# A Comprehensive Review of Spiking Neural Networks: Interpretation,
Optimization, Efficiency, and Best Practices

Kai Malcolm  Josue Casco-Rodriguez

###### Abstract

Biological neural networks continue to inspire breakthroughs in neural network
performance. And yet, one key area of neural computation that has been under-
appreciated and under-investigated is biologically plausible, energy-efficient
spiking neural networks, whose potential is especially attractive for low-
power, mobile, or otherwise hardware-constrained settings. We present a
literature review of recent developments in the interpretation, optimization,
efficiency, and accuracy of spiking neural networks. Key contributions include
identification, discussion, and comparison of cutting-edge methods in spiking
neural network optimization, energy-efficiency, and evaluation, starting from
first principles so as to be accessible to new practitioners.

Machine Learning, Spiking Neural Networks, Sparsity, Convex Optimization,
Geometry, Discrete, Spike, Energy, Efficiency, Interpretable, Interpretation,
Normalization, Initialization, Architecture, Accuracy

##  1 Introduction

Spiking neural networks (SNNs) offer a perspective on machine learning
different than that of traditional artificial neural networks (ANNs): SNNs
encode data in a series of discrete time spikes, modeled after the action
potentials created by neurons in biological brains. The original development
of ANNs was also inspired by the human brain, which consists of nearly 100
billion neurons, each of which can have up to 15,000 synapses. Despite this
potentially massive parameter space, the human brain only consumes about 20 W
and takes up under 3 pounds. For comparison, today’s top-performing language
models can require over 300 kWh of power (Patterson et al., 2021). Biological
neural networks’ impressive energy- and space-efficiency is due in part to the
fact that most neurons are usually inactive, meaning that the brain is a
highly sparse neural network. SNNs attempt to combine more biologically
plausible inputs, learning methods, and architectures in order to match the
performance of ANNs while also achieving the energy efficiency and robustness
of the human brain. Currently, the development of SNNs is a growing field, but
lacks clear benchmarks and tool sets, complete optimization of nonlinear
spiking dynamics, and robust mechanisms for encoding targets as binary spike
trains (Skatchkovsky et al., 2021). The main goal of this literature review is
to chronicle recent advances in SNN optimization, with particular focused
applied to biologically plausible mechanisms and advances in the optimization
of non-differentiable spike trains.

From an optimization point of view, SNNs pose a unique and interesting
challenge, as they lack interpretable models, straight-forward gradient
calculation, and also methods that leverage the inherent advantages of SNNs
(Li et al., 2021) (Mancoo et al., 2020). We have compiled a meta-analysis of
recent SNN advances, providing a quantitative comparison between given
implementations when possible. Inherent to this process is the identification
of standard datasets to act as benchmarks between different implementations.
Another focus of this work is to propose how to better showcase differences
and advancements between recent innovations. For performance comparisons, we
have identified the most frequently used benchmark datasets so that we can
rank existing SNN architectures in an accessible and understandable format. In
a similar vein of work, we have included which tools and frameworks were used
to create each model, as there does not exist a standardized set of tools for
SNN development (see Table 1). We believe such a characterization of tools
available to researchers would be a valuable resource for unifying the field,
making SNNs more accessible to new researchers.

##  2 Foundational Neurobiology

This section lays the groundwork for the biological analog from which SNNs
originate, focusing on how neurons in the human brain transmit information.

###  2.1 Underlying Neuroanatomy

Neurons. The neuron is the fundamental atomic unit of the brain. In humans,
there here are three primary categories of neurons: sensory, motor, and inter-
neurons. Neurons usually consist of a bulky soma (i.e., cell body), branching
dendrites (which propagate stimuli received from adjacent cells to the soma),
and a long axon that forms synaptic junctions with the dendrites of adjacent
neurons.

Synapses. Synapses are the spaces between axon terminals and dendrites through
which a presynpatic neuron sends a biological signal to the postsynaptic
neuron. Synaptic activity typically consists of either chemical or electrical
signals. In the chemical case, presynaptic neuron activation triggers the
release of chemical neurotransmitters that bind to receptors lodged in the
cell membrane of the postsynpatic neuron, which then cause the propagation of
an electrical signal throughout the postsynaptic neuron. In the electrical
case, the pre- and post-synpatic neurons are connected by gap junctions which
pass the current from one neuron to the other, with the benefit of quicker
transmission times (Barnett & Larkman, 2007).

Action Potentials. Action potentials (APs) are the discrete signals (spikes)
through which biological neurons communicate. Each AP is a continuous chain of
depolarization events: one segment along the axon experiences a rise and fall
of membrane potential (voltage across the membrane), and this rise and fall is
propagated along the axon. More specifically, APs typically originate from
voltage-gated ion channels within a cell’s membrane, such that when the cell’s
membrane potential increases enough to exceed an intrinsic biological
threshold, the channels open, allowing charged ions to enter the cell driven
by passive gradients, e.g. both a concentration gradient and an electro-
chemical gradient due to difference in charges and ion concentrations inside
and outside the cell. Note that the influx of charged ions is equivalent to a
current flowing into the cell, and this rapid increase in membrane potential
is commonly referred to as depolarization. The resulting diffusion of ions
across the cell membrane creates a positive feedback loop for the electro-
chemical gradient: incoming ions raise the cell’s membrane potential at
different points along the membrane, prompting downstream channels to open and
allow more ions to enter. Once enough ions have entered the cell, the membrane
potential reverses, which prompts the opening of different set of voltage-
gated ion channels, this time resulting in an outward flow of ions, repeating
the same process as before and bringing the cell membrane back to its resting
potential. The drastic drop in membrane potential is known as hyper-
polarization (Barnett & Larkman, 2007), (Luo & Rudy, 1991).

Figure 1: An illustration by Barnett & Larkman (2007) showing the different
phases of depolarization and polarization as they relate to action potentials
and voltage-gated ion channels.

During action potentials, the membrane potential undergoes very sharp
exponential rise and fall within only a few milliseconds. The action potential
always drops the neuron’s resting potential below the normal resting state
potential, resulting in a refractory period (typically about 2 milliseconds)
in which the neuron cannot undergo another AP. As will be discussed shortly,
these biological mechanisms are the motivation and mathematical framework for
today’s neural networks. Important terminology borrowed from biology includes
the idea of a spike train (the time indices at which spikes occurred for a
given neuron) and firing (the creation/propagation of an AP). Note that APs
are synonymous with spikes for this discussion (Barnett & Larkman, 2007).

Figure 2: Plotted measurements of (b) an observed spike train and (c) an
isolated action potential by Wattanapanitch et al. (2007).

###  2.2 Dynamical Models of the Brain

Hodgkin-Huxley Model. One of the most popular models for describing the
initiation and propagation of action potentials in neurons is the Hodgkin-
Huxley (HH) model, as shared by Hodgkin & Huxley (1952). The HH model is a
conductance-based dynamical system (see Figure 3), in which each neuron’s
membrane is modeled simply as a series of parallel capacitive and resistive
electrical components, as shown in Equation 1. Such abstractions of the
intricate inner workings of neurons allow for compartmentalized modeling of
many neurons.

Figure 3: The equivalent circuit of a section of a neuron’s membrane, as
described by the Hodgkin-Huxley model (Thanapitak, 2012).

