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Hi David,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?

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 |

First | 4 |

Second | 3 |

Third | 2 |

Fourth | 1 |

TOTAL | 4*3*2*1=24 |

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:

Factorial | Numerical Answer |

1! | 1 |

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.

Best regards,

Ricky J. Sethi.

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