LTMD: Learning Improvement of Spiking Neural Networks with Learnable Thresholding Neurons and Moderate Dropout
Authors: Siqi Wang
Tee Hiang Cheng
Meng-Hiot Lim
Venue: NeurIPS 2022 Accept (2022)
Keywords: Spiking neural networks
SNN
Spiking neuronal mechanism
SNN learning enhancement

Abstract: Spiking Neural Networks (SNNs) have shown substantial promise in processing spatio-temporal data, mimicking biological neuronal mechanisms, and saving computational power. However, most SNNs use fixed model regardless of their locations in the network. This limits SNNs’ capability of transmitting precise information in the network, which becomes worse for deeper SNNs. Some researchers try to use specified parametric models in different network layers or regions, but most still use preset or suboptimal parameters. Inspired by the neuroscience observation that different neuronal mechanisms exist in disparate brain regions, we propose a new spiking neuronal mechanism, named learnable thresholding, to address this issue. Utilizing learnable threshold values, learnable thresholding enables flexible neuronal mechanisms across layers, proper information flow within the network, and fast network convergence. In addition, we propose a moderate dropout method to serve as an enhancement technique to minimize inconsistencies between independent dropout runs. Finally, we evaluate the robustness of the proposed learnable thresholding and moderate dropout for image classification with different initial thresholds for various types of datasets. Our proposed methods produce superior results compared to other approaches for almost all datasets with fewer timesteps. Our codes are available at https://github.com/sq117/LTMD.git.

LTMD: Learning Improvement of Spiking Neural Networks with Learnable Thresholding Neurons and Moderate Dropout
Siqi Wang*
Tee Hiang Cheng
Electrical and Electronic Engineering Nanyang Technological University Singapore siqi0020e.ro ntu. htw.:02eeenuu.uuu.eee. edu.
Electrical and Electronic Engineering Nanyang Technological University Singapore ethchengentu.edus edw.::tttuuuuuuuggg....e
Meng-Hiot Lim
Electrical and Electromic Engineering Nanyang Technological Univensity Singapore emblim@ntu.edu..
Abstract
Spiking Neural Networks (SNNs) have shown substantial promise processing spatio-temporal data, mimicking biological neuronal mechanisms, and saving com- putational power. However, most SNNs use fixed model regardless of their locations in the network. This limits SNNs' capability of transmitting precise information in the network, which becomes worse for deeper SNNs. Some researchers try to specified parametric models in different network layers regions, but most still use preset or suboptimal parameters. Inspired by the neuroscience observation that different neuronal mechanisms exist in disparate brain regions, propose new spiking neuronal mechanism, named learnable thresholding, to address this issue. Utilizing learnable threshold values, learnable thresholding enables flexible neuronal mechanisms across layers, proper information flow within the network, and fast network convergence. In addition, propose moderate dropout method to serve as an enhancement technique to minimize inconsistencies between inde- pendent dropout runs. Finally, we evaluate the robustness of the proposed learnable thresholding and moderate dropout for image classification with different initial thresholds for various types of datasets. Our proposed methods produce superior results compared to other approaches for almost all datasets with fewer timesteps. Our codes are available at /nttt::///eee /tt///gttttu com/ http///11111/MMMeeppe git.
Introduction
Modelled after the impulse communication between biological neurons, spiking neural network (SNN) is class of neural network with neurons exhibiting distinctive binary output property. SNNs have shown great potential in event-driven data processing, computation reduction, and network biological plausibility enhancement. However, unoptimized learning algorithm and complex neuronal dynamics make it challenging to construct high-performance SNN models.
Current SNN training approaches can generally be divided into two categories: ANN-to-SNN conversion and SNN direct training algorithms based on gradient descent. Both ways only focus
"corresponding author
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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on adjusting synaptic connections to minimize the enror between a model's output and target values with the same neuronal model being used across the whole network. However, has been found from neuroscience observations that biological neurons' dynamics vary with their relative locations in the brain especially for primates [1.) Therefore, using the same neuronal model across the whole SNN may limit the SNN's s expression ability as compared to allowing heterogenous neuronal models to be used in the same SNN. Although some researchers have proposed many techniques to balance presynaptic inputs and neuronal behaviors, for example, threshold regularization [ and spike-based normalization E, they are either targeting at neuronal state distribution or only applicable for ANN-to-SNY conversion algorithms.
Dropout 6 a commonly used regularization methodology to train neural networks. As dropping neurons are randomly selected, each run of network will lead to the input interacting with different sub-models, which will cause the results in different runs to deviate unpredictably. Consequently, regularization technique is needed to minimize the inconsistencies between output probability distributions during the training phase to improve the performance.
