IEEE Circuits and Systems Magazine - Q2 2022 - 9

where ( )ut is the membrane potential, C is the membrane
capacitance, R is the input resistance, ()i to
current driving the neural state, ()i tj
from the j-th synaptic input, and wj
ito
is the external
is the input current
represents the strength
itj
,
of the j-th synapse. Both () and () are functions of
spike trains, as given in Equation 2. When R " 3 formula 3
is reduced to an IF model. In both IF and LIF models, a neuron
is supposed to fire a spike, whenever the membrane
potential u reaches a certain value y referred to as the firing
threshold. Immediately after the spike, the neuron state
is reset to a new value ures
1 y and holds at that level for
the time interval representing the refractory period.
The majority of the neuromorphic systems utilize IF and
LIF neurons as they are easier to implement and are computationally
efficient. The LIF model has been extended with
one or more adaptation variables to account for different
firing patterns. A well-known model is the Izhikevich model,
which can produce firing patterns experimentally verified
on neocortical and thalamic neurons [26]. However, it is
not clear what roles the different firing patterns are playing
in learning and cognition, and those additional adaptation
variables increase the model complexity. Therefore, they
are less used in machine intelligence applications.
B. Neuron Dynamics in Spike
Response Model (SRM)
The aforementioned IF and LIF models are over-simplified
by considering the synaptic connection as a time-invariant
device with a constant efficacy w and assume that the
membrane potential reset as an instantaneous procedure.
A more realistic neuron model considers the dynamics in
the neuron and synapse behavior.
The arrival of a presynaptic spike triggers the synaptic
electric current flowing into the biological neuron
[20]. It causes a change in the membrane potential of the
synapse, which is referred to as post-synaptic potential
(PSP). In a general form, the time course of jth PSP can
be described as the convolution of the presynaptic spike
train ()
Stj
and a kernel function () scaled by a weightKtj
coefficient wj
PSPjj jj
as the following:
()
tw St sK ss
wK tt
==jj
tt
/
#
jl
1
where tj
l is the time of spikes on the input ().Stj
K(t) can
be an exponential, a dual exponential or an alpha kernel
defined by following equations:
Kt e
() =
()
t
= x
-
Kt t e- t
x
x
Kt Ve e
SECOND QUARTER 2022
() ()0 ms
tt
=-xx
-(5)
(6)
(7)
3
()
()
(),
jl
d
(4)
1
0.8
0.6
0.4
0.2
Their spike responses are illustrated in Figure 2.
The reset of the membrane potential is no longer instantaneous.
Instead, it is modeled as a negative potential
induced by the output spike train ()
Sto
kernel function h(t),
() () ()
Rt =- =- o
#3
St sh ss ht tl(),
ttlo1
o
d
/
going through a
(8)
where to
l is the time of spikes on the output spike train.
Usually h(t) is a kernel given in Equation 5.
The way to interpret the neuron dynamics is as a convolution
of the impulse response of a filter with the input
spike train as in Equations 4 and 8, and is referred to as
the Spike Response Model (SRM).
The membrane potential is the combined effect of
PSP(t) and R(t) as shown in Figure 3. Using SRM representation,
it can be represented as an integral over the past
input and kernel responses. A typical SRM model is defined
as the following [12]:
N
Vt wK tt Vh tto
= jo1 ll
mj
j
where ()
Vtm
=- -1
()
()
t
j
th /
() // ll (9)
tt
is the membrane potential, K(t) and h(t) are
two convolution kernels associated to synaptic dynamics
and membrane potential reset events. When ()
Vtm
exceeds the threshold Vth, the neuron generates a spike
output whose time is indicated by the spike time
t .o
l
By using a kernel K(t) with arbitrary shape, the SRM
model provides complicated dynamics and rich temporal
information. The simplified LIF model in Equation 3 is a
special case of the SRM model, where the K(t) and h(t)
are two low-pass filters. The SRM model shows that the
membrane potential is a function based on not only the
current but also the past input spikes, which explains the
neuron's ability to respond to temporal patterns.
The kernels in the SRM model can be implemented as
discretized digital filters. Using the Z-transform [30], [31],
they can be represented as a Linear Constant-Coefficient
Difference (LCCD) equation in the following form:
P
yt
=- +==
11
ab
pyt
pxtp
q
[]
Q
6 @ // (10)
p
q [],
Exponential Kernel
Alpha Exponential Kernel
Dual Exponential Kernel
020406080 100
Time
Figure 2. Exponential, Alpha and dual exponential Kernels.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
9
PSP