$I=C_{m}\frac{dV_{m}}{dt}+g_{K}(V_{m}-V_{K})+g_{Na}(V_{m}-V_{Na})+g_{l}(V_{m}-V_{l})$
(1)

The HH model treats each neuron’s total current flow $I$ as the sum of four
separate current sources described as functions of the membrane potential
$V_{m}$. The first source $C_{m}\frac{dV_{m}}{dt}$ arises from the cell
membrane separating the neuron’s internal contents (negatively charged) from
the external environment (positively charged), resulting in the membrane
acting as a capacitor with capacitance $C_{m}$. The other three sources take
the form $g_{i}(V_{m}-V_{i})$, where $g_{i}$ is the electrical conductance of
ion channel $i$ and $V_{i}$ is the membrane potential threshold at which the
direction of current flow in ion channel $i$ reverses (e.g. the electrical
gradient is now acting in the opposite direction due to the resulting build up
of charge). The specific terms are $i\in[K,Na,l]$, where $K$ and $Na$
represent potassium and sodium ions, and $l$ represents membrane potential
leakage instead of a specific ion channel. Note that other types of currents
may also be included, but these are by far the most common. Recall from the
earlier section on neuroanatomy, the first set of opening channels
corresponding to the rapid depolarization were the sodium channels, and the
second set of channels responsible for bring the cell back to baseline were
the potassium channels (Hodgkin & Huxley, 1952), (Barnett & Larkman, 2007).

Through simple models of neural electrophysiology, scientists can rigorously
model the cell membrane, track how it approaches its firing threshold, and
therefore understand how best to model the creation and propagation of spikes
across neurons. Such understandings yield crucial insights upon which the
fundamental ideas of information processing in neural networks can be built.
As will be shown, the key similarities between artificial and biological
spiking neurons are membrane leakage and communication via action potentials
(e.g. spiking activity).

##  3 Biological Plausibility and Interpretability

This section aims to show various insights into how spiking neural network
(SNN) models can be not only human-interpretable, but also biologically
plausible in silico analogues to biological neural networks.

###  3.1 Biological Plausibility

Traditional artificial neural networks (ANNs) use floating point numbers to
encode the information transmission between neurons, whereas biological
neurons communicate via action potentials. Given that action potentials occur
and decay quickly, they can be approximated as discrete, binary spikes; when
taking sparsity into account, information may be transmitted via highly sparse
binary vectors instead of relying on arbitrarily precise floating point
numbers. However, exactly how neural spike trains encode information is still
debated. The two most popular information encoding schemes are discussed
below.

Rate coding. One of the most popular spike-train information-encoding models
is the rate-coding framework, which represents stimuli intensity as a firing
rate. If the input stimuli consists of an image, each pixel’s intensity could
be converted to the firing rate of a Poisson spike train. Rate coding has been
experimentally discovered in the visual cortex as well as the motor cortex
(Guo et al., 2021); however, rate coding is not the only method through which
neural systems encode information (Olshausen & Field, 2006), since it requires
long processing periods and therefore has a poor rate of information
transmission.

Latency coding (TTFS). An increasingly popular method of information-encoding
is the time-to-first-spike (TTFS) model, which only requires a single spike
from each neuron in a group to fully encode stimuli. Thus, information can be
encoded in a binary fashion over time. Specifically, TTFS-encoding negatively
correlates stimulus intensity with the time that a neuron takes to spike once,
and hence is also known as latency-coding. The biological plausibility of
TTFS-like neural coding has been verified by the dependence on the first spike
in systems such as the retina, much of the auditory system, and many responses
to tactile stimulus that require fast reactions (Guo et al., 2021).

Figure 4: An illustration by Eshraghian et al. (2021) showing three common
methods for encoding real-valued data as spike trains. Unlike rate- and
latency-coding, delta modulation is primarily used only at the sensor-level to
represent time-varying stimuli.

Neural coding comparisons. While both rate- and latency-encoding are believed
to be used in biological neural computation (Olshausen & Field, 2006), SNN
practitioners need only be concerned with the advantages of each method: rate
coding has a higher error tolerance and provides more information with which
gradient learning can converge, while latency coding is significantly more
power-efficient and fast than rate coding (since each neuron only spikes once
per input) (Eshraghian et al., 2021). The choice of optimal coding scheme
could depend on whether the dataset is static (e.g., images) or dynamic (e.g.,
speech or video data). However, multiple works have confirmed that
TTFS/latency-coding is the best choice for achieving maximum accuracy while
still keeping a low energy consumption (Guo et al., 2021) (Eshraghian et al.,
2021) (Kundu et al., 2021) (Perez-Nieves & Goodman, 2021). Figure 4 depicts
the encoding mechanisms of both rate- and latency-encoding.

Leaky Integrate-and-Fire Modeling. Spiking neural networks are usually modeled
and understood as networks of Leaky Integrate-and-Fire (LIF) neurons whose
membrane potentials $\mathbf{V}$ evolve at timestep $t$ according to some
variation of the following dynamics:

$\mathbf{V}[t]=\lambda\mathbf{V}[t-1]+\mathbf{WX}[t]-\mathbf{S}[t]V_{th},$ (2)

where $\lambda\in[0,1)$ is a membrane leakage constant, $\mathbf{X}$ is an
input (i.e., an external stimuli to the network or the spiking activity from
another neuron), $\mathbf{W}$ is a weight matrix, and the binary spiking
function $\mathbf{S}[t]=\begin{cases}1,&\text{if $\mathbf{V}[t]>V_{th}$}\\\
0,&\text{otherwise}\end{cases}$ is a function of the activation threshold
voltage $V_{th}$ (Eshraghian et al., 2021).

###  3.2 Spiking Neural Networks as Convex Optimizers

Mancoo et al. (2020) tackle SNN biological plausibility from first principles.
After showing that SNNs solve quadratic optimization problems, the authors
tested their SNNs on a multi-dimensional classification problem, and found
that SNNs had irregular spike patterns, a balance of excitation and
inhibition, robustness to (Gaussian noise) perturbations, and low firing
rates. Furthermore, the authors found that SNNs were robust to cell death
(since each neuron corresponds to only one piece of the feasible set
boundary), have population firing rates that negatively correlate with the
number of neurons.

The authors begin by representing a generic SNN with $N$ neurons and $K$
inputs as follows:

$\dot{\mathbf{V}}(t)=-\lambda\mathbf{V}(t)+\mathbf{Fc}(t)+\mathbf{\Omega
s}(t)+\mathbf{I}_{bg}(t),$ (3)

where $\mathbf{V}(t)\in\mathbb{R}^{N}$ represents the voltage of each neuron,
$\lambda$ is the time-constant of membrane leakage,
$\mathbf{F}\in\mathbb{R}^{N\times K}$ is a feedforward matrix projecting the
inputs $\mathbf{c}(t)\in\mathbf{R}^{K}$ onto the $\mathbb{R}^{N}$ voltage
space, $\mathbf{\Omega}\in\mathbf{R}^{N\times N}$ represents the weights of
recurrent connections, $\mathbf{s}(t)\in\mathbb{R}^{N}$ is a vector containing
the neural spike trains, and $\mathbf{I}_{bg}(t)\in\mathbb{R}^{N}$ represents
background currents or noise.

