Tuesday, June 26, 2012

scintillating scotoma 3.a

I went ahead and figured out how to do the cortex transform. there are many papers describing the equations; some good recent ones (like this one) which are basically reviews at the same time that they are tweaking one or another aspect of the basic model. It's not really that complicated; the log-polar transform is very similar, except that the angles are calculated outside the logarithm. The space-V1 transform is the logarithm of a complex number representing the spatial coordinates plus the limit of the foveal confluence. The paper I linked above describes what further steps can be taken to get the transform more precise, accounting for meridional anisotropies. they go further, but I stopped there. The basic model was proposed by E.L. Schwartz in 1977, and hasn't changed much since then; I'm using Schira et al's version with their shear equation, and some parameters they cite in another paper.

This is similar to the second plot from the last post, but you will notice the geometry is different, as it gets narrower towards the fovea (lower part). Colors indicate time in minutes as shown by the colorbar. The grid drawn in the background isn't labeled, but it's easy to understand if you've seen these before. The lines going up and down are, from left to right, the superior vertical meridian, the left superior 45 degree meridian, the left horizontal meridian, and so on. From bottom to top, the left-right lines are spaced 5 degrees of visual angle apart. You don't see the first one until about 30mm up. The origin in this plot, (0,0), is where the foveal representation converges with V2 and V3, the foveal confluence.

This is interesting, the foveal confluence. I probably had heard of this before and forgotten it. I actually stated to E* yesterday that I didn't know what was on the other side of the foveal edge of V1, though I knew that the edges are flanked all the way around by V2. In fact, V1, V2, and V3 foveae all meet in the same place. This is apparently a relatively poorly understood region of visual cortex; imaging and physiology studies have focused on the more peripheral regions. The reason is that it can't be certain of what is being studied if one looks closely at the confluence, since the three areas are mixed together in a fashion that is still not well understood. I'm going to read more about this (the main writing on it is by the same group as the paper I cited at top; this one explains things up front).

Okay, so that map. What can I do with it, now that I have the coordinates right (or as close to right as I can)? Yes: I can measure the rate of progression of the wave in cortical distance over time. Awesome. I don't have the best method worked out just yet, but here's my approximation, summarized in the last figure below:

On the left, we have the same coordinates as in the figure above. The plotted line is the mean, over time, of the recorded scotoma regions. This is not a great measure of position of the waves, since as they got further out and larger, I couldn't trace them completely, and because it took time to trace them, so at a given epoch a trace might be in one place, or another, and that shows up here as a back-and-forth wave, on top of whatever sort of limiting bias is imposed by the screen size, etc. Still, it's okay. We know this because of the next plot: On the right, we have the distance of that waggly trace (from its starting point near the bottom of the left plot) as a function of time. A straight line. That's not why we know it's okay; it's because of the slope of this line: 2.76 mm/min. This is extremely slow, but exactly in the realm of cortical spreading depression. Not going to give references on that (need to save some work for an actual paper on this business), but they're there. Pretty sure I'm doing this right.

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