Learning Rules in Spiking Neural Networks: A Survey

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Highlights
• Research highlight 1
Guided by a hierarchical classification of SNN learning rules, we present a
comprehensive survey of learning rules in spiking neural networks.
• Research highlight 2
We discus and compare their characteristics, advantages, limitations, and
performance on several popular datasets.
• Research highlight 3
We review practical a p p l i c a t i o n s of SNNs to bette r p r e s e n t the
SNN research landscape.
Figure: A three-level hierarchical taxonomy of learning rules in SNNs.

Learning Rules in Spiking Neural Networks: A Survey
Zexiang Yia, Jing Lianb, Qidong Liuc, Hegui Zhud, Dong Liange, Jizhao
Liua,∗
aSchool of Information Science and Engineering, Lanzhou
University, Lanzhou, 730000, Gansu, China
bSchool of Electronics and Information Engineering, Lanzhou Jiaotong
University, Lanzhou, 730070, Gansu, China
cSchool of Computer and Artificial Intelligence, Zhengzhou
University, Zhengzhou, 450001, Henan, China
dCollege of Sciences, Northeastern University, Shenyang, 450001, Liaoning, China
eDepartment of Computer Science and Technology, Nanjing University of Aeronautics
and Astronautics, Nanjing, 211106, Jiangsu, China
Abstract
Spiking neural networks (SNNs) are a promising energy-efficient alternative
to artificial neural networks (ANNs) due to their rich dynamics, capability to
process spatiotemporal patterns, and low-power consumption. The complex
intrinsic properties of SNNs give rise to a diversity of their learning rules
which are essential to functional SNNs. This paper is aimed at presenting a
comprehensive overview of learning rules in SNNs. Firstly, we introduce the
basic concepts of SNNs and commonly used neuromorphic datasets. Then,
guided by a hierarchical classification of SNN learning rules, we present a
comprehensive survey of these rules with discussions on their characteristics,
advantages, limitations, and performance on several datasets. Moreover,
we review practical applications of SNNs, including event-based vision and
∗Corresponding author
Email address: liujz@lzu.edu.cn (Jizhao Liu)
Preprint submitted to Neurocomputing January 11, 2023

audio signal processing. Finally, we conclude this survey with a discussion
on challenges and promising future research directions in this area.
Keywords: spiking neural networks, pulse-coupled neural networks,
neuromorphic computing, learning rules, image classification.
1. Introduction
The human brain is the best-known intelligent system, performing func-
tions such as perception, reasoning, and control with a power consumption
of nearly 20W[1]. There have been many artificial intelligence (AI) models
inspired by the brain. Rosenblatt [2] proposed the perceptron to realize bi-
nary classification of input patterns. Lecun et al. [3] applied convolutional
neural networks (CNNs) to handwritten digit recognition. Recently, neural
networks based on attention mechanisms[4] have invoked a new wave of re-
search. Although ANNs have achieved great success in many fields[5], they
have the following drawbacks: 1) the training and inference of ANNs con-
sume huge amounts of energy[6]; 2) limitations such as poor robustness and
catastrophic forgetting [7]. However, these problems do not exist in the brain
where neurons have complex spatiotemporal dynamics, communicating and
processing information through discrete spikes. Additionally, neurons are
connected in a hierarchical way, forming different functional neural networks
with diverse plasticity. How to introduce the above characteristics to con-
struct more energy-efficient and robust AI models is an open problem in AI
research.
2

1.1. Unique Characteristics of spiking neural networks
Spiking neural networks (SNNs) mimic the way the human brain process
information by taking advantage of discrete and asynchronous spikes, thus
they are believed to have the capability to process spatiotemporal information
efficiently[8, 9, 10]. The basic building blocks of SNNs are spiking neurons.
Due to the description of the generation of spikes at different levels of bio-
fidelity, there is a diversity of spiking neuron models such as Hodgkin-Huxley
(H-H) model[11], Izhikevich mmodel (IM)[12]. Additionally, by utilizing the
time dimension, spiking neurons can represent information in a sparse and
robust way[13].
SNNs are bridges between brain science (BS) and AI. On the one hand,
neuroscientists use SNNs to simulate biological neural networks to deepen
their understanding of the brain[14, 15, 16]; On the other hand, AI re-
searchers draw inspiration from BS to build energy-efficient and robust neural
networks[17, 18]. However, Due to the non-differential nature of spikes, train-
ing efficient and high-performance SNNs has remained a major difficulty[19,
20, 21].
1.2. Motivation
In recent years, SNNs have attracted enormous research interest. There
has been an upward trend in SNN-related papers. To visualize this trend, we
present Fig.1, which illustrates the number of SNN papers that are available
on the Web of Science since 2015.
Learning is an essential part of SNNs that adapts a network to perform
specific tasks, such as classification or object detection. Fig.2 shows some of
the important learning rules over the past two decades. SpikeProp[22] is the
3

2015 2016 2017 2018 2019 2020 2021 2022
Year
0
200
400
600
800
1000
1200
1400
No. Papers
Journal
Conference
Review
Figure 1: The number of SNN papers published after 2015.
earliest spike-based backpropagation in multilayer SNNs. Tempotron[23] can
perform binary classification tasks in analogy to perceptron. Remote super-
vised method (ReSuMe)[24] and spike pattern association neuron (SPAN)[25]
are classical spike sequence learning rules. Masquelier and Thorpe [26] apply
spike-timing-dependent plasticity (STDP) to multilayer neural networks in-
spired by ventral visual pathways to enable unsupervised feature learning. In
2015 and beyond, SNNs have been dominated by deep networks. Cao et al.
[27] propose converting a pre-trained ANN to an SNN. Kheradpisheh et al.
[28] use STDP to train deep spiking CNNs layer by layer. SuperSpike[29]
and spatio-temporal backpropagation (STBP)[30] train multilayer SNNs via
surrogate gradients. Spike-element-wise (SEW) ResNet[19] is proposed to
combat degradation problem in deep SNNs. Bu et al. [31] realize ultra-low-
latency inference in ANN-converted SNNs.
Several survey papers [32, 33, 34, 35, 36, 37, 38, 39, 40, 41] have so far
reviewed recent advances in SNNs. However, some of these papers have a
4

Converting a
pre-trained
ANN
Surrogate
gradient Low latency
conversion
2000 2005 2010 2015 2020
SpikeProp
Tempotron
STDP+Shallow Networks
ReSuMe
SPAN STDP+Deep networks SEW ResNet
Shallow SNNs Deep SNNs
Figure 2: The evolution of learning rules in SNNs.
limited scope of learning rules, for instance, [36, 34, 41] focus on supervised
learning, [32, 33] have an emphasis on learning rules in multilayer SNNs.
Moreover, most of the above surveys only cover the papers published un-
til 2021. Nonetheless, many important breakthroughs in learning rules in
SNNs have occurred since 2022. Additionally, none of the surveys cover
pulse-coupled neural networks (PCNNs), which are cortex models exhibit-
ing synchronous oscillation behavior and have been widely applied to image
processing[42, 43, 44].
1.3. Contribution
This paper surveys the advances in learning rules in SNNs, providing
insights into both technical and performance in a systematic way.
The key contributions of this article are summarized as follows. First, we
provide a systematic review of the evolution of learning rules in SNNs, where
many of them have not been reviewed in previous surveys, and present com-
parisons between the state-of-the-art using results reported on several pop-
5

ular datasets. Second, we review practical applications of SNNs, including
event-based vision and audio signal processing. Third, we discuss a number
of challenges and promising future research directions.
1.4. Organization
The rest of this survey is structured as follows. We start by providing
the basic concepts of SNNs including neuron and networks models, synaptic
plasticity, and neural coding. Next, in Section 3, we review the commonly
used neuromorphic datasets. Section 4 presents a hierarchical classification
of learning rules in SNNs and analyzes the research trend and their character-
istics, advantages, limitations, and performance on several datasets. Section
5 reviews practical applications of SNNs. Finally, Section 6 discusses some
challenges and directions of this field.
2. Basic concepts of SNNs
SNNs involve more neuroscience-related concepts than ANNs. To better
analyze the learning rules in SNNs, basic concepts of SNNs are introduced in
this section, including neuron and network models, synaptic plasticity, and
neural coding.
2.1. Neuron models
There are billions of neurons in the human brain which have a basic
structure as shown in Fig.3. Dendrite is the input terminal. The cell body
integrates incoming spikes received by different branches of dendrites and
emits a spike when its membrane potential reaches the threshold. Spikes
travel along the axons to other neurons via synapses.
6