Meanwhile, the following is an example of a quadratic optimization problem (or
linear if $\lambda=0$) (Boyd et al., 2004):

$\min_{\begin{subarray}{c}\mathbf{y}\\\ \mathbf{Fx-
Gy}\leq\mathbf{T}\end{subarray}}{\left(E(\mathbf{y})=\frac{\lambda}{2}\mathbf{y}^{T}\mathbf{y}+\mathbf{b}^{T}\mathbf{y}\right)},$
(4)

where $\mathbf{y,b}\in\mathbb{R}^{M}$ are the optimization variable and bias,
$\mathbf{x}\in\mathbb{R}^{K}$ is an input to the problem, and the remaining
variables $\mathbf{F}\in\mathbb{R}^{N\times K}$,
$\mathbf{G}\in\mathbb{R}^{N\times M}$, $\mathbf{T}\in\mathbb{R}^{N}$, and
$\lambda>0$ are the parameters of the optimization problem. Note that the
optimization problem can be geometrically interpreted as describing a series
of $N$ linear inequalities,
$\mathbf{G}_{i}^{T}\mathbf{y}\geq\mathbf{F}_{i}^{T}\mathbf{x}-T_{i}$, where
the column vectors $\mathbf{G}_{i}$ and $\mathbf{F}_{i}$ are the $i$-th rows
of $\mathbf{G}$ and $\mathbf{F}$, respectively. Each inequality constraint
divides the solution space $\mathbb{R}^{M}$ into two half-planes, with one
side satisfying the inequality (i.e., being feasible) and the other not (i.e.,
being infeasible). Taking the intersection of the feasible set of each
inequality constraint yields the feasible set of the optimization problem, as
shown in Figure 5 (Mancoo et al., 2020).

Figure 5: An illustration by Mancoo et al. (2020) of a quadratic optimization
problem in $\mathbb{R}^{2}$ with an input $x\in\mathbb{R}^{3}$. Note how each
dimension of the input $\mathbf{x}$ divides the solution space
$\mathbb{R}^{2}$ into two half-planes, one feasible and one infeasible.

One algorithm to solve the optimization problem described by Equation 4 is
gradient descent:

$\mathbf{\dot{y}}=-\frac{\partial
E}{\partial\mathbf{y}}=-\lambda\mathbf{y}-\mathbf{b}$ (5)

However, in order to ensure that $\mathbf{y}$ does not leave the feasible set
$\left\\{\mathbf{y}\mid\mathbf{Fx-Gy}\leq\mathbf{T}\right\\}$, a term
$\mathbf{Ds}(t)$ can be added to $\mathbf{\dot{y}}$ such that the new term
reflects $\mathbf{y}$ back into the feasible set whenever $\mathbf{y}$ touches
a boundary of the feasible set. Specifically, $\mathbf{s}(t)\in\mathbb{R}^{N}$
can be modeled a vector of delta-function bouncing events, while the matrix of
bounce directions $\mathbf{D}\in\mathbb{R}^{M\times N}$ can be modeled as
always bouncing $\mathbf{y}$ orthogonal to the boundary it touched (i.e.,
$\mathbf{D}\propto\mathbf{G}$):

$\mathbf{\dot{y}}=-\lambda\mathbf{y}-\mathbf{b}+\mathbf{Ds}(t)$ (6)

Connecting the quadratic problem formulation to SNNs proceeds by defining the
constraint from Equation 4 as
$\mathbf{V}=\mathbf{Fx}-\mathbf{Gy}\leq\mathbf{T}$, where $\mathbf{V,T}$
represent the neurons’ voltages and voltage thresholds:

$\displaystyle\mathbf{\dot{V}}$ $\displaystyle=\mathbf{F\dot{x}-G\dot{y}}$ (7)
$\displaystyle=\mathbf{F\dot{x}}+\lambda\mathbf{Gy+Gb-GDs}(t)$ (8)
$\displaystyle=-\lambda\mathbf{V+F}(\lambda\mathbf{x+\dot{x}})-\mathbf{GDs}(t)+\mathbf{Gb},$
(9)

where the derivation follows from using the definitions of $\mathbf{\dot{V}}$
and $\mathbf{\dot{y}}$ from Equations 3 and 6. The new formulation of
$\mathbf{\dot{V}}$ is related to Equation 3 in that the inputs
$\mathbf{c}(t)=\lambda\mathbf{x+\dot{x}}$, the recurrent connections
$\mathbf{\Omega=-GD}$, and the background currents or noise
$\mathbf{I}_{bg}(t)=\mathbf{GD}$. The resulting reformulation shows that SNNs
solve quadratic ($\lambda>0$) or linear ($\lambda=0$) problems.

##  4 Optimization

This section presents the fundamental theory and recent findings in SNN
optimization. Key challenges include performing gradient descent on non-
differentiable spiking actions and leveraging the intrinsic characteristics of
neuromorphic computation towards more efficient SNN training methods.

###  4.1 Background

Backpropagation Through Time. Two of the most prevalent methods for training
SNNs are shadow training and backpropagation through time (BPTT). The former
involves training a SNN as an approximation of a pre-trained ANN; although
such an approach is beneficial for reaching competitive performance in static
tasks such as image classification, shadow training will not be the focus of
this review due to its inherent inefficiency (requiring two networks to be
trained) and its under-utilization of the temporal dynamics of SNNs
(Eshraghian et al., 2021). The latter method (BPTT) stems from recurrent
neural network (RNN) optimization, and becomes necessary for SNN training due
to each neuron’s membrane potential acting as a form of memory as it decays
over time, unlike the activation of ANN neurons.

Following the notation of Eshraghian et al. (2021), the expression for error
as a function of SNN weights, $\frac{\partial L}{\partial\mathbf{W}}$, takes
the following form when using BPTT:

$\frac{\partial L}{\partial\mathbf{W}}=\sum_{t}\frac{\partial
L[t]}{\partial\mathbf{W}}$ (10) $\frac{\partial
L}{\partial\mathbf{W}}=\sum_{t}\sum_{s\leq t}\frac{\partial
L[t]}{\partial\mathbf{W}[s]}\frac{\partial\mathbf{W}[s]}{\partial\mathbf{W}}$
(11) $\frac{\partial L}{\partial\mathbf{W}}=\sum_{t}\sum_{s\leq
t}\frac{\partial L[t]}{\partial\mathbf{W}[s]},$ (12)

where $L[t]$ and $\mathbf{W}[s]$ are the loss and weights at their respective
timesteps $t$ and $s$. Equation 11 follows from Equation 10 since
$\mathbf{W}[s]$ directly determines $L[s]$ (referred to as an immediate
influence) while also influencing $L[t]$ for all $t>s$ (referred to as a prior
influence). Equation 11 then simplifies to Equation 12 since a recurrent
system constrains the weights $\mathbf{W}$ to be shared along all timesteps
($\mathbf{W}[0]=\mathbf{W}[1]=...=\mathbf{W}$), meaning that
$\frac{\partial\mathbf{W}[s]}{\partial\mathbf{W}}=1$.

Surrogate Gradients. Unlike ANNs, SNNs use spikes to perform computation.
Letting $\mathbf{S}$ denote spiking activity, calculation of the gradient
$\frac{\partial L}{\partial\mathbf{W}}$ takes the form of $\frac{\partial
L}{\partial\mathbf{S}}$ multiplied by additional terms (via the chain rule)
that constitute $\frac{\partial\mathbf{S}}{\partial\mathbf{W}}$. However,
since the spiking activity $\mathbf{S}$ as a function of synaptic weights
$\mathbf{W}$ involves a Heaviside step function, the derivative
$\frac{\partial\mathbf{S}}{\partial\mathbf{W}}$ contains an ill-behaved Dirac-
delta function. As a relaxation of the non-smooth spiking nonlinearity,
surrogate gradients are used as an alternative to the derivative of the
Heaviside step function during backpropagation (Neftci et al., 2019). The
choice of surrogate gradient for SNN optimization is not unique, but recent
work has shown that SNNs are robust to the choice of surrogate gradient (Zenke
& Vogels, 2021).

