Information Technology Reference
In-Depth Information
Our model addresses most of these problems because of the modifications de-
scribed next and the inherent high quality of the code sets now in our possession.
A neuronal unit
i
is represented by a codeword
Ci
from a DNA code; these code-
words only cross-hybridize to their own complements
Di
, not to any other code-
words or their complements. These single strands can be permanently affixed to a
solid medium (such as those used for DNA microarrays) so as to form a DNA chip
A, by what is now standard biotechnology [8,26]. The chip A contains two spots for
each neuron
i
(one spot for positive activation and one spot for negative activation,
located far apart enough to avoid cross-hybridization), each with as many single
strands (
Ci
or
Di
) as necessary for a given resolution on the activation levels (for
example, three decimal digits will require
M
000 copies of the corresponding
oligonucleotide attached at each spot). A positive activation level of the unit
i
is
given at any time
t
by the concentration of the particular double stranded DNA
species
Ci
-
Di
, whereas a negative activation is given by the concentration of its
complementary labeled word
Di
-
Ci
. An optical census of double-stranded DNA (or
their Watson-Crick complements) can be taken on this chip by using fluorescent
tags (e.g., SYBR green attached to the double-stranded pair
Ci
-
Di
) [8, 36] that will
reveal the current activation levels of the various units
x
i
=
1
,
(
)
at a given time
t
,if
normalized to
M
. We will denote by
m
the length of the
m
-mer oligonucleotides
Ci
in the code set. Note that a complementary copy of the activation vector
x
t
at a
given time
t
can be made by simply heating the chip to an appropriate temperature
exceeding the maximum melting temperature of all pairs
Ci
-
Di
, then washing the
complementary single-stranded representation
x
(
t
)
into a temporary tube T.
stipulated by
the Hopfield model requires longer strands representing the synaptic weights
Wij
from neuron
j
into neuron
i
to be attached to the oligonucleotide
Ci
that represents
it. For this design, we will assume that the weights are integer valued for simplicity
(similar designs could be used for rational values to approximate any other values).
The weights are themselves copies of the complementary DNA oligomers
Di
for
neuron
i
extended by -
Wij
- copies of
D
i separated by a restriction site r, so they
can be eventually separated into their original pieces. For example, a weight
Wij
The transition from an activation state
x
(
t
)
to another state
x
(
t
+
1
)
=
3
circjrcj rcj
. Zero weights are not represented,
i.e., the absence of any molecular representation containing
Ci
and
Cj
means that
Wij
would be expressed in DNA as
Wij
=
0. These weights will likewise be permanently affixed to another chip W in
a similar manner to chip A. (We will use the same symbol
Wij
to denote the weight
and its molecular representation for simplicity. The context makes clear which one
is being referred to.)
A Hopfield network transitions from total activation vector
x
=
(
t
)
to activation vec-
tor
x
, where
s
is a saturation function (typically a sigmoid, but
here assumed to be just a linear approximation as the identity map squashed to 0
for negative values and 1 otherwise) and
Wx
(
t
+
1
)=
s
(
Wx
(
t
))
(
)
=[
]
t
is the product of matrix
W
Wij
(
)
and activation vector
x
, i.e., each unit
i
computes its weighted net input by the
scalar product of row
i
and column
x
t
and
saturates it using
s
(i.e., activation
values outside a unit's range are squashed down within the activation range). The
matrix product is here realized by a two-step process, first make a complementary
(
)
t