Dendrite
Cell body Synapse
Axon
Dendrite
Figure 3: Diagram of a neuron. It can be divided into three parts: dendrite, cell body,
and axon.
To emulate the generation of spikes with different levels of bio-fidelity
and computational cost, a variety of spiking neuron models have been pro-
posed. For simplicity and mathematical tractability, leaky integrate-and-fire
(LIF)[45], spike response model (SRM)[46] and PCNN neuron[14] models are
widely used in SNNs. To better formalize these models, Tab.1 summarizes
the main notations used in the following equations.
LIF model dates to 1907[45], when the mechanism of generating spikes
had not yet been revealed, so neurons were modeled as a parallel circuit of
resistance Rand a capacitance Cas shown in Fig.4(a). When the input
current Iis injected into the capacitor, its voltage Vrises. Meanwhile,
charges on the capacitor will leak through the resistor. A spike will be
emitted whenever Vreaches the threshold Vth, which is then reset to the
resting voltage Vrest. Fig.4(b) depicts the dynamics of an LIF neuron under
constant input, which can formally be described in differential form as:
τm
dV
dt =−(V−Vrest) + RI +S(t) (Vrest −Vth) (1)
S(t) = X
tj≤t
δt−tj
s(2)
7

Table 1: Notation list.
Notation Description
τ,R,CTime constant, input resistance and capacitor of a neuron, respectively
FFeeding input of a neuron
LLinking input of a neuron
VMembrane potential of a neuron
EDynamic threshold of a neuron
SOutput spike train of a neuron
tj
sTime of jth spike of output spike train
SiInput spike train from ith synapse
tj
iTime of jth spike from ith synapse
C
R
IV
Vrest
Vth
(a)
0 25 50 75 100 125 150 175 200
0.0
0.1
0.2
I
0 25 50 75 100 125 150 175 200
Time steps
0.0
0.5
1.0
V
Vth
Vrest
Spike
(b)
Figure 4: The LIF model. (a) Diagram of the LIF model circuits. (b) Neuronal dynamics
of a LIF neuron under the constant current.
8

where τm=RC is the time constant of the membrane. When working
with differential equations, it is convenient to denote a spike as a Dirac delta
function δ(t), so the postsynaptic spike train S(t) can be presented as a sum
of Dirac functions at different output spike times tj
s.
Synapse transmits spikes via neurotransmitter which acts like a low pass
filter with synaptic weight. Dendrite integrates the weighted and filtered
synaptic current, obtaining the total input current I. The dynamics of these
operations are given by:
τs
dI
dt =−I+X
i
wisi(t) (3)
Si(t) = X
ti≤t
δt−tj
i(4)
where τsis the synapse time constant. tj
idenote the presynaptic spike times
of ith afferent. The sum runs over all presynaptic neurons i.wiand si(t) are
the corresponding synaptic weights and presynaptic spike trains, respectively.
It is customary to simulate SNNs in discrete time using Euler’s method.
To reduce computational costs, the synaptic filter effect is often ignored, but
there are also researchers[47] incorporating this filter dynamics to improve
the convergence of SNNs training. In addition, there are some variants of the
LIF Model such as quadratic integrate-and-fire (QIF) and integrate-and-fire
(IF) models.
SRM uses spike response kernels to model a neuron’s membrane potential
in response to its input and out spikes[46]. A commonly used form is given
as follows:
V(t) = X
i
wiX
ti≤t
εt−tj
i+X
tj≤t
ηt−tj
s+Vrest (5)
9

where εand ηare the input and output spike response kernel, respectively.
SRM neuron models are appealing as they can add other features simply
by embedding them into the kernel. In addition, since the membrane po-
tential is explicitly expressed, the simulation of SRM models is often event-
based, thus less cost and time-consuming than LIF models.
The PCNN neuron is a two-compartmental model[42], show as in Fig.5.
The dendrite tree has two distinct inputs, the primary input termed feeding
Spike generatorModulation
Linking input L Dynamic threshold E
1
Feeding input FS
IL
IF
β
VE
V
Dendrite tree
Figure 5: Schematic of a PCNN neuron. It consists of three parts: dendrite tree, modu-
lation, and spike generator.
input F, and the auxiliary input termed linking input L. They are repre-
sented as leaky integrator given by
τf
dF
dt =−F+If(6)
τl
dL
dt =−L+Il(7)
where τf,τf,Ifand Ilare the time constants and synaptic currents of the
feeding and linking input, respectively. The synaptic currents can be spikes,
10

constants, analog time-varying signals, or any combination.
The linking input modulates the feeding input by multiplication resulting
in the membrane potential V.
V=F(1 + βL) (8)
where beta is the linking strength.
The spike generator will emit a spike whenever the membrane potential
crosses the threshold E. Unlike LIF models resetting the membrane poten-
tial, the PCNN neuron feeds back to the threshold, which is another leaky
integrator given by
τe
dE
dt =−E+VES(t) (9)
where S(t) denotes the output spike train of the PCNN neuron. τeis the
time constant and VEis the amplitude gain.
2.2. Network models
Neurons are connected to form neural networks following some connec-
tion patterns, as shown in Fig.6. There are three types of feedforward con-
nections, convolutional, local, and full connection. Convolutional and local
connections both have local visual receptive fields, but in convolutional con-
nection, the weights are shared between all receptive fields, while in local
connection, each receptive field has its own set of weights, which is more
biologically plausible[48]. Full connection layers are often used to classify ex-
tracted features. Recurrent connections consist of self-recurrent, lateral, and
feedback connections. Zhang and Li [49] added self-recurrent connections to
implement local memory. Diehl and Cook [50] used feedback connections to
implement winner-take-all mechanism. Cheng et al. [17] introduced lateral
11

connections to improve the recognition robustness against noise. The last
type of connection is the residual connection which can solve the degrada-
tion problem in deep ANNs and is also the key to deep SNNs[19, 20].
Lateral
Residual connection
Full Convolution/Local
Self-recurrent
Recurrent connectionFeedforward connection
Feedback
Network block
Figure 6: Connection patterns in SNNs.
Network models can be constructed using the above connection patterns.
We review three types of network models as follows. The PCNN is a cortex
model originated from the simulation of synchronous oscillation behavior in
the primary cortex of cats[42]. The classical PCNN and its variants such
as the spiking cortical model (SCM)[51] have been widely used in image
segmentation[52, 53], fusion[54, 55, 56], enhancement[57, 58, 59], invariant
texture retrieval[60], and many other image processing tasks[61, 62, 63]. Al-
though a vast body of works utilize PCNNs for image processing, less atten-
tion has been paid to learning rules for PCNNs as we will see in the following.
Based on the characteristics of ventral visual flow, Riesenhuber and Poggio
[64] proposed the HMAX model. Serre et al. [65] expanded HMAX into
the field of computer vision. Masquelier and Thorpe [26] proposed a spik-
ing version of HMAX equipped with STDP for unsupervised feature learning.
12

Single layer networks (SLNs), multilayer Perceptron (MLP), recurrent neural
networks (RNNs), and Convolutional neural networks (CNNs) are network
models widely used in deep learning[66]. VGG and ResNet are popular neural
architectures for deeper CNNs which have been adopted by a large number
of high-performance SNNs.
2.3. Synaptic plasticity
Synaptic plasticity refers to the modulation of synaptic weights[67]. In
1949, Hebb [68] proposed Hebb’s postulate, and It can be simply stated as
”neurons that fire together, wire together[69]. ” Subsequent finding of long-
term potentiation[70] provided experimental evidence for Hebb’s postulate.
LTP together with long-term depression (LTD) regulates synaptic weights
bidirectionally, serving as the synaptic basis for learning and memory. The
induction of LTP and LTD is spike-timing dependent. Studies[71, 72] demon-
strated that the relative timing of the pre and postsynaptic spikes determine
the direction and magnitude of synaptic modification. This phenomenon is
known as spike-timing-dependent plasticity (STDP)[73]. Fig.7(a) shows two
neurons connected by a synapse, and Fig.7(b) illustrates the relationship be-
tween the amount of change in synaptic weights and the timing of pre and
postsynaptic spikes.
Fitting experimental data with exponential function, we can formalize
STDP as
∆w=


