### Tunnels, Funnels and Spirals

Neurofuture found an excellent (but very esoteric) lecture by mathematician Jack Cowan on the origins of various patterns of visual hallucinations as a property of noise in a particular type of neural lattice. The lecture is available here (in 64 and 256 kBit mp4 streams). Jack Cowan has dual appointments in both mathematics and neurology at the University of Chicago, and his approach to computational modeling of neural dynamics clearly falls under the umbrella of the "dynamic systems" viewpoint.

Although much of the lecture is geared towards advanced physicists and mathematicians, and I cannot claim to have understood all of it, the essential points of the lecture seem to be:

1) The retinotopic maps of V1 are actually transformed from retinal coordinates via a complex logarithm. The end result of this transformation is that concentric circles in the visual field would be represented by vertical lines in the cortex, whereas radial lines would become horizontal lines of activity in cortex.

2) The development of orientation and spatial frequency maps in V1 can be simulated with some fairly "simple" (maybe simple to Jack Cowan, but not to me!) self-organizing functions, such that "orientation and spatial frequency are the zeroth and first order spherical harmonics" and "the coefficient of the first order representation is just the dot product of the vector of feature preferences with the vector of stimulus features, where the vectors have components given by the first order spherical harmonics. Similar representation can be found for directional motion, binocular disparity, and color."

3) This theory predicts several phenomena which have been verified experimentally, such as aftereffect interference patterns when a subject views a display of white noise, after fixating a display of a single spatial frequency. This theory also predicts that, at rest, neural activity manifests a 1/f pattern of noise, which can be observed in RT data from a variety of behavioral experiments.

The bottom line of Cowan's simulations are that the functional organization of cortex is quasi-periodic, and thus shares many of the characteristics of quasi-crystals (pictured at the start of this post, and which may or may not look similar to the orientation tuning of hypercolumns in V1, pictured here). Therefore, many of the mathematical techniques used in crystal physics can apply to understanding neural noise.

Although much of the lecture is geared towards advanced physicists and mathematicians, and I cannot claim to have understood all of it, the essential points of the lecture seem to be:

1) The retinotopic maps of V1 are actually transformed from retinal coordinates via a complex logarithm. The end result of this transformation is that concentric circles in the visual field would be represented by vertical lines in the cortex, whereas radial lines would become horizontal lines of activity in cortex.

2) The development of orientation and spatial frequency maps in V1 can be simulated with some fairly "simple" (maybe simple to Jack Cowan, but not to me!) self-organizing functions, such that "orientation and spatial frequency are the zeroth and first order spherical harmonics" and "the coefficient of the first order representation is just the dot product of the vector of feature preferences with the vector of stimulus features, where the vectors have components given by the first order spherical harmonics. Similar representation can be found for directional motion, binocular disparity, and color."

3) This theory predicts several phenomena which have been verified experimentally, such as aftereffect interference patterns when a subject views a display of white noise, after fixating a display of a single spatial frequency. This theory also predicts that, at rest, neural activity manifests a 1/f pattern of noise, which can be observed in RT data from a variety of behavioral experiments.

The bottom line of Cowan's simulations are that the functional organization of cortex is quasi-periodic, and thus shares many of the characteristics of quasi-crystals (pictured at the start of this post, and which may or may not look similar to the orientation tuning of hypercolumns in V1, pictured here). Therefore, many of the mathematical techniques used in crystal physics can apply to understanding neural noise.

## 5 Comments:

Thanks for making me feel stupid.

:)

I don't get it completely either. I nearly laughed out loud at this statement: "the coefficient of the first order representation is just the dot product of the vector of feature preferences with the vector of stimulus features, where the vectors have components given by the first order spherical harmonics."

Oh, that's all? !@#!

I am not sure that sentence would make complete sense to me even if I did have a PhD in math.

That's a nice lecture. What he's talking about are harmonics in a hyperspherical space. Actually all this stuff is very similar to what Steve Grand was doing with his Lucy robot.

Oh, thanks bob. Now it's all cleared up for me.

:P

I'll have to go to school tomorrow where I have something better than a dial-up connection and check out the math, but not to be a pain, the math doesn't sound as if it requires more than undergrad math. Looking at the thumbnails of the lecture, it appears that he was addressing mathematicians rather than psychologist. I suspect that if the intended audience were psychologists, he would have simplified the math as I do.

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