Axiomatize This !

“Analyze This”, is a very funny 1999 sendup of the Mafia and psychiatry with Robert DeNiro and Billy Crystal.  For some reason the diagram on p. 7 of Barrett O’Neill’s book “Elementary Differential Geometry” revised 2nd edition 2006 made me think of it.

O’Neill’s  book was highly recommended by the wonderful “Visual Differential Geometry and Forms” by Tristan Needham — as “the single most clear-eyed, elegant and (ironically) modern treatment of the subject available — present company excpted !”

So O’Neill starts by defining a point  as an ordered triple of real numbers.  Then he defines R^3 as a set of such points along with the ability to add them and multiply them by another real number.

O’Neill then defines tangent vector (written v_p) as two points (p and v) in R^3 where p is the point of application (aka the tail of the tangent vector) and v as its vector part (the tip of the tangent vector).

All terribly abstract but at least clear and unambiguous until he says — “We shall always picture v_p as the arrow from point p t0 the point p + v”.

The picture is a huge leap and impossible to axiomatize (e.g. “Axiomatize This”).   Actually the (mental) picture came first and gave rise to all these definitions and axioms.

The picture is figure 1.1 on p. 7 — it’s a stick figure of a box shaped like an orange crate sitting in a drawing of R^3 with 3 orthogonal axes (none of which is or can be axiomatized).  p sits at one vertex of the box, and p + v at another.  An arrow is drawn from p to p + v (with a barb at p + v) which is then labeled v_p.  Notice also, that point v appears nowhere in the diagram.

What the definitions and axioms are trying to capture is our intuition of what a (tangent) vector really is.

So on p. 7 what are we actually doing?  We’re looking at a plane in visual R^3 with a bunch of ‘straight’ lines on it.  Photons from that plane go to our (nearly) spherical eye which clearly is no longer a plane.  My late good friend Peter Dodwell, psychology professor at Queen’s University in Ontario, told me that the retinal image actually preserves angles of the image (e.g. it’s conformal). 1,000,000 nerve fibers from each eye go back to our brain (don’t try to axiomatize them).   The information each fiber carries is far more processed than that of a single pixel (retinal photoreceptor) but that’s another story, and perhaps one that could be axiomatized with a lot of work.

100 years ago Wilder Penfield noted that blood flowing through a part of the brain which was active looked red rather than blue (because it contained more oxygen).  That’s the way the brain appears to work.  Any part of the brain doing something gets more blood flow than it needs, so it can’t possibly suck out all the oxygen the blood carries.  Decades of work and zillions researchers have studied the mechanisms by which this happens.  We know a lot more, but still not enough.

Today we don’t have to open the skull as Penfield did, but just do a special type of Magnetic Resonance Imaging (MRI) called functional MRI (fMRI) to watch changes in vessel oxygenation (or lack of it) as conscious people perform various tasks.

When we look at that simple stick figure on p. 7, roughly half of our brain lights up on fMRI, to give us the perception that that stick figure really is something in 3 dimensional space (even though it isn’t).  Axiomatizing that would require us to know what consciousness is (which we don’t) and trace it down to the activity of billions of neurons and trillions of synapses between them.

So what O’Neill is trying to do, is tie down the magnificent Gulliver which is our perception of space with Lilliputian strands of logic.

You’ve got to admire mathematicians for trying.

The pericyte controls local cerebral blood flow

Actively firing neurons get all the blood flow they need. More in fact. And this is the entire basis of functional magnetic resonance imaging (fMRI). At long, long last we may be close to understanding exactly how this happens.

Almost 100 years ago Wilder Penfield operating on unanesthetized patients with epilepsy to find the epileptic focus and remove it, noted that when a patient had a seizure on the table, veins became red, because so much blood flowed to the active area that it couldn’t absorb all the oxygen contained in the hemoglobin of the red cells, so they stayed red. Penfield was not a sadist, the brain contains no pain fibers, and so the skull could be opened using just local anesthetics. 

Exactly the same thing happens locally when neurons become active firing lots of action potentials. The functional MRI signal is due to the difference in magnetic susceptibility of the iron atom in hemoglobin when it is binding oxygen and when it isn’t.

So how does a firing neuron tell blood vessels it needs more flow?  A superb paper [ Proc. Natl. Acad. Sci. vol. 117 pp. 27022 – 27033 ’20 ]–https://www.pnas.org/content/pnas/117/43/27022.full.pdf probably explains exactly how this happens.  

The pericyte is a cell which is found outside cerebral capillaries and very small arteries.  It isn’t like a rubber band around the vessel (that’s for smooth muscle).  It’s like our bony spine with ribs coming from it, so the spine lies on the long axis of the vessel with the ribs coming down and wrapping (partially) around the vessel.

Pericytes in the brain and the retina are found primarily where two capillaries join each other according to the paper (which provides a convincing picture).

Neurons firing impulses release potassium into the extracellular space.  The endothelial cells of brain capillaries sense this and open up the inwardly rectifying potassium channel KIR2.1, exposing the outside to the resting potential of potassium which is quite negative (e. g the endothelial cell hyperpolarizes in response to neuronal activity.  The signal propagates upstream THROUGH the endothelial cells (because they are coupled together by gap junctions). 

Enter the pericytes which are electrically coupled to the underlying capillary endothelium by gap junctions, so they can receive the endothelial hyperpolarizing signal directly.  This causes the pericyte process receiving the signal to relax opening up the capillary or small artery increasing blood flow.  The authors followed this by watching intracellular calcium changes in pericytes, and noted that individual processes (ribs in the analogy above) could respond individually.  This is how a pericyte straddling the junction of two capillaries will open just the one which is hyperpolarized by neural activity.  

An incredibly elegant mechanism.  Of course with something so dramatic the work needs to be repeated. 

It is a pleasure to write something not involving the pandemic virus and our response to it.