In this paper, we propose a new learnable thresholding mechanism with a moderate dropout method to enhance the learning of modulated SNNs. The learnable thresholding mechanism can be integrated with the backpropagation-basee SNN direct training algorithm so that neuronal parameters can be changed dynamically and self-optimize during training. We adapt the DenseNet architecture and modify the encoding and decoding layers to make the network suitable for diverse datasets. A new 'moderate dropout' technique developed to minimize inconsistencies between network sub-models generated in different runs. We demonstrate the proposed methods can enhance the SNN's stability and performance through evaluation based on static and neuromorphic datasets. To the best of our knowledge, our proposed SNNs achieve the best or comparable accuracies for all the datasets tested compared to other state-of-the-art SNNs.
2 Related Work
2.1 ANN-to-SNN Conversion
aims convert trained ANNs to SNNs by using rate coding scheme instead of ReLU activation to represent data flowing inside the network (9. An ANN of the same structure will be trained first, followed by adjusting the trained synapses for spatio-temporal data. Many constraints such as bias term needs to be excluded, and only average pooling should be used, etc. need to be incorporated in the pre-training [IO]. However, this indirect training approach poses lot of problems, such as temporal information is disregarded during ANN training, long delay incurred for encoding presynaptic inputs, etc., which seriously limits the network's application spectrum. Furthermore, this training method is inappropriate for neuromorphic datasets since only spatial data can be utilized.
ANN-to-SNN conversion
2.2 Direct-Trained Deep SNNs and Neuronal Model
The other method is to train SNNs directly based on gradient descent by backpropagating errors [I] in both spatial and temporal dimensions. Various strategies have been proposed to realize gradient value calculation for non-differentiable spike activation. Some recent works apply pseudo derivative algorithm to replace the non-differentiable spike triggering part of membrane potential curves with an auxiliary function and report satisfactory results [12
13 14.. As both spatial and temporal information are used in network training, dataset restrictions will not blight direct training method as in the ANN-to-SNN conversion approach. In addition, simulation latency can be significantly shortened because more efficient data encoding schemes can be freely used with gradient descent.
The direct supervised learning methods train SNNs based on event driven data and gradient descent, which scalable and thus suitable for deep SNNs. In [1]] IO, 16]. several approaches to build deep SNN with direct learning methods are described. It is pointed out in [9] that information expressiveness loss attributes to step functional activation causes deep SNNs to suffer performance
degradation. Multiple deep network architectures such as VGG [17] ResNet [8]] 9. and DenseNet [19] have been leveraged to SNNs so that spiking neurons can maintain sufficient information for transmission.
2

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Timkel
Mesters
C
230
30
130
303
480
180
80
(a) Information transmission between neurons.
(b) Potential responses with different thresholds.
Figure 1: Illustration of (a) spiking neuron membrane potential update in LIF model. (b) neuron
responses to the same input when using different threshold values.
Spiking neuronal model plays a crucial role controlling data transmission in SNNs. Inspired
by biological neurons, some researchers add lateral effect to spiking neurons [20] 13 Some use
parametric neuronal models for different tasks by making time constant be learnable (2 22 23]. One group proposes a network which contains neurons of adaptive thresholds [24] and another
group makes the potential leaky process to be learnable [25). Better accuracies and higher network robustness are demonstrated with these neuronal model improvements.
2.3 Dropout
One critical issue in using backpropagation to update synaptic connections is the gradient vanishing and exploding problem especially for deep networks. To reduce the degradation effect, numerous optimization techniques such as regularization (3 and normalization [4] [6] have been designed.
Dropout is one of the widely applied regularization methods to prevent overfitting in deep networks. Neurons will be retained randomly with certain probability and form multiple sub-models for training, after which the trained network will be generated based on all these sub-models. Some researchers modify dropout to make it suitable for spatio-temporal data and successfully apply it on SNNs. However, as spiking neurons have both spatial and temporal connections, this results in sub-models to exhibit high level of inconsistencies in different sub-model runs.
3 Methods
In this section, we introduce the learnable thresholding mechanism and moderate dropout and expound the proposed SNN.
3.1 Leaky Integrate-and-Fin Model with Learnable Thresholding
Spiking neurons the basic elements for information processing and transmission via alternating the membrane states in SNNs. Many neuronal models have been proposed by neuroscientists to simulate biological neurons' behaviors computers. Due to the complexity of neuronal dynamics in real nervous systems, there exists trade-offs between biological plausibility and computational cost in computer simulation. The Leaky Integrate-and-Firg (LIF) is simple and pre-eminent mathematical model for modelling such neuronal behaviors as potential update, spike emission, and state reset. can be described as
du(t) = u(t) + (t))
(I)
di
where is time constant which represents how fast neuron's 's potential will decay with time, u(t) and (t) are the membrane potential and integrated presynaptic input at time f, respectively, this
work, we set neurons' initial states and reset potentials to be zero. Fig. 1a illustrates neuron's
different behaviors under presynaptic inputs.