Figure 6: An illustration by Neftci et al. (2019) showing how surrogate
gradients can be used to smoothen the loss landscape of SNNs.

###  4.2 Advancements in Surrogate Gradient Descent

Differentiable Spike. Li et al. (2021) have recently brought attention to the
finite difference gradients (FGD) as a novel method of computing surrogate
gradients. With loss $L$ and network parameters $\mathbf{W}$, approximation of
the true gradient $\nabla_{\mathbf{W}}L$ via the FGD
$\hat{\nabla}_{\varepsilon,\mathbf{w}}L$ is defined as:

$\hat{\nabla}_{\varepsilon,\mathbf{w}}L=\frac{1}{2\varepsilon}\begin{bmatrix}L(\mathbf{w}_{1}+\varepsilon\mathbf{e}_{1})-L(\mathbf{w}_{1}-\varepsilon\mathbf{e}_{1})\\\
L(\mathbf{w}_{2}+\varepsilon\mathbf{e}_{2})-L(\mathbf{w}_{2}-\varepsilon\mathbf{e}_{2})\\\
...\\\
L(\mathbf{w}_{k}+\varepsilon\mathbf{e}_{k})-L(\mathbf{w}_{k}-\varepsilon\mathbf{e}_{k})\end{bmatrix},$
(13)

where $\textbf{w}\in\mathbb{R}^{k}$ is the flattened vector of $\mathbf{W}$,
$\mathbf{e}_{i}$ is the standard basis vector with 1 at its $i$-th index, and
$\varepsilon$ is a hyperparameter. As $\varepsilon\rightarrow 0$,
$\hat{\nabla}_{\varepsilon,\mathbf{w}}L$ becomes the true gradient
$\nabla_{\mathbf{W}}L$. Li et al. (2021) have shown that, although using very
small $\varepsilon\in[0.1,0.001]$ successfully approximates the true gradient
in artificial neural networks, SNN optimization is more sensitive to the
magnitude of $\varepsilon$, since the optimal values of $\varepsilon$ must be
low enough to faithfully approximate the characteristics of the SNN gradient
while being high enough so as to have a smoothening effect on the
discontinuous SNN loss landscape.

Li et al. (2021) then proceed to define the Differential Spike (Dspike)
surrogate gradient with temperature parameter $b$ and input $x$:

$\text{Dspike}(x,b)=\begin{cases}1,&\text{if $x>1$}\\\
\frac{\text{tanh}(b(x-0.5))+\text{tanh}(\frac{b}{2})}{2\cdot\text{tanh}(\frac{b}{2})}&\text{if
$0\leq x\leq 1$}\\\ 0,&\text{if $x<0$}\end{cases}$ (14)

The authors train each of their SNNs by, in each epoch $e$, first computing
the FDG $\hat{\nabla}_{\varepsilon,\mathbf{w}}L$ of the first SNN layer, then
choosing the value of $b\in\\{b^{e-1},b^{e-1}+\Delta b,b^{e-1}-\Delta b\\}$
(where $\Delta b$ is a step size hyperparameter and $b^{e-1}$ is the optimal
$b$ from the previous epoch) to maximize the cosine similarity between
$\hat{\nabla}_{\varepsilon,\mathbf{w}}L$ and $\nabla_{b,\mathbf{w}}L$, and
finally optimizing the entire SNN using Dspike gradient descent with the newly
optimized value of $b$.

Information-Maximization and ESG. More recently, Guo et al. (2022) have
introduced Information-Maximization Loss (IM-Loss) and the Evolutionary
Surrogate Gradient (ESG) for differentiable spike activity estimation. The
authors derive the IM-Loss by first arguing that one key reason why SNN
performance lags behind ANN performance is because of the loss of information
produced by quantizing membrane voltages into binary spiking functions.
Representing the tensor of membrane potentials as $\mathbf{V}$ and the tensor
of spiking activity as $\mathbf{S}$, the authors seek to maximize the mutual
information between $\mathbf{v}$ and $\mathbf{S}$ by maximizing the entropy of
$\mathbf{S}$:

$\operatorname*{arg\,max}_{\mathbf{V},\mathbf{S}}I(\mathbf{V};\mathbf{S})=H(\mathbf{S})-H(\mathbf{S}|\mathbf{V})=H(\mathbf{S}),$
(15)

where $H(\mathbf{S}|\mathbf{V})$ is argued to be 0 due to the binary nature of
$\mathbf{S}$. Since each element of $\mathbf{S}$ is either 0 or 1, the entropy
of $\mathbf{S}$ is maximized when each element of $\mathbf{S}$ is equally
likely to be 0 or 1. Using the observation that $\mathbf{V}$ is often Gaussian
distributed and therefore has a median equal to its mean $\mathbf{\bar{V}}$,
Guo et al. (2022) propose the IM-Loss as:

$\mathcal{L}_{IM}=\frac{1}{L}\sum_{l=0}^{L}(\bar{V}^{(l)}-V_{th})^{2},$ (16)

where $V_{th}$ is the spike activation threshold voltage. The authors’ final
loss function for SNN training was
$\mathcal{L}_{Total}=\mathcal{L}_{CE}+\lambda\mathcal{L}_{IM}$, where
$\mathcal{L}_{CE}$ is cross-entropy loss and $\lambda=2$.

Similarly to Dspike (Li et al., 2022), Guo et al. (2022) use a differentiable
spike activation function $\varphi(x)$ whose sole parameter $K(i)$ evolves as
a function of the epoch index $i\in[0,N-1]$:

$\displaystyle\varphi(x)$
$\displaystyle=\frac{1}{2}\text{tanh}(K(i)(x-V_{th}))+\frac{1}{2}$ (17)
$\displaystyle K(i)$
$\displaystyle=\frac{1}{9}((10^{\frac{i}{N}}-1)K_{max}+(10-10^{\frac{i}{N}})K_{min})$
(18)

Since $K(i)$ grows monotonically as a function of the epoch index $i$, the
Evolutionary Surrogate Gradient $\varphi(x)$ aims to improve SNN convergence
by beginning as a function with a very wide gradient (i.e., allowing many
weights to be updated) and then gradually transitioning to a function that
more closely resembles the true spiking function.