A+exp −∆t
τ+∆t > 0
−A−exp ∆t
τ−∆t < 0
(10)
where A+and A−are the modulation magnitudes for LTP and LTD, re-
spectively. τ+and τ−are the corresponding time constants of the learning
13

Post
Pre
Δw= f (tpre, tpost)
(a)
-100 0 100
-60
0
-100
Dt (ms)
Dw (%)
LTP
LTD
(b)
Figure 7: STDP. (a) Two neurons are connected by a synapse equipped with STDP. (b)
Relationship between spike timing and synaptic weight change. In a time window of tens
of milliseconds, when the presynaptic spike is earlier (later) than the postsynaptic spike,
the weight increases (decreases), resulting in LTP (LTD)
window, and ∆t=tpost −tpre is the time difference between a pair of pre
and postsynaptic spikes. In STDP modeling studies, Song et al. [73] found
that synapses modified by STDP compete with one other, resulting in a bi-
modal distribution of synaptic weights. Guyonneau et al. [74] showed that
a neuron with STDP-modified synapse stimulated by a repeatedly presented
spike pattern will be selective to it and decrease response latency. STDP also
enables neurons to learn visual features in an unsupervised way[26, 50, 75].
Increasing experimental observations demonstrate that neuromodulators
play a vital role in synaptic plasticity. They can change the polarity[76] or
adjust the time window of STDP[77]. Fr´emaux and Gerstner [78] proposed
three-factor rules to incorporate the influence of neuromodulators. Inspired
by these observations, researchers model neuromodulator effects to imple-
14

ment bio-plausible supervised learning[79] and reinforcement learning[80] for
image recognition.
2.4. Neural coding
Stimulus, such as light or odors are converted to spikes for neural pro-
cessing, a process known as neural coding. Currently, there are three main
coding methods applied in SNNs: rate, temporal and direct coding.
Rate coding converts input stimulus into Poisson-distributed spike trains,
with firing rates proportional to the input intensity. To reduce the compu-
tational cost, the binomial distribution is commonly used instead of Poisson
distribution[17].
As rate coding represents information via firing rate, i.e. spike count over
a short time, a single spike contains little information. temporal coding is
concerned with the timing of a spike. A common temporal coding method is
time-to-first-spike (TTFS) coding[81] in which a larger input intensity corre-
sponds to an earlier spike. Therefore, TTFS coding requires fewer spikes to
encode information than rate coding, resulting in a lower power consumption
and inference latency[82].
Rueckauer et al. [83] suggested that the variability in rate coding impairs
the performance of SNNs. So many studies[84, 85] used a trainable spik-
ing neuron layer to convert analog input that can be regarded as synaptic
currents into output spike strains, which is called direct coding[86].
3. Neuromorphic datasets
The performances of SNNs are often evaluated on existing ANN-oriented
datasets, for example CIFAR-10[87] and ImageNet[88], which are static-
15

image datasets containing no temporal information. Before feeding them
into SNNs, researchers often convert such frame-based data to spike trains
using coding methods described in Section 2.4. However, these ANN-oriented
datasets can’t exploit the spatiotemporal processing capability of SNNs[89].
To this end, researchers have gathered neuromorphic datasets inspired by the
biological visual system.
Neuromorphic datasets are recorded by dynamic vision sensors (DVS)
which capture the changes in the sensing field using two channels. The On
channel for intensity increases and the Off channel for intensity decreases.
Currently, neuromorphic datasets can be divided into two categories: DVS-
converted and DVS-captured[90]. DVS-converted datasets are converted
from traditional datasets, such as MNIST and CIFAR-10. Researchers use a
DVS camera to record static images by moving the image or camera. Both
N-MNIST[91] and CIFAR10-DVS[92] are acquired by this method. In con-
trast, DVS-captured datasets are recorded via real-world motion. DVS128
Gesture[93] is a typical example. We present an overview of the main char-
acteristics of well-known neuromorphic datasets in Tab.2. In the following,
we review these datasets in detail.
Table 2: Summary of well-known neuromorphic datasets.
Datase Year Data category #classes URL
N-MNIST 2015 DVS-converted 10 http://www.garrickorchard.com/datasets
N-Caltech101 2015 DVS-converted 101 http://www.garrickorchard.com/datasets
CIFAR10-DVS 2017 DVS-converted 10 https://figshare.com/s/d03a91081824536f12a8
DVS128 Gesture 2019 DVS-captured 11 http://research.ibm.com/dvsgesture
ASL-DVS 2019 DVS-captured 24 https://github.com/PIX2NVS/NVS2Graph
N-MNIST: The N-MNIST dataset[91] is converted from the MNIST
16

dataset. Researchers first displayed MNIST examples on an LCD monitor,
and then moved the ATIS sensor mounted on a pan-tilt unit to record the
image. It includes a training set with 60,000 samples and a test set with
10,000 samples.
N-Caltech101: The N-Caltech101 dataset[91] is a neuromorphic version
of the N-Caltech101 dataset. It was recorded using the same method as the
N-MNIST dataset and contains 9146 images in 101 classes.
CIFAR10-DVS: The CIFAR10-DVS dataset[92] is a spiking version of
the CIFAR-10 dataset. The dataset was recorded by moving images in front
of a DVS camera. It consists of 10,000 examples in 10 classes, with 1000
examples in each class.
DVS128 Gesture: The DVS128 Gesture dataset[93] was recorded by a
DVS128 camera and contained 11 kinds of hand gestures from 29 subjects
under 3 kinds of illumination conditions.
ASL-DVS: The ASL-VDS dataset[94] was recorded in an office environ-
ment with low environmental noise and constant illumination. It contains 24
classes of gestures corresponding to 24 English letters.
4. Learning rules in SNNs
In this section, we first introduce the hierarchical classification of SNN
learning rules which gives a overview of these rules. Then, we analyze the
research trends in SNN learning rules. Finally, we review these rules in a hier-
archical way and summarize the state-of-the-art performance for comparison
on several datasets.
17

4.1. Hierarchical classification of SNN learning rules
In order to help illustrate an overall structure for SNN learning rules, we
categorized the existing learning rules hierarchically, which is presented in
Fig.8. Previous papers [35, 95] classify SNN learning rules according to the
usage of data label, i.e. supervised and unsupervised learning [35] or the
biological realism and plasticity scale of learning rules [95]. We differ from
these papers by considering the working principles of the SNN learning rules.
The details of each group of learning rules will be discussed later.
18

STDP-based
ANN for pre-training
(ReLU neuron) SNN for inference
(IF neuron)
W-H rule-based
Likelihood gradient Voltage gradient
Spike time
t1
t2
t3
t4=f (t1, t2, t3)
Neurons
Spike
generator
PS(t)
Membrane potential
Activation
Time
LTP
Time difference
Weight
change
LTD
Sd(t)
Vmax
Bio-plausible algorithms
L=f (Vmax)
Gradient-based algorithms
Other bio-plausible algo.
LTD LTP SBP
So(t)
tmax
Vth
Forward
propagation
Backpropagation
0.5
0.8
0.1
0.2
0.3
0.6
Voltage
Activation gradientTiming gradient
ANN pre-training
Direct training ANN-coupled training
ANN layer
Input
Output
ANN layer
SNN layer
SNN layer
Forward pass
Backward pass
Weight sharing
Figure 8: Illustration of the three-level hierarchical classification of learning rules in SNNs. Bold only, bold italic, and italic
only style means the first, second, and third level of the hierarchy, respectively
19