To suit computer simulation, we need to temporally discretize the LIF model and spiking neurons' states. The following numerical representation of a spiking neuron's 's potential states can derived from Eq.
1. u kut-
1-1.1 (1
of (-1, +
W Wyo fn-1
(2)
jEl(n-1)
0 1.
flu!"))
(3)
3

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where un is the membrane potential of postsynaptic neuron located in layer n. at time t, K is the decay factor, oj of:"11 is the output spike from presynaptic neuron j. and ((n 1) represents all presynaptic neurons connected to target i. f(u) is the step functional activation to determine if the target neuron will fire at current timestep, and [()) = 1 when u 2 Via. otherwise [(u) = 0. Eq. describes the membrane potential update and Eq. introduces how spiking neurons emit new spikes. In the LIF model, a postsynaptic neuron's 's state in the previous timestep and presynaptic inputs from preceding layers -pmpoolly govern the target neuron's behaviors.
As SNNs need to use backpropagation to update their parameters, we need to cireumvent the obstacle of nondifferentiable activation function f(u). Hence, we adopt the idea of pseudo derivative and designate an auxiliary rectangular function to serve as the approximated gradient:
('( {0,
1, u Vial < 0.5 otherwise
By doing so, SNNs is endowed with the parameter optimization capability through backpropaga- tion.
Our learnable thresholding mechanism is developed on the basis of LIF model by specifically considering the diversities among spiking neurons even if they are in the same network. This means neuron's response depends not only its internal state but also the threshold level. In order to investigate the effects of different threshold values on neuronal behaviors, we assume that two presynaptic neurons are connected with one postsynaptic neuron through synapses of weight wi and Wy, respectively. When the postsynaptic neuron receives input spikes from preceding layers, its internal state will change accordingly based the weighted input spike trains and the preset thresholds. As illustrated in Fig. la two presynaptic neurons are triggered at t = (40, 160) and {200, 240} respectively. According Eq. 2 the weighted input to postsynaptic neuron is [tt) = w1 (ō(t 40) + 6(t - 160)) + wy (8(t 200) + +8t - 240)). where d(t) represents the Dirac delta function. Fig. 1b depicts the postsynaptic neurons' responses to the same input [(t) when using different threshold values, showing a trend of decreasing output spike number and lower system sensitivity to the external stimulations as threshold value becomes higher. The high sensitivity to the input helps the system to track small and instant input signals while low sensitivity can enhance the system's 's noise tolerance. Thus, considering the effects of various threshold levels spiking neurons' behaviors, our learnable thresholding mechanism can achieve optimal sensitivity using the following modified neuron response function:
-
- g(u M
(5)
where M is the threshold value for neurons in the nthh layer. g(u, m) f(u m). and f(x) is the same step function with zero valued threshold and follows Eq. [
The main idea of learnable thresholding is to analytically find the comprehensive gradient of the loss function, VL(W, .M) = Low BM
BC ac J. and then simultancously update synaptic connections W and the neuron's threshold M until convergence. From Eq. and partial derivatives of the loss function ac aw and ac can be calculated by 8M7
Oc oW"
T
OC 0 Outin Jotn
(6)
Outin OWn
t=l
dC
OC 0
dM
Do aMa
aa The term aM" in Eq. represents the derivative of a neuron's 's output with respect to its threshold.
- (u -M") f' (u M whose value is determined by the auxiliary function's 's gradient defined Eq. It can be seen from the above formulae that there no restriction on the threshold value M. If the threshold becomes extremely low or high, a neuron will keep firing or stay silent when receiving the input from presynaptic neurons, causing difficulties in transmitting information. To ensure threshold values to stay within an appropriate region, we create new parameter i to define M using hyperbolic tangent relation, formulated as M tanh (k). This stabilizes neuronal activity by avoiding having too many too few neurons to fire due to extreme
It can be shown that aar=

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threshold levels. With this, Eq. 7 can be expressed as:
OC
T
Oc Oo OM
T
OL do f' tanh (1*)) sech?
2 (A*)
(8)
Dk
d RM Dke
With this enhancement, a neuron's 's threshold M can be trained iteratively using the backpropagation method and takes a value within the range (-1,1). Furthermore, the gradient of the hyperbolic tangent function becomes very small when 1 becomes infinitely large or small, ensuring that there will not be any sudden big change on the threshold value and thus achieves the stabilization effect. Although some recent works have treated threshold level as variable to introduce heterogeneity to SNN, grim prerequisites such as neuronal states transformation [26] or pre-defined firing count [27] are required. Some other STDP-based approaches only work on excitatory neurons (28), leading to slow convergence and heavy computational load to individually update every neuron's 's threshold. However, the proposed strategy provides an easy approach grant neuronal thresholds with learning capability under light computational load and can be freely migrated to other gradient descent-basee SNNs, which exactly the aim of this paper.
