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.

Visual Differential Geometry and Forms — Take 3

Visual Differential Geometry and Forms is a terrific math book about which I’ve written

here — https://luysii.wordpress.com/2021/07/12/a-premature-book-review-and-a-60-year-history-with-complex-variables-in-4-acts/

and

here — https://luysii.wordpress.com/2021/12/04/a-book-worth-buying-for-the-preface-alone-or-how-to-review-a-book-you-havent-read/

but first some neurology.

In the old days we neurologists figured out what the brain was doing by studying what was lost when parts of the brain were destroyed (usually by strokes, but sometimes by tumors or trauma).  Not terribly logical, as pulling the plug on a lamp plunges you in darkness, but the plug has nothing to do with how the lightbulb or LED produces light.  It was clear that the occipital lobe was important — destroy it on both sides and you are blind — https://en.wikipedia.org/wiki/Occipital_lobe but it’s only 10% of the gray matter of the cerebral cortex.

The information flowing into your brain from your eyes is enormous.  The optic nerve connecting the eyeball to the brain has a million fibers, and they can fire ‘up to 500 times a second.  If each firing (nerve impulse) is a bit, then that’s an information flow into your brain of a gigaBit/second.   This information is highly processed by the neurons and receptors in the 10 layers of the retina.  There are many different cell types — cells responding to color, to movement in one direction, to a light stimulus turning on, to a light stimulus turning off, etc. etc.   Over 30 cell types have been described, each responding to a different aspect of the visual stimulus.

So how does the relatively small occipital lobe deal with this? It doesn’t.At least half of your the brain responds to visual stimuli.  How do we know?   It’s complicated, but something called functional Magnetic Resonance Imaging (fMRI) picks up increased neuronal activity (primarily by the increase in blood flow it causes).

Given that half your brain is processing what you see, it makes sense to use it to ‘see’ what’s going on in Mathematics.  This is where Tristan Needham’s books come in.

If you’ve studied math at the college level with some calculus you shouldn’t have much trouble.  But you definitely need to look at Euclid as it’s used heavily throughout. Much use is made of similar triangles to derive relationships.

I’ll assume you’ve read the first two posts mentioned above.  Needham’s description of curvature and torsion of curves in 3 dimensional space is terrific.  They play a huge role in relativity, and I was able to mouth the formulas for them but they remained incomprehensible to me, as they are just symbols on the page.  Hopefully the discussion further on in the book will let me ‘see’ what they are when it comes to tensors.

He does skip about a bit passing Euclid when he tell how people living t on a 2 dimensional surface could tell it wasn’t flat (p. 19).  It involves numerically measuring the circumference of a circle.  But this involves a metric and putting numbers on lines, something Euclid never did.

Things really get interesting when he started talking about how Newton found the center of curvature of an arbitrary curve.  Typically Needham doesn’t really define curve, something obvious to a geometer, but it’s clear the curve is continuous.  Later he lets it slip that the curve is differentiable (without saying so).

So what did Newton do?   Start with a point p and find its normal line.  Then find point q near p on the curve and find its normal line and see where they intersect.  The center of curvature at p is the point of intersection of the normals as the points get closer and closer to p.

This made wonder how Newton could find the normal to an arbitrary continuous curve.  It would be easy if he knew the tangents, because Euclid very early on (Proposition 11 Book 1) tells you how to construct the perpendicular to a straight line.  It is easy for Euclid to find the tangent to a circle at point p — it’s just the perpendicular to the line formed between the center of the circle (where you put one point of the compass used by Newton) and the circle itself (the other point of the compass.

But how does Newton find the tangent to an arbitrary continuous curve?  I couldn’t find any place that he did it, but calc. 101 says that you just find the limit of secants ending at p as the other point gets closer and closer.  Clearly this is a derivative of sorts.

Finally Needham tells you that his curves in 3 dimensions are differentiable in the following oblique way.  On p. 106 he says that “each infinitesimal segment (of a curve) nevertheless lies in a plane.”  This tells you that the curve has a tangent, and a normal to it at 90 degrees (but not necessarily in the plane).  So it must be differentiable (oblique no?).   On p. 107 he differentiates the tangent in the infinitesimal plane to get the principal normal (which DOES  lie in the plane).  Shades of Bishop Berkeley (form Berkeley California is named) — differentiating the ghost of a departed object.

Addendum: 28 March ’22:  It’s was impossible for me to find a definition of surface even reading the first 164 pages.  Needham highly recommends a book I own “Elementary Differential Geometry (revised 2nd edition) by Barrett O’Neill calling it “the single most clear-eyed elegant, and (ironically) modern treatment of the subject  . . .  at the undergraduate level”.  The first edition was 1966.   In the preface to his book O’Neill says “One weakness of classical differential geometry is its lack of any adequate definition of surface”.    No wonder I had trouble.

So it’s great fun going through the book, get to “Einstein’s Curved Spacetime” Chapter 30 p. 307, “The Einstein FIeld Equation (with Matter) in Geometrical Form” complete with picture p. 326.

