LM-HT SNN: Enhancing the Performance of SNN to ANN Counterpart through Learnable Multi-hierarchical Threshold Model
Authors: Zecheng Hao
Xinyu Shi
Yujia Liu
Zhaofei Yu
Tiejun Huang
Venue: NeurIPS 2024 poster (2024)
Keywords: Spiking Neural Networks
Learnable Multi-hierarchical Threshold Model
STBP Training
Primary area: neuroscience_and_cognitive_science
TL;DR: We propose an advanced LM-HT model, which can enhance the performance of SNNs to the level of ANNs and further establish a bridge between the vanilla STBP and quantized ANNs training.

Abstract: Compared to traditional Artificial Neural Network (ANN), Spiking Neural Network (SNN) has garnered widespread academic interest for its intrinsic ability to transmit information in a more energy-efficient manner. However, despite previous efforts to optimize the learning algorithm of SNNs through various methods, SNNs still lag behind ANNs in terms of performance. The recently proposed multi-threshold model provides more possibilities for further enhancing the learning capability of SNNs. In this paper, we rigorously analyze the relationship among the multi-threshold model, vanilla spiking model and quantized ANNs from a mathematical perspective, then propose a novel LM-HT model, which is an equidistant multi-threshold model that can dynamically regulate the global input current and membrane potential leakage on the time dimension. The LM-HT model can also be transformed into a vanilla single threshold model through reparameterization, thereby achieving more flexible hardware deployment. In addition, we note that the LM-HT model can seamlessly integrate with ANN-SNN Conversion framework under special initialization. This novel hybrid learning framework can effectively improve the relatively poor performance of converted SNNs under low time latency. Extensive experimental results have demonstrated that our model can outperform previous state-of-the-art works on various types of datasets, which promote SNNs to achieve a brand-new level of performance comparable to quantized ANNs. Code is available at https://github.com/hzc1208/LMHT_SNN.

LM-HT SNN: Enhancing the Performance of SNN to ANN Counterpart through Learnable Multi-hierarchical Threshold Model
Zecheng Hao', Xinyu Shil?, Yujia Liu', Zhaofei Yu! & Tiejun Huang!.
School of Computer Science, Peking University
Institute for Artificial Intelligence, Peking University
Abstract
Compared to traditional Artificial Neural Network (ANN), Spiking Neural Net- work (SNN) has garnered widespread academic interest for its intrinsic ability to transmit information in more energy-efficient manner. However, despite previous efforts to optimize the learning algorithm of SNNs through various methods, SNNs still lag behind ANNs in terms of performance. The recently proposed multi- threshold model provides more possibilities for further enhancing the learning capability of SNNs. In this paper, we rigorously analyze the relationship among the multi-threshold model, vanilla spiking model and quantized ANNs from a mathe- matical perspective, then propose a novel LM-HT model, which is an equidistant mulli-threshole model that can dynamically regulate the global input current and membrane potential leakage on the time dimension. The LM-HT model can also be transformed into vanilla single threshold model through reparameterization, thereby achieving more flexible hardware deployment. In addition, we note that the LM-HT model can seamlessly integrate with ANN-SNN Conversion framework under special initialization. This novel hybrid learning framework can effectively improve the relatively poor performance of converted SNNs under low time latency. Extensive experimental results have demonstrated that our model can outperform previous late-of-theeart works various types of datasets, which promote SNNs to achieve a new level of performance comparable to quantized ANNs. Code is available hhttt::///// git.gtth.bu..... com com/h2cc208/LMHT..
Introduction
Recognized the third generation of artificial neural networks [33], Spiking Neural Network (SNN) is increasingly receiving significant academic attention due to its enormous potential in biological plausibility and high energy efficiency. As the information transmission between the pre-synaptic and post-synaptic layers relies on the discrete spike signal, which will be only emitted when the membrane potential of the corresponding neuron exceeds the firing threshold, SNNs have a unique event-driven property compared to conventional Artificial Neural Network (ANN). By utilizing this property, researchers have pointed out that SNNs can achieve significant advantages in terms of energy consumption neuromorphic hardware [35] 5 37). Currently, SNNs have further fulfilled role multiple application scenarios including object detection
[22], natural language processing
32 and 3D recognition [24]
back-propagatie (STBP) with surrogate gradients is currently the most mainstream supervised learning algorithm suitable for SNNs. Although previous works have attempted to further enhance the learning ability of SNNs by delving into various optimization strategies. including
"Corresponding author; yuzf12@ @epk.eduun
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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gradient adjustment 29 8 14 and structural improvement 54 51 44 performance gap between ANNs and SNNs.
550,
there is still a certain
42 is considered as another potential way improve the performance of SNNs. In this scenario, multiple levels of the firing threshold enable SNNs to transmit richer information at each time-step. Unfortunately, we think that current related works have not accurately recognized the mathematical essence of multi-threshold models as well as their relationship with ANNs and SNNs. In this paper, we innovatively propose a learnable malti-hierarchical and equidistant threshold model based on global imput information, which is called LM-HT model. On the one hand, we note that our LM-HT model can equivalently represent the information of the vanilla model over multiple consecutive time-steps within a single step. Furthermore, we can convert the LM-HT model into vanilla single threshold model through a layer-by-laye reparameterization scheme. On the other hand, the STBP method based on the LM-HT model can be transformed into the training modes of the vanilla STBP and quantized ANNs under different parameter initialization conditions, respectively. The main contribution of this work has been summarized as follows:
Recently, the STBP learning algorithm based on multi-threshole models 41
We point out that the essence of the equidistant multi-threshold model is to simulate the spike firing situation of the vanilla spiking model within specific time windows. Specially, when the input current follows completely uniform distribution on the time dimension, its spike firing rate is mathematically equivalent to the activation output of quantized ANNs.
We propose an advanced LM-HT model, which can enhance the performance of SNNs to the level of ANNs and be transformed into vanilla single threshold model losslessly during the inference stage. By adopting different parameter initialization schemes, the LM-HT model can further establish bridge between the vanilla STBP and quantized ANNs training.
We further design brand-new hybrid training framework based on the LM-HT model, which is enable to effectively improve the performance degradation problem of traditional ANN-SNN Conversion methods regardless of the time latency degree involved.
Experimental results have indicated that our model can fulfill state-of-the-ar learning performance for various types of datasets. For instance, we achieve the top-1 accuracy of 81.76% for CIFAR-100, ResNet-19 within merely 2 time-steps.
2
Related Works
STBP supervised training. STBP is the most prevailing recurrent-mode learning algorithm in the field of SNN direct training. Wu et al. [47) tackled the non-differentiable problem existed in the spike firing process by utilizing surrogate gradients and achieved gradient smoothing calculation between layers. Deng el al. [8] and Guo el al. 14] respectively proposed brand-new target loss functions by analyzing the temporal distribution of the spike sequence and membrane potential. Furthermore, various cemporal-dependen batch normalization layers [54] [0, [15) and advanced spiking neuron models [5] 44 have been pointed out, which enhances the capability and stability of SNN learning. The researchers also designed a variety of residual blocks [T) and Transformer structures [55] 49
suitable for SNNs, promoting the development of STBP training towards the domains of deep and large-scale models. In addition, some variant and extended learning methods based on STBP have also received widespread attention. Temporal Coding (36] and Time-to- -or-ttoF--rt-ppiii (TTFS)
algorithm conduct one-time back-propagation based on the specific firing moment. Meng et al. introduced the idea of online learning into vanilla STBP algorithm, which significantly saves training memory overhead by eliminating the gradient chains between different time-steps. Fang et al. [12 proposed a spiking neuron model that supports parallel computing in forward propagation, which also provides inspiration for this work.
34
ANN-SNN Conversion. ANN-SNN Conversion is another widely used method for obtaining high- performance SNNs with limited computational resources, which establishes a mathematical mapping relationship between activation layers and the Integrate- and-Fire (IF) models. Cao et al. [3] first
proposed two-step conversion learning framework, which replaces the activation functions of pre-trained ANNs with the IF models layer by layer. On this basis, Han el al. [16) and Li et al. (28
classified and summarized the relevant errors existed in the conversion process. Deng el al. 7 and
Bu et al. ( further reduced the conversion errors through deriving the optimal values for the bias term and initial membrane potential. For the critical conversion error caused by uneven spike firing