Though setting a lower threshold level may have a similar effect as having higher synaptic weights, changing threshold values has other benefits, for example, the potential decay curve will not be stretched and the SNN will not overly depend on a subset of synapses for optimization. In this work, we apply the learnable thresholding mechanism to all neurons and let neurons in the same layer share the same threshold to reduce total learnable parameters incurred.
3.2 Input Encoding and Network Architecture
Since SNNs are inherently suitable for processing event-driven data, spike trains in neuromorphic datasets can be directly fed into SNNs without barrier. However, the situation becomes challenging for static datasets due to lack of temporal information. Rate coding scheme is widely applied in ANN-to-SNN conversion and some recent works (29] (6] aim to encode input into serial spikes generated with certain firing probability that is proportional to the original value. One spiking neuron can encode maximally T + values into distinguishable spike trains of firing probabilities {0, 4,. ..... 1} within simulation window T. Therefore, the rate coding scheme suffers long latency due to the inevitably long simulation period for maintaining high precision among inputs of a wide range. In this work, the first layer serves as the encoding layer that directly receives input data and then, converts data into spike signals before transmitting to the second layer. In other words, there is no value-to-spike pre-processing step, inputs from dataset can be liberally fed into the network even if they are not in spike forms. In this way, we can significantly shorten the simulation window, which makes possible apply our SNN for analysis within several timesteps.
DenseNet is proposed in 19 where network layers are densely connected so that data from preceding layers can be transmitted to subsequent layers directly through dense connections between neurons. In this work, we leverage the idea and build our SNNs with DenseNet structure to retain information from degradation caused by the step activation. Our proposed learnable thresholding method is applied to all neurons in the network and the normal dropout in fully-connected layers is replaced by our moderate dropout. A detailed structural illustration of our network is shown in Fig. 2
3.3 Output Decoding and Lateral Interaction
In some works, the first firing neuron in the output layer indicates the predicted result of the network.
However, as neurons' states are temporally discretized in simulation, there exists a situation in which more than one output neurons will fire together at the same timestep. To avoid such a situation, we choose to neuron's 's potential value instead of firing state for prediction. In this manner, network will choose the neuron with the highest internal state as output even if multiple neurons are triggered simultancously.
neuron's internal state is determined by both environmental stimulus and local effects of its neighboring neurons. Excitatory and inhibitory interactions within the receptive field are termed as the lateral interaction property of neurons. Lateral interaction was first introduced to explain an optical phenomenon that color contrast will be exaggerated to form simultaneous contrast by visual systems when two color blocks of slightly different gray levels start to connect with each other. Lateral interactions can avoid information overload by dampening input from some neurons and enhancing
5

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x (Nap
Dense block
Dense block
sampling
Transition layer
Conv
Conv
Conv
Moderate
dropout
1x1
Spiking
Spiking neuron
Spiking
FC layer
Adaptive pooling
4
Avg pooling
Conv pooling 4
Conv x
Conv
Spiking
1
Spiking
neuroe
Spiking
Spiking neuren
Spiking
Nye
Deuron
Rub
Figure 2:
Network architecture of the proposed SNNs. The schematic represents data transmission in one timestep, Nec, Nas. and o; o;(Ora) (Otas) ) are the number of fully-connected layers, total dense block number, and layer iterations in each dense block.
input from others. In this respect, it is used to sharpen the image and enhance system noise tolerance capability. In this work, we consider the lateral interactions among neurons in fully-connected layers enhance system robustness and plausibility. neuron's 's state dominated by two components: potential update triggered by previous state or preceding neurons, and interaction with its neighboring neurons in the same layer. Membrane potential of spiking neurons in fully- callycconnected layers and neuron's s output can be derived from Eq. and as follows:
f.
1-1. ku
f(u ((u)-
Σ jElge(n-1)
WiJo +W, 1-1.
(9)
Judin (f(ut*), otherwise
E
(10)
where the term W 1-1, represents lateral interaction between the target neuron and its surrounding neurons, and (n(nn 1) is set of neurons connected to the target neuron in fully- fully-connected layer (n - 1). Wry is a matrix of lateral interaction learnable weights, whose size defines the width of the receptive field. As aforementioned, the proposed network
potential spike output in different layers, following Eq. [0] By integrating the lateral interaction mechanism, the robustness of the network will be ameliorated because the noise effect vanishes when neuron interactions in larger area are considered, especially for neuromorphic datasets that suffer from greater noise disturbances.