###  4.3 Leveraging Sparsity

Perez-Nieves & Goodman (2021) present a novel approximation of surrogate
gradient descent that leverages spike train sparsity for remarkable
improvements in computational efficiency and speed. The authors begin with the
assumption that neurons operate according to a simplified Leaky Integrate and
Fire model:

$V_{i}^{(l)}[t]=\begin{cases}F(V_{i}^{(l)}[t-1],x[t]),&\text{if
$V_{i}^{(l)}[t]<V_{th}$}\\\ V_{r},&\text{if $V_{i}^{(l)}[t]\geq
V_{th}$}\end{cases}$ (19)

$F$ represents the dynamics governing the voltage $V_{i}^{(l)}$ of the $i$-th
neuron of layer $l$, $V_{th}$ is a voltage threshold, and $V_{r}$ is the
resting potential of the neuron after spiking. At each time, neurons spiking
according to $S_{i}^{(l)}[t]=f(V_{i}^{(l)}[t])$, where $f(v)\in\\{0,1\\}$
equals $1$ if and only if $v>V_{th}$. The authors use gradient descent via
backpropagation through time (BPTT) with the SuperSpike surrogate gradient
$g(V)=\frac{1}{(\beta\lvert V-V_{th}\rvert+1)^{2}}$ (Zenke & Ganguli, 2018;
Zenke & Vogels, 2021). In order to make gradient optimization more efficient,
the authors introduce an approximation of the surrogate gradient:

$\frac{dS_{i}^{(l)}[t]}{dV_{i}^{(l)}[t]}=\begin{cases}g(V_{i}^{(l)}[t]),&\text{if
neuron $i^{(l)}$ is active}\\\ 0,&\text{otherwise}\end{cases}$ (20)

where the $i$-th neuron of layer $l$ is defined as active if and only if

$\lvert V_{i}^{(l)}-V_{th}\rvert<B_{th}$ (21)

By approximating the surrogate gradient as being nonzero only for neurons that
are active at any timestep $t$, the resulting derivation of BPTT yields an
optimization strategy that approximates weight updates very well, having a
backward pass speedup of up to 150x and saving up to 85% more memory compared
to vanilla BPTT on datasets such as Fashion-MNIST (Xiao et al., 2017)
(processed via latent coding), Neuromorphic-MNIST (Orchard et al., 2015), and
Spiking Heidelberg Digits (Cramer et al., 2022) (Perez-Nieves & Goodman,
2021).

###  4.4 Learning Rules

One of the most common learning rules for training SNNs is backpropagation
modeled as the reverse of a simple forward pass. However, backpropagation
methods, such as backpropagation through time (BPTT), are not biologically
plausible (Hunsberger, Eric, 2018), are memory-intensive, and are not fully
compatible with neuromorphic hardware (Zhu et al., 2022), instead
necessitating computationally intensive training on normal hardware. This
section explores recent extensions or alternatives to backpropagation.

Spike-Timing Dependent Plasticity. Another alternative to BPTT that focuses on
spike timing is spike-timing dependent plasticity (STDP), a learning framework
whose origins are not in optimization, but rather in experimental observations
of neural plasticity. Unlike event-driven optimization, STDP is unsupervised
and does not use gradient descent or backpropagation, instead determining the
synaptic connection strength between each neuron solely as a function of the
relative timings between pre- and post-synaptic neuronal spikes (Eshraghian et
al., 2021). Such learning rules are commonly known as Hebbian learning methods
(Caporale & Dan, 2008) and are characterized by the principle that neurons
that fire together wire together (Eshraghian et al., 2021).

$\Delta w_{ij}=\begin{cases}a^{+}w_{ij}(1-w_{ij}),&\text{if $t_{j}-t_{i}\leq
0$}\\\ a^{-}w_{ij}(1-w_{ij}),&\text{if $t_{j}-t_{i}>0$}\end{cases}$ (22)

One example of STDP learning rules for SNNs is shown in Equation 22, where
$a^{+}$ and $a^{-}$ are learning rates, $t_{i}$ and $t_{j}$ are the post- and
pre-synaptic spike times of two neurons, and $w_{ij}$ is the strength of
connection between the two neurons (Kheradpisheh et al., 2018). A recent
extension of this STDP is R-STDP, or reward-modulated STDP, where
reward/punishment signals are used in conjunction with traditional STDP to
train the weights (Rathi & Roy, 2021b).

Event-Driven Optimization. One potential alternative to traditional
backpropagation through time for SNNs is event-driven optimization,
specifically backpropagation with respect to spike timing (Bohte et al.,
2002). Such optimization methods involve calculating time-based gradients that
indicate whether a spike should happen earlier or later with respect to time.
Zhu et al. (2022) have recently been the first to successfully train SNNs on
the CIFAR-100 dataset using event-driven methods.

Online Training Through Time. Xiao et al. (2022) recently proposed another
alternative to traditional backpropagation through time. Their method,
referred to as online training through time (OTTT), diverges from traditional
SNN optimization by not using a surrogate gradient. Instead, the authors
leverage the fact that the gradient of spiking activity is naturally zero
almost everywhere to approximate
$\frac{\partial\mathbf{V}^{(l)}[t+1]}{\partial\mathbf{S}^{(l)}[t+1]}\frac{\partial\mathbf{S}^{(l)}[t+1]}{\partial\mathbf{V}^{(l)}[t]}\approx
0$, where $\mathbf{V}^{(l)}[t]$ and $\mathbf{S}^{(l)}[t]$ are the voltage and
spiking activity of layer $l$ at time $t$. As a result, each layer’s voltage
evolves simply as a function of its membrane leakage constant $\lambda<1$:
$\frac{\partial\mathbf{V}^{(l)}[t+1]}{\partial\mathbf{V}^{(l)}[t]}=\lambda\mathbf{I}$.
As a result, Xiao et al. (2022) are able to use a loss function and a gradient
whose values are instantaneous (i.e., only depending on the timestep $t$).
Further analysis formulates OTTT as a three-factor Hebbian learning rule, thus
building a pathway towards biologically plausible training on neuromorphic
devices.

Implicit Differentiation. As another alternative to feedforward SNNs trained
via BPTT, Xiao et al. (2021) bring to attention feedback spiking neural
networks (FSNNs), which are SNNs with recurrent connections, and propose
implicit differentiation on the equilibrium state (IDE) as a method to train
them. The authors define the equilibrium state $\mathbf{a}^{*}$ of a network
$f_{\theta}$ with parameters $\theta$ as the state where
$\mathbf{a}^{*}=f_{\theta}(\mathbf{a}^{*})$. Implicit differentiation on the
equation satisfies $(\mathbf{I}-\frac{\partial
f_{\theta}(\mathbf{a}^{*})}{\partial\mathbf{a}^{*}})\frac{d\mathbf{a}^{*}}{d\theta}=\frac{\partial
f_{\theta}(\mathbf{a}^{*})}{\partial\theta}$. Writing the objective function
evaluated at $\mathbf{a}^{*}$ as $\mathcal{L}(\mathbf{a}^{*})$, and letting
$g_{\theta}(\mathbf{a})=f_{\theta}(\mathbf{a})-\mathbf{a}$, differentiation of
$\mathcal{L}(\mathbf{a}^{*})$ with respect to the parameters $\theta$ can be
expressed as:

$\frac{\partial{\mathcal{L}(\mathbf{a}^{*})}}{\partial\theta}=-\frac{\partial{\mathcal{L}(\mathbf{a}^{*})}}{\partial\mathbf{a}^{*}}\left(\mathbf{J}^{-1}_{g_{\theta}}|_{\mathbf{a}^{*}}\right)\frac{\partial
f_{\theta}(\mathbf{a}^{*})}{\partial\theta},$ (23)

where $\left(\mathbf{J}^{-1}_{g_{\theta}}|_{\mathbf{a}^{*}}\right)$ is the
inverse Jacobian of $g_{\theta}$ evaluated at $\mathbf{a}^{*}$. Solving the
inverse Jacobian requires solving an alternative linear system
$\left(\mathbf{J}^{T}_{g_{\theta}}|_{\mathbf{a}^{*}}\right)\mathbf{x}+\left(\frac{\partial\mathcal{L}(\mathbf{a}^{*})}{\partial\mathbf{a}^{*}}\right)^{T}=0$,
which can be done using second-order quasi-Newton methods or by using a fixed-
point update scheme
$\mathbf{x}=\left(\mathbf{J}^{T}_{f_{\theta}}|_{\mathbf{a}^{*}}\right)\mathbf{x}+\left(\frac{\partial\mathcal{L}(\mathbf{a}^{*})}{\partial\mathbf{a}^{*}}\right)^{T}$,
since
$\left(\mathbf{J}^{T}_{g_{\theta}}|_{\mathbf{a}^{*}}\right)=\left(\mathbf{J}^{T}_{f_{\theta}}|_{\mathbf{a}^{*}}\right)-\mathbf{I}$
(Xiao et al., 2021). Updating the parameters $\theta$ can then occur via
gradient descent based on $\frac{\partial\mathcal{L}}{\partial\theta}$.

