<s>
Hopfield	B-Algorithm
networks	I-Algorithm
serve	O
as	O
content-addressable	O
(	O
"	O
associative	O
"	O
)	O
memory	O
systems	O
with	O
binary	O
threshold	O
nodes	B-Algorithm
,	O
or	O
with	O
continuous	O
variables	O
.	O
</s>
<s>
Hopfield	B-Algorithm
networks	I-Algorithm
also	O
provide	O
a	O
model	O
for	O
understanding	O
human	O
memory	O
.	O
</s>
<s>
The	O
Ising	O
model	O
of	O
a	O
recurrent	B-Algorithm
neural	I-Algorithm
network	I-Algorithm
as	O
a	O
learning	O
memory	O
model	O
was	O
first	O
proposed	O
by	O
Shun'ichi	O
Amari	O
in	O
1972	O
and	O
then	O
by	O
William	O
A	O
.	O
</s>
<s>
Large	O
memory	O
storage	O
capacity	O
Hopfield	B-Algorithm
Networks	I-Algorithm
are	O
now	O
called	O
Dense	O
Associative	O
Memories	O
or	O
modern	B-General_Concept
Hopfield	I-General_Concept
networks	I-General_Concept
.	O
</s>
<s>
The	O
units	O
in	O
Hopfield	B-Algorithm
nets	I-Algorithm
are	O
binary	O
threshold	O
units	O
,	O
i.e.	O
</s>
<s>
Discrete	O
Hopfield	B-Algorithm
nets	I-Algorithm
describe	O
relationships	O
between	O
binary	O
(	O
firing	O
or	O
not-firing	O
)	O
neurons	O
.	O
</s>
<s>
At	O
a	O
certain	O
time	O
,	O
the	O
state	O
of	O
the	O
neural	B-Architecture
net	I-Architecture
is	O
described	O
by	O
a	O
vector	O
,	O
which	O
records	O
which	O
neurons	O
are	O
firing	O
in	O
a	O
binary	O
word	O
of	O
bits	O
.	O
</s>
<s>
(	O
Note	O
that	O
the	O
Hebbian	O
learning	B-Algorithm
rule	I-Algorithm
takes	O
the	O
form	O
when	O
the	O
units	O
assume	O
values	O
in	O
.	O
)	O
</s>
<s>
In	O
this	O
way	O
,	O
Hopfield	B-Algorithm
networks	I-Algorithm
have	O
the	O
ability	O
to	O
"	O
remember	O
"	O
states	O
stored	O
in	O
the	O
interaction	O
matrix	O
,	O
because	O
if	O
a	O
new	O
state	O
is	O
subjected	O
to	O
the	O
interaction	O
matrix	O
,	O
each	O
neuron	O
will	O
change	O
until	O
it	O
matches	O
the	O
original	O
state	O
(	O
see	O
the	O
Updates	O
section	O
below	O
)	O
.	O
</s>
<s>
The	O
connections	O
in	O
a	O
Hopfield	B-Algorithm
net	I-Algorithm
typically	O
have	O
the	O
following	O
restrictions	O
:	O
</s>
<s>
Hopfield	O
also	O
modeled	O
neural	B-Architecture
nets	I-Architecture
for	O
continuous	O
values	O
,	O
in	O
which	O
the	O
electric	O
output	O
of	O
each	O
neuron	O
is	O
not	O
binary	O
but	O
some	O
value	O
between	O
0	O
and	O
1	O
.	O
</s>
<s>
Notice	O
that	O
every	O
pair	O
of	O
units	O
i	O
and	O
j	O
in	O
a	O
Hopfield	B-Algorithm
network	I-Algorithm
has	O
a	O
connection	O
that	O
is	O
described	O
by	O
the	O
connectivity	O
weight	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
can	O
be	O
formally	O
described	O
as	O
a	O
complete	O
undirected	O
graph	O
,	O
where	O
is	O
a	O
set	O
of	O
McCulloch	B-Algorithm
–	I-Algorithm
Pitts	I-Algorithm
neurons	I-Algorithm
and	O
is	O
a	O
function	O
that	O
links	O
pairs	O
of	O
units	O
to	O
a	O
real	O
value	O
,	O
the	O
connectivity	O
weight	O
.	O
</s>
<s>
Updating	O
one	O
unit	O
(	O
node	O
in	O
