<s>
Spiking	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
(	O
SNNs	O
)	O
are	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
that	O
more	O
closely	O
mimic	O
natural	O
neural	B-General_Concept
networks	I-General_Concept
.	O
</s>
<s>
In	O
addition	O
to	O
neuronal	B-Algorithm
and	O
synaptic	B-Application
state	O
,	O
SNNs	O
incorporate	O
the	O
concept	O
of	O
time	O
into	O
their	O
operating	O
model	O
.	O
</s>
<s>
The	O
idea	O
is	O
that	O
neurons	B-Algorithm
in	O
the	O
SNN	O
do	O
not	O
transmit	O
information	O
at	O
each	O
propagation	O
cycle	O
(	O
as	O
it	O
happens	O
with	O
typical	O
multi-layer	O
perceptron	B-Algorithm
networks	I-Algorithm
)	O
,	O
but	O
rather	O
transmit	O
information	O
only	O
when	O
a	O
membrane	O
potential	O
–	O
an	O
intrinsic	O
quality	O
of	O
the	O
neuron	O
related	O
to	O
its	O
membrane	O
electrical	O
charge	O
–	O
reaches	O
a	O
specific	O
value	O
,	O
called	O
the	O
threshold	O
.	O
</s>
<s>
When	O
the	O
membrane	O
potential	O
reaches	O
the	O
threshold	O
,	O
the	O
neuron	O
fires	O
,	O
and	O
generates	O
a	O
signal	O
that	O
travels	O
to	O
other	O
neurons	B-Algorithm
which	O
,	O
in	O
turn	O
,	O
increase	O
or	O
decrease	O
their	O
potentials	O
in	O
response	O
to	O
this	O
signal	O
.	O
</s>
<s>
A	O
neuron	O
model	O
that	O
fires	O
at	O
the	O
moment	O
of	O
threshold	O
crossing	O
is	O
also	O
called	O
a	O
spiking	B-Algorithm
neuron	I-Algorithm
model	O
.	O
</s>
<s>
The	O
most	O
prominent	O
spiking	B-Algorithm
neuron	I-Algorithm
model	O
is	O
the	O
leaky	O
integrate-and-fire	O
model	O
.	O
</s>
<s>
In	O
the	O
integrate-and-fire	O
model	O
,	O
the	O
momentary	O
activation	O
level	O
(	O
modeled	O
as	O
a	O
differential	O
equation	O
)	O
is	O
normally	O
considered	O
to	O
be	O
the	O
neuron	O
's	O
state	O
,	O
with	O
incoming	O
spikes	B-Algorithm
pushing	O
this	O
value	O
higher	O
or	O
lower	O
,	O
until	O
the	O
state	O
eventually	O
either	O
decays	O
or	O
-	O
if	O
the	O
firing	O
threshold	O
is	O
reached	O
-	O
the	O
neuron	O
fires	O
.	O
</s>
<s>
Various	O
decoding	O
methods	O
exist	O
for	O
interpreting	O
the	O
outgoing	O
spike	B-Algorithm
train	I-Algorithm
as	O
a	O
real-value	O
number	O
,	O
relying	O
on	O
either	O
the	O
frequency	O
of	O
spikes	B-Algorithm
(	O
rate-code	O
)	O
,	O
the	O
time-to-first-spike	O
after	O
stimulation	O
,	O
or	O
the	O
interval	O
between	O
spikes	B-Algorithm
.	O
</s>
<s>
Many	O
multi-layer	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
are	O
fully	B-Architecture
connected	I-Architecture
,	O
receiving	O
input	O
from	O
every	O
neuron	O
in	O
the	O
previous	O
layer	O
and	O
signalling	O
every	O
neuron	O
in	O
the	O
subsequent	O
layer	O
.	O
</s>
<s>
Although	O
these	O
networks	O
have	O
achieved	O
breakthroughs	O
in	O
many	O
fields	O
,	O
they	O
are	O
biologically	O
inaccurate	O
and	O
do	O
not	O
mimic	O
the	O
operation	O
mechanism	O
of	O
neurons	B-Algorithm
in	O
the	O
brain	O
of	O
a	O
living	O