IEEE Circuits and Systems Magazine - Q2 2022

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2022

IEEE Circuits and Systems Magazine - Q2 2022 - Cover1
IEEE Circuits and Systems Magazine - Q2 2022 - Cover2
IEEE Circuits and Systems Magazine - Q2 2022 - 1
IEEE Circuits and Systems Magazine - Q2 2022 - 2
IEEE Circuits and Systems Magazine - Q2 2022 - 3
IEEE Circuits and Systems Magazine - Q2 2022 - 4
IEEE Circuits and Systems Magazine - Q2 2022 - 5
IEEE Circuits and Systems Magazine - Q2 2022 - 6
IEEE Circuits and Systems Magazine - Q2 2022 - 7
IEEE Circuits and Systems Magazine - Q2 2022 - 8
IEEE Circuits and Systems Magazine - Q2 2022 - 9
IEEE Circuits and Systems Magazine - Q2 2022 - 10
IEEE Circuits and Systems Magazine - Q2 2022 - 11
IEEE Circuits and Systems Magazine - Q2 2022 - 12
IEEE Circuits and Systems Magazine - Q2 2022 - 13
IEEE Circuits and Systems Magazine - Q2 2022 - 14
IEEE Circuits and Systems Magazine - Q2 2022 - 15
IEEE Circuits and Systems Magazine - Q2 2022 - 16
IEEE Circuits and Systems Magazine - Q2 2022 - 17
IEEE Circuits and Systems Magazine - Q2 2022 - 18
IEEE Circuits and Systems Magazine - Q2 2022 - 19
IEEE Circuits and Systems Magazine - Q2 2022 - 20
IEEE Circuits and Systems Magazine - Q2 2022 - 21
IEEE Circuits and Systems Magazine - Q2 2022 - 22
IEEE Circuits and Systems Magazine - Q2 2022 - 23
IEEE Circuits and Systems Magazine - Q2 2022 - 24
IEEE Circuits and Systems Magazine - Q2 2022 - 25
IEEE Circuits and Systems Magazine - Q2 2022 - 26
IEEE Circuits and Systems Magazine - Q2 2022 - 27
IEEE Circuits and Systems Magazine - Q2 2022 - 28
IEEE Circuits and Systems Magazine - Q2 2022 - 29
IEEE Circuits and Systems Magazine - Q2 2022 - 30
IEEE Circuits and Systems Magazine - Q2 2022 - 31
IEEE Circuits and Systems Magazine - Q2 2022 - 32
IEEE Circuits and Systems Magazine - Q2 2022 - 33
IEEE Circuits and Systems Magazine - Q2 2022 - 34
IEEE Circuits and Systems Magazine - Q2 2022 - 35
IEEE Circuits and Systems Magazine - Q2 2022 - 36
IEEE Circuits and Systems Magazine - Q2 2022 - 37
IEEE Circuits and Systems Magazine - Q2 2022 - 38
IEEE Circuits and Systems Magazine - Q2 2022 - 39
IEEE Circuits and Systems Magazine - Q2 2022 - 40
IEEE Circuits and Systems Magazine - Q2 2022 - 41
IEEE Circuits and Systems Magazine - Q2 2022 - 42
IEEE Circuits and Systems Magazine - Q2 2022 - 43
IEEE Circuits and Systems Magazine - Q2 2022 - 44
IEEE Circuits and Systems Magazine - Q2 2022 - 45
IEEE Circuits and Systems Magazine - Q2 2022 - 46
IEEE Circuits and Systems Magazine - Q2 2022 - 47
IEEE Circuits and Systems Magazine - Q2 2022 - 48
IEEE Circuits and Systems Magazine - Q2 2022 - 49
IEEE Circuits and Systems Magazine - Q2 2022 - 50
IEEE Circuits and Systems Magazine - Q2 2022 - 51
IEEE Circuits and Systems Magazine - Q2 2022 - 52
IEEE Circuits and Systems Magazine - Q2 2022 - 53
IEEE Circuits and Systems Magazine - Q2 2022 - 54
IEEE Circuits and Systems Magazine - Q2 2022 - 55
IEEE Circuits and Systems Magazine - Q2 2022 - 56
IEEE Circuits and Systems Magazine - Q2 2022 - 57
IEEE Circuits and Systems Magazine - Q2 2022 - 58
IEEE Circuits and Systems Magazine - Q2 2022 - 59
IEEE Circuits and Systems Magazine - Q2 2022 - 60
IEEE Circuits and Systems Magazine - Q2 2022 - 61
IEEE Circuits and Systems Magazine - Q2 2022 - 62
IEEE Circuits and Systems Magazine - Q2 2022 - 63
IEEE Circuits and Systems Magazine - Q2 2022 - 64
IEEE Circuits and Systems Magazine - Q2 2022 - 65
IEEE Circuits and Systems Magazine - Q2 2022 - 66
IEEE Circuits and Systems Magazine - Q2 2022 - 67
IEEE Circuits and Systems Magazine - Q2 2022 - 68
IEEE Circuits and Systems Magazine - Q2 2022 - Cover3
IEEE Circuits and Systems Magazine - Q2 2022 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q1
https://www.nxtbookmedia.com