the	O
graph	O
simulating	O
the	O
artificial	B-Algorithm
neuron	I-Algorithm
)	O
in	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
is	O
performed	O
using	O
the	O
following	O
rule	O
:	O
</s>
<s>
Updates	O
in	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
can	O
be	O
performed	O
in	O
two	O
different	O
ways	O
:	O
</s>
<s>
Bruck	O
shed	O
light	O
on	O
the	O
behavior	O
of	O
a	O
neuron	O
in	O
the	O
discrete	O
Hopfield	B-Algorithm
network	I-Algorithm
when	O
proving	O
its	O
convergence	O
in	O
his	O
paper	O
in	O
1990	O
.	O
</s>
<s>
A	O
subsequent	O
paper	O
further	O
investigated	O
the	O
behavior	O
of	O
any	O
neuron	O
in	O
both	O
discrete-time	O
and	O
continuous-time	O
Hopfield	B-Algorithm
networks	I-Algorithm
when	O
the	O
corresponding	O
energy	O
function	O
is	O
minimized	O
during	O
an	O
optimization	O
process	O
.	O
</s>
<s>
The	O
discrete	O
Hopfield	B-Algorithm
network	I-Algorithm
minimizes	O
the	O
following	O
biased	O
pseudo-cut	O
for	O
the	O
synaptic	O
weight	O
matrix	O
of	O
the	O
Hopfield	B-Algorithm
net	I-Algorithm
.	O
</s>
<s>
The	O
complex	O
Hopfield	B-Algorithm
network	I-Algorithm
,	O
on	O
the	O
other	O
hand	O
,	O
generally	O
tends	O
to	O
minimize	O
the	O
so-called	O
shadow-cut	O
of	O
the	O
complex	O
weight	O
matrix	O
of	O
the	O
net	O
.	O
</s>
<s>
Hopfield	B-Algorithm
nets	I-Algorithm
have	O
a	O
scalar	O
value	O
associated	O
with	O
each	O
state	O
of	O
the	O
network	O
,	O
referred	O
to	O
as	O
the	O
"	O
energy	O
"	O
,	O
E	O
,	O
of	O
the	O
network	O
,	O
where	O
:	O
</s>
<s>
Hopfield	O
and	O
Tank	O
presented	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
application	O
in	O
solving	O
the	O
classical	O
traveling-salesman	O
problem	O
in	O
1985	O
.	O
</s>
<s>
Since	O
then	O
,	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
has	O
been	O
widely	O
used	O
for	O
optimization	O
.	O
</s>
<s>
The	O
idea	O
of	O
using	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
in	O
optimization	O
problems	O
is	O
straightforward	O
:	O
If	O
a	O
constrained/unconstrained	O
cost	O
function	O
can	O
be	O
written	O
in	O
the	O
form	O
of	O
the	O
Hopfield	O
energy	O
function	O
E	O
,	O
then	O
there	O
exists	O
a	O
Hopfield	B-Algorithm
network	I-Algorithm
whose	O
equilibrium	O
points	O
represent	O
solutions	O
to	O
the	O
constrained/unconstrained	O
optimization	O
problem	O
.	O
</s>
<s>
Initialization	O
of	O
the	O
Hopfield	B-Algorithm
networks	I-Algorithm
is	O
done	O
by	O
setting	O
the	O
values	O
of	O
the	O
units	O
to	O
the	O
desired	O
start	O
pattern	O
.	O
</s>
<s>
Therefore	O
,	O
in	O
the	O
context	O
of	O
Hopfield	B-Algorithm
networks	I-Algorithm
,	O
an	O
attractor	O
pattern	O
is	O
a	O
final	O
stable	O
state	O
,	O
a	O
pattern	O
that	O
cannot	O
change	O
any	O
value	O
within	O
it	O
under	O
updating	O
.	O
</s>
<s>
Training	O
a	O
Hopfield	B-Algorithm
net	I-Algorithm
involves	O
lowering	O
the	O
energy	O
of	O