thing	O
.	O
</s>
<s>
The	O
biologically	O
inspired	O
Hodgkin	O
–	O
Huxley	O
model	O
of	O
a	O
spiking	B-Algorithm
neuron	I-Algorithm
was	O
proposed	O
in	O
1952	O
.	O
</s>
<s>
This	O
model	O
describes	O
how	O
action	B-Algorithm
potentials	I-Algorithm
are	O
initiated	O
and	O
propagated	O
.	O
</s>
<s>
Communication	O
between	O
neurons	B-Algorithm
,	O
which	O
requires	O
the	O
exchange	O
of	O
chemical	O
neurotransmitters	O
in	O
the	O
synaptic	B-Application
gap	O
,	O
is	O
described	O
in	O
various	O
models	O
,	O
such	O
as	O
the	O
integrate-and-fire	O
model	O
,	O
FitzHugh	O
–	O
Nagumo	O
model	O
(	O
1961	O
–	O
1962	O
)	O
,	O
and	O
Hindmarsh	B-Algorithm
–	I-Algorithm
Rose	I-Algorithm
model	I-Algorithm
(	O
1984	O
)	O
.	O
</s>
<s>
Information	O
in	O
the	O
brain	O
is	O
represented	O
as	O
action	B-Algorithm
potentials	I-Algorithm
(	O
neuron	O
spikes	B-Algorithm
)	O
,	O
which	O
may	O
be	O
grouped	O
into	O
spike	B-Algorithm
trains	I-Algorithm
or	O
even	O
coordinated	O
waves	O
of	O
brain	O
activity	O
.	O
</s>
<s>
A	O
fundamental	O
question	O
of	O
neuroscience	O
is	O
to	O
determine	O
whether	O
neurons	B-Algorithm
communicate	O
by	O
a	O
rate	O
or	O
temporal	O
code	O
.	O
</s>
<s>
Temporal	O
coding	O
suggests	O
that	O
a	O
single	O
spiking	B-Algorithm
neuron	I-Algorithm
can	O
replace	O
hundreds	O
of	O
hidden	O
units	O
on	O
a	O
sigmoidal	O
neural	B-Architecture
net	I-Architecture
.	O
</s>
<s>
The	O
idea	O
is	O
that	O
neurons	B-Algorithm
may	O
not	O
test	O
for	O
activation	O
in	O
every	O
iteration	O
of	O
propagation	O
(	O
as	O
is	O
the	O
case	O
in	O
a	O
typical	O
multilayer	O
perceptron	B-Algorithm
network	O
)	O
,	O
but	O
only	O
when	O
their	O
membrane	O
potentials	O
reach	O
a	O
certain	O
value	O
.	O
</s>
<s>
When	O
a	O
neuron	O
is	O
activated	O
,	O
it	O
produces	O
a	O
signal	O
that	O
is	O
passed	O
to	O
connected	O
neurons	B-Algorithm
,	O
raising	O
or	O
lowering	O
their	O
membrane	O
potential	O
.	O
</s>
<s>
In	O
a	O
spiking	B-Algorithm
neural	I-Algorithm
network	I-Algorithm
,	O
a	O
neuron	O
's	O
current	O
state	O
is	O
defined	O
as	O
its	O
membrane	O
potential	O
(	O
possibly	O
modeled	O
as	O
a	O
differential	O
equation	O
)	O
.	O
</s>
<s>
A	O
neural	B-Architecture
network	I-Architecture
model	I-Architecture
based	O
on	O
pulse	O
generation	O
time	O
can	O
be	O
established	O
.	O
</s>
<s>
Using	O
the	O
exact	O
time	O
of	O
pulse	O
occurrence	O
,	O
a	O
neural	B-General_Concept
network	I-General_Concept
can	O
employ	O
more	O
information	O
and	O
offer	O
better	O
computing	O
properties	O
.	O
</s>
<s>
SNNs	O
consider	O
space	O
by	O
connecting	O
neurons	B-Algorithm
only	O
to	O
nearby	O
neurons	B-Algorithm
so	O
that	O
they	O
process	O
input	O
blocks	O
separately	O
(	O
similar	O
to	O
CNN	B-Architecture
using	O
filters	O
)	O
.	O
</s>
<s>
This	O
avoids	O
the	O
additional	O
complexity	O
of	O
a	O
recurrent	B-Algorithm
neural	I-Algorithm
network	I-Algorithm
