Possibly because of the movie *Amélie* and possibly because people like to quote statistics or calculations without knowing what they mean, it’s fairly easy to find some absurd claims about the number of synapses or “neural connections” in the human brain and how they compare to the number of stars/atoms/something else in the universe.

Near the end of *Amélie*, the narrator says,

At the same moment, at the La Villette garden, Felix L’Herbier discovers that the number of possible connections in a human brain is superior to the number of atoms in the universe.

The first thing you might think is that “neural connections” means “synapses”, which seems reasonable to me, and the second thing you would then think might be, “Uhh, that would mean there are more synapses than *atoms* in the brain, which is absurd.” Of course it’s absurd, and false, but the key word in that statement is easy to overlook: *possible*. Because it says “possible connections”, it takes some more investigation and calculation to see what the statement means and if it’s even true.

The short answer is: Yes, it’s technically correct, and also fabulously meaningless. The long answer is provided by Ricky J. Sethi on madsci.org. The word “possible” converts this from a biology question into a combinatorics problem:

But, if you look for potential, rather than actual, connections, the story changes. If you just look at the number of possible connections between neurons, then it’s simply adding up each consecutive term. For example, if you had 5 neurons and wanted to know the total number of connections possible between them you’d simply add each decreasing term as in: 4+3+2+1 = 10. Think of this as drawing a pentagram and then connecting up each point to every other point and counting the number of lines (or connections). There’s a equally simple formula to let you add up all n terms: [n*(n+1)]/2 (e.g., for n=4, this gives, not surprisingly, 10). So, for a billion, this is [billion*(billion+1)]/2 which is about 5 x 10^17. That’s not so much, is it?

However, the story changes even more when you consider the total number of unique connections; i.e., the total number of unique neuronal “networks” that are possible. Before getting into the “billion” neurons case, let’s again look at a simpler example to illustrate this. Say you have 5 neurons, each capable of making a connection with each of the others. What is the total number of unique connections (or networks) possible for this system? Here, instead of just counting up the number of lines in the pentagram, we’ll treat each path that connects one point to another as the variable and attempt to calculate the number of unique paths that connect all points to all other points. Incidentally, I guess you could also think of this as a variation of the famous combinatorial traveling salesman problem (for our case, you could think of it as sending a action potential from one neuron to a final one and seeing how many different sequences of activation are possible; e.g., the message can travel from neuron 5 to neuron 1 via 5-3-2-4-1, 5-2-4-3-1, etc.). So, let’s get right to it…

Well, the first neuron can make a connection with each of the other 4 neurons. For each of these connections, the 2nd neuron can then only make connections with the remaining 3. So the total number of unique networks so far are 4*3 = 12. For each of these 12 connection possibilities (or permutations), the 3rd neuron in the chain can then only make connections with the remaining two. This brings the total up to 12*2 = 24. And finally, for each of these 24 possible connections, the 4th neuron can only make a single additional connection with the last neuron. So the total number of unique possible connections, or networks, are 24*1 = 24 connections.

[…]

There’s actually a mathematical way of summarizing this by using the factorial notation. Using that, we see the total number of connections was (5-1)! = 4! = 4*3*2*1 = 24. This is a generalization of the standard formula for finding the number of possible permutations for n-1 elements, namely (n-1)!. So, using this notation, if we substitute the figure of a billion+1 neurons, we get (billion)!. This number is enormous. To get some idea of it’s magnitude, I used Mathematica to create the following table:

[He pastes a table going from 1! to 100,000!, showing that the latter is ~10^{456,000}. If that weren’t unfathomably large enough, he continues in the next paragraph…]This seems to imply that 1,000,000,000! is about 3 x 10^5,000,000,000 (my computer just hung when I tried to get it to estimate (billion)!… there’s a reason they’re using DNA computing to solve this! :). This is obviously much bigger than the total amount of known matter.