states	O
that	O
the	O
net	O
should	O
"	O
remember	O
"	O
.	O
</s>
<s>
This	O
allows	O
the	O
net	O
to	O
serve	O
as	O
a	O
content	B-Data_Structure
addressable	I-Data_Structure
memory	I-Data_Structure
system	O
,	O
that	O
is	O
to	O
say	O
,	O
the	O
network	O
will	O
converge	O
to	O
a	O
"	O
remembered	O
"	O
state	O
if	O
it	O
is	O
given	O
only	O
part	O
of	O
the	O
state	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
we	O
train	O
a	O
Hopfield	B-Algorithm
net	I-Algorithm
with	O
five	O
units	O
so	O
that	O
the	O
state	O
(	O
1	O
,	O
−1	O
,	O
1	O
,	O
−1	O
,	O
1	O
)	O
is	O
an	O
energy	O
minimum	O
,	O
and	O
we	O
give	O
the	O
network	O
the	O
state	O
(	O
1	O
,	O
−1	O
,	O
−1	O
,	O
−1	O
,	O
1	O
)	O
it	O
will	O
converge	O
to	O
(	O
1	O
,	O
−1	O
,	O
1	O
,	O
−1	O
,	O
1	O
)	O
.	O
</s>
<s>
There	O
are	O
various	O
different	O
learning	B-Algorithm
rules	I-Algorithm
that	O
can	O
be	O
used	O
to	O
store	O
information	O
in	O
the	O
memory	O
of	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
.	O
</s>
<s>
It	O
is	O
desirable	O
for	O
a	O
learning	B-Algorithm
rule	I-Algorithm
to	O
have	O
both	O
of	O
the	O
following	O
two	O
properties	O
:	O
</s>
<s>
Local	O
:	O
A	O
learning	B-Algorithm
rule	I-Algorithm
is	O
local	O
if	O
each	O
weight	O
is	O
updated	O
using	O
information	O
available	O
to	O
neurons	O
on	O
either	O
side	O
of	O
the	O
connection	O
that	O
is	O
associated	O
with	O
that	O
particular	O
weight	O
.	O
</s>
<s>
These	O
properties	O
are	O
desirable	O
,	O
since	O
a	O
learning	B-Algorithm
rule	I-Algorithm
satisfying	O
them	O
is	O
more	O
biologically	O
plausible	O
.	O
</s>
<s>
Storkey	O
also	O
showed	O
that	O
a	O
Hopfield	B-Algorithm
network	I-Algorithm
trained	O
using	O
this	O
rule	O
has	O
a	O
greater	O
capacity	O
than	O
a	O
corresponding	O
network	O
trained	O
using	O
the	O
Hebbian	O
rule	O
.	O
</s>
<s>
The	O
weight	O
matrix	O
of	O
an	O
attractor	O
neural	B-Architecture
network	I-Architecture
is	O
said	O
to	O
follow	O
the	O
Storkey	O
learning	B-Algorithm
rule	I-Algorithm
if	O
it	O
obeys	O
:	O
</s>
<s>
This	O
learning	B-Algorithm
rule	I-Algorithm
is	O
local	O
,	O
since	O
the	O
synapses	O
take	O
into	O
account	O
only	O
neurons	O
at	O
their	O
sides	O
.	O
</s>
<s>
The	O
Network	O
capacity	O
of	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
model	O
is	O
determined	O
by	O
neuron	O
amounts	O
and	O
connections	O
within	O
a	O
given	O
network	O
.	O
</s>
<s>
Furthermore	O
,	O
it	O
was	O
shown	O
that	O
the	O
recall	O
accuracy	O
between	O
vectors	O
and	O
nodes	B-Algorithm
was	O
0.138	O
(	O
approximately	O
138	O
vectors	O
can	O
be	O
recalled	O
from	O
storage	O
for	O
every	O
1000	O
nodes	B-Algorithm
)	O
(	O
Hertz	O
et	O
al.	O
,	O
1991	O
)	O
.	O
</s>
<s>
When	O
the	O
Hopfield	B-Algorithm
model	I-Algorithm
does	O
not	O
recall	O
the	O
right	O
pattern	O
,	O
it	O
is	O
possible	O
that	O
an	O
intrusion	O
has	O
taken	O
place	O
,	O