##  5 Energy Efficiency

This section focuses on how SNNs garner impressive energy savings, covering
the mathematical underpinnings as well as documenting notable gains in memory
and complexity reduction.

###  5.1 Low-Power Applications and Constraints

SNNs hold particular promise for low-power and implantable applications, such
as brain-machine interfaces (BMIs), where there exists a strict power
requirement: chips directly in contact with brain tissue cannot emit more than
10 mW of energy before permanently damaging brain cells by heating them by
over 1 degree Celsius (Dethier et al., 2011). SNNs promise to reduce energy
consumption and thus heat output, making them a promising choice for implants
(Liao et al., 2022). However, one can only benefit from SNNs when running on
hardware built to take advantage of the inherent benefits of SNNs. such as
their sparsity and inclination for parallel computation. Neuromorphic
processors deviate from the traditional von Neumann architecture of today’s
general purpose CPUs by employing massive parallelism, asynchronous event-
driven operations, and a more highly-distributed, easily-accessible memory
that reduces the number of computations—and thus, the energy
consumption—required for information processing (Guo et al., 2021). As found
by Dethier et al. (2011), neuromoprhic hardware implementations can result in
SNNs which consume as little as 0.1 mW, well below the aforementioned maximum
power threshold of 10 mW for BMIs.

###  5.2 Mechanisms for Energy Savings

Reduced Activity. Energy consumption has been shown to be reduced by up to 3
orders of magnitude when compared to conventional ANN implementations, largely
due to spiking activity, sparsity, and static data suppression (event-driven
processing) (Ceolini et al., 2020) (Davies et al., 2021). Neurons producing
discrete spikes in sparsely-connected networks result in spatio-temporally
sparse activity, where activity can be defined as the following:

$A=\frac{100a}{BTN},$ (24)

where $a$ is the number of active neurons, $B$ is the batch size, $T$ is the
total number of time steps, and $N$ is the number of neurons in the layer
(Perez-Nieves & Goodman, 2021). As one would intuitively guess, the level of
activity is inversely proportional to the sparsity of the inputs: as more
neurons become active such that the fraction of neurons firing, $a/N$,
increases, the network is by definition less sparse. The energy-efficiency of
spike-based computation becomes especially apparent when considering that only
a fraction of neurons in real brains use rate-coding, whereas all others opt
for much sparser representations of activity (Olshausen & Field, 2006).

Estimating Computational Expenditure. Computational energy for neural networks
is typically assessed in two ways: the number and type of operations
performed, and the frequency of memory access. For the former, many papers
have quantified the energy cost of AC (accumulation) and MAC (multiply and
accumulate) operations for various CMOS processors. In ANNs, the summation of
the weighted inputs to a neuron require a MAC operation (e.g. one MAC
operation per input), whereas for SNNs, spikes rely on the much cheaper AC
operation per input. For comparison, one conservative estimate is that three
AC operations equate to one MAC operation (Liao et al., 2022). Note that SNNs
need to update their membrane potentials every time step, accounting for an
additional MAC operation per each neuron, as opposed to ANNs where there is a
MAC operation per input to each neuron.

Memory Bottleneck. As for memory, the continual increase of neural network
parameter complexity produces an increasing difficulty of storing everything
on-chip: when accessing memory that is further away (for example, there is a
substantial jump in delay for accessing larger memory bases; instead of
requiring data from the CPU’s cache, having to read from RAM, or the SSD), the
time and energy required grows, thus presenting a key bottleneck in the
implementation of all neural network architectures (Sharma et al., 2022). Many
popular learning methods in today’s SNNs (e.g., BPTT variants) are
biologically implausible, since they require access to past states to use as
inputs (i.e., data must be pulled from memory): thus, the memory consumed
scales with time, limiting BPTT to small temporal dependencies (Eshraghian et
al., 2021). Another example of this hardware bottleneck is that, in dense back
propagation situations, there can only be a few thousand time steps, as
gradients must be computed at each of those time steps. Notably, many of these
gradients have little effect on the output; Perez-Nieves & Goodman (2021)
present a backpropagation technique that only computes the gradients of active
neurons (i.e., only calculates the gradients that meaningfully affect the
output), which facilitates training on GPUs, whereas traditional BPTT runs out
of memory before it can converge. As Perez-Nieves & Goodman (2021) point out,
this effectively means that such SNN implementations can be trained with more
data for a longer amount of time when compared to previous generations of
SNNs, given the same computing resources.

###  5.3 Observed Energy Efficiency Gains

An observed issue with reporting the energy efficiency or decrease in
memory/energy consumption by SNNs throughout the literature was that there
were no standardized measurements for quantifying the energy savings offered
by SNNs. A number of works reported the number of estimated MAC and AC/add
operations that their implementation required, frequently contrasted against
the number of said operations required for ANN equivalents; while this method
is simpler to estimate on the researcher’s behalf, it remains unstandardized,
as different researchers estimate the number of operations in different ways,
and the number of operations is fundamentally a proxy for true energy
consumption. As for the works that do report the expected energy consumption,
different teams make different assumptions about how many AC operations are
equal to one MAC operation, and thus introduce a wide range of reported energy
savings. We recommend that researchers take the conservative approach, using
the methodology suggested by Horowitz (2014) of using 3 additions as the
equivalent to 1 MAC operation. Note that while this is of course a proxy for
actual energy consumption, the amount of energy consumed is also dependent on
the actual hardware used to train and deploy the model.

HIRE-SNN. HIRE-SNN VGG11 (a direct-coded SNN) compared against a rate-coded
SNN and an ANN achieved 4.6x and 10x less energy consumption, respectively
(Kundu et al., 2021).

Sparse Spiking Gradient Descent. Perez-Nieves & Goodman (2021) introduced a
method that speeds up backward execution by up to two orders of magnitude, and
thus they expected that this method would likewise reduce the energy
consumption by a similar factor. By constraining back propagation to active
neurons, 98% of the neurons (i.e., 98% of the gradient calculations) can be
skipped. Such accelerations yield much faster back propagation times, lower
memory requirements, and less energy consumption on GPU hardware while not
affecting test accuracy. These laudable improvements are possible because the
surrogate gradient used to approximate the derivative of the spiking function
is larger when the membrane potential is closer to the threshold, resulting in
most of the gradient being concentrated on active neurons (Perez-Nieves &
Goodman, 2021).

SNN Brain-Machine Interface. Liao et al. (2022) compared ANNs and SNNs for an
implantable BMI that decodes finger velocity. They found that an ANN required
529K total operations and 2116K memory accesses, and an SNN trained to
approximate the ANN similarly required 293K total operations and 2615K memory
accesses. However, the authors’ proposed SNN required just 36K total
operations and only 199K memory accesses. Their proposed SNN produced
reductions of 95% and 90% in the number of total operations required, and 91%
and 93% in the number of memory accesses, respectively (Liao et al., 2022).