In summary, the “possible connections” calculation is a math problem, not a biology or neuroscience problem, which makes it *totally meaningless*.

You know what else can produce an answer that’s greater than the number of atoms in the universe? The number of “possible connections” between stars in the Milky Way. There are supposedly about 200 billion neurons in the human brain and about 200–400 billion stars in the Milky Way. So those two numbers are actually quite close. To calculate the number of “possible connections” between neurons in a human brain is a similar calculation (with similar significance, viz., nearly zero) to the number of “possible connections” between stars in the Milky Way. It’s just a combinatorics problem that exists only in the abstract. Are there *actually* any connections or networks of any kind between those stars? Well, possibly far away, but none that we know of, and certainly not 200 billion factorial. Are there *actually* 200 billion factorial connections or networks between the neurons of the human brain? No, and it isn’t possible for there to ever be anything remotely approaching that.

Here’s another thing that would give you a number larger than the number of atoms in the universe: the number of “possible connections (networks)” among all the humans on Earth. Who cares? Those “connections” or “networks” (whatever that would mean) don’t actually exist. We could imagine Professor X in Cerebro looking at all the humans on Earth and drawing every possible line segment and series of line segments connecting their images in every possible combination. What would this mean about humans or society or biology or the universe? Absolutely nothing.

Now, we do know how many connections exist between individual neurons (about 10^{14} synapses), but we have no idea how many unique pathways actually exist between series of two or more neurons = how many ways there are to connect every individual neuron to every other neuron via actually existing series of synapses. It isn’t 200 billion factorial, which would be the maximum mathematically allowable number, but it isn’t just 10^{14}. These would more properly be called pathways or networks than connections; I think it’s best to limit the definition of “neural connection” to the single synapse level.

Let’s look at some comparisons with real-life numbers that actually mean something, just for fun. The number of synapses in the human brain is approximately 10^{14} (100 trillion). This is the most common definition of “neural connection” and the only one I was aware of until I encountered this article about Stanford neuroscientist Stephen Smith. Smith says,

One synapse, by itself, is more like a microprocessor—with both memory-storage and information-processing elements—than a mere on/off switch. In fact, one synapse may contain on the order of 1,000 molecular-scale switches. A single human brain has more switches than all the computers and routers and Internet connections on Earth.

So let’s multiply the number of synapses in a human brain by the maximum number of molecular “switches” that a synapse could have, giving us 10^{14} * 10^{3} = 10^{17}.

How would that compare to the number of atoms in the (observable) universe? The observable universe contains around 10^{80} atoms, so nothing currently known about the human brain comes close to that.

How about the number of atoms in our Sun alone? There are about 10^{57} atoms in *our Sun alone*, dwarfing the number of synapses or molecular switches in all humans who have ever lived by dozens of orders of magnitude (10^{17} per human * 100 billion humans = 10^{28} molecular switches, or only 10^{25} synapses, in the history of our species).

When I first saw *Amélie*, I remember thinking how absurd its claim sounded, and I remember thinking right then or shortly afterward that I would bet that not only was it false, but if you changed “the universe” to “our Sun” it would still be false. But I didn’t say anything, which is too bad, because I would have been right.

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Some food for thought (pun intended) on neurological phenomenon and the nature of ‘universal thought’: The Thoughtful Universe: http://porkopolis.blogspot.com/2013/03/the-thoughtful-universe.html

One wonders whether this perennial discussion ever takes actual physical constraints into account?

There must be a physical upper limit to possible connections. Are they all long enough to reach every other neuron? Is there enough space in the skull to accommodate all of the tissue and intertwining that would be involved?

Or should all this fantastical fun be reduced through division by whatever significant factor admits of the facts that you can only put so many bits of thread of any necessary length into a child’s shoebox and still have every one of them touch every other?

This is a very good point, and one I hadn’t considered. I have absolutely no idea how to even begin calculating the degree of physical/practical constraint on the number of neurons a given neuron can synapse with.

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