since	O
semantically	O
related	O
items	O
tend	O
to	O
confuse	O
the	O
individual	O
,	O
and	O
recollection	O
of	O
the	O
wrong	O
pattern	O
occurs	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
model	O
is	O
shown	O
to	O
confuse	O
one	O
stored	O
item	O
with	O
that	O
of	O
another	O
upon	O
retrieval	O
.	O
</s>
<s>
Ulterior	O
models	O
inspired	O
by	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
were	O
later	O
devised	O
to	O
raise	O
the	O
storage	O
limit	O
and	O
reduce	O
the	O
retrieval	O
error	O
rate	O
,	O
with	O
some	O
being	O
capable	O
of	O
one-shot	O
learning	O
.	O
</s>
<s>
The	O
Hopfield	B-Algorithm
model	I-Algorithm
accounts	O
for	O
associative	O
memory	O
through	O
the	O
incorporation	O
of	O
memory	O
vectors	O
.	O
</s>
<s>
In	O
associative	O
memory	O
for	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
,	O
there	O
are	O
two	O
types	O
of	O
operations	O
:	O
auto-association	O
and	O
hetero-association	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
that	O
Hopfield	O
's	O
network	O
model	O
utilizes	O
the	O
same	O
learning	B-Algorithm
rule	I-Algorithm
as	O
Hebb	O
's	O
(	O
1949	O
)	O
learning	B-Algorithm
rule	I-Algorithm
,	O
which	O
basically	O
tried	O
to	O
show	O
that	O
learning	O
occurs	O
as	O
a	O
result	O
of	O
the	O
strengthening	O
of	O
the	O
weights	O
by	O
when	O
activity	O
is	O
occurring	O
.	O
</s>
<s>
Rizzuto	O
and	O
Kahana	O
(	O
2001	O
)	O
were	O
able	O
to	O
show	O
that	O
the	O
neural	B-Architecture
network	I-Architecture
model	I-Architecture
can	O
account	O
for	O
repetition	O
on	O
recall	O
accuracy	O
by	O
incorporating	O
a	O
probabilistic-learning	O
algorithm	O
.	O
</s>
<s>
By	O
adding	O
contextual	O
drift	O
they	O
were	O
able	O
to	O
show	O
the	O
rapid	O
forgetting	O
that	O
occurs	O
in	O
a	O
Hopfield	B-Algorithm
model	I-Algorithm
during	O
a	O
cued-recall	O
task	O
.	O
</s>
<s>
Hopfield	O
would	O
use	O
Pitts	O
's	O
dynamical	O
rule	O
in	O
order	O
to	O
show	O
how	O
retrieval	O
is	O
possible	O
in	O
the	O
Hopfield	B-Algorithm
network	I-Algorithm
.	O
</s>
<s>
Hopfield	B-Algorithm
networks	I-Algorithm
are	O
recurrent	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
with	O
dynamical	O
trajectories	O
converging	O
to	O
fixed	O
point	O
attractor	O
states	O
and	O
described	O
by	O
an	O
energy	O
function	O
.	O
</s>
<s>
In	O
the	O
original	O
Hopfield	B-Algorithm
model	I-Algorithm
of	O
associative	O
memory	O
,	O
the	O
variables	O
were	O
binary	O
,	O
and	O
the	O
dynamics	O
were	O
described	O
by	O
a	O
one-at-a-time	O
update	O
of	O
the	O
state	O
of	O
the	O
neurons	O
.	O
</s>
<s>
Dense	O
Associative	O
Memories	O
(	O
also	O
known	O
as	O
the	O
modern	B-General_Concept
Hopfield	I-General_Concept
networks	I-General_Concept
)	O
are	O
generalizations	O
of	O
the	O
classical	O
Hopfield	B-Algorithm
Networks	I-Algorithm
that	O
break	O
the	O
linear	O
scaling	O
relationship	O
between	O
the	O
number	O
of	O
input	O
features	O
and	O
the	O
number	O
of	O
stored	O
memories	O
.	O
</s>
<s>
The	O
key	O
theoretical	O
idea	O
behind	O
the	O
modern	B-General_Concept
Hopfield	I-General_Concept
networks	I-General_Concept
is	O
to	O
use	O
an	O
energy	O
function	O
and	O
an	O
update	O
rule	O
that	O
is	O
more	O
sharply	O
peaked	O
around	O
the	O
stored	O
memories	O
in	O
the	O
space	O
of	O
neuron	O
’s	O
configurations	O
compared	O
to	O
the	O
classical	O
Hopfield	B-Algorithm
Network	I-Algorithm
.	O
</s>
<s>
A	O
simple	O
example	O
of	O
the	O
modern	B-General_Concept
Hopfield	I-General_Concept
network	I-General_Concept
can	O
be	O
written	O
in	O
terms	O
of	O
binary	O
variables	O
that	O
represent	O
the	O
active	O
and	O
inactive	O
state	O
of	O
the	O
model	O
neuron	O
.In	O
this	O
formula	O
the	O
weights	O
represent	O
the	O
matrix	O
of	O
memory	O
vectors	O
(	O
index	O
enumerates	O
different	O
memories	O
,	O
and	O
index	O
enumerates	O
the	O
content	O
of	O
each	O
memory	O
corresponding	O
to	O
the	O
-th	O
feature	O
neuron	O
)	O
,	O
and	O
the	O
function	O
is	O
a	O
rapidly	O
growing	O
non-linear	O
function	O
.	O
</s>
<s>
In	O
the	O
limiting	O
case	O
when	O
the	O
non-linear	O
energy	O
function	O
is	O
quadratic	O
these	O
equations	O
reduce	O
to	O
the	O
familiar	O
energy	O
function	O
and	O
the	O
update	O
rule	O
for	O
the	O
classical	O
binary	O
Hopfield	B-Algorithm
Network	I-Algorithm
.	O
</s>
<s>
Modern	B-General_Concept
Hopfield	I-General_Concept
networks	I-General_Concept
or	O
dense	O
associative	O
memories	O
can	O
be	O
best	O
understood	O
in	O
continuous	O
variables	O
and	O
continuous	O
time	O
.	O
</s>
<s>
For	O
Hopfield	B-Algorithm
Networks	I-Algorithm
,	O
however	O
,	O
this	O
is	O
not	O
the	O
case	O
-	O
the	O
dynamical	O
trajectories	O
always	O
converge	O
to	O
a	O
fixed	O
point	O
attractor	O
state	O
.	O
</s>
<s>
Classical	O
formulation	O
of	O
continuous	O
Hopfield	B-Algorithm
Networks	I-Algorithm
can	O
be	O
understood	O
as	O
a	O
special	O
limiting	O
case	O
of	O
the	O
modern	B-General_Concept
Hopfield	I-General_Concept
networks	I-General_Concept
with	O
one	O
hidden	O
layer	O
.	O
</s>
<s>
This	O
completes	O
the	O
proof	O
that	O
the	O
classical	O
Hopfield	B-Algorithm
Network	I-Algorithm
with	O
continuous	O
states	O
is	O
a	O
special	O
limiting	O
case	O
of	O
the	O
modern	B-General_Concept
Hopfield	I-General_Concept
network	I-General_Concept
(	O
)	O
with	O
energy	O
(	O
)	O
.	O
</s>
<s>
Biological	O
neural	B-Architecture
networks	I-Architecture
have	O
a	O
large	O
degree	O
of	O
heterogeneity	O
in	O
terms	O
of	O
different	O
cell	O
types	O
.	O
</s>
<s>
This	O
section	O
describes	O
a	O
mathematical	O
model	O
of	O
a	O
fully	O
connected	O
modern	B-General_Concept
Hopfield	I-General_Concept
network	I-General_Concept
assuming	O
the	O
extreme	O
degree	O
of	O
heterogeneity	O
:	O
every	O
single	O
neuron	O
is	O
different	O
.	O
</s>
