contact structure on a 3-dimensional space \(M\) consists of a plane at each point satisfying some conditions. That is, at each point in the space, we have a plane sitting there. But not just any plane at each point. The planes have to vary smoothly from point to point — having such smoothly varying planes forms a (smooth) *plane field*. But moreover, the plane field, which we can call \(\xi\), is required to be *non-integrable*.

There are various ways to explain non-integrability. To “integrate” a 2-plane field is to find a smooth surface \(S\) in space so that, at every point of \(S\), the tangent plane to \(S\) is given by the plane of \(\xi\) there. At every point \(p\) of the 2-dimensional surface \(S\), the tangent plane is a 2-dimensional plane, which we write as \(T_p S\). If we write \(\xi_p\) for the plane of \(\xi\) at the point \(p\), then the integrability condition can be written as \(\xi_p = T_p S\).

Well that’s what integrability means (roughly) — \(\xi\) is integrable if you can always find a surface tangent to \(\xi\) in this way.

But a contact structure is just the opposite: you can *never* find a surface tangent to it in this way! The planes of the plane field \(\xi\) somehow twist and turn so much that you can’t every find a surface tangent to it. You can always find a surface tangent to \(\xi\) at a single point, and you might even be able to find a surface which is tangent to \(\xi\) at *some* of its points, (perhaps even *along a curve* on \(S\)), but you’ll never be able to find a surface which is tangent to \(\xi\) at all its points.

(If you’re familiar with differential forms, then the plane field \(\xi\) can be described (locally, at least) as the kernel of a 1-form, \(\xi = \ker (\alpha)\), and then the non-integrability condition is that \(\alpha \wedge d\alpha\) is a volume form. If you’re not familiar with differential forms, don’t worry.)

Contact structures can be hard to visualise. Here is a picture of one contact structure on 3-dimensional space:

You’ll note that, if you consider going from left to right in this picture in a straight line, you can actually stay tangent to the contact planes. A curve like this is called a *Legendrian* curve. Let’s call the curve/line \(C\). But the planes *twist* around \(C\) as you travel along \(C\). This is a characteristic property of contact structures (and in fact, with a few extra technicalities, can be made into an equivalent characterisation).

Another example of a contact structure is a *projectivised tangent bundle*. Let’s say what this means. (Actually we’ll only consider one such contact structure: on the projectivised tangent bundle of a plane.)

Consider a 2-dimensional plane; let’s call it \(P\). Let’s even be concrete and call it the \(xy\)-plane, complete with coordinates. So all the points on \(P\) can be written as \((x,y)\).

Now, lay \(P\) flat on the ground, in 3-dimensional space. (More precisely, embed it into \(\mathbb{R}^3\).) We would usually denote points in 3-dimensional space by \((x,y,z)\), but I want to suggestively call the third coordinate \(\theta\), because it will denote an *angle*. In any case, the points of \(P\) now lie horizontally along \(\theta = 0\); so they lie at the points \((x,y,0)\) in 3-dimensional space.

Now in 3-dimensional space, through every point of \(P\) there is a vertical line. For instance, through the point \((1,2,0)\) of \(P\) is a line, and the points on this line are all the points of the form \((1,2,\theta)\).

And now the “projective” part of the situation comes in. Pick a point on the plane \(P\): let’s say \((1,2,0)\) again. Now consider *lines* on \(P\) through this point. There are *many* such lines; in fact, infinitely many. But we can specify a line by specifying its *direction*. And that direction can be specified by an *angle* \(\theta\). We could have various conventions to measure the angle \(\theta\), but let’s do it in the standard way: \(\theta\) is the angle (measured anticlockwise) from the positive \(x\)-direction, round to the line.

Now at each point \(p = (x,y, \theta)\) in 3-dimensional space, we’ll define a plane \(\xi_p\) as follows. The plane \(\xi_p\) contains the vertical line (i.e. in the \(\theta\) direction) through \(p\); and it also contains a horizontal line through \(p\) in the direction given by the angle \(\theta\). The result is as shown below.