3.4 Moderate Dropout
Noting that highly deviatory sub-models generated by dropout will be integrated and scaled, which may inhibit the constructed SNN's performance. In our SNN, the last-layer neurons' time-averaged potentials can be treated the probabilities of classifying input pattern each potential class. As such, we propose the moderate dropout that aims to minimize inconsistencies between different sub-models' output probability distributions. The Kullback-Leible (KL) divergence score is used to measure the inconsistence in the output probability distributions different network runs as illustrated in Fig. 3 With P and Q being the averaged probability distributions over the entire simulation period on the same probability space, KL divergence is defined as
P(x) Q(T)
(P Q) 100-PP
P(x) log
(11)
We compare sub-models' output distributions and incorporate the divergence into the overall loss of the network minimize the inconsistencies between output distributions. Therefore, the overall loss function comprises the loss actual output distributions against the target labels and the KL divergence loss between different distributions output by the same model with moderate dropout.
can observed from Eq. that the KI. divergence, depicting the relative entropy of with respect Q, is asymmetric in nature, such that (P || Q) of DxL(Q || P). In this respect, we involve

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Speiial synapie
Dropped syntppse
Tomporal symups
Drupped temponal synapes
Retsined neuron
Drupped
22
t1
Ourput
KL divergence
t2
Figure 3: Illustration of KL divergence between different runs of SNN sub-models. Each row of frames represents one sub-model's temporally unfolded architecture after neuron dropping. and Q are two averaged sub-model output distributions over the entire simulation period T.
both divergences in the network loss calculation. Furthermore, we use one additional parameter y to control the impact of moderate dropout in the overall loss function as follows:
1 +Cr+ (o)
Loverali
(K || PJ)
(12)
From the discussion dropout, it can be that the probability in which neurons obscured from the fully culle-coneected layers are stipulated before training and will be kept fixed throughout the training phase when drawing from the Bernoulli distribution. However, this does not take into account the membrane potential values of the neurons before deciding to drop from the network, leading to possible scenarios where essential neurons with high potential are temporarily eliminated from the network. Especially in our SNNs, the potential values are directly used to represent the neurons' states in fully-connected layers. The dropping probability of neurons should be accorded some flexibility for alteration taking into consideration the neuronal membrane potential values so that neurons processing larger membrane values can have smaller dropping probabilities. If neuron i in fully-connected layer has a membrane value of u,", and u," lies in a distribution with uf (umhin: umag)) where unin and umar correspond to the minimum and maximum potential values of neurons in that layer respectively, we can derive the following equation to transform and scale up the potential value uZ and use it to compute the corresponding dropping probability p!"
u
p Pdrop
(13)
U 2 mean
Eq. 13 ensures that a neuron's transformed dropping probability is inversely proportional to its membrane potential and the average dropping probability of all neurons in the same layer equals the drop rate Pdrops which has been defined before training. In this regard, neuron dropping probabilities can be altered throughout the training phase, with larger potentials denoting smaller probabilities, rather than being preset before training in conventional dropout. With the creation of dntential-depender probability tensor, this can be subsequently placed into the Bermoulli distribution to generate binary masks for different sub-models. KL divergence between the different output distributions from sub- model runs is incorporated into the loss function according to Eq. 12 with the overall objective being minimize the loss, leading to increased consistencies among different runs of the network.
7

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Table 1: SNN structural parameters.
Dataset
Nab
01
0/
03
04
N
fc
Static
3
12
12
24
2
Neuromorphic
4
6
12
24
16
2
Table 2: Classification accuracies comparison among proposed methods and state-of-the-art records on various datasets. Number of timesteps and the highest accuracies achieved are highlighted in italics (in parentheses) and bold.
Model
Method
MNIST
CIFARTO
N-MNIST DVS-CIFARTO
ANN-to-SNN
98.37%
82.95
ANN-1o-SNN
77.43%
ANN-to-SNN
95.72%
33
ANN-to-SNN
89.32%
34
ANN-to-SNN
99.10%
35
ANN-to-SNN
99.44%
88.82%
4
ANN-to-SNN
91.55%
36
ANN-to-SNN
93.63%
37
Random Forest
31.00%
38
SKIM
92.87% (360)
39
HATS
52.40%
40
DART
65.78%
Streaming rollout ANN
66.75%
3
Direct training
98.74% (300)
15
Hybrid direct training
99.28% (175)
Direct training
99.42% (30)
50.70% (30)
98. 78% (30)
Hybrid direct training
99.49% (400)
98.88% (500)
30
Direct training
99.62% (400)
10
Direct training
90.53% (/2)
99.53% (8)
60.50% (8)
16
Direct training
93.16% (6)
67.80% (10)
6
Direct training
99.59% (50)
90.95% (100)
99.09% (100)
13
Direct training
99.50% (20)
99.45% (20)
43
Direct training
99.46% (25)
99.39% (25)
44
Direct training
96.30% (120)
32.2% (80)
9
Direct training
70.2% (8)
LTMD
Direct training
99.60% (4)
94.19% (4)
99.65% % (15)
73.30% (7)
Experiment
We evaluate our proposed learnable thresholding and moderate dropout by testing our deep SNNs of DenseNet architecture for classification tasks using both static and neuromorphic datascts. We demonstrate the superiority of our methods in terms of inference accuracies and simulation timesteps by comparing it with other state-of-the-art SNN approaches.
