THE HUMAN BRAIN AS A GRAPH

Among the cells in the brain are those called neurons, which are
communication cells that can signal one another via connections
(synapses). Moreover, each synapse has a strength or weight that
affects the degree of influence a signal coming from one cell has on
the receiving cell. The neurons and the synapses between them thus
form an enormously large directed and weighted graph.  How large?
Well, there are at least 10^11 (100,000,000,000 or 100 billion)
neurons in the human brain, and on the order of 10^15 synapses.  That
is, n = |V| ~=~ 10^11 and m = |E| ~=~ 10^15.  That's big!!!

Aside from it's incredible size, why is such a graph interesting?  (i)
it's our brain, and supposedly the basis for who we are; (ii) it
changes over time: old neurons die, new ones grow, new connections
form, weights change with experience; (iii) it is an active structure,
sending information around inside itself, rather than a more typical
graph that is simply a passive data storage that a separate algorithm
acts upon; (iv) is is massively parallel: activity goes on in billions
of neurons at the same time, unlike a typical graph algorithm that
travels along only one edge (between two vertices) at a time.

Thinking about the brain this way has stimulated research in computer
science, to pursue computational models of massive parallel
processing. There are two rather distinct areas that have come about:
(i) artificial neural networks, and (ii) abstract parallel processing
machines. The former (ANNs) are more closely patterned on actual
neuronal behaviors, and often are used as models of the brain, as well
as in artificial intelligence, with a special focus on learning or
training based on experience. The latter provide designs for parallel
computer architectures aimed at achieving much greater processing
speed than conventional computers have. We will examine the latter
type in the next week or two; for now we will take a (brief) look at
artificial neural networks, after an even briefer look at the brain.

Most of our neurons and synapses are contained in the ``neocortex''
(present in mammals); it wraps around a smaller and evolutionarily
older part of the brain sometimes called the "smell brain", "crocodile
brain", or "reptile brain".

The rough outline of the human brain seen from the left is:

        __________________
       /       */#         )
f     (    F  */#  P      / \
r      \_____*/#_________/   \
o            _-         (  O  )
n           /      T     \   /
t          (______________\_/
                    \    C  )
                     \   __)
                      \ \

                 Fig 1

where F = frontal lobe, P = parietal lobe, T = temporal lobe, and
O = occipital lobe. The lower tip of the smell brain is just visible
below the temporal lobe, where it joins the spinal cord; also visible
is a bit of the cerebellum, C.  The four lobes shown form the
neocortex; inside, at the top of the spinal cord (not shown) are the older
parts of the brain (hidden behind the temporal lobe).

From the top:

            _______________________
           /                       \
          |                         \
f         |                         /
r          \_______________________/
o           _______XXXXX___________
n          /                       \
t         |                         \
          |                         /
           \_______________________/

                    Fig 2

The top view shows that the brain (neocortex) comes in two very
similar "hemispheres", left and right, each of which look like Fig 1
above when seen from the side.  Thus there are two frontal lobes, two
parietal lobes, etc. The two hemispheres are connected by
a thick band of axon fibers XXXXX called the corpus callosum.

The strip *** along the back edge of F, and the strip ### along the
front edge of P, are called the motor strip and the somatosensory
strip, resp. The motor strip consists of neurons whose signals proceed
down the spinal cord to muscles, causing them to contract; the
frontal lobes (left and right) are thought to be involved in planning,
and thus it seems not unreasonable that motor commands (enacting a
plan) would emanate from there.  The somatosensory strip receives sensory
signals traveling from the body up through the spinal cord and smell
brain and other inner structures and into the parietal lobe.

A striking feature is that of the "crossed" brain: all of the motor
and sensory signals that travel to and from the brain via the
spinal cord cross over to the opposite side. That is, the left half of
the body (below the neck) is connected (both for sensory and for motor
signals) to the right side of the brain, and vice versa.

The two eyeballs are situated just under the left and right frontal
lobes.  The "crossing" situation for vision is more complicated than
for the motor and somatosensory strips just mentioned. Each eye (at
the back, or retina) sends via its light-sensitive neurons (rods and
cones) signals to either the right or left occipital lobe, depending
not on which eye but rather on whether the light comes from the right
or left of center of gaze. That is, half of the retina in each eye
projects to the left occipital lobe, and half to the right occipital
lobe. Specifically, light from the right of center falls on the left
half of the retina in each eye, and then signals are sent from there
to the left side of the brain, and similarly for the right. (However,
the corpus callosum transmits this information from each side also to
the other side.)  Destruction of, say, the left occipital lobe,
results in complete loss of visual awareness of the right visual
field, and vice versa.

It is known that the occipital lobe performs only the "early" visual
processing, such as edge orientation, and that this information in
turn is passed to other areas such as the temporal lobe where a
presumed (but ill-understood) process of integration occurs, affording
high-level determination of what is seen (eg, a house or a face). The
temporal lobe is also involved in memory and in processing of auditory
information.  The parietal lobe is thought to process highly abstract
ideas such as mathematics.

Overall, however, although many volumes of detailed information about
the brain is well estrablished, how it manages to do what it does is
still in many respects uncertain.  Even vision, the most heavily
studied brain function, remains cloudy in at least one fundamental
respect: at what point, and as part of what process, does visual
awareness (actual subjective experience of seeing) occur?  Not only is
this not known, but no one has even managed to formulate a clear guess
as to what it could be.  The easiest notion to form (and then reject!)
is that some sort of image is created in the brain, a bit like a TV
set (but then "who" is looking at it?).

Now let us return to individual neurons.  A neuron is a highly
specialized cell, with the usual cell body with its nucleus as well as
a long "axon" that acts a bit like a current-conducting wire.  The
axon usually splits into many "tips" that can come close to other
neurons.
                                               _______
                                              |       |
                                      _____   |       =====
     __________                      /        |_______|
    |          |               /  __/       ^      another neuron
    |          | signal -->   /  /       synapse
    |          ===============-----         
    |          |    axon      \   \      
    |__________|               |   \__

     cell body

When the proper electro-chemical conditions occur in the cell body, an
electrical current ("action potential") is initiated there, which
travels from the cell body all the way along the axon to the axonal
tips where it causes molecules known as neurotransmitters to be
released.  If there is another neuron close enough, some of the
neurotransmitters will come into contact with it.  This close
proximity that allows such contact is called a synapse.  A synapse can
be such that, when the neurotransmitter makes contact, the contacted
cell becomes more likely to fire (excitatory synapse) or less likely
(inhibitory synapse).

The cerebellum (C, in Fig 1) consists mostly of inhibitory synapses,
which seems not unreasonable given that its apparent function is to
provide fine motor control (as in piano-playing or reaching for
something); damage to the cerebellum leads to loss of this control so
that motions tend to be exaggerated as in overreaching.

One neuron, on average, will be connected to (have synapses with) on
the order of 5000 others.