More Later.

Functional MRI research is a scientific sewer

First a primer about the science underlying functional Magnetic Resonance Imaging (fMRI). Chemists use MRI all the time, but they call it Nuclear Magnetic Resonance. Docs and researchers quickly changed the name to MRI because no one would put their head in something with Nuclear in the name.

There are now noninvasive methods to study brain activity in man. The most prominent one is called BOLD (Blood Oxygen Level Dependent), and is based on the fact that blood flow increases way past what is needed with increased brain activity. This was actually noted by Wilder Penfield operating on the brain for epilepsy in the 1930s. When a patient had a seizure on the operating table (they could keep things under control by partially paralyzing the patient with curare) the veins in the area producing the seizure turned red. Recall that oxygenated blood is red while the deoxygenated blood in veins is darker and somewhat blue. This implied that more blood was getting to the convulsing area than it could use.

BOLD depends on slight differences in the way oxygenated hemoglobin and deoxygenated hemoglobin interact with the magnetic field used in magnetic resonance imaging (MRI). The technique has had a rather checkered history, because very small differences must be measured, and there is lots of manipulation of the raw data (never seen in papers) to be done. 10 years ago functional magnetic imaging (fMRI) was called pseudocolor phrenology.

Some sort of task or sensory stimulus is given and the parts of the brain showing increased hemoglobin + oxygen are mapped out. As a neurologist as far back as the 90s, I was naturally interested in this work. Very quickly, I smelled a rat. The authors of all the papers always seemed to confirm their initial hunch about which areas of the brain were involved in whatever they were studying. Science just isn’t like that. Look at any issue of Nature or Science and see how many results were unexpected. Results were largely unreproducible. It got so bad that an article in Science 2 August ’02 p. 749 stated that neuroimaging (e.g. functional MRI) has a reputation for producing “pretty pictures” but not replicable data. It has been characterized as pseudocolor phrenology (or words to that effect). Keep reading you’re about to find out just why this was.

What was going on? The data was never actually shown, just the authors’ manipulation of it. Acquiring the data is quite tricky — the slightest head movement alters the MRI pattern. Also the difference in NMR signal between hemoglobin without oxygen and hemoglobin with oxygen is small (only 1 – 2%). Since the technique involves subtracting two data sets for the same brain region, this doubles the error.

Under two years ago, it was shown that 70% of people having functional MRIs (fMRIs) were asleep during the test, and that until then fMRI researchers hadn’t checked for it. For details please see
https://luysii.wordpress.com/2014/05/18/how-badly-are-thy-researchers-o-default-mode-network/. You don’t have to go to med school, to know that the brain functions quite differently in wake and sleep.

Recent work shows that functional MRI work is even worse. A devastating report in [ Proc. Natl. Acad. Sci. vol. 113 pp. 7699 – 7600, 7900 – 7905 ’16 ] showed that certain common settings in 3 software pacakages (SPM, FSL, AFNI) used to analyze fMRI data gave false positive results ‘up to’ 70% of the time. Some 3,500 of the 40,000 fMRI studies in the literature over the past 20 years used these settings. The paper also notes that a bug (now corrected after being used for 15 years) in one of them also led to false positive results.

Here’s a bit more detail on what they did. It turns out that analyzing one voxel (essentially a single MRI pixel) at a time produces valid results. The problem comes when multiple voxels (clusters) are analyzed together. Clusterwise inference considers both the strength of activity at spots throughout the brain as well as the size of the spots. When a parameter called the cluster defining threshold (CDT) is set too low, the analysis is more likely to be false positive. This was true for all 3 packages tested. Parametric statistical methods produce the problem (not for voxels but for clusters). It relies on Gaussian Random Field Theory (RFT) for clusters , which depends on two other assumptions (1) the spatial autocorrelation function has a squared exponential shape — e.g. Gaussian (2) the spatial smoothness of the fMRI signal is constant over the brain. Neither of these assumptions is correct. Those of you who’ve read Nassim Nicholas Taleb about the stock market know about ‘fat tails’. It turns out that the spatial correlation function has them. Here’s what a fat tail is all about. Human height goes fall quite nicely into a Gaussian distribution. There are 7 and 8 footers about but they are rare. If the human height distribution wasn’t Gaussian but had a fat tail, we’d see 12 and 15 footers.

If that wasn’t bad enough,the following is even worse (in my opinion). 40% of 241 recent fMRI studies didn’t report using well known methods for correcting for multiple testing. They may have done so, but every biomedical paper, and drug study says so explicitly. Not only that but drug studies are required to explicitly state the hypothesis (or hypotheses) they are testing.