##  6 Best Practices

This section aims to present the best practices in SNN training and
evaluation: widely used benchmarks, baseline standards, architectures, and
frameworks.

###  6.1 Benchmarks/Datasets

Background. Throughout the development of ANNs, certain datasets—such as the
MNIST (Deng, 2012) and CIFAR datasets (Krizhevsky & Hinton, 2009)—have served
as reliable benchmarks for all researchers to compare against. The development
of SNNs has come as a subset of the growing neuromorphic engineering
landscape, wherein advances made in software and hardware must be accompanied
by advances in benchmarks such as standardized neuromorphic datasets. A key
prerequisite to SNN implementation is to encode the input data as spikes:
thus, event-based sensors (such as the ATIS vision sensors) that record pixel-
level spike-equivalents of the data in the form of inherently neuromorphic
datasets have gained traction, as have software conversions of traditional
datasets into neuromorphic form. Note that, as discussed earlier, the precise
mechanism through which to encode data as binary spike trains is not unique,
with the primary two modes of doing so being rate-coding and latency-coding.

Neuromorphic Datasets. Many imaging datasets now have neuromorphic
counterparts, wherein the changes in intensity for each pixel (measured from 3
vantage points via sensors) are represented as spikes occurring at the given
pixel locations (Iyer et al., 2021). The most common neuromorphic datasets in
the literature include the N-MNIST dataset (spiking version of the classic
MNIST dataset), DVS Gestures (video data of hand gestures recorded under
different lighting conditions), and SHD (spiking version of the Heidelberg
Digits in the form of audio data converted via a simulated cochlear model). A
number of other neuromorphic vision/imaging datasets exist, such as ASL-DVS,
DAVIS, MVSEC, POKER DVS, and DSEC, and other common audio datasets simulating
cochlear activity include SSC and N-TIDIGITS (Eshraghian et al., 2021).

Non-Neuromorphic Datasets. Despite the proliferation of neuromorphic datasets,
they remain niche and unstandardized, indicating that many practitioners
prefer to convert non-neuromorphic datasets to spiking inputs themselves. That
said, there are a number of commonly-used datasets, particularly the MNIST and
CIFAR-10 datasets. Of the total benchmarks reviewed, 26.6% of all models were
compared using the MNIST dataset (33.1% when including MNIST variants such as
N-MNIST and F-MNIST), and CIFAR-10 was the second most common with 30.5% of
models being benchmarked against it. Notably, while more papers choose to use
MNIST, the papers reviewed that used CIFAR tended to have many more models per
paper. It is important to acknowledge that many SNN implementations utilized
in neuroengineering applications are trained and deployed directly onto
private collected data (for instance, EEG data collected from an implant
within a rhesus monkey) as opposed to a standardized benchmark, and thus these
were not included in the prior calculations.

Recommendations. We recommend that aspiring researchers use either the MNIST
dataset or CIFAR-10, as they remain the most common and thus have the
benchmarking network effect. However, in a further effort to standardize
results across the field, we also recommend using the publicly available
N-MNIST or DVS Gestures datasets, as these would help ensure repeatability
since these latter two datasets are already converted to a neuromorphic input
form.

Baselines. One primary result of our literature review is a compilation of
some of the latest state of the art baselines in SNN optimization, which we
have compiled in Table 1.

###  6.2 Architectures and Frameworks

Software. There exists a plethora of development options for implementing
SNNs, each with subtle differences, but many attempting to fill the same
niche. Common frameworks include BindsNET, Norse, Nengo, SpykeTorch, and
SNNToolBox. BindsNET is built on top of PyTorch as is focused on simulating
SNNs for general machine learning and reinforcement learning purposes (Hazan
et al., 2018). Norse similarly expands PyTorch with SNNs (El-Allami et al.,
2020). Nengo is built on top of Keras/TensorFlow, and has specific modules for
deep learning and simulation backends (e.g. on OpenCL, FPGAs, Loihi)
(Rasmussen, 2019). SpykeTorch is also built from PyTorch, and is focused on
simulating convolutional SNNs with one spike per neuron, rank-order
information encoding, and learning via STDP or R-STDP (Mozafari et al., 2019).
Finally, SNNToolBox (SSN-TB) is a framework solely focused on transforming
traditional ANNs to SNNs and supports a number of different encoding schemes:
SNN-TB can be used with models from Keras/TF, PyTorch, and other popular deep
learning libraries, and provides an interface for both simulation backends
(pyNN, brian2, etc.) and deployment (via SpiNNaker or Loihi) (Rueckauer et
al., 2017). Many modern machine learning frameworks do not offer ways to
accelerate the convolution of binary spike activation, and furthermore
struggle to implement custom continuous-time differential expressions given
the inherent discrete execution of their built-in operations (Li et al., 2022)
(Guo et al., 2022).

Hardware. Another equally important aspect of SNN implementation is the
hardware itself: when implemented on non-neuromorphic hardware (i.e.,
traditional von-Neumann architectures), the benefits of SNNs are less
pronounced since the hardware is physically incapable of taking advantage of
the massive parallelism offered by SNNs. By far, the most common hardware for
SNNs is the Loihi chip by Intel (whose successor, the Loihi 2, was recently
released in late 2021). The Loihi is a neuromorphic many-core processor that
supports on-chip learning, making it a viable option for both inference and
deployment. The Loihi has 128 cores, simulates up to 131,072 neurons with
130,000,000 synapses possible; furthermore, each Loihi can be put in parallel
with up to 16,384 other chips, reaping the benefits of massive parallelism by
allowing the number of effective on-chip cores to be 4,096 (Davies et al.,
2018). Another common implementation tool for SNNs is SpiNNaker, a million-
core, open-access, ARM-based neuromorphic computing platform developed and
housed in the University of Machester: SpiNNaker is used for simulation and
testing for applications that do not require on-site implementations (Furber
et al., 2014).

Architectural Techniques. The final, but just as important, subsection of
methodologies required to enable SNNs to reach their potential as universal
approximators (Zhang & Zhou, 2022) consists of how practitioners can best
structure, initialize, or normalize the connections between spiking layers to
maximize performance. As a prominent example, Fang et al. (2021) have recently
developed a spike-element-wise (SEW) residual network that faithfully
implements key properties of ANN residual networks (He et al., 2016) in SNNs:
namely, the ability to learn an identity function and the ability to train
very deep SNN architectures (over 100 layers) by virtue of mitigating
vanishing/exploding gradient problems. New work (Rossbroich et al., 2022) in
SNN initialization has extended weight initialization methods that target a
specific variance for neuronal activity—such as Glorot and He initialization
(Glorot & Bengio, 2010; He et al., 2015)—to spiking networks. Specifically,
Rossbroich et al. (2022) present a new SNN initialization formula that
surpasses He initialization for deep SNNs (i.e., more than 7 layers) by simply
targeting a specific membrane potential variance. As for SNN normalization,
one of the more fundamental works is threshold-dependent batch normalization
(tdBN) (Zheng et al., 2021), which extends traditional batch normalization
(BN) (Ioffe & Szegedy, 2015) to the additional temporal dimension brought by
SNNs and makes the normalized pre-activation variance dependent on the firing
threshold $V_{th}$. Newer developments include batch normalization through
time (BNTT) (Kim & Panda, 2021) and temporal effective batch normalization
(TEBN) (Duan et al., 2022), which both aim to mitigate temporal co-variate
shift of spiking activity.

