Hexagonal orientation maps in V1

Interesting paper from Se-Bum Paik and Dario Ringach in this month’s issue of Nature Neuroscience on the origins of the orientation map in V1. Dr. Ringach has been developing a model of V1 orientation selectivity for a number of years now, the statistical connectivity hypothesis, based on the idea that the retinotopic map in V1 is at the source of orientation selectivity. In an article in the Journal of Neurophysiology, he showed that by simply pooling from a small number of LGN afferents which are physically close (and thus have close receptive field centers) together on the retinotopic map in V1, you end up with orientation-selective subunits, as shown in the image below. Note that the resulting receptive fields are not perfect matches for V1 simple cells; in particular their aspect ratio is off by a factor 2. Rather, the hypothesis is that the orientation bias stemming from the retinotopic map seeds the development of orientation selectivity in V1 which is refined by Hebbian mechanisms.

In the latest paper, the authors contend that this statistical connectivity mechanism can explain the existence of orientation selectivity maps in V1. Orientation columns form striking patterns visible through optical imaging; an example of such an orientation map is shown at the top, taken from Kandel and Schwartz.

Perhaps the key finding in the paper is that the orientation map has a very peculiar, non-random spatial pattern. The autocorrelation of the maps, shown above, show 6 distinct peaks, across several subjects and species. That means that underneath the beautiful patterns of pinwheels and wild transitions lies an underlying hexagonal grid, not unlike the reptiles illustrated by M.C. Escher.

But where does this hexagonal map come from? The authors propose that the map is ultimately inherited from the hexagonal tiling pattern of retinal ganglion cells. To generate orientation columns, they propose that the ON- and OFF- mosaics have slightly different periods and relative angles. As you can see below, two superimposed hexagonal grids with slightly different periods and angle cause Moiré interference patterns. This is explained eloquently in the News and Views article by Spencer L Smith (incidentally, curator of the excellent Labrigger blog) and Ikuto T Smith. By the statistical connectivity hypothesis, the orientation selectivity of a simple cell is seeded by the pooling of a handful of retinal ganglion cells in close physical proximity to one another. The Moiré interference pattern coupled with statistical connectivity causes pairs of ON- and OFF- retinal ganglion cells (dipoles) of similar orientation to cluster together spatially.

With different ratios of mismatch between the periods and angles of the ON- and OFF- grids, different orientation maps are generated. In the limit of highly mismatched maps, no orientation columns are generated, yet cells still have orientation selectivity, and the authors propose this could explain the lack of orientation columns in rodents.

I think that the hexagonal bias finding is rock solid. However, the authors have not explicitly shown that their model accounts well for pinwheels, point discontinuities in the orientation map, so I’m a little skeptical about the Moiré hypothesis.

If you look at figure above, you will notice that as you go around the orientation discontinuities caused by the Moiré pattern orientation selectivity shifts by a full 360 degrees; you might call this a spin 1 pinwheel. This is shown below, at the left: notice that for every color there is the same color on the opposite side of the centerpoint (for example, red at the top and the bottom). Classic pinwheels (below, right) have 1/2 spin, however; you might dub them Fermi wheels. Only in Supplementary Figure 4 do the authors show that with noise in the hexagonal grid, Fermi wheels are generated. This does make me weary, however, that the presence of spin-1/2 pinwheels is sensitive to noise levels, the assumed size of the integration pool, and smoothing parameters.

More generally, the Moiré interference hypothesis implies that the orientation map has a very particular topological structure, of which the hexagonal tiling pattern is only one aspect. In the noiseless case, at least, it seems that the pinwheels should be all of the same sign; that is, starting from the top of the pinwheel, and going around in the counterclockwise direction, the shift in orientation selectivity should be positive (counterclockwise) for all pinwheels, or negative (clockwise) for all pinwheels, but it can’t be mixed for a given orientation map.

My second comment regards a finding from Gilbert that the orientation map and the retinotopic maps are correlated with each other in a particular way. Namely, they found, as shown below, that around pinwheels, not only does orientation shift rapidly, but the retinotopic maps shift rapidly as well. The authors of the current study have assumed smooth retinotopy, and it’s not clear how well the map structure implied by the Moiré hypothesis should be conserved in the presence of retinotopic faults.

Overall, though, I really like that the model makes many specific predictions about both the ON- and OFF mosaics in the retina and the orientation map in V1, which I’m sure will be verified or refuted in due time, and I highly recommend that you take a look at it.


Paik SB, & Ringach DL (2011). Retinal origin of orientation maps in visual cortex. Nature neuroscience, 14 (7), 919-25 PMID: 21623365

Smith SL, & Smith IT (2011). Life imitates op art. Nature neuroscience, 14 (7), 803-4 PMID: 21709673

Ringach DL (2004). Haphazard wiring of simple receptive fields and orientation columns in visual cortex. Journal of neurophysiology, 92 (1), 468-76 PMID: 14999045

Das A, & Gilbert CD (1997). Distortions of visuotopic map match orientation singularities in primary visual cortex. Nature, 387 (6633), 594-8 PMID: 9177346

8 thoughts on “Hexagonal orientation maps in V1

  1. The most obvious prediction is that there should be regular points of low orientation specificity. The evidence doesn’t support this.

      1. Move a constant sized RF to each position in the moire; the points of lowest specificity are midway *between* pinwheels.

  2. Thanks for the precisions. Indeed, the PLoS One article, which I hadn’t read in detail, deals with the Das and Gilbert finding quite nicely.

    I couldn’t find any reference to the instability of the -1 pinwheels in the Nature Neurosci article, though you say in your comment that it is noted in the article; am I crazy? I thought I had read it pretty throughly including the supp. info.

  3. Hi Patrick,

    Thanks for the coverage!

    You correctly point out the model goes beyond predicting hexagonal structure and, not surprisingly, this is the topic of our upcoming manuscript.

    Now, let me clarify: the model generates -1 and+1/2 pinwheels, with the -1 pinwheels being unstable to noise and splitting up in two -1/2 pinwheels as noted in the article. The model is in fact consistent with Das and Gilbert. See Fig 4 in our prior publication in PLoS:



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