(	O
RNN	O
)	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
impulse	O
neurons	B-Algorithm
are	O
more	O
powerful	O
computational	O
units	O
than	O
traditional	O
artificial	B-Algorithm
neurons	I-Algorithm
.	O
</s>
<s>
Spike-based	O
activation	O
of	O
SNNs	O
is	O
not	O
differentiable	O
thus	O
making	O
it	O
hard	O
to	O
develop	O
gradient	B-Algorithm
descent	I-Algorithm
based	O
training	O
methods	O
to	O
perform	O
error	O
backpropagation	B-Algorithm
,	O
though	O
a	O
few	O
recent	O
algorithms	O
such	O
as	O
NormAD	O
and	O
multilayer	O
NormAD	O
have	O
demonstrated	O
good	O
training	O
performance	O
through	O
suitable	O
approximation	O
of	O
the	O
gradient	O
of	O
spike	B-Algorithm
based	O
activation	O
.	O
</s>
<s>
Pulse-coupled	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
(	O
PCNN	B-Algorithm
)	O
are	O
often	O
confused	O
with	O
SNNs	O
.	O
</s>
<s>
A	O
PCNN	B-Algorithm
can	O
be	O
seen	O
as	O
a	O
kind	O
of	O
SNN	O
.	O
</s>
<s>
The	O
expressions	O
for	O
both	O
the	O
forward	O
-	O
and	O
backward-learning	O
methods	O
contain	O
the	O
derivative	O
of	O
the	O
neural	O
activation	O
function	O
which	O
is	O
non-differentiable	O
because	O
neuron	O
's	O
output	O
is	O
either	O
1	O
when	O
it	O
spikes	B-Algorithm
,	O
and	O
0	O
otherwise	O
.	O
</s>
<s>
This	O
all-or-nothing	O
behavior	O
of	O
the	O
binary	O
spiking	O
nonlinearity	O
stops	O
gradients	O
from	O
“	O
flowing	O
”	O
and	O
makes	O
LIF	O
neurons	B-Algorithm
unsuitable	O
for	O
gradient-based	B-Algorithm
optimization	I-Algorithm
.	O
</s>
<s>
Due	O
to	O
their	O
relative	O
realism	O
,	O
they	O
can	O
be	O
used	O
to	O
study	O
the	O
operation	O
of	O
biological	B-General_Concept
neural	I-General_Concept
circuits	I-General_Concept
.	O
</s>
<s>
Starting	O
with	O
a	O
hypothesis	O
about	O
the	O
topology	B-Architecture
of	O
a	O
biological	O
neuronal	B-General_Concept
circuit	I-General_Concept
and	O
its	O
function	O
,	O
recordings	B-Application
of	O
this	O
circuit	O
can	O
be	O
compared	O
to	O
the	O
output	O
of	O
the	O
corresponding	O
SNN	O
,	O
evaluating	O
the	O
plausibility	O
of	O
the	O
hypothesis	O
.	O
</s>
<s>
When	O
using	O
SNNs	O
for	O
image	O
based	O
data	O
we	O
need	O
to	O
convert	O
static	O
images	O
into	O
binary	O
spike	B-Algorithm
trains	I-Algorithm
coding	O
.	O
</s>
<s>
Temporal	O
coding	O
generates	O
one	O
spike	B-Algorithm
per	O
neuron	O
in	O
which	O
spike	B-Algorithm
latency	O
is	O
inversely	O
proportional	O
to	O
the	O
pixel	O
intensity	O
.	O
</s>
<s>
Rate	O
coding	O
converts	O
pixel	O
intensity	O
into	O
a	O
spike	B-Algorithm
train	I-Algorithm
where	O
the	O
number	O
of	O
spikes	B-Algorithm
is	O
proportional	O
to	O
the	O
pixel	O
intensity	O
.	O
</s>
<s>
Phase	O
coding	O
encodes	O
temporal	O
information	O
into	O
spike	B-Algorithm
patterns	O
based	O
on	O
a	O
global	O
oscillator	O
.	O
</s>
<s>
Burst	O
coding	O
transmits	O
the	O
burst	O
of	O
spikes	B-Algorithm
in	O
a	O
