X-Message-Number: 15332
Date: Sat, 13 Jan 2001 07:41:34 -0500
From: Thomas Donaldson <>
Subject: and more on computers and brains

Hi again!

More on just how many connections become possible between neurons. This note
will take a little different approach. Suppose we have a set of N neurons.
As I mentioned, the neurites may extend quite far, while dendrites tend
to lie more close to their neuron. 

This means that a neuron will generally send messages much farther than
its dendrites extend. On the other hand, its dendrites can RECEIVE messages
from more than one neuron. The role of exponentiality in the working of
our brain therefore needs more than superficial comment. First, the
average length of an axon isn't a good parameter, since that length varies
a good deal. It's more appropriate to consider the distribution of axons
of different lengths among neurons, a figure which I do not have and is
likely to differ with the exact kind of neurons. Pyramidal neurons are
likely to have longer axons, other kinds (except possibly the Purkinje
cells in our cerebellum). Other kinds of interneurons may be quite short
in the length of their axons. 

The exponentiality of increase caused by generation of one new neuron 
would depend on just how far its axon and its neurites might extend. 
To limit our thinking to simple cases, I shall suppose that its axon 
extends over (say) 1/10th of the entire brain. This gives the number
of possible connections it may form in our brain... smaller than that
of the total brain, but a large number still. It also suggests taht
if we had a brain with N neurons, then the possible connections come
to (0.1) * (N!). (Please understand that this is VERY approximate;
it is an attempt to estimate behavior based almost on theory alone).

The main point to remember here is that such a figure, even multiplied
by a relatively small restriction (1/10, 1/100, etc) remains extremely
large. The factorial of a large number is very very large.

The importance of this comes from one simple truth, believed by almost
all neuroscientists: our memories consist of connections between many
neurons. The number of POSSIBLE connections gives the possible memories
we may have. To the extent that our connections belong to us alone, 
we may think of them as our own special memories. (In reality, we're
likely to find that an individual neuron is limited by the connections
it can form, so that "unique" connections are again limited; for the
purpose of this discussion, I am ASSUMING that those limits can be
ignored in a rough estimate). Moreover, a NEW neuron will form new
connections, again with a wide range of possibilities.

The discussion by others, as I understand it, looks at the connections
of one particular brain. Clearly if we knew them beforehand we could
predict the behavior of that particular brain, and even make a machine
to imitate it, at least for a short while. (As the brain learned, those
connections would change). In that sense, someone at some particular
time might well be imitatable by a computer. However unless the computer
ALSO had similar features those of brains, it could not continue this
imitation for long. Does the possession of those similar notion of 
attributes to brains mean that a Turing machine, EVEN FORGETTING THE
ISSUE OF TIME, could imitate the behavior of such a machine? 

		Best wishes and long long life for all,

			Thomas Donaldson

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