4.2. Analysis and trends
Before we dive into individual rules, we summarize the most represen-
tative SNN learning rules in Tab.3, sorted according to their publication
dates, for analysis of the evolution and trends in this field. LIF and SRM are
mainly used neuron models for their simplicity and mathematical tractabil-
ity. Direct coding has become popular due to its advantages over rate coding
in accuracy[86]. Temporal coding is recently limited to certain rules such as
timing gradient. Notably, there has been a clear trend in formulating efficient
learning rules such as ANN pre-training and activation gradient for deeper
SNNs (>10 layers) to perform challenging tasks, such as image classification
on ImageNet and CIFAR-100.
20

Table 3: Summary of some representative learning rules in SNNs
Ref. Year Venue Neuron Model Coding Method Learning Rule Network Model Dataset
[22] 2002 Neurocomp. SRM Temporal Timing gradient MLP Iris
[23] 2006 Nat. Neurosci. LIF — Voltage gradient SLN —
[96] 2006 Neural Comput. SRM — Likelihood gradient SLN —
[26] 2007 PLoS Comput. Biol. IF Temporal STDP-based HMAX Caltech-101
[97] 2009 Neural Netw. SRM Temporal Timing gradient MLP Iris
[24] 2010 Neural Comput. LIF; H-H; IM — W-H rule-based SLN —
[98] 2012 PLoS One SRM — W-H rule-based SLN N/A
[99] 2013 PLoS One LIF; IM — W-H rule-based SLN N/A
[100] 2014 Neurocomp. SRM Temporal STDP-based MLP Iris
[50] 2015 Front. Comput. Neurosci. LIF Rate STDP-based SLN MNIST
[101] 2015 Neural Comput. SRM — Likelihood gradient MLP N/A
[27] 2015 IJCV IF Rate ANN pre-training CNN CIFAR-10; Neovision2
[102] 2015 IJCNN IF Rate ANN pre-training CNN MNIST
[103] 2015 IJCNN LIF — Voltage gradient SLN —
[104] 2016 Science LIF — Voltage gradient SLN TIDIGITS
[105] 2016 arXiv IF Direct ANN pre-training CNN CIFAR-10
[106] 2016 Neurocomp. IF Temporal STDP-based HMAX 3D-Ob ject
[107] 2016 Front. Neurosci. LIF Rate Voltage gradient CNN MNIST; N-MNIST
[83] 2017 Front. Neurosci. LIF Direct Conversion method VGG-16 MNIST; CIFAR-10; ImageNet
[108] 2018 IEEE T-NNLS IF Temporal Timing gradient MPL MNIST
[109] 2018 NeurIPS IF; QIF — Activation gradient RNN —
[28] 2018 Neural Netw. IF Temporal STDP-based CNN Caltech-101; ETH-80; MNIST
[80] 2018 IEEE T-NNLS IF Temporal STDP-based HMAX Caltech-101; ETH-80;
[110] 2018 Neural Netw. LIF Direct Voltage gradient CNN MNIST
21

Table 3 continued from previous page
Ref. Year Venue Neuron Model Coding Method Learning Rule Network Model Dataset
[30] 2018 Front. Neurosci. LIF Rate Activation gradient CNN MNIST; N-MNIST
[85] 2019 AAAI LIF Direct Activate gradient CNN N-MNIST; CIFAR10-DVS; CIFAR-10
[111] 2019 IEEE T-Cyb. LIF — Voltage gradient SLN —
[112] 2019 IEEE T-CDS LIF Rate STDP-based CNN Caltech-101; MNIST
[48] 2019 Neural Netw. LIF Rate STDP-based SLN MNIST
[113] 2019 Front. Neurosci. IF Rate ANN pre-training VGG-16; ResNet-20/34 CIFAR-10; ImageNet
[82] 2020 Int. J. Neural Syst. IF Temporal Timing gradient MLP Caltech-101; MNIST
[114] 2020 ICASSP SRM Temporal Timing gradient MLP MNIST
[115] 2020 CVPR IF Rate ANN pre-training VGG-16; ResNet-20/34 CIFAR-10; CIFAR-100; ImageNet
[116] 2020 ICLR IF Rate ANN pre-training VGG-16; ResNet-20/34 CIFAR-10; CIFAR-100; ImageNet
[117] 2021 IEEE T-NNLS IF Direct ANN-coupled training CifarNet; AlexNet CIFAR-10; ImageNet
[17] 2021 IJCAI LIF Rate Activation gradient CNN MNIST; Fashion-MNIST
[84] 2021 ICCV LIF Direct Activation gradient CNN CIFAR10; CIFAR10-DVS; DVS128 Gesture
[118] 2021 AAAI IF Temporal Timing gradient VGG-16; GoogleNet MNIST; CIFAR-10; ImageNet
[119] 2021 ICLR IF Direct ANN pre-training VGG-16; ResNet-20 CIFAR-10; CIFAR-100; ImageNet
[120] 2021 ICML IF Direct ANN pre-training VGG-16; ResNet-20/34; RegNetX CIFAR-10; CIFAR-100; ImageNet
[121] 2021 IEEE T-NNLS IF Direct ANN pre-training ResNet-50/110 CIFAR-10; CIFAR-100; ImageNet
[122] 2021 IJCAI IF Direct ANN pre-training VGG-16; PreActResNet-18/34 MNIST; CIFAR-10; CIFAR-100
[123] 2021 Sci. Adv. LIF Rate Other bio-plau. algo. MLP MNIST; NETtalk; DVS128 Gesture
[19] 2021 NeurIPS LIF; IF Direct Activation gradient ResNet-18/34/50/101/152 ImageNet; DVS128 Gesture; CIFAR10-DVS
[20] 2021 arXiv LIF Direct Activation gradient ResNet-104/482 CIFAR10-DVS; ImageNet
[124] 2022 AAAI IF Direct ANN pre-training VGG-16; ResNet-18/20 CIFAR-10; CIFAR-100; ImageNet
[31] 2022 ICLR IF Direct ANN pre-training VGG-16; ResNet-18/20/34 CIFAR-10; CIFAR-100; ImageNet
[125] 2022 ICLR LIF Direct Activation gradient VGG-11;ResNet-19/34 CIFAR-100; ImageNet; CIFAR10-DVS
22

4.3. Bio-plausible rules
Driven by considerations of biological plausibility, bio-plausible rules pri-
marily focus on implementation via experimentally observed biological phe-
nomena such as STDP, coherent oscillation[126], and self-backpropagation[127].
We classify these rules into STDP-based, Widrow-Hoff (W-H) rule-based[128]
and other bio-plausible rules according to the underlying principles they are
derived from.
4.3.1. STDP-based rules
As mentioned in Section 2.3, STDP observed in biology experiments can
enable a neuron to learn visual features in an unsupervised way. Thus STDP-
based rules have been studied in many research. Masquelier and Thorpe [26]
proposed an HMAX-based SNN trained with unsupervised STDP to extract
visual features from a temporal coded image. Many subsequent works were
based on this method. Tavanaei and Maida [129] incorporated probabilis-
tic STDP to improve performance. Kheradpisheh et al. [106] expanded this
method to perform robust invariant object recognition tasks. Unsupervised
STDP can extract repeated features; however, it has difficulty in detect-
ing rare but diagnostic features. To this end, Mozafari et al. [80] introduce
reward signals to STDP, called reward-modulated STDP (R-STDP), to im-
prove feature extraction ability.
There are several ways to formalize supervised STDP. Wang et al. [100]
combined STDP and anti-STDP to implement supervised learning for the
output layer of a two-layer SNN. When a neuron emits spikes correctly, STDP
is applied, otherwise, anti-STDP is applied. Beyeler et al. [130] used super-
visory neurons to send excitatory signals to target output neurons, making
23