small-time	O
duration	O
,	O
increasing	O
the	O
reliability	O
of	O
synaptic	B-Application
communication	O
between	O
neurons	B-Algorithm
.	O
</s>
<s>
A	O
diverse	O
range	O
of	O
application	B-Application
software	I-Application
can	O
simulate	O
SNNs	O
.	O
</s>
<s>
Brian	B-Application
–	O
developed	O
by	O
Romain	O
Brette	O
and	O
Dan	O
Goodman	O
at	O
the	O
École	O
Normale	O
Supérieure	O
;	O
</s>
<s>
GENESIS	B-Application
(	O
the	O
GEneral	O
NEural	O
SImulation	O
System	O
)	O
–	O
developed	O
in	O
James	O
Bower	O
's	O
laboratory	O
at	O
Caltech	O
;	O
</s>
<s>
NEST	B-Application
–	O
developed	O
by	O
the	O
NEST	B-Application
Initiative	O
;	O
</s>
<s>
Simulations	O
show	O
that	O
arrays	O
of	O
ferroelectric	O
nanosynapses	O
can	O
autonomously	O
learn	O
to	O
recognize	O
patterns	O
in	O
a	O
predictable	O
way	O
,	O
opening	O
the	O
path	O
towards	O
unsupervised	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
Akida	O
is	O
a	O
completely	O
digital	O
event-based	O
neural	O
processing	O
device	O
with	O
1.2	O
million	O
artificial	B-Algorithm
neurons	I-Algorithm
and	O
10	O
billion	O
artificial	O
synapses	B-Application
developed	O
by	O
BrainChip	O
.	O
</s>
<s>
Neurogrid	O
is	O
a	O
board	O
that	O
can	O
simulate	O
spiking	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
directly	O
in	O
hardware	O
.	O
</s>
<s>
SpiNNaker	B-General_Concept
(	O
Spiking	B-Algorithm
Neural	I-Algorithm
Network	I-Algorithm
Architecture	O
)	O
uses	O
ARM	B-Architecture
processors	I-Architecture
as	O
the	O
building	O
blocks	O
of	O
a	O
massively	B-Operating_System
parallel	I-Operating_System
computing	I-Operating_System
platform	O
based	O
on	O
a	O
six-layer	O
thalamocortical	O
model	O
.	O
</s>
<s>
(	O
University	O
of	O
Manchester	O
)	O
The	O
SpiNNaker	B-General_Concept
system	O
is	O
based	O
on	O
numerical	O
models	O
running	O
in	O
real	O
time	O
on	O
custom	O
digital	O
multicore	O
chips	O
using	O
the	O
ARM	B-Architecture
architecture	I-Architecture
.	O
</s>
<s>
A	O
single	O
chip	O
can	O
simulate	O
16,000	O
neurons	B-Algorithm
with	O
eight	O
million	O
plastic	O
synapses	B-Application
running	O
in	O
real	O
time	O
.	O
</s>
<s>
Classification	O
capabilities	O
of	O
spiking	O
networks	O
trained	O
according	O
to	O
unsupervised	B-General_Concept
learning	I-General_Concept
methods	O
have	O
been	O
tested	O
on	O
the	O
common	O
benchmark	O
datasets	O
,	O
such	O
as	O
,	O
Iris	O
,	O
Wisconsin	O
Breast	O
Cancer	O
or	O
Statlog	O
Landsat	O
dataset	O
.	O
</s>
<s>
Based	O
on	O
the	O
idea	O
proposed	O
in	O
Hopfield	O
(	O
1995	O
)	O
the	O
authors	O
implemented	O
models	O
of	O
local	O
receptive	O
fields	O
combining	O
the	O
properties	O
of	O
radial	O
basis	O
functions	O
(	O
RBF	O
)	O
and	O
spiking	B-Algorithm
neurons	I-Algorithm
to	O
convert	O
input	O
signals	O
(	O
classified	O
data	O
)	O
having	O
a	O
floating-point	O
representation	O
into	O
a	O
spiking	O
representation	O
.	O
</s>