4.1 Empirical Evaluation
Due to the simplicity of MNIST, we specially create an 5-layer fully-connected SNN (input-convl Serrffffffffooooe -fc2-ouupu)) for this dataset. All the other datasets are tested with decp-layer SNNs of DenseNet architecture. Detailed structures of the DenseNet SNNs we build are listed in Tab. [
We implement the proposed methods our deep SNNs and their accuracies for classification tasks on both static MNIST, CIFAR10 datasets, and neuromorphic N-MNIST, DVS-CIFAR10 datasets. We the initial threshold parameter 0.5 and moderate dropout impact factor ? 0.5 for all the datasets. Dataset description, network parameters, and setting details are given in Appendix. The results we obtained are shown Tab. 2 can be seen that our SNNs outperforms other state-of- the-art SNNs for all datasets except MNIST. Although our inference accuracy for MNIST is slightly lower than 30), our SNN achieves a comparable result with much fewer timesteps needed. This makes proposed SNN suitable for fast data analysis.

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Table 3: Inference results using proposed methods.
Dataset
LIF
LT
LTMD
MNIST (step- = -4
99.53%
99.57%
99.60%
CIFARIO (step
92.88%
93.51%
93.75%
N-MNIST (step = 15)
99.55%
99.58%
99.65%
DVS-CIFARIO (step- 71.30%
72.30%
73.30%
99.6
99.7
70
299.5
265
10
30
30
50
160
240
320
400
10
20
35
65
30
50
1.0
1.50
200
250
Fipuchs
Fpuecho
Epochs
Eppcch:c
(a) MNIST.
(b) CIFAR10.
(c) N-MNIST.
(d) DVS-CIFARIO.
Figure 4: Inference accuracies with proposed methods implemented. Moving averages of 5 or 20 epoches used in (a)/(c) and (b)((d) Light-color curves the original data.
The number of timesteps used in previous works and ours are compared in Tab. 2 It can be seen from the table that our SNNs use much fewer timesteps in training and inferencing compared to other SNNs (whether based on ANN-to-SNN conversion backpropagation) and our proposed methods can significantly improve the performance of SNNs in many few timesteps. As result, our proposed SNNs not only have lower respond latency but also need less memory space due to the reduced temporal states of neurons.
4.2 Simulation Study
In this section, we conduct a series of comprehensive simulation studies to evaluate the effects of the proposed learnable thresholding and moderate dropout. We train SNNs with LIF and learnable thresh- olding neurons respectively, followed by integrating moderate dropout. Then, we test and compare the proposed SNNs' inference capabilities. can be observed from Tab. and Fig. 4 accuracies and convergence speed improve after using learnable thresholding, and are further enhanced with moderate dropout.
We show the change in network inference result during training to evaluate the effect of different initial threshold values on our network performance. Fig. shows the high robustness of our proposed SNNs that comprise learnable thresholding neurons.
To analyze how thresholds change during network convergence, in Fig. connected SNN with different initial thresholds to classify CIFAR1O dataset and plot the threshold update curves of neurons in different layers during training. can be seen from the figure that although different initial threshold values are used, the neurons' threshold curves tend to converge after training with learnable thresholding. This shows the robustness of SNNs with learnable thresholding implemented to the initial threshold setting. Additionally, it can be seen that threshold
8-layer fully-
95
65
65
505
291
LIF
LP
35
240
400
200
50
250
(a) CIFARIO with & (b) CIFARIO with A: 0. (c) DVS- OVS-CIFARIO with (d) DVS- CVS-CIFARIO with 0.5.
k = 0.5.
k= O.
Figure 5: Inference accuracies of LIF and learnable thresholding on different datasets with various initial thresholds.

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0.7
0.8
0.8
Threstold
0.4
Threshold
0.6
Thgesbole
0.6
0.4
004
0.2
0.2
0.2
50
100
200
250
150
250
50
100
150
200
250
Epochs
Epochs
Epochs
0.8
0.8
0.8
Thresholl
Threshold
Threshold
0.6
0.6
0.6
0.4
0.4
0.44
8
0.8
0.2
0.2
0.2
0.35
0.1
50
100
200
150
250
50
100
200
Epochs
Epochs
Epochs
Figure 6: Thresholds change of different layers during network training with different initial threshold parameter k.
Table 4: Performance evaluation of LTMD with ResNet and VGG architectures.
SNN architecture
LIF
LT
LTMD
ResNet19
91.97%
93.59%
93.78%
VGG16
91.68%
92.10%
92.35%
values increase the network goes deeper, indicating that information can pass through shallow- layer neurons to maintain sufficient data flow and be filtered by deep-layer neurons to extract useful information for classification.