them spike at the desired firing rate. Illing et al. [131] implemented super-
vised learning via target post spike trace. Synaptic weights update at every
presynaptic spike times. When the actual post spike trace is lower than
the target value, the corresponding weight increases, and decreases instead.
Hao et al. [79] introduced dopamine-modulated STDP (DA-STDP) combined
with synaptic scaling to realize supervised learning.
As early works to perform digit recognition on MNIST using unsuper-
vised STDP, Querlioz et al. [132, 133] proposed a single-layer network with
lateral inhibition and dynamic threshold, yielding an accuracy of 93.5% with
300 neurons. They used memristors as synapses. Although the rectangular
STDP time-window was used for modulation of synaptic weights in these
works, the memristive devices can implement biological STDP learning rule
easily[134, 135]. Diehl and Cook [50] increased the network scale and im-
proved its biological plausibility, yielding an accuracy of 95%. It was further
improved by Saunders et al. [48, 136] with local connections, resulting in
reduced parameters and training time. To take advantage of the feature ex-
traction capabilities of CNNs, Xu et al. [137] proposed deep CovDenseSNN
which uses spiking neurons to learn features extracted by CNNs.
Recent works adopted CNN-like neural architecture for SNNs. Kherad-
pisheh et al. [28] used STDP to train a three-layer spiking CNN layer-by-
layer. The complexity of learned features increases along the network hierar-
chy. A support vector machine (SVM) was used for feature classification and
achieved an accuracy of 98.6% on MNIST. Mozafari et al. [138] improved the
biological plausibility of this model by placing SVM with a layer of decision-
making neurons trained with R-STDP. Unlike previous works, SpiCNN [112]
24

used rate coding with Poisson distribution and an output layer trained with
supervised STDP.
4.3.2. W-H rule-based rules
The W-H rule[128] is a classical learning rule proposed for ANN neurons
defined as
∆wi=ηxi(yd−yo) (11)
where xiis the input of the ith synapse. yoand ydare the actual and desired
output of the neuron, respectively. ηis the learning rate. The W-H rule
requires no need for gradient calculation, so there is a bunch of SNN learning
rules that present a spiking analogy to this rule.
ReSuMe[24] interprets the W-H rule through two biological processes:
STDP and anti-STDP, which is illustrated in Fig.9. It updates the synaptic
weights according to
dwi(t)
dt = [sd(t)−so(t)] a+Z+∞
0
T(s)si(t−s)ds(12)
where sd(t) and so(t) are the desired and actual output spike trains, respec-
tively. ais a constant used for speeding up the convergence of learning. T(s)
is the STDP-like learning window. The convolution of the learning window
and the ith input spike train R+∞
0T(s)si(t−s)ds represents the trace of the
spike train.
Unlike the spike-driven update of the synaptic weights in ReSuMe, the
perceptron-based spiking neuron learning rule (PBSNLR)[139] transforms
a spiking neuron to a perceptron at fixed points in time, where the mem-
brane potential in the misclassification intervals is utilized for synaptic up-
dating. Inspired by the biological property that the synaptic delay is not
25

Input spikes
Trace
Desired output spikes
Actual output spikes
Synaptic weight
Time
Figure 9: Illustration of the ReSuMe rule[24]. The amount of modification of an excitatory
synaptic weight is proportional to the trance induced by the convolution of input spikes
and the learning window. The STDP process is triggered by the desired output spikes,
while the anti-STDP process is triggered by the actual output spikes.
constant, Taherkhani et al. [140] proposed a synaptic delay learning rule,
delay-learning ReSuMe (DL-ReSuMe), which can improve the learning accu-
racy and convergence speed. To overcome the one-way adjustment problem
in DL-ReSuMe, Zhang et al. [141] proposed synaptic weight-delay plasticity
for ReSuMe (ReSuMe-DW) and PBSNLR (PBSNLR-DW) that can decrease
the delay to increase the membrane potential at desired spike times.
Inspired by ReSuMe, the chronotron I-learning[98] is a heuristic learning
rule, which adjusts the synaptic weights proportionally to their corresponding
synaptic currents. SPAN[25] uses convolving kernels to convert input, actual,
and desired spike trains into analog signals. Unlike SPAN, precise-spike-
driven (PSD) synaptic plasticity[99] only convolves input spike trains.
These rules make neurons fire spikes at desired times. By emitting differ-
ent spike trains, neurons can classify input spike patterns. However, they do
26

not extend well to deep networks which are vital for solving complex tasks.
4.3.3. Other bio-plausible rules
STDP modifies synaptic weights depending on the pre and postsynaptic
neuronal activity, which lacks global teaching signals for the whole network.
To this end, researchers use neural activity-encoded errors to drive synaptic
changes that can be backpropagated to upstream layers. Zhang et al. [123]
introduced self-backpropagation (SBP) into a three-layer SNN to reduce the
computational cost without affecting accuracy. Payeur et al. [142] proposed
burst-dependent synaptic plasticity to achieve hierarchical credit assignment
in multi-layer SNNs.
Xie et al. [143] found that STDP cannot be applied to recurrent synap-
tic connection in PCNNs, which obstruct the learning in PCNNs. Inspired
by the neural activity-dependent property of STDP, they proposed spike-
synchronization-dependent plasticity (SSDP) rule to improve the spike syn-
chronization. Experimental results showed that SSDP-based PCNNs can
get better segmentation performance. In addition, they designed a novel
memristor-based circuit model of SSDP.
4.4. Gradient-based rules
Application of powerful gradient descent formalism to SNNs is compli-
cated by the hard non-linearity of spike generation mechanism: small changes
in synaptic weights can cause large changes in the output spike train, i.e.
spikes times or counts. Specifically, the gradient of the output spike train
with respect to the synaptic weights ∇wS(t) is zero except at spikes times
where it is ill-defined[29]. According to the insights on how to circumvent the
27

problem, gradient-based rules can be classified into three categories, direct
training, ANN pre-training, and ANN-coupled training.
4.4.1. Direct training
Direct training approaches can be categorized into likelihood, voltage,
timing, and activation gradient approaches according to the state variable
used for the optimization of the objective function.
Likelihood gradient. The threshold nonlinearity can be smoothed via stochas-
ticity which makes it possible to perform gradient descent to maximize the
likelihood of generating desired output spike trains. There are various meth-
ods to introduce stochasticity, such as stochastic threshold[96] or synapse[144]
Pfister et al. [96] used likelihood gradient to train single-layer networks with
temporal coding. This approach recently has been extended to multilayer
networks[145, 101, 144, 146]. However, it has not been applied to deep net-
works for complex tasks.
Voltage gradient. The voltage of a neuron at time instants is differentiable
with respect to its synaptic weights under certain conditions or approxima-
tions which facilitates the formulation of gradient-base rules.
Tempotron[23] learns to classify binary spike patterns by minimizing the
distance between the firing threshold and shunted membrane potential max-
imal on error trials. It changes the synaptic weights according to
∆wi=η(y−ˆy)X
tj
i<tmax
Ktmax −tj
i(13)
where ηis the learning rate. tmax denotes the time of maximal membrane
potential value. tj
idenotes the jth spike time of the ith synapse. K(t) is
28

the normalized postsynaptic potential (PSP), which is a double exponential.
y∈ {0,1}is the label of input patterns. ˆydenotes the prediction of the
neuron, which is determined by
ˆy=


1V(tmax)≥Vth
0V(tmax)< Vth
(14)
The limitation of tempotron to binary classifications was overcome by
multi-spike tempotron (MST)[104] which solves multi-classification problems
by mapping input spike patterns to desired numbers of output spikes. MST
introduces the spike-threshold-surface (STS) function, which maps critical
thresholds to the numbers of emitted spikes, to formulate a continuous ob-
jective function differentiable with respect to neuron’s synaptic weights.
To reduce the computational costs of MST, Yu et al. [111] utilized the
linear assumption for threshold crossing[22] to derive the efficient threshold-
driven plasticity (TDP) algorithm. Subsequent efficient multispike learning
(EML) [147], joint weight-delay plasticity (TDP-DL) [148] further improve
MST from various perspectives. Unlike the rules in Section 4.3.2, these multi-
spike learning rules can train a neuron to emit a desired number of spikes
which is proportional to the number of underlying clues, without specifying
the spike times. Thus this type of learning is termed aggregate-label learning
[104]. In contrast to MST and TDP, membrane-potential driven aggregate-
label learning (MPD-AL) [149] constructs the error functions based on the
membrane potential instead of the critical thresholds.
The above-mentioned rules can’t be applied to multilayer networks. To fa-
cilitate image classification tasks via these rules, a common practice is encod-
ing static images into sparse spike representations, such as S1C1-SNN[150],
29

