Some thoughts about neurons

Thought #1
It seems that the basis for deferred pain (eg when my rib muscles hurt and I experience heartburn, or my hip muscles hurt and I experience abdominal cramps) is that the nerve bundles coming from those voluntary muscles, pass close to other autonomic nerve bundles from the abdominal muscles, with the proximity of the bundles causing cross talk and confusion at the other end of the nerve signals, in the brain.
With cross talk being so prevalent for simple muscle->brain signaling, how could the nerves of the brain possibly work - packed in as they are. Why isn't cross-talk a problem in brain->brain signaling? The obvious answer is that there is a different kind of  'insulation'  (signal isolation) for nerves in the brain than for nerves in the peripheral system. But another possibility is that cross talk is exactly how the brain works and that lack of signal isolation is how the brain works. Might the brain use very broad band filtering? How to test this?

Thought #2
In conjunction with the previous thought, one wonders if signaling during so-called "sensation" corresponds to some sort of raw data, such that it is pure signal until it gets to the brain and becomes "information". Alternatively the sensory nerves could be creating different types of signals, mapping to meaningful differences in the information - so the brain passes and mixes but is not solely responsible for creating information.
I know that is pretty vague but I believe there is a version of cognitive theory supporting the idea that thinking and information processing begin peripherally, not just centrally. How to test this?

Thought #3
Increasingly I have come to believe that "completing the incomplete" is a primary function in cognition, supported at the level of individual nerves or, at least, very low level entities. I get this belief in its purest narrative form of Truism 7, which says: that which is blocked [read "incomplete'"] will become unblocked:

X*::X 

But I also get this belief when thinking about Berkeley's analysis of the moon illusion and the perceived magnitude being affected by dimming and de-focusing of the light [which is more extreme when the moon is at the horizon than over head]. Years ago, I noticed the same phenomenon when viewing an object through a gauzy curtain (I thought a seagull looked like an albatross) and, still later, realized the same is true when an object is viewed behind a bush or some other kind of grating that partially occludes the view. In other words, the visual boundary caused by the occlusion of an object produces a perception of object magnification.
When I think about these things, I imagine an abstract field of view, with a boundary dividing what is seen of an object in the scene, versus what is occluded - and in my imagination the incomplete side of the boundary is "hot" and uncomfortable. We don't like it, we want it to go away, we strive to remove the boundary in our thoughts. This feels very similar to truism 7 but at different scales - a bit of visual scene versus an entire expectation of story line.

Knowing that ethical behavior can (sometimes, per Bloom's psychology experiments with infants) be generated by narrative preference for Truism 7, and that the moon illusion is a byproducts of reasonably low level cognitive processing, is quite suggestive. It would mean that ethical behavior is a byproduct of a low level sensation processing requirement. This begins to put ethics and -say- visual edge detection into the same neuro-psychological framework.

Update: I find that #3 is called Friston's principle of free energy

Derivatives of the associated function

I gave this up in graduate school because I did not know how to program. I think I could do it today but cannot remember the details of the first lemmas:

Let theta be angle between tangent at t and the chord from t to t+s on a [arc length parameterized] convex curve. Let r be the distance between points at t and t+s. Let psi be the angle between the chord and the tangent at t+s.

To find nth derivative in r of cos(theta) as r->0.

Let D = d()/dr, Dn the n-th such derivative and let L = lim as r->0. The problem is to find L*Dn( cos(theta) )

Lemma 1:

D(theta) = -tan(psi)/r

D(psi) = D(theta) - kappa*sec(psi)

where kappa is the curvature at t+s

Dn+1(kappa) = -Dn(kappa)*sec(psi)

Lemma 2:

L(theta) = 0
L(psi) = PI
L(Dn(kappa)) = ki  (a definition)

One needs to apply l'Hospitals rule and do a bit of algebra to solve for each Dn(theta). I find
L(D1(theta)) = -k0/2

L(D2(theta)) = k1/3 [or is it -(k0/2)^2??]

L(D3(theta)) = k0*k1/2

I propose that the successive answers are formulas in the ki's and you need a computer to find them, lacking cleverness.

Speculation: the successive answers, given in terms of polynomials in the ki's, form a Tauberian collection of polynomials, sufficient to reconstruct the complete original kappa - and hence the curve [usually!] If I had been a smart mathematician, I would have proved "Tauberian" without the explicit calculations.

First Halloween in Woods Hole

Made my pumpkins, bought the candy, got no visitors. But I had fun with the pumpkins.

On Buzzards Bay Ave:

On Gardiner Rd:

 That's a bluefish, right?

An interesting follow up to this is that two days later, someone stole this pumpkin. I cannot imagine why, but hope it is because of the nice design.