The proposed learnable thresholding scheme is based on gradient descent method to update neuronal thresholds, which is always functionable when the activation gradient exists among network layers. And the moderate dropout scheme a network stability enhancement strategy to be implemented replace the original normal dropout in the model. Consequently, these proposed methods restricted to SNNs based on DenseNet architecture. We further conduct ablation study to evaluate effect of the proposed LTMD on network performance with ResNet and VGG architectures. The experimental results are shown in Tab. 4 Consistent performance improvement triggered by the proposed methods be observed from this table, which verifies the generalized effectiveness of LTMD.
Conclusion
In this work, we propose learnable thresholding endow spiking neurons with the ability of self- optimizing their threshold values during training and moderate dropout to enhance model stability by minimizing inconsistencies between output probability distributions in different sub-model runs. We construct deep SNNs based on DenseNet architecture and incorporate the aforementioned methods and obtaining promising results. Our proposed SNNs achieve higher accuracies for almost all the test datasets with significantly fewer timesteps required, which reflect their excellent classification capability and low system latency. We demonstrate higher robustness and faster convergence the initial threshold for the learnable thresholding mechanism compared to using normal LIF neurons. The proposed methods' generalized effectiveness also illustrated. Last not least, we show that SNNs with learnable thresholding neurons have higher thresholds in deep layers after training to constraint the number of firing neurons, which proves the neuronal heterogeneity and helps to reduce the amount of computation.
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] Our methods will induce a bit
more computational load, please see Appendix A.4
(c) Did you discuss any potential negative societal impacts of your work? [No] There is no potential negative societal impacts of this work
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section
(b) Did you include complete proofs of all theoretical results? [Yes] See Section 3.1 3.3 and 3.4
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material URL)? [Yes] See the pseudo code provided in Appendix(A..
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Hyperparameters settings are provided in Appendix A.2.3 A.2.6
A.2.7 Full details can be found in code.
(c) Did you report error bars (e.g.. with respect to the random seed after running experi- ments multiple times)? [Yes] We ran experiments for multiple times and please see Tab. in Appendix AA.3 for details.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Number and type of GPUs used in this work are listed in Tab. in Appendix A.2.8
4. If you using existing assets (e.g., code, data, models) curating/releasing assets...
(a) If your work uses existing assets, did you cite the creators? [N/A]
(b) Did you mention the license of the assets? [N/A]
(c) Did you include any assets either in the supplemental material as URL? [N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A]
(c) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "y_80si1DO9Q", "reviewer": "Reviewer_khhY", "summary": "The authors use dynamic thresholding to incorporate heterogeneity in the neuronal parameters of spiking neurons. Dynamic thresholding utilizes learnable threshold values and enables flexible neuronal mechanisms across layers with proper information flow within the network, and fast network convergence. The authors also propose a moderate dropout method to serve as an enhancement technique to minimize inconsistencies between independent dropout runs. The authors show that these methods increase the performance and robustness of the model for image classification. ", "questions": "Since the authors introduce a dynamic thresholding technique, it would be great if the authors could show the increase in computational load due to this. \n\nAlso, a comparative study on the training times for the DTMD with the static thresholding case would be also helpful. A table showing the percentage change in performance on complex datasets with the increase in training time/computational load (measured using MACs/FLOPs)\n\nThe illustration showing the KL divergence of different runs of SNN sub-models is a bit unclear and not properly explained in the main text. It would be great if the authors could furnish some more details. Especially, the authors are claiming the output of the spiking neural network as a probability distribution, on which they are taking the KL divergence which seems unclear.\n\nThe authors use the “KL divergence between the different output distributions from sub-network runs is incorporated into the loss function according to Eq. 12, with the overall objective being to minimize the loss, leading to increased consistencies among different runs of the network.” However, the authors add the two KL divergences to get the overall loss. However, this is still not a metric as it still does not satisfy the triangle inequality i.e., it does not measure the distance between the two distributions. Thus, I believe effectively trying to minimize the symmetrized KL divergence counteracts and nullifies each other. It would be great if the authors could show an ablation study with and without using the symmetrized KL divergence and using each of the asymmetric KL divergences also.\n", "limitations": "The authors did not discuss the limitations of their work or the potential negative social impact of their work in the paper. ", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\nThe paper addresses a key problem that plagues current SNN research - i.e., homogeneity in the neuronal parameters. Though some recent papers have shown increased performance and robustness when heterogeneity is introduced in the neuronal parameters, the process of hyperparameter optimization is complex especially when the number of neurons is large. This paper uses a backpropagation-based method to find the optimal hyperparameters for dynamic thresholding which is great.