This is probably why in the early days, fMRI researchers always confirmed their original hypothesis. They could test the massive fMRI for statistical rarity, and since the data was so large, find it and post hoc propter hoc publish it. Possibly they did so out of ignorance, but even so this is inexcusable

The Silence is Deafening

A while back I wrote a post concerning a devastating paper which said that papers concerning the default mode of brain activity (as seen by functional magnetic resonance imaging { fMRI } ) had failed to make sure that the subjects were actually awake during the study (and most of them weren’t). The post is copied here after the ****

Here’s a paper from July ’14 [ Proc. Natl. Acad. Sci. vol. 111 pp. 10341 – 10346 ’14 ] Functional brain networks are typically mapped in a time averaged sense, based on the assumption that functional connections remain stationary in the resting brain. Typically resting state fMRI (default network == rsfMRI) is sampled at a resolution of 2 seconds or slower.

However the human connectome project (HCP) has high-quality rsfMRI data at subsecond resolution (using multiband accelerated echo planar imaging. This work used a sliding window approach mapping the evolution of functional brain networks over a continuous 15 minute interval at subsecond resolution in 10 people. I wrote the lead author 21 July ’14 to ask how he knew the subjects weren’t asleep during this time.

No response. The silence is deafening.

Another more recent paper [ Proc. Natl. Acad. Sci. vol. 111 pp. 14259–14264 ’14 ] had interesting things to say about brain maturation in attention deficit disorder/ hyperactivity — here’s the summary

It was proposed that individuals with attention-deficit/hyperactivity disorder (ADHD) exhibit delays in brain maturation. In the last decade, resting state functional imaging has enabled detailed investigation of neural connectivity patterns and has revealed that the human brain is functionally organized into large-scale connectivity networks. In this study, we demonstrate that the developing relationships between default mode network (DMN) and task positive networks (TPNs) exhibit significant and specific maturational lag in ADHD. Previous research has found that individuals with ADHD exhibit abnormalities in DMN–TPN relationships. Our results provide strong initial evidence that these alterations arise from delays in typical maturational patterns. Our results invite further investigation into the neurobiological mechanisms in ADHD that produce delays in development of large-scale networks.

I wrote the lead author a few days ago to ask how he knew the subjects weren’t asleep during this time.

No response. The silence is deafening.

***

Addendum 22 Nov ’14 — In a huge review of resting state MRI (Neuron vol. 84 pp. 681 – 696 ’14) The following appeared —
“The issue of inadvertent sleep has only recently gained prominence, and the field has not yet developed consensus on how to deal with this issue.” Well, silence is no longer an option.

If you Google “default mode network” you get 32 million hits in under a second. This is what the brain is doing when we’re sitting quietly not carrying out some task. If you don’t know how we measure it using functional mMRI skip to the #### and then come back. I’m not a fan of functional MRI (fMRI), the pictures it produces are beautiful and seductive, and unfortunately not terribly repeatable.

If [ Neuron vol. 82 pp. 695 – 705 ’14 ] is true than all the work on the default network should be repeated.

Why?

Because they found that less than half of 71 subjects studied were stably awake after 5 minutes in the scanner. E.g. they were actually asleep part of the time.

How can they say this?

They used Polysomnography — which simultaneously measures tons of things — eye movements, oxygen saturation, EEG, muscle tone, respiration pulse; the gold standard for sleep studies on the patients while in the MRI scanner.

You don’t have to be a neuroscientist to know that cognition is rather different in wake and sleep.

Pathetic.

####

There are now noninvasive methods to study brain activity in man. The most prominent one is called BOLD, and is based on the fact that blood flow increases way past what is needed with increased brain activity. This was actually noted by Wilder Penfield operating on the brain for epilepsy in the 30s. When the patient had a seizure on the operating table (they could keep things under control by partially paralyzing the patient with curare) the veins in the area producing the seizure turned red. Recall that oxygenated blood is red while the deoxygenated blood in veins is darker and somewhat blue. This implied that more blood was getting to the convulsing area than it could use.

BOLD depends on slight differences in the way oxygenated hemoglobin and deoxygenated hemoglobin interact with the magnetic field used in magnetic resonance imaging (MRI). The technique has had a rather checkered history, because very small differences must be measured, and there is lots of manipulation of the raw data (never seen in papers) to be done. 10 years ago functional magnetic imaging (fMRI) was called pseudocolor phrenology.

Some sort of task or sensory stimulus is given and the parts of the brain showing increased hemoglobin + oxygen are mapped out. As a neurologist, I was naturally interested in this work. Very quickly, I smelled a rat. The authors of all the papers always seemed to confirm their initial hunch about which areas of the brain were involved in whatever they were studying. Science just isn’t like that. Look at any issue of Nature or Science and see how many results were unexpected. Results were largely unreproducible. It got so bad that an article in Science 2 August ’02 p. 749 stated that neuroimaging (e.g. functional MRI) has a reputation for producing “pretty pictures” but not replicable data. It has been characterized as pseudocolor phrenology (or words to that effect).

What was going on? The data was never actually shown, just the authors’ manipulation of it. Acquiring the data is quite tricky — the slightest head movement alters the MRI pattern. Also the difference in NMR signal between hemoglobin without oxygen and hemoglobin with oxygen is small (only 1 – 2%). Since the technique involves subtracting two data sets for the same brain region, this doubles the error.