SCNN[151], CNN-TDP[152] and UMP-TDP[152]. This approach can achieve
comparable performance to deep SNNs on small-scale datasets, such as MNIST
and Fashion-MNIST.
Normalized approximate descent (NormAD)[103] was derived under the
consumption of sparse spike trains, which approximate the gradient of mem-
brane potential at a given time instant with respect to synaptic weights.
This approximation method has been effectively applied to multilayer SNNs
with a weight-fixed convolutional layer[110], and spiking convolutional auto-
encoders which yield 99.08% accuracy on MNIST dataset[153]. Similarly,
Zhang et al. [95, 154] force the neuron to reset only at desired output spike
times[139, 155] thus enabling easy calculation of voltage gradients. Unlike
obtaining voltage gradient at specific times, Lee et al. [107] ignored discon-
tinuities of membrane potential at spike times and treated the output of a
neuron as a linear function of its inputs which has filtered by the membrane.
They trained a spiking CNN, achieving an accuracy of 99.31%.
Timing gradient. The timing gradient approaches focus on the neuron’s out-
put spike times and compute the gradients with respect to the neuron’s
synaptic weights. Due to the event-driven nature of spikes, these approaches
allow for efficient even-based network simulation.
SpikeProp[22] is the first algorithm applying backpropagation to tempo-
rally coded networks of spiking neurons firing at most one. Its formula is as
follows:
∆wl
ij =−η∂L
∂tl
j
∂tl
j
∂V l
j(tl
j)
∂V l
j(tl
j)
∂wl
ij
(15)
where ηis the learning rate, a positive constant. Lis the loss function.
Vl
jand tl
jdenote the membrane potential and spike of neuron jin layer l,
30

respectively. wl
ij is the synaptic weight from neuron iin l−1 layer to neuron
jin llayer. The key challenge is to solve the partial derivative ∂tl
j/∂V l
j(tl
j)
in Eq. 15. SpikeProp assumes that the membrane potential of a neuron
increases linearly in a small enough region around the firing time as shown
in Fig.10. Thus, ∂tl
j/∂V l
j(tl
j) can be expressed as
∂tl
j
∂V l
j(tl
j)=−1
∂V l
j(tl
j)/∂tl
j
=−1
Piwl
ij
∂K (tl
j−tl−1
i)
∂tl
j
(16)
where K(t) is used to describe the PSP of neuron jgenerated by the input
spike tl−1
i, which also denotes the output spike time of neuron iin layer l−1.
Time
tj
Voltage
Vth
Figure 10: Illustration of linear approximation for threshold crossing[22]. It assumes that
the membrane potential of a neuron increases linearly in a small enough region around the
firing time. The sold line denotes the membrane potential when threshold crossing.
Multi-SpikeProp[97] overcomes the limitation of one spike in SpikeProp
and improves performance. Without approximation, Mostafa [108] used LF
neurons to obtain an analytical expression relating input and output spike
times. Comsa et al. [114, 156] used SRM with biologically realistic alpha
function for membrane dynamics and derived exact gradients with respect
31

to input spike times and synaptic weights. Kheradpisheh and Masquelier
[82] found that the first spike time of an IF neuron can approximate ReLU
activation, they thus derived a new learning algorithm for multilayer SNNs
with TTFS coding and achieved an accuracy of 97.4% on MNIST, which is
comparable to [108, 114].
Recent works extend the timing gradient approach to train deeper SNNs.
To overcome gradient explosion and dead neuron problem encountered in
timing gradient-based method, Zhang et al. [157] proposed a rectified linear
postsynaptic potential function (ReL-PSP) based neuron model where mem-
brane potential increases linearly prior to postsynaptic spike time. They
trained convolutional SNNs, yielding 99.4% and 90.1% accuracy on MNIST
and Fashion-MNITST, respectively. Zhou et al. [118] applied the method
proposed by [108] to spiking VGG and GoogleNet with deep ANN training
techniques such as batch normalization (BN), achieving an accuracy of 68.6%
on ImageNet.
Activation gradient. Activation gradient approaches smooth the hard non-
linearity due to the none-or-one response of spiking neurons via modification
of their activation function in forward or backward path.
The hard thresholds of spiking neurons can be placed with soft ones in
forward propagation of SNNs. Huh and Sejnowski [109] introduced active
zone where the synaptic current is activated gradually. They trained recur-
rent SNNs with backpropagation through time (BPTT) to perform dynamics
tasks such as predictive coding. Liu et al. [158] found that all PCNN models
can’t explain the phenomenon that biological neurons stimulated by periodic
signals exhibit chaotic behavior. They thus proposed a continuous-coupled
32

neural network (CCNN) model to solve the problem. CCNN is a mean-field
model where the step function used in PCNN is replaced by the sigmoid
function which facilitates standard BP for training. CCNN also achieves
better results in image segmentation tasks than state-of-the-art visual cortex
models. These methods modify the binary output of spiking neurons, which
is less friendly for low-power hardware implementation.
An alternative approach is to use surrogate gradients for backpropaga-
tion. SuperSpike[29] and STBP[30] were the early works applying gradient
gradients to train multilayer SNNs. STBP uses the explicitly iterative LIF
model. An SNN with iterative LIF can be unrolled in time as an RNN, which
facilitates utilizing BPTT for the wight update. STBP updates the synaptic
weights as follows
∆wl
ij =−ηX
t
∂L
∂St,l
j
∂St,l
j
∂V t,l
j
∂V t,l
j
∂I t,l
j
∂I t,l
j
∂wl
ij
(17)
where ηis the learning rate. Lis the loss function. St,l
j,Vt,l
j, and It,l
jdenote
the output spike, the membrane potential, and the input current of neuron
jat time tin layer l, respectively. wl
ij is the synaptic weight from neuron
iin l−1 layer to neuron jin llayer. In Eq. 17, the derivative of the
spike function ∂St,l
j/∂V t,l
jis not well defined. To enable gradient descent,
a surrogate derivative is used for approximation. A common family of the
surrogate gradient is the rectangular function[30] given by
h(V) = 1
asign |V−Vth|<a
2(18)
where ais a hyper-parameter determining the shape for gradient estimation.
Shrestha and Orchard [159] considered the temporal dependence between
spikes and interpreted the surrogate gradient as the probability density func-
33

tion. Wu et al. [85] extended STBP to deep convolutional SNNs with neuron
normalization(NeuNorm). Zheng et al. [160] proposed threshold-dependent
batch normalization (tdBN) for alleviating the gradient vanishing or explo-
sion problem and maintaining the firing rate. Combined with STBP, tdBN re-
alizes direct training of spiking ResNet-50 on ImageNet. As a time-dependent
variant of tdBN, temporal effective batch normalization (TEBN) [161] can
regularize the temporal distribution. To overcome the degradation problem
in deep SNNs, Fang et al. [19] proposed SEW ResNet which can realize iden-
tity mapping . Feng et al. [162] proposed a multi-level firing (MLF) unit to
combat the gradient vanishing problem. Hu et al. [20] scaled SNNs up to
104 layers on ImageNet and 482 layers on CIFAR-10, achieving 76.02% and
91.9% accuracy, respectively. The above studies have fully demonstrated the
scalability of the surrogate gradient method.
The shape and smoothness of surrogate gradients have an impact on per-
formance. To avoid the heuristic choice of surrogate gradients, Li et al. [163]
proposed differentiable spike (Dspike) that can find the optimal shape and
smoothness for surrogate gradients. Deng et al. [125] proposed temporal effi-
cient training (TET) to solve the SNN generalization problem which results
from incorrect surrogate gradients, obtaining a remarkable accuracy of 83%
on CIFAR10-DVS.
The neuronal heterogeneity is a critical property of biological neural net-
works which can be incorporated into SNNs. Fang et al. [84] proposed a
parametric LIF (PLIF) model which has learnable time constants. [164] pro-
posed a learnable threshold scheme during training. Yao et al. [165] proposed
a unified gated LIF (GLIF) neuron, wherein bio-features in different neural
34