\n\nThough the performance of the model is promising, there are some key weaknesses that caught my attention.\nWeakness:\nThe concept of introducing heterogeneity is not novel as several recent papers have introduced the concept and shown great results. \nThe paper shows results on small datasets like MNIST, CIFAR10, NMNIST - similar results have already been shown in recent related works.\nThe authors introduce two very different concepts - dynamic thresholding and moderate dropout which seems to be two independent modifications both of which are supposed to greatly increase the computational load\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "GuvDx05YSD", "reviewer": "Reviewer_n9SC", "summary": "In this work, the authors inspired by the neuroscience observation that different neuronal mechanisms exist in disparate brain regions to propose a new spiking neuronal mechanism, named dynamic thresholding. It is clear that the goal of this study is to mimic the brain's ability to build different types of neurons that facilitate the transmission of information in a deep spiking model. Overall, this work is interesting.\n\n---------------------------------------------------------------\nThe author has addressed all my concerns and I would like to upgrade my rating.", "questions": "1. One of my main concerns is that the author gives a github link in the abstract that provides the code, but the anonymous link to github should have this generated (i.e., https://anonymous.4open.science/). I am not sure if this issue violates the double-blind policy. If it does not violate it, I think this article should be WEAK ACCEPT.\n2. The authors should explain the problem that the proposed model on the N-MNIST dataset in Table 2 has a slightly improvement in both performance and timesteps.", "limitations": "The authors have clarified.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The new dynamic thresholding mechanism.\n2. Moderate random deactivation technique based on DenseNet structure is used to enhance SNN modeling capability.\n\nWeakness:\n1. Comparison and analysis with other dynamic thresholding techniques is lacking.\n2. The authors claim that output decoding can enhance the robustness of the system, but there are no experiments related to robustness in the manuscript. The reviewer suggests that if possible the authors need to simply supplement the SNR based input noise experiments to verify the robustness of the system. Unless the robustness claimed by the authors is not in the traditional sense.\n3. The moderate dropout uses KL fused into the loss function, but the cross-entropy loss of the classification task already includes this component. The reviewer suggests that  the authors elaborate further.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "8QZtFjku0o7", "reviewer": "Reviewer_BGk9", "summary": "The authors proposed a learnable (authors called dynamic) threshold scheme and a moderate dropout method for SNNs. The effectiveness\nof the proposed approach is reflected by classification tasks and DenseNet. Based on the experimental results, the proposed method requires fewer time steps for achieving state-of-the-art classification performance. ", "questions": "Please see weaknesses. ", "limitations": "1. The proposed approach is not a dynamic threshold scheme. I suggest the authors call it a learnable threshold scheme. \n2. The authors should compare other existing dynamic threshold schemes to the proposed learnable threshold, e.g., r1 and r2.\n3. The validation only based on DesneNet is very limited. The authors should provide more experimental results to show the effectiveness of the proposed approach. \n\n\nr1: Yunzhe Hao, Xuhui Huang, Meng Dong, and Bo Xu. A biologically plausible supervised learning method for spiking neural networks using the symmetric stdp rule. Neural Networks, 121:387–395, 2020.\nr2: TaeyoonKim,SumanHu,JaewookKim,JoonYoungKwak,JongkilPark,SuyounLee,InhoKim,Jong- Keuk Park, and YeonJoo Jeong. Spiking neural network (snn) with memristor synapses having non-linear weight update. Frontiers in computational neuroscience, 15:22, 2021.", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. Looking into authentic bio-plausible dynamic threshold schemes is a promising research direction for SNNs.\n2. Even though the authors did not develop an actual dynamic threshold scheme, assigning different thresholds to different SNN neurons is interesting. \n\nWeaknesses:\n\n-1. If I understand correctly, the proposed threshold scheme is not dynamically changed during inference. But, it is just learnable parameters during a training process. If so, I do not think it is a \"dynamic\" threshold scheme, which is just a learnable threshold scheme. \n\n-2. The validation mainly relies on DesneNet. But the authors used 8-layer FC SNN for MNIST. How does the DesneNet work with the MNIST dataset? Does the proposed method work with other architectures well, e.g., ResNet, VGG?  \n\n-3. lines 282: \"much fewer time steps needed\" does not mean \"suitable for processing real-time data.\" It depends on how much time spend on each time step. The logic does not hold here. The authors should report the processing speed for each time step.\n\n-4. There are so many \"-\" in table 2. The authors did not explain why which I think is very necessary. \n\n-5. The authors claimed \"less memory space\" but without providing corresponding proofs. \n\nIn summary, I think the validation is not convincing. If the proposed learnable threshold scheme only works with DenseNet, it is just an application. To make it more compelling, the authors should show the generalization of the proposed method. In addition, the authors should also show the experimental results of other tasks besides classification tasks. However, based on the current manuscript, I cannot see any generality of the proposed approach.  \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept