Mathematically Describing Neuronal Connections in the Brain

Students in my classes have started to catch on to the fact that I tend to dress up for lectures I am particularly excited about. For a topic I'm indifferent toward or lukewarm about, I wear my standard dress shirt, slacks, and dress shoes. To prepare for a subject I like a little bit more, that's when I throw on a sports jacket, and possibly a nice belt. But get me all hot and bothered, and that's when I break out...the ties.

Not surprisingly, then, I wear a three-piece suit when talking about neurons. These sexy little suckers act as the basic cells of communication throughout your brain and throughout your nervous system, relaying electrical transmissions all the way down your axons to the synaptic gap, those terminal buttons precariously poised on the precipice of a protoplasmic kiss, until finally, excruciatingly, those tiny vesicles of chemical bliss burst from their vile durance, recrudescent, crushing out the last throb of the last ecstasy man or monster has ever known.

...Let me catch my breath...Where was I? Oh yes - neurons. Besides their role in transmitting electrical and chemical signals throughout the brain, they also exist in astonishingly high numbers, with somewhere on the order of tens of billions of neurons packed into a single brain. On top of this, each one can share hundreds or thousands of connections with other neurons, leading to a staggering number of potential synaptic connections. The mind boggles.

To provide the full mathematical treatment of understanding neurons, we are joined again by Keith "The Rookie" Bartley, whose interest in synaptic connections was recently piqued by an introductory cognitive science course. Along the way Keith touches on mathematics, the Turing Test, Friends, oatmeal, the uncanny valley, and where genitals are represented in the brain, providing a theoretical basis for why foot massages can lead to greater chances for successful coitus.


As a TA for an Introduction to Cognitive Science course, one of our instructors briefly discussed the concept of Block's "Aunt Bertha Machine" and how the Turing test alone required more bits of memory than there are atoms in the universe. (If you aren't familiar with Ned Block's work click here).  Below is a section of one of his slides:

Volume: (15*10^9 light-years)^3 = (15*10^9*10^16 meters)^3
Density: 1 bit per (10^-35 meters)^3
Total storage capacity: 10^184 bits < 10^200 bits < 2^670 bits
Critical Turing Test length: 670 bits < 670 characters < 140 words < 1 minute

Difficult as it is for some people to conceptualize and subsequently deal with such induced feelings of simultaneous intelligence, stupidity, and insignificance pervading recapitulations of their own life's meaning, I would like to warn those people that the following information about the capacity of the human brain is likely to do much worse. READ RESPONSIBLY.

For the human brain, the possible number of combinations and permutations of neural connections has been purported to vastly exceed the number of elementary particles in the Universe. Consider for a moment that the brain has 85,000,000,000 neurons, we'll round that up to the previously estimated 100,000,000,000 for hypothetical simplicity, each with a capacity for up to 10,000 synaptic "connections". 

1! = 1
10! = 3,628,800
100! = 9.33 x 10^157
1000! = 4.02 x 10^2,567
***This is where Google's calculator starts to report infinity***
For bigger factorials, we'll have to use Stirling's approximation.

So using Stirling's approximation....

10,000,000,000! ~ 2.33 x 10^95,657,055,186
100,000,000,000! ~ 3.75 x 10^1,056,570,551,815

But wait, each of the 100 billion neurons can have 10,000 synaptic connections, so…

10^11 * 10,000 = 1e26

So rather...

100,000,000,000,000,000,000,000,000! ~ REALLY BIG NUMBER

But Keith! Come on, this looks like a gross misrepresentation of the limits of human cognition?

Our theories about the brain are much more modular in scope, but at the same time, distributed enough to adapt. These two points function as a much better descriptor of the networked brain. It's reasonable to see, in our post-Scopes trial times, that humans' brains developed around how they are utilized in their day-to-day lives, and perhaps more importantly, the phenomenal ability to constantly adapt, even in the wake of extreme trauma. The development and existence of a single "grandmother neuron", is misrepresentative of our degree of sustainability in the brain. Neurons for your grandmother have to be very distributed, so that when some of your brains cells die off, as they often do, it's important you don't forget the old woman that squeezes your cheeks when she sees you, lest you punch her in the face for assault. Jennifer Aniston would no longer instill memories of how much time you wasted watching reruns of Friends on TBS every afternoon for 4 years, and Halle Berry would no longer remind you of how Hollywood reduced one of the greatest antiheroines in the history of DC Comics to a mannequin in spandex with a speech impediment. 

At the same time, however, that three pounds of oatmeal between your ears still retains a relative degree of modularity in regional function, which is a reason why your brain is compartmentalized in various folds (gyri) and crevices (sulci). When you have an itch from what is in fact a really small bug bite, your desired area to itch is very distributed across skin because the signals in the brain are themselves both distributed and modular. A diagram often used to demonstrate this modularity is the cortical homunculus.

Straight out of the backwoods of the uncanny valley, this diagram demonstrates the relative intensity and location that each section along your somatosensory cortex corresponds with on your body. As for the proximity of feet relative to your genitals, well that might just explain a lot about some guys now wouldn't it.