Computational Intelligence - May 2016 - 59
(2)
v (i 2)(t) = v (i 21) (t) + v (i 22) (t) + v inh (t) (8)
with
v (i 21)(t) =
/ w ij(21) f (ij1) (t - t j)
Latency
Coding
(9)
j
and
(22)
i
v ( t) =
/w
(22) (2)
ij f ij
(t - t j )
Image
(10)
Spatio Temporal
Spatio-Temporal
Pattern
Garbor Filters
j!i
Therefore, the network connectivity
mainly includes two types of connections: First, lateral connections between
neurons in the same layer. Second, interlayer connections from input layer to
Layer I and from Layer I to Layer II.
D. temporal population Coding
The information about stimulation is
encoded by the time of spikes generated by a specific population of neurons,
and each input pattern is coded by a
particular group of neurons. This work
employs the temporal population coding to mimic sensory encoding process.
Here, we take visual signal as an example to show how real-world stimuli can
be encoded into single-spike spatiotemporal patterns as shown in Fig. 3. A
grayscale image is fed into Garbor filters
[32] and the output are converted into
neural firings corresponding to the following equation.
t i = f (s i) = t max - ln (a · s i + 1) (11)
where ti is the firing time of neuron i,
tmax is the width of encoding window, a
is a scaling factor, and si is the intensity
of output of Garbor filter. As a result,
each spike codes orientation components of the image and the latency
denotes the weight of the corresponding component.
Figure 3 Encoding scheme. A grayscale image is convolved with Garbor filters to extract orientation related features and then converted into a spike pattern by latency coding method.
forms of synaptic plasticity, STDP is
believed to be the underlying mechanism
for learning and information storage in
the brain [33]. It assumes that repeated
presynaptic spikes contribute to the
closely following postsynaptic action
potential and lead to long-term potentiation (LTP) of the synapse, whereas an
inverse temporal relation results in longterm depression (LTD) of the same synapse. Therefore, the change of the synapse
is defined as a function of the relative
timing of pre- and postsynaptic spikes,
which is called the STDP function as
shown in the following equation:
Z +
]] a · exp ` s+ j if s 1 0
x
Dw ij = [
-s
a
exp
` - j if s 2 0
]
·
x
\
(12)
where wij is the synaptic weight from
neuron j to neuron i, a + and a - are
amplitudes of exponential functions, and
s = tj - ti denotes the time difference
between pre- and postsynaptic spikes.
The STDP function (also called learning window) is illustrated in Fig. 4.
As precise spike timing and the interval
between pre- and postsynaptic firing
were discovered, learning with millisecond precision has intrigued intensive
interest. The temporally asymmetric form
of Hebbian learning induced by temporal
correlations between pre- and postsynaptic spikes is called STDP. Similar to other
Z +
]] a · exp ( s+ ) ·G (s)
if s 1 0
x
Dw ij = [
] - a - · exp ( --s ) ·G (s) if s 2 0
x
\
(13)
with
Pre
Post
e. StDp and tempotron
learning rule
In this model, we apply two different types of STDP processes: fast STDP
in Layer I and slow STDP in Layer II.
Neurophysiological experiments have
found that synaptic modifications varies
with different decaying time constants
of postsynaptic N-methyl-D-aspartate
(NMDA) receptors [34], [35], the predominant molecular device for controlling neural plasticity. In the STM
model, STDP learning mediated by fast
and slow NMDA receptors (fast:
x fast ~ 25 ms, slow: x slow ~ 150 ms ) is
referred to as fast and slow STDP,
respectively (Fig. 5). Fast STDP in Layer
I regulates neurons firing with a temporal distance less than gamma cycles,
while slow STDP in Layer II results in
synaptic modification between neurons
firing with a greater temporal distance.
By multiplying a simplified activation
function of NMDA channel, the modified STDP can be rewritten as
tj
G (s ) = '
∆w ti
a+
0
s
-a-
Figure 4 Spike-timing-dependent plasticity
(STDP).
1 if | s | # x NMDA
0 if | s | 2 x NMDA
(14)
where the decaying constant x NMDA =
x fast is set for fast STDP and x NMDA =
x slow for slow STDP.
Among existing spike-timing based
learning approaches, the tempotron rule
is a biologically plausible supervised synaptic learning scheme compatible with
temporal codes [17]. Tempotron rule is
used to train the network to reproduce
may 2016 | IEEE ComputatIonal IntEllIgEnCE magazInE
59
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