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% now chapter 11 (4-6-95)

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{\bf NEURAL NETWORKS}
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The boundary
between different scientific disciplines is not always clear cut.
For example, one might claim 
that the Lorenz model belongs to atmospheric science,  
or that percolation, since it can be used to discuss the
properties of porous materials, is really a geoscience problem. 
Arguments over such matters are generally not very useful,
but they do illustrate that the lines between disciplines are
not sharply defined.  Indeed, the ideas developed in one area can
often lead to important breakthroughs when applied to other fields.
In the book we consider in some detail three cases in which it appears that
such cross fertilization has, or may, prove extremely fruitful.
These topics are
currently active areas of research so even the tentative conclusions
we will draw may change considerably in the next few years.
Nevertheless, the computational methods we employ, which are
extensions of approaches encountered earlier in the
book, have already led to useful new insights.  

The first problem in this chapter is
from biology and concerns protein structure.
Proteins are long
flexible molecules that can fold into many different shapes.
The problem is to understand the folding process so that one can
predict the folded structure of a newly encountered protein, and 
thereby learn how to design molecules with a desired shape.
The second example is from geoscience and concerns earthquakes and
a possible connection with phase transitions.
The third problem deals with the brain and some proposals concerning the
way (human) memory might work.  These proposals are based on
what are known as neural networks, and we will now give a brief glimpse of
the science they entail.

The Ising model of magnetism (which is the topic of chapter 8)
consists of a large number of very simple units, 
i.e., spins,
which are connected together in a very simple manner.
By ``connected'' we mean that the orientation of any given
spin, $s_i$, is influenced by the direction of
other, neighboring, spins.
We found that the behavior of an isolated spin
was unremarkable.  Things only became
{\em really} interesting when we considered the behavior of a 
large number of spins and allowed them to interact.  
In that case we found that under the
appropriate conditions some remarkable things could occur,
including the singular behavior associated with a phase transition.
Neural network models of the brain share some of these features.
The human brain consists of an extremely large number ($\sim 10^{12}$)
of basic units
called neurons, each of which is connected to many other 
neurons in a relatively simple manner.     
A biologically complete discussion of neurons and how they function
is a long story, which we don't want to go into here.  The bottom line
is that it seems reasonable to model a neuron as an Ising spin, which
has a value of either ``up'' or ``down.''  A neuron can then store
information, and by allowing neurons to interact with each other,
much as Ising spins in a ferromagnet interact with each other, the network
as a whole can be made to function as a memory.

This memory is a bit different from the usual type of a computer
memory.  In the case of a neural network, it is
a {\em content addressable memory}.  It is made to
operate by presenting it with a pattern which resembles the pattern one is 
trying to remember;
the system then ``recalls'' the appropriate configuration. 
This seems similar to a human memory; for example, given
part of a phone number or address, a human memory is often able to
recall the rest of the information.
The details of how such a network can be made to
function as a memory, along with how to implement it in
a simulation are all covered in the text.  Examples like those shown
in the figure are then given.  Here an image which resembles the
letter ``A''is presented to the network, which then recalls a complete
image of the letter.

\begin{figure}
\centerline{\hbox{
\psfig{figure=a_close.eps,width=2.5in}
\psfig{figure=a_close_final.eps,width=2.5in}
}}
\caption{
{\em
Left: spin arrangement close to the letter ``A'', but with
a few spins flipped to the wrong state.
Right: spin configuration after one Monte Carlo pass through the network.
}
}
\label{a_close}
\end{figure}

Many other issues can be considered, including how
information is stored in the network, the effect of ``damage'' on the
performance, the concept of metastability,
and the capacity of the network.  Other related models,
such as perceptrons, are also discussed.

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