behaviors are fused by the gating factor. These methods can improve the
learning of SNNs, thus obtaining better accuracy with fewer time steps
Combined with BPTT, the surrogate gradient approach can achieve spa-
tiotemporal credit assignment in very deep SNNs[20]. However, the large
computational graph from the unrolled networks requires tremendous hard-
ware resources during training. Local learning scheme provides an alterna-
tive to assign spatiotemporal credit in SNNs. Deep continuous local learn-
ing (DECOLLE) [166] uses layer-wise local readouts with random and fixed
weights. Ma et al. [167] proposed a local learning scheme, wherein each layer
is trained with a local auxiliary classifier. Inspired by the teacher-student
learning approach, Yang et al. [168] proposed local tandem learning (LTL)
to transfer the feature representation of a teacher ANN to a student SNN
through layer-wise loss function. These local learning rules provide a com-
petitive performance while consuming less computational costs than BPTT.
4.4.2. ANN pre-training
Leveraging the fact that the firing rate of IF models can approximate
the activation value of ReLU functions, ANN pre-training approaches first
train an ANN with constraints, then convert it into an SNN with or without
post-conversion techniques fine-tuning the parameters.
The first ANN pre-training method was proposed by Cao et al. [27] which
imposes constraints on CNN such as using ReLU activation function and
removing biases to avoid negative outputs. Diehl et al. [102] suggested us-
ing weight normalization or threshold balancing to reduce conversion error,
which is exactly equivalent mathematically[113]. Rueckauer et al. [105, 83]
proposed spike subtraction, also called soft reset, to alleviate information loss
35

caused by resetting and implemented spiking equivalents of common opera-
tions such as BN which allow conversion of deeper CNNs including VGG-16
and GoogLeNet Inception-V3. Sengupta et al. [113] argued that removing
the constraints in ANN training[83] suffers significant accuracy loss in the
conversion process and proposed spike-norm to extend converted SNNs to
residual architectures. RMP-SNN[115] used soft reset and a threshold bal-
ancing method that alleviates the firing rate vanishing problem. Hu et al.
[121] proposed a compensation mechanism to reduce discretization errors and
firstly built a converted SNN with more than 100 layers. Ding et al. [122]
placed ReLU with rate norm layer (RLN) for ANN training, enabling direct
conversion without setting thresholds manually.
In recent works, most studies focus on a theoretical analysis of conversion
errors which facilitate methods reducing them and thus inference latency of
the converted SNNs. Deng and Gu [119] added a threshold and shift in ReLU
activation function to reduce conversion error. Li et al. [120] proposed SNN
calibration which calibrates the parameters in converted SNN to match the
activations in ANN. Bu et al. [124] used optimal initialization of membrane
potentials to implement expected error-free conversion. Bu et al. [31] went
deeper into error analysis and proposed a quantization clip-floor-shift activa-
tion function to replace the ReLU, achieving ultra-low latency (4 time steps)
of converted SNNs.
There are also some interesting works that explore post-conversion fine-
tuning of converted SNNs to reduce latency and increase accuracy. Rathi
et al. [116] proposed a spiking-timing dependent surrogate function to train
a converted SNN which converges within a few epochs and requires fewer
36

time steps. Rathi and Roy [169] jointly optimized membrane leak and fir-
ing threshold along with synaptic weights to reduce latency and increases
activation sparsity. Wu et al. [21] proposed a progressive tandem learning
(PTL) framework that compensates the conversion errors layer-wise by tan-
dem learning (TL)[117] with an adaptive training scheduler.
The ANN pre-training approach leverages the superior performance of
ANNs while avoiding tremendous hardware overhead for direct training of
SNNs, and thus quicker implementation on low-power neuromorphic hard-
ware. However, this approach only works for static datasets so far and can’t
exploit the temporal dynamics of SNNs which makes them the third gener-
ation of neural networks[170].
4.4.3. ANN-coupled training
Unlike copying the weights from a well-trained ANN in the ANN pre-
training approach, ANN-couple training consists of an SNN and an ANN with
shared weights. The SNN feeds forward spike trains, while the ANN back
propagates errors to update the shared weights. Wu et al. [117] first proposed
this idea termed tandem learning in which the networks are coupled layer-
wise. They demonstrated its effectiveness on both static and neuromorphic
datasets. Kheradpisheh et al. [171] used a proxy ANN to backpropagate
the errors of an SNN on the basis of rate-coded LF neurons approximating
ReLU. They outperformed [117] on the CIFAR-10 dataset with an accuracy
of 93.11%.
37

4.5. Performance Comparison
So far, the majority of works on SNNs have used image classification
datasets as benchmarks. To shed more light on the performance of SNN
learning rules, we summarize the state-of-the-art results of existing methods
tested on two types of datasets, i.e. static and neuromorphic datasets, which
are presented in Tab.4 and 5, respectively.
The static datasets in Tab.4 include MNIST, CIFAR-10/100, and Im-
ageNet. Due to the simplicity of MNIST, the recent accuracies reported
on MNIST are pretty high (>99%). Therefore, CIFAR-10/100 and Ima-
geNet have become more popular to evaluate deep SNNs. Recent results
reported on these datasets are dominated by ANN pre-training[31, 120] and
activation gradient approach[20, 125]. Though ANN pre-training methods
are better than activation gradient-based methods in terms of accuracy, ANN
pre-training methods need more time steps for inference.
Concerning the neuromorphic datasets, we include N-MNIST, CIFAR10-
DVS, and DVS128 Gesture in Tab.5. Similar to MNIST, the accuracies
reported on N-MINST are above 99% in recent publications. The CIFAR10-
DVS is a challenging dataset. TEBN [161] reported the best accuracy of
84.90% on CIFAR10-DVS with neuromorphic data augmentation[172]. No-
tably, almost all methods are based on activation gradient since they can
exploit the temporal dynamics of the neuromorphic datasets.
38

Table 4: State-of-the-art results on the neuromorphic datasets, in-
cluding MNIST, CIFAR-10/100, and ImageNet. — denotes the
number of time steps is not reported in the paper or is not applica-
ble to the method. T denotes the number of time steps. †denotes
local learning rules.
Dataset Reference Year Learning Rule Accuracy T
MNIST
Diehl’s method [50] 2015 STDP-based 95.00 —
Lee’s method [107] 2016 Voltage gradient 98.77 —
SDNN [28] 2018 STDP-based 98.40 —
STBP [30] 2018 Activation gradient 99.42 —
Zhou’s method [118] 2021 Timing gradient 99.33 —
RNL [122] 2021 ANN pre-training 99.46 —
PLIF [84] 2021 Activation gradient 99.72 8
STDBP [157] 2022 Timing gradient 99.40 —
CIFAR-10
NeuNorm [85] 2019 Activation gradient 90.53 12
RMP-SNN [115] 2020 ANN pre-training 93.39 512
TL [117] 2021 ANN-coupled training 90.98 8
tdBN [160] 2021 Activation gradient 93.16 6
Zhou’s method [118] 2021 Timing gradient 92.68 —
PLIF [84] 2021 Activation gradient 93.50 8
Dspike [89] 2021 Activation gradient 94.25 6
Calibration [120] 2021 ANN pre-taining 95.79 256
Proxy [171] 2022 ANN-coupled training 93.11 60
TET [125] 2022 Activation gradient 94.50 6
QCFS [31] 2022 ANN pre-taining 96.08 32
MLF [162] 2022 Activation gradient 94.25 4
TEBN [161] 2022 Activation gradient 95.60 6
LTL [168] 2022 Activation gradient†95.28 32
39

