|MadSci Network: Neuroscience|
I have read many times (in Ornstein's *Right Mind*, Buzan's *Use Both Sides of Your Brain* [citing Moscow U's Pyotr Anokhin], e.g.) that "there are more potential connections between the billion or so neurons in the human brain than there are atoms in the universe" (Ornstein p. 95). How can this be true? Doesn't each "potential connection" correspond to more than one atom?Hi David,
I guess the key to answering this question is the term "potential connections". I haven't read the books you cite (although I have read another of Ornstein's books) so they might have different reasons for that statement but here are my thoughts on the matter. I think that they were probably talking about the possible connections, and not the actual connections. If they were actual connections, your reasoning is definitely correct... the connections are mediated by structures made of atoms hence this would be self-contradictory.
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. The table below summarizes this nicely:
|Neuron||Number of 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:
|10!||3.62879 x 10^6|
|100!||9.33262 x 10^157|
|1000!||4.023872600770937 x 10^2567|
|10000!||2.8462596809170545 x 10^35659|
|100000!||2.82422940796034787 x 10^456573|
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. For comparison, one mole of a gas contains 6.022 x 10^23 atoms (for a monatomic gas) and that takes up about 22.4 litres of volume (assumptions for one mole of an ideal gas at STP). This is much smaller than the huge number of possible connections (based on the assumption of a billion neurons) even after extending this over all space (to get a more accurate idea of the total mass of the universe, please check Prof. Ned Wright's excellent Cosmology Tutorial)! Well, I hope this helps make sense of that seemingly paradoxical statement. If you have any further questions, or corrections, please don't hesitate to drop me a line.
Ricky J. Sethi.
Try the links in the MadSci Library for more information on Neuroscience.