Of Math Tribes, Brains, and Beauty...

Just want to quickly pass along this new fun "n-Category Cafe" post which includes links back to two other rich reads (that I haven't fully digested yet), one being from David Mumford. It all has to do once again with mathematicians and the experience of beauty (from a neuroscience perspective):

https://golem.ph.utexas.edu/category/2015/10/four_tribes_of_mathematicians.html

Research, Statistics, Oy veyyy

via NIH/WikimediaCommons

Almost 40 years ago in grad school I railed a bit about the non-random and small sample sizes of so many journal-published studies, particularly in the social and medical sciences… I felt like a lone wolf in the wilderness though.
Eventually, in 2005 these sorts of concerns became center-stage when John Ioannidis published his oft-cited paper, "Why Most Published Research Findings Are False."

Now, Katherine Button, Ioannidis, and others have published "Power failure: why small sample size undermines the reliability of neuroscience," further delineating such problems, as specific to up-and-coming neuroscience:
http://www.guardian.co.uk/science/sifting-the-evidence/2013/apr/10/unreliable-neuroscience-power-matters

A few lines therefrom:

"There is growing interest in the need to improve reliability in science… Many of the most hyped scientific discoveries eventually cannot be replicated...
"A major factor that influences the reliability of science is statistical power. We cannot measure everyone or everything, so we take samples and use statistical inference to determine the probability that the results we observe in our sample reflect some underlying scientific truth."

The article goes on to discuss various problems with statistical samples, false positives, and false negatives, and also making mention of publication bias, before concluding, "The current reliance on small, low-powered studies is wasteful and inefficient, and it undermines the ability of neuroscience to gain genuine insight into brain function and behaviour. " And from the research article's abstract: "Improving reproducibility in neuroscience is a key priority and requires attention to well-established but often ignored methodological principles."

Anyway, read the whole Guardian piece, or if you have access, the original journal article in Nature Neuroscience.

Mathematics Underlying Brain Structure

Study from British neuroscientists looks at the mathematical nature of neuronal growth:

http://www.kurzweilai.net/simple-mathematical-pattern-describes-shape-of-neuron-jungle

Excerpt:

"Neurons look remarkably like trees, and connect to other cells with many branches that effectively act like wires in an electrical circuit, carrying impulses that represent sensation, emotion, thought and action...

"Over 100 years ago, Santiago Ramon y Cajal, the father of modern neuroscience… proposed that neurons spread out their branches so as to use as little wiring as possible to reach other cells in the network. Reducing the amount of wiring between cells provides additional space to pack more neurons into the brain, and therefore increases its processing power.
"New work by UCL neuroscientists has revisited this century-old hypothesis using modern computational methods. They show that a simple computer program that connects points with as little wiring as possible can produce tree-like shapes that are indistinguishable from real neurons — and also happen to be very beautiful."

Neuro-mathematics...

The human brain is probably the 'final frontier'...

Blogger Jason Goldman reviews some of what we know/believe about mathematical dysfunction of the brain here, based in part on case-studies of patients with different brain lesions and on fMRI studies:

http://tinyurl.com/2dz2m3a

This was actually the 4th in a series of related posts Jason did that can be looked up here:

http://tinyurl.com/266o57k

"Pattern Decorrelation"

THERE'S a term I was previously unfamiliar with... It refers, in this instance, to certain complex mechanisms in the brain which have been mathematically modeled to account for workings of the olfactory system. Read about it in this Web article entitled, "Explaining Scent Mathematically," referencing some recent corroborative work of Swiss neurobiologists and mathematicians, which helps explain how olfactory neural circuits are structured so as to discriminate different odors (decorrelation also helps explain the neural circuitry of the visual system as well):

http://www.physorg.com/news199367766.html


Wikipedia entry on "decorrelation" here: http://en.wikipedia.org/wiki/Decorrelation