##  7 Conclusion

Our review has covered the fundamental principles and the cutting-edge methods
for reaching competitive performance in spiking neural networks (SNNs) while
also improving SNN memory and energy efficiency. This literature search has
identified the most common benchmarks for aspiring SNN practitioners to use,
the current state of the art performances on those benchmarks, and the
trajectories of recent advances in SNN optimization. SNNs offer an exciting
new technology, particularly for low-power, implanted, mobile, or other
hardware-constrained applications, such as brain-machine interfaces and
wearable technologies. Our understanding of SNNs as the third generation of
neural networks continues to grow; with such growth comes not only the
potential to improve neural network performance while reducing energy costs,
but also the potential to further develop our understanding of neuroscience
and neural computation.

## Acknowledgements

We thank Anastasios Kyrillidis for his guidance and support during the writing
process.

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##  Appendix A You _can_ have an appendix here.

You can have as much text here as you want. The main body must be at most $8$
pages long. For the final version, one more page can be added. If you want,
you can use an appendix like this one, even using the one-column format.

##  Appendix B Performance Comparisons

Model |  Architecture |  Learning Method |  Training Framework |  Dataset |  Accuracy
---|---|---|---|---|---
(Perez-Nieves & Goodman, 2021) |  3 layer SNN |  Backpropagation |  Pytorch CUDA Extension |  SHD |  77.5
(Perez-Nieves & Goodman, 2021) |  3 layer SNN |  Sparse Backpropagation |  Pytorch CUDA Extension |  SHD |  71.7
(Diehl & Cook, 2015) |  2 layer SNN |  Unsupervised |  BRIAN |  MNIST |  95.0
(Liu & Yue, 2017) |  3 layer SNN |  Unsupervised |  BRIAN |  MNIST |  81.9
(Kulkarni et al., 2017) |  3 layer SNN |  Supervised |  BRIAN |  MNIST |  98.0
(Kheradpisheh et al., 2018) |  6 layer SNN |  Supervised |  BRIAN |  MNIST |  98.4
(Guo et al., 2021) |  2 layer SNN |  TTFS Encoding, STDP |  Not Listed |  MNIST |  88.6
(Guo et al., 2021) |  2 layer SNN |  Phase Encoding, STDP |  Not Listed |  MNIST |  88.2
(Guo et al., 2021) |  2 layer SNN |  Burst Encoding, STDP |  Not Listed |  MNIST |  88.4
(Guo et al., 2021) |  2 layer SNN |  Rate Encoding, STDP |  Not Listed |  MNIST |  87.5
(Rathi & Roy, 2021a) |  2 layer SNN |  Unsupervised STDP |  BRIAN |  MNIST |  93.2
(Rathi & Roy, 2021a) |  4 layer SNN |  Unsupervised STDP |  BRIAN |  MNIST, TI46 |  98.0
(Perez-Nieves & Goodman, 2021) |  3 layer SNN |  Backpropagation |  Pytorch CUDA Extension |  N-MNIST |  92.7
(Perez-Nieves & Goodman, 2021) |  3 layer SNN |  Sparse Backpropagation |  Pytorch CUDA Extension |  N-MNIST |  97.4
(Perez-Nieves & Goodman, 2021) |  3 layer SNN |  Backpropagation |  Pytorch CUDA Extension |  F-MNIST |  82.2
(Perez-Nieves & Goodman, 2021) |  3 layer SNN |  Sparse Backpropagation |  Pytorch CUDA Extension |  F-MNIST |  80.1
(Perez-Nieves & Goodman, 2021) |  6 layer SNN |  Sparse Backpropagation |  Pytorch CUDA Extension |  F-MNIST |  82.7
(Perez-Nieves & Goodman, 2021) |  CSNN |  Backpropagation |  Pytorch CUDA Extension |  F-MNIST |  86.9
(Perez-Nieves & Goodman, 2021) |  CSNN |  Sparse Backpropagation |  Pytorch CUDA Extension |  F-MNIST |  86.7
(Guo et al., 2021) |  2 layer SNN |  Rate Encoding, STDP |  Not Listed |  F-MNIST |  68.3
(Guo et al., 2021) |  2 layer SNN |  TTFS Encoding, STDP |  Not Listed |  F-MNIST |  71.3
(Guo et al., 2021) |  2 layer SNN |  Phase Encoding, STDP |  Not Listed |  F-MNIST |  71.4
(Guo et al., 2021) |  2 layer SNN |  Burst Encoding, STDP |  Not Listed |  F-MNIST |  71.3
(Salaj et al., 2020) |  CSNN + RSNN |  BPTT |  Not Listed |  DVS |  97.1
(Kaiser et al., 2020) |  CSNN |  Surrogate GD |  Not Listed |  DVS |  97.5
(Kaiser et al., 2020) |  CSNN |  Surrogate GD |  Not Listed |  DVS |  97.8
(Sironi et al., 2018) |  SVM |  Time Surfaces + SVM |  Not Listed |  DVS |  95.2
(She et al., 2021) |  CSNN |  STDP |  Not Listed |  DVS |  96.2
(She et al., 2022) |  mMND (SNN) |  STDP |  Not Listed |  DVS |  96.6
(She et al., 2022) |  mMND (SNN) |  BPTT |  Not Listed |  DVS |  98.0
(Wade et al., 2010) |  3 layer SNN |  Supervised |  BRIAN |  TI46 |  95.3
(Zhang et al., 2015) |  Reservoir SNN |  Supervised |  BRIAN |  TI46 |  99.8
(Rathi & Roy, 2021a) |  3 layer SNN |  Unsupervised STDP |  BRIAN |  TI46 |  96.0
(Li et al., 2022) |  ResNet-19 |  DSpike |  Not Listed |  CIFAR-10 |  94.2
(Guo et al., 2022) |  ResNet-19 |  IM-Loss, ESG, & tdBN |  Not Listed |  CIFAR-10 |  95.4
(Guo et al., 2022) |  VGG-16 |  IM-Loss, ESG, & tdBN |  Not Listed |  CIFAR-10 |  93.8
(Guo et al., 2022) |  CIFARNet |  IM-Loss, ESG, & tdBN |  Not Listed |  CIFAR-10 |  92.2
(Kundu et al., 2021) |  VGG5 |  Not Listed |  Pytorch |  CIFAR-10 |  87.5
(Kundu et al., 2021) |  ResNet12 |  Not Listed |  Pytorch |  CIFAR-10 |  90.3
(Kundu et al., 2021) |  ResNet12 |  Not Listed |  Pytorch |  CIFAR-100 |  58.9
(Kundu et al., 2021) |  VGG11 |  Not Listed |  Pytorch |  CIFAR-100 |  65.1
(Li et al., 2022) |  ResNet-19 |  DSpike |  Not Listed |  CIFAR-100 |  72.2
(Guo et al., 2022) |  VGG-16 |  IM-Loss, ESG, & tdBN |  Not Listed |  CIFAR-100 |  70.1
Table 1: Summary of common architectures, learning methods, frameworks,
datasets, and achieved accuracies.


channels and local measurements. However, broadcasting is a rather expensive
resource. In realistic laboratory settings, we might have access to only
restricted sets of broadcasting channels and local measurements. It is thus
crucial to find the most fruitful local measurements for the BLMPP protocol in
realistic restricted settings. It seems likely that, in general, rank-I PVMs
might no longer be the best local measurements one could use, but this
requires further study.

## Acknowledgements

This work is funded by the National Research Foundation, Prime Minister’s
Office, Singapore and the Ministry of Education, Singapore under the Research
Centres of Excellence programme.

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