<s>
An	O
artificial	B-Algorithm
neuron	I-Algorithm
is	O
a	O
mathematical	O
function	O
conceived	O
as	O
a	O
model	O
of	O
biological	O
neurons	O
,	O
a	O
neural	B-Architecture
network	I-Architecture
.	O
</s>
<s>
Artificial	B-Algorithm
neurons	I-Algorithm
are	O
elementary	O
units	O
in	O
an	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
.	O
</s>
<s>
The	O
artificial	B-Algorithm
neuron	I-Algorithm
receives	O
one	O
or	O
more	O
inputs	O
(	O
representing	O
excitatory	O
postsynaptic	O
potentials	O
and	O
inhibitory	O
postsynaptic	O
potentials	O
at	O
neural	O
dendrites	O
)	O
and	O
sums	O
them	O
to	O
produce	O
an	O
output	O
(	O
or	O
,	O
representing	O
a	O
neuron	O
's	O
action	B-Algorithm
potential	I-Algorithm
which	O
is	O
transmitted	O
along	O
its	O
axon	B-Algorithm
)	O
.	O
</s>
<s>
Usually	O
each	O
input	O
is	O
separately	O
weighted	B-General_Concept
,	O
and	O
the	O
sum	O
is	O
passed	O
through	O
a	O
non-linear	O
function	O
known	O
as	O
an	O
activation	B-Algorithm
function	I-Algorithm
or	O
transfer	O
function	O
.	O
</s>
<s>
The	O
transfer	O
functions	O
usually	O
have	O
a	O
sigmoid	B-Algorithm
shape	I-Algorithm
,	O
but	O
they	O
may	O
also	O
take	O
the	O
form	O
of	O
other	O
non-linear	O
functions	O
,	O
piecewise	B-Algorithm
linear	O
functions	O
,	O
or	O
step	O
functions	O
.	O
</s>
<s>
Non-monotonic	O
,	O
unbounded	O
and	O
oscillating	O
activation	B-Algorithm
functions	I-Algorithm
with	O
multiple	O
zeros	O
that	O
outperform	O
sigmoidal	O
and	O
ReLU	B-Algorithm
like	O
activation	B-Algorithm
functions	I-Algorithm
on	O
many	O
tasks	O
have	O
also	O
been	O
recently	O
explored	O
.	O
</s>
<s>
The	O
artificial	B-Algorithm
neuron	I-Algorithm
transfer	O
function	O
should	O
not	O
be	O
confused	O
with	O
a	O
linear	O
system	O
's	O
transfer	O
function	O
.	O
</s>
<s>
Artificial	B-Algorithm
neurons	I-Algorithm
can	O
also	O
refer	O
to	O
artificial	O
cells	O
in	O
neuromorphic	O
engineering	O
that	O
are	O
similar	O
to	O
natural	O
physical	O
neurons	O
.	O
</s>
<s>
For	O
a	O
given	O
artificial	B-Algorithm
neuron	I-Algorithm
k	O
,	O
let	O
there	O
be	O
m+1	O
inputs	O
with	O
signals	O
x0	O
through	O
xm	O
and	O
weights	O
wk0	O
through	O
wkm	O
.	O
</s>
<s>
The	O
output	O
is	O
analogous	O
to	O
the	O
axon	B-Algorithm
of	O
a	O
biological	O
neuron	O
,	O
and	O
its	O
value	O
propagates	O
to	O
the	O
input	O
of	O
the	O
next	O
layer	O
,	O
through	O
a	O
synapse	O
.	O
</s>
<s>
Depending	O
on	O
the	O
specific	O
model	O
used	O
they	O
may	O
be	O
called	O
a	O
semi-linear	O
unit	O
,	O
Nv	B-Algorithm
neuron	I-Algorithm
,	O
binary	B-Algorithm
neuron	I-Algorithm
,	O
linear	O
threshold	O
function	O
,	O
or	O
McCulloch	O
–	O
Pitts	O
(	O
MCP	O
)	O
neuron	O
.	O
</s>
<s>
Simple	O
artificial	B-Algorithm
neurons	I-Algorithm
,	O
such	O
as	O
the	O
McCulloch	O
–	O
Pitts	O
model	O
,	O
are	O
sometimes	O
described	O
as	O
"	O
caricature	O
models	O
"	O
,	O
since	O
they	O
are	O
intended	O
to	O
reflect	O
one	O
or	O
more	O
neurophysiological	O
observations	O
,	O
but	O
without	O
regard	O
to	O
realism	O
.	O
</s>
<s>
Artificial	B-Algorithm
neurons	I-Algorithm
are	O
designed	O
to	O
mimic	O
aspects	O
of	O
their	O
biological	O
counterparts	O
.	O
</s>
<s>
However	O
a	O
significant	O
performance	O
gap	O
exists	O
between	O
biological	O
and	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
.	O
</s>
<s>
In	O
particular	O
single	O
biological	O
neurons	O
in	O
the	O
human	O
brain	O
with	O
oscillating	O
activation	B-Algorithm
function	I-Algorithm
capable	O
of	O
learning	O
the	O
XOR	O
function	O
have	O
been	O
discovered	O
.	O
</s>
<s>
Axon	B-Algorithm
–	O
The	O
axon	B-Algorithm
gets	O
its	O
signal	O
from	O
the	O
summation	O
behavior	O
which	O
occurs	O
inside	O
the	O
soma	O
.	O
</s>
<s>
The	O
opening	O
to	O
the	O
axon	B-Algorithm
essentially	O
samples	O
the	O
electrical	O
potential	O
of	O
the	O
solution	O
inside	O
the	O
soma	O
.	O
</s>
<s>
Once	O
the	O
soma	O
reaches	O
a	O
certain	O
potential	O
,	O
the	O
axon	B-Algorithm
will	O
transmit	O
an	O
all-in	O
signal	O
pulse	O
down	O
its	O
length	O
.	O
</s>
<s>
In	O
this	O
regard	O
,	O
the	O
axon	B-Algorithm
behaves	O
as	O
the	O
ability	O
for	O
us	O
to	O
connect	O
our	O
artificial	B-Algorithm
neuron	I-Algorithm
to	O
other	O
artificial	B-Algorithm
neurons	I-Algorithm
.	O
</s>
<s>
Unlike	O
most	O
artificial	B-Algorithm
neurons	I-Algorithm
,	O
however	O
,	O
biological	O
neurons	O
fire	O
in	O
discrete	O
pulses	O
.	O
</s>
<s>
Each	O
time	O
the	O
electrical	O
potential	O
inside	O
the	O
soma	O
reaches	O
a	O
certain	O
threshold	O
,	O
a	O
pulse	O
is	O
transmitted	O
down	O
the	O
axon	B-Algorithm
.	O
</s>
<s>
at	O
which	O
an	O
axon	B-Algorithm
fires	O
converts	O
directly	O
into	O
the	O
rate	O
at	O
which	O
neighboring	O
cells	O
get	O
signal	O
ions	O
introduced	O
into	O
them	O
.	O
</s>
<s>
The	O
faster	O
a	O
biological	O
neuron	O
fires	O
,	O
the	O
faster	O
nearby	O
neurons	O
accumulate	O
electrical	O
potential	O
(	O
or	O
lose	O
electrical	O
potential	O
,	O
depending	O
on	O
the	O
"	O
weighting	B-General_Concept
"	O
of	O
the	O
dendrite	O
that	O
connects	O
to	O
the	O
neuron	O
that	O
fired	O
)	O
.	O
</s>
<s>
It	O
is	O
this	O
conversion	O
that	O
allows	O
computer	O
scientists	O
and	O
mathematicians	O
to	O
simulate	O
biological	O
neural	B-Architecture
networks	I-Architecture
using	O
artificial	B-Algorithm
neurons	I-Algorithm
which	O
can	O
output	O
distinct	O
values	O
(	O
often	O
from	O
−1	O
to	O
1	O
)	O
.	O
</s>
<s>
Research	O
has	O
shown	O
that	O
unary	B-Algorithm
coding	I-Algorithm
is	O
used	O
in	O
the	O
neural	O
circuits	O
responsible	O
for	O
birdsong	O
production	O
.	O
</s>
<s>
Another	O
contributing	O
factor	O
could	O
be	O
that	O
unary	B-Algorithm
coding	I-Algorithm
provides	O
a	O
certain	O
degree	O
of	O
error	O
correction	O
.	O
</s>
<s>
There	O
is	O
research	O
and	O
development	O
into	O
physical	O
artificial	B-Algorithm
neurons	I-Algorithm
–	O
organic	O
and	O
inorganic	O
.	O
</s>
<s>
For	O
example	O
,	O
some	O
artificial	B-Algorithm
neurons	I-Algorithm
can	O
receive	O
and	O
release	O
dopamine	O
(	O
chemical	O
signals	O
rather	O
than	O
electrical	O
signals	O
)	O
and	O
communicate	O
with	O
natural	O
rat	O
muscle	O
and	O
brain	O
cells	O
,	O
with	O
potential	O
for	O
use	O
in	O
BCIs/prosthetics	O
.	O
</s>
<s>
Low-power	O
biocompatible	O
memristors	O
may	O
enable	O
construction	O
of	O
artificial	B-Algorithm
neurons	I-Algorithm
which	O
function	O
at	O
voltages	O
of	O
biological	O
action	B-Algorithm
potentials	I-Algorithm
and	O
could	O
be	O
used	O
to	O
directly	O
process	O
biosensing	O
signals	O
,	O
for	O
neuromorphic	O
computing	O
and/or	O
direct	B-Application
communication	I-Application
with	I-Application
biological	I-Application
neurons	I-Application
.	O
</s>
<s>
Organic	O
neuromorphic	O
circuits	O
made	O
out	O
of	O
polymers	B-Language
,	O
coated	O
with	O
an	O
ion-rich	O
gel	O
to	O
enable	O
a	O
material	O
to	O
carry	O
an	O
electric	O
charge	O
like	O
real	O
neurons	O
,	O
have	O
been	O
built	O
into	O
a	O
robot	O
,	O
enabling	O
it	O
to	O
learn	O
sensorimotorically	O
within	O
the	O
real	O
world	O
,	O
rather	O
than	O
via	O
simulations	O
or	O
virtually	O
.	O
</s>
<s>
Moreover	O
,	O
artificial	O
spiking	O
neurons	O
made	O
of	O
soft	O
matter	O
(	O
polymers	B-Language
)	O
can	O
operate	O
in	O
biologically	O
relevant	O
environments	O
and	O
enable	O
the	O
synergetic	O
communication	O
between	O
the	O
artificial	O
and	O
biological	O
domains	O
.	O
</s>
<s>
The	O
first	O
artificial	B-Algorithm
neuron	I-Algorithm
was	O
the	O
Threshold	B-Algorithm
Logic	I-Algorithm
Unit	I-Algorithm
(	O
TLU	O
)	O
,	O
or	O
Linear	O
Threshold	O
Unit	O
,	O
first	O
proposed	O
by	O
Warren	O
McCulloch	O
and	O
Walter	O
Pitts	O
in	O
1943	O
.	O
</s>
<s>
Since	O
the	O
beginning	O
it	O
was	O
already	O
noticed	O
that	O
any	O
boolean	O
function	O
could	O
be	O
implemented	O
by	O
networks	O
of	O
such	O
devices	O
,	O
what	O
is	O
easily	O
seen	O
from	O
the	O
fact	O
that	O
one	O
can	O
implement	O
the	O
AND	O
and	O
OR	O
functions	O
,	O
and	O
use	O
them	O
in	O
the	O
disjunctive	B-Application
or	O
the	O
conjunctive	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Researchers	O
also	O
soon	O
realized	O
that	O
cyclic	O
networks	O
,	O
with	O
feedbacks	O
through	O
neurons	O
,	O
could	O
define	O
dynamical	O
systems	O
with	O
memory	O
,	O
but	O
most	O
of	O
the	O
research	O
concentrated	O
(	O
and	O
still	O
does	O
)	O
on	O
strictly	O
feed-forward	B-Algorithm
networks	I-Algorithm
because	O
of	O
the	O
smaller	O
difficulty	O
they	O
present	O
.	O
</s>
<s>
One	O
important	O
and	O
pioneering	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
that	O
used	O
the	O
linear	O
threshold	O
function	O
was	O
the	O
perceptron	B-Algorithm
,	O
developed	O
by	O
Frank	O
Rosenblatt	O
.	O
</s>
<s>
The	O
representation	O
of	O
the	O
threshold	O
values	O
as	O
a	O
bias	O
term	O
was	O
introduced	O
by	O
Bernard	O
Widrow	O
in	O
1960	O
–	O
see	O
ADALINE	B-Algorithm
.	O
</s>
<s>
In	O
the	O
late	O
1980s	O
,	O
when	O
research	O
on	O
neural	B-Architecture
networks	I-Architecture
regained	O
strength	O
,	O
neurons	O
with	O
more	O
continuous	O
shapes	O
started	O
to	O
be	O
considered	O
.	O
</s>
<s>
The	O
possibility	O
of	O
differentiating	O
the	O
activation	B-Algorithm
function	I-Algorithm
allows	O
the	O
direct	O
use	O
of	O
the	O
gradient	B-Algorithm
descent	I-Algorithm
and	O
other	O
optimization	O
algorithms	O
for	O
the	O
adjustment	O
of	O
the	O
weights	O
.	O
</s>
<s>
Neural	B-Architecture
networks	I-Architecture
also	O
started	O
to	O
be	O
used	O
as	O
a	O
general	O
function	O
approximation	O
model	O
.	O
</s>
<s>
The	O
best	O
known	O
training	O
algorithm	O
called	O
backpropagation	B-Algorithm
has	O
been	O
rediscovered	O
several	O
times	O
but	O
its	O
first	O
development	O
goes	O
back	O
to	O
the	O
work	O
of	O
Paul	O
Werbos	O
.	O
</s>
<s>
The	O
transfer	O
function	O
(	O
activation	B-Algorithm
function	I-Algorithm
)	O
of	O
a	O
neuron	O
is	O
chosen	O
to	O
have	O
a	O
number	O
of	O
properties	O
which	O
either	O
enhance	O
or	O
simplify	O
the	O
network	O
containing	O
the	O
neuron	O
.	O
</s>
<s>
Crucially	O
,	O
for	O
instance	O
,	O
any	O
multilayer	B-Algorithm
perceptron	I-Algorithm
using	O
a	O
linear	O
transfer	O
function	O
has	O
an	O
equivalent	O
single-layer	O
network	O
;	O
a	O
non-linear	O
function	O
is	O
therefore	O
necessary	O
to	O
gain	O
the	O
advantages	O
of	O
a	O
multi-layer	O
network	O
.	O
</s>
<s>
Below	O
,	O
u	O
refers	O
in	O
all	O
cases	O
to	O
the	O
weighted	B-General_Concept
sum	O
of	O
all	O
the	O
inputs	O
to	O
the	O
neuron	O
,	O
i.e.	O
</s>
<s>
This	O
function	O
is	O
used	O
in	O
perceptrons	B-Algorithm
and	O
often	O
shows	O
up	O
in	O
many	O
other	O
models	O
.	O
</s>
<s>
It	O
can	O
be	O
approximated	O
from	O
other	O
sigmoidal	B-Algorithm
functions	I-Algorithm
by	O
assigning	O
large	O
values	O
to	O
the	O
weights	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
output	O
unit	O
is	O
simply	O
the	O
weighted	B-General_Concept
sum	O
of	O
its	O
inputs	O
plus	O
a	O
bias	O
term	O
.	O
</s>
<s>
A	O
number	O
of	O
such	O
linear	B-Algorithm
neurons	I-Algorithm
perform	O
a	O
linear	B-Architecture
transformation	I-Architecture
of	O
the	O
input	O
vector	O
.	O
</s>
<s>
A	O
number	O
of	O
analysis	O
tools	O
exist	O
based	O
on	O
linear	O
models	O
,	O
such	O
as	O
harmonic	O
analysis	O
,	O
and	O
they	O
can	O
all	O
be	O
used	O
in	O
neural	B-Architecture
networks	I-Architecture
with	O
this	O
linear	B-Algorithm
neuron	I-Algorithm
.	O
</s>
<s>
See	O
:	O
Linear	B-Architecture
transformation	I-Architecture
,	O
Harmonic	O
analysis	O
,	O
Linear	O
filter	O
,	O
Wavelet	O
,	O
Principal	B-Application
component	I-Application
analysis	I-Application
,	O
Independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
,	O
Deconvolution	B-Algorithm
.	O
</s>
<s>
A	O
fairly	O
simple	O
non-linear	O
function	O
,	O
the	O
sigmoid	B-Algorithm
function	I-Algorithm
such	O
as	O
the	O
logistic	O
function	O
also	O
has	O
an	O
easily	O
calculated	O
derivative	O
,	O
which	O
can	O
be	O
important	O
when	O
calculating	O
the	O
weight	O
updates	O
in	O
the	O
network	O
.	O
</s>
<s>
It	O
was	O
previously	O
commonly	O
seen	O
in	O
multilayer	B-Algorithm
perceptrons	I-Algorithm
.	O
</s>
<s>
However	O
,	O
recent	O
work	O
has	O
shown	O
sigmoid	O
neurons	O
to	O
be	O
less	O
effective	O
than	O
rectified	B-Algorithm
linear	I-Algorithm
neurons	O
.	O
</s>
<s>
The	O
reason	O
is	O
that	O
the	O
gradients	O
computed	O
by	O
the	O
backpropagation	B-Algorithm
algorithm	O
tend	O
to	O
diminish	O
towards	O
zero	O
as	O
activations	O
propagate	O
through	O
layers	O
of	O
sigmoidal	O
neurons	O
,	O
making	O
it	O
difficult	O
to	O
optimize	O
neural	B-Architecture
networks	I-Architecture
using	O
multiple	O
layers	O
of	O
sigmoidal	O
neurons	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
,	O
the	O
rectifier	B-Algorithm
or	O
ReLU	B-Algorithm
(	O
Rectified	B-Algorithm
Linear	I-Algorithm
Unit	I-Algorithm
)	O
is	O
an	O
activation	B-Algorithm
function	I-Algorithm
defined	O
as	O
the	O
positive	O
part	O
of	O
its	O
argument	O
:	O
</s>
<s>
This	O
activation	B-Algorithm
function	I-Algorithm
was	O
first	O
introduced	O
to	O
a	O
dynamical	O
network	O
by	O
Hahnloser	O
et	O
al	O
.	O
</s>
<s>
It	O
has	O
been	O
demonstrated	O
for	O
the	O
first	O
time	O
in	O
2011	O
to	O
enable	O
better	O
training	O
of	O
deeper	O
networks	O
,	O
compared	O
to	O
the	O
widely	O
used	O
activation	B-Algorithm
functions	I-Algorithm
prior	O
to	O
2011	O
,	O
i.e.	O
,	O
the	O
logistic	O
sigmoid	O
(	O
which	O
is	O
inspired	O
by	O
probability	O
theory	O
;	O
see	O
logistic	O
regression	O
)	O
and	O
its	O
more	O
practical	O
counterpart	O
,	O
the	O
hyperbolic	O
tangent	O
.	O
</s>
<s>
A	O
commonly	O
used	O
variant	O
of	O
the	O
ReLU	B-Algorithm
activation	B-Algorithm
function	I-Algorithm
is	O
the	O
Leaky	O
ReLU	B-Algorithm
which	O
allows	O
a	O
small	O
,	O
positive	O
gradient	O
when	O
the	O
unit	O
is	O
not	O
active	O
:	O
</s>
<s>
The	O
following	O
is	O
a	O
simple	O
pseudocode	B-Language
implementation	O
of	O
a	O
single	O
TLU	O
which	O
takes	O
boolean	O
inputs	O
(	O
true	O
or	O
false	O
)	O
,	O
and	O
returns	O
a	O
single	O
boolean	O
output	O
when	O
activated	O
.	O
</s>
<s>
An	O
object-oriented	B-Language
model	O
is	O
used	O
.	O
</s>