Starting from \(p\) (and the plane there), if you move vertically upward you get to other points of the form \(p’ = (x,y,\theta’)\), with the same \(x,y\) coordinates but different \(\theta\) coordinates. The plane at \(p’\) still contains a vertical line, but the horizontal line has rotated from angle \(\theta\) to angle \(\theta’\). Thus, as you move upwards along a vertical curve, the planes spin around the vertical curve — just as shown in the animation.

It’s a contact structure. Indeed, you can even, if you want, *identify* the point \((x,y,\theta)\) with the line through \((x,y)\) in the plane \(P\) with direction given by \(\theta\). In this way, the points in 3-dimensional space correspond to the lines in the plane through various points, and this is the thing referred to as the “projectivised tangent bundle”. (Strictly speaking though, a line at angle \(\theta\) and a line at angle \(\theta + \pi\) point in the same direction, so we should identify points \((x,y,\theta) \sim (x,y,\theta+\pi)\).)

###### What does this have to do with the brain?

Well I’m no neurophysiologist, but the claim is that the neurons in the visual cortex can be regarded functionally as exactly this kind of contact structure. This is not to say that the neurons are planes, or spin around quite like the picture above. But it is to say that neurons in some ways, *functionally*, behave like this contact structure.

When you look at an image, the photoreceptors in your eye send signals into your brain. These signals are processed, at a low level, in your visual cortex. They are then processed at a higher level, extracting features, objects and eventually reaching the level of consciousness as the unified visual field which is part of ordinary human experience. However, here we are only interested in the lower-level processing, which extracts basic information from the image projected on the retina. This low-level processing extracts features like which areas of the visual field are light and dark, the shapes of light and dark areas, and importantly for us here, the *orientation* of any lines or curves that we see.

The particular area of interest in the visual cortex seems to be an area called “V1”. This area of the brain contains many structures. It contains several “horizontal” layers 1-6, each divided into sublayers; the most important is apparently the sublayer 4C. We’ll call this the “cortical layer”, as it’s the one important for our purposes.

Now it turns out that different points on this cortical layer relate to different points on the retina. Each point in your visual field corresponds gets projected to a different point on your retina, which (roughly speaking) connects to a different point in the cortical layer. The map from the retina to the cortical layer is called a *retinotopy*. In fact, beautifully, this map from the retina (which is a surface at the back of your eye) to the cortical layer (Which is a surface in your brain) is a map which appears to *preserve angles* (but not lengths). In other words, *the retinotopy is a conformal map*.

Even better, the cells of the cortex are organised into structures called *columns* and *hypercolumns*. Along each hypercolumn, the cells detect curves which point in the same *orientation*. So there are not only cells which are specialised to detect images arriving at particular points on your retina; there are also cells which are specialised to detect a curve at a particular *in a particular orientation*.

Functionally, then, the visual cortex behaves like a contact structure. The neurons aren’t arranged in a contact structure, but they behave like one. And this means that various processes in low-level visual processing can be understood in terms of contact geometry.

In particular, the “association field” can be understood in terms of contact geometry, as perhaps also can certain hallucinations — including those seen under the influence of psychedelics like LSD.

Well, it’s definitely the most psychedelic application of contact geometry I’ve seen.

Some further references:

- Bressloff et al, Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex, Phil. Trans. R. Soc. Lond. B (2001) 356, 299–300 (Also here.)
- David J Field, Anthony Hayes, Robert F Hess, Contour Integration by the Human Visual System: Evidence for a Local “Association Field”, Vision Res. Vol. 33, No. 2, 173–193, 1993. (Also here.)
- Alessandro Sarti, Giovanna Citti, Jean Petitot, The symplectic structure of the primary visual cortex, Biol Cybern (2008) 98:33–48