Table 4 continued from previous page
Dataset Reference Year Learning Rule Accuracy T
CIFAR-100
RMP-SNN [115] 2020 ANN pre-training 70.58 1024
DIET-SNN [169] 2021 ANN pre-training 69.67 5
S-ResNet [121] 2021 ANN pre-training 70.62 350
RNL [122] 2021 ANN pre-training 75.10 —
Calibration [120] 2021 ANN pre-training 77.30 128
TET [125] 2022 Activation gradient 74.72 6
QCFS [31] 2022 ANN pre-training 79.62 32
TEBN [161] 2022 Activation gradient 78.76 6
LTL [168] 2022 Activation gradient†76.08 32
ImageNet
RMP-SNN [115] 2020 ANN pre-training 73.09 2048
S-ResNet [121] 2021 ANN pre-traning 73.77 350
tdBN [160] 2021 Activation gradient 67.05 6
Zhou’s method [118] 2021 Timing gradient 68.80 —
SEW-ResNet [19] 2021 Activation gradient 69.26 4
MS-ResNet [20] 2021 Activation gradient 76.02 5
Calibration [120] 2021 ANN pre-training 77.50 256
TET [125] 2022 Activation activation 68.00 4
QCFS [31] 2022 ANN pre-training 74.22 256
TEBN [161] 2022 Activation gradient 68.28 4
GLIF [165] 2022 Activation gradient 60.09 6
5. Applications of SNNs
SNNs have potential advantages in representing and processing spatiotem-
poral patterns due to their inherent temporal dynamics. Thus, a vast body
of studies explored the applications of SNNs in spatiotemporal tasks, such as
40

Table 5: State-of-the-art results on the neuromorphic datasets, including N-MNIST,
CIFAR10-DVS, and DVS128 Gesture. — denotes the number of time steps is not re-
ported in the paper. T denotes the number of time steps. †denotes neuromorphic data
augmentation.
Dataset Reference Year Learning Rule Accuracy T
N-MNIST
Lee’s Method [107] 2018 Voltage gradient 98.74 —
STBP [30] 2018 Activation gradient 98.78 —
SLAYER [159] 2018 Activation gradient 99.20 —
NeuNorm [85] 2019 Activation gradient 99.53 —
TL [117] 2021 ANN-coupled traning 99.31 —
PLIF [84] 2021 Activation gradient 99.61 10
LTMD [164] 2022 Activation gradient 99.65 15
CIFAR10-DVS
NeuNorm [85] 2019 Activation gradient 60.50 —
TL [117] 2021 ANN-coupled traning 65.59 —
tdBN [160] 2021 Activation gradient 67.80 10
PLIF [84] 2021 Activation gradient 74.80 20
Dspike [89] 2021 Activation gradient 75.40 10
SEW-ResNet [19] 2021 Activation gradient 74.40 16
TET [125] 2022 Activation gradient 77.33/83.17†10
TEBN [161] 2022 Activation gradient 84.90†10
GLIF [165] 2022 Activation gradient 78.10 16
DVS128 Gesture
SLAYER [159] 2018 Activation gradient 93.64 —
tdBN [160] 2021 Activation gradient 96.87 40
PLIF [84] 2021 Activation gradient 97.57 20
SEW-ResNet [19] 2021 Activation gradient 97.92 16
MLF[162] 2022 Activation gradient 97.29 40
41

event-based vision and audio signal processing. In this section, we introduce
the applications of SNNs in these two fields.
Compared to conventional frame cameras, brain-inspired event cameras
have several advantages, such as low power and high temporal resolution
[9]. The SNNs are well-suited for processing the spatiotemporal data gen-
erated by event cameras due to their temporal dynamics. Orchard et al.
[173] proposed an SNN for visual motion estimation, in which LIF neurons
with synaptic delays implement motion-sensitive receptive fields. Haessig
et al. [174] proposed an SNN-based direction sensitive (DS) unit for optical
flow estimation. Paredes-Valles et al. [10] incorporated STDP learning to
a hierarchical SNN which exhibits motion selectivity after training. Except
for the above low-level vision applications, SNNs can also perform high-level
vision tasks, such as recognition. The HMAX-inspired HFirst model [173]
utilizes the spike timing provided by event cameras for the character recog-
nition task. Xiao et al. [175] improved HFirst by integrating a tempotron
classifier. Recent work [159, 84, 30, 117] apply spike-based backpropagation
to training deep SNNs, obtaining high accuracy on complex neuromorphic
datasets created by event sensors [91, 92].
SNN-based acoustic models have shown great potential for energy-efficient
and high performance auditory information processing tasks, such as auto-
matic sound classification (ASC) [176, 177, 178], automatic speech recogni-
tion (ASR) [179], and sound source localization (SSL) [180]. Several works
[176, 177, 178] have succes apply tempotron-based learning for ASC tasks. In
these works, conventional feature extraction methods, such as mel-frequency
cepstral coefficient (MFCC) and self-organizing map (SOM), are used for gen-
42

erating spike patterns, which are then classified by SNNs trained with the
tempotron-based learning rule. Tavanaei and Maida [181, 182] applied STDP
to features extraction from raw speech signals. The extracted spike features
are then post-processed into real-valued feature vectors and classified by tra-
ditional classifier, such as SVM. The biological plausibility of these models
is further improve by [183], wherein a fully SNN-based ASC framework is
presented by combining competitive STDP learning and tempotron-based
classification. Recent studies apply deep SNNs to auditory systems. Pan
et al. [180] utilized surrogate gradient learning for SSL tasks. Deep SNNs
trained with tandem learning rule have been explored for speech separation
[21] and large vocabulary ASR [179].
6. Conclusion and future research directions
We presented a comprehensive survey of SNN learning rules, in which
we reviewed the most representative learning rules in SNNs and provided
discussions on their characteristics, advantages, limitations, and performance
on several popular datasets. Besides, we introduced practical applications of
SNNs in event-based vision and audio signal processing. Here, we further
discuss a few challenges and promising research directions which may foster
real-world applications in the field.
1. Learn lessons from deep learning. As we can see from the paper, the
performance of SNNs has improved a lot by utilizing deep learning techniques,
such as BPTT and BN. Even though there is still a gap between ANNs and
SNNs in terms of accuracy, deep learning techniques have great potential
to may further improve the performance of SNNs. On the one hand, the
43

unique characteristics of SNNs, which may increase their performance, can
be explored via deep learning techniques. For example, recent works show
that neural architecture search (NAS)[184] can be exploited for finding better
SNN architectures[185, 186]. On the other hand, the temporal dimension and
spike representation of SNNs should be considered when applying training
techniques in deep learning to SNNs. For example, training deep SNNs using
BPTT is constrained to working on dense tensors, limiting training speed and
efficiency[187]. Standard BN in SNNs does not show scalability to large-scale
datasets[188].
2. Draw inspirations from brain science. Biological observations and
mechanisms of the human brain are natural references to creating intelli-
gence. For example, burst-dependent plasticity can solve the credit assign-
ment problem in hierarchical networks[142]. Dendritic spines[189] can facili-
tate weight optimization for pruning[190]. LIF neurons with lateral interac-
tions [17] have shown improved robustness to noisy inputs. However, current
SNNs mainly utilize single neuron dynamics such as LIF, while network-level
dynamics, for instance, coherent oscillations[126] and chaos[191], are less ex-
plored in SNNs[158]. Although PCNNs are more biologically plausible [14]
and the coupling mechanism have the potential for robust pattern recogni-
tion [17, 158], the learning rules for them, as this survey revealed, have not
been well developed. We expect that exploring learning rules in PCNNs may
help us understand how the brain works and further improve the robustness
of existing methods.
3. Algorithm-hardware co-design for energy-efficient neuromorphic sys-
tems. In this methodology, design goals are achieved by exploiting the syn-
44

ergism between algorithms and hardware platforms. The SNNs are spa-
tiotemporal networks that are not well-suited for simulation on conventional
hardware. Thus, it is necessary to explore the efficient implementation of
the SNNs and their learning rules. On the one hand, spike-based compu-
tation in SNNs can be utilized for low-power neuromorphic chips[192, 193].
On the other hand, novel devices may leverage more brain-like systems due
to their unique merits. For instance, the memristor has nonlinearity, non-
volatility, and is compatible with the CMOS technology[194]. It can emulate
the complex nonlinear dynamics of biological neurons and synapses[135, 195,
196, 134] and learning in SNNs[197, 198]. We expect this paradigm to gain
increased popularity in the near future.
Declaration of competing interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study is supported by the Natural Science Foundation of Gansu
Province (No. 21JR7RA510) and the National Natural Science Foundation
of China (No. 61906174).
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