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sequences, multiple optimization strategies have been proposed successively, including memorizing the residual membrane potential [17], firing negative spikes 4S 25 calibrating offset spikes [18] and hybrid finetuning training 455] Currently, ANN-SNN Conversion has been further applied to the
training of large-scale visual and language models 46 32
The current proposed multi-threshold models can be generally divided into two categories: one emits signed spikes [22] 52 43), while the other emits multi-bit spikes [27 41 42 24]. However, these works generally consider using multi-threshold models to reduce ANN-SNN Conversion errors and lack further theoretical analysis. In this paper, we have the foresight to recognize the mathematical equivalence relationship between equidistant mulli-threshen models and quantized ANNs under the conditions of using the soft-reset mechanism and uniform input current, achieving the current optimal performance in the domain of STBP learning.
Spiking neural models with multi-threshold.
3 Preliminaries
The spiking neuron models for SNNs. The Saaky-Integrate and--ir- (LIF) model is one of the most commonly used models in the current SNN community. The following equations have depicted the dynamic procedure of the LIF model in a discrete form:
(t) = (u 1) + (t), 7'(1) = W sTIF (t))
(1) (1) (1)0 (ur(() [1, m/rp(t) 2 0¹
(2)
1o otherwise
[I describes the charging process: Vt [1, (t) and w/ 1) respectively represent
the membrane potential before and after the charging at the /-th time-step. I' (1) denotes the input current and Xbrr characterizes the leakage degree of the membrane potential. When XLu 1, the LIF model will degenerate into a more specialized model called the IF model. ( depicts the reset and firing process: (t)) indicates the spike emitting situation and 0 is the firing threshold. Here adopt the soft-reset mechanism, which means that the reset amplitude of the membrane potential is equal the value of 0'.
STBP learning algorithm for SNNs. The gradient calculation mode of STBP is inspired by the back- pokkaaoppattiinn Through Time (BPTT) algorithm in Recurrent Neural Network (RNN), which will propagate along the spatial and temporal dimensions of SNNs simultancously. Following equations have described the specific propagation process:
UL
UC (tl (00 1) (stup((-1) (1-1) (t- @ml ( (1m1p((-1))
Oc
(t/ +
(3)
@mlup(t-1)
spatial dimension
temporal dimension
Here L denotes the target loss function. From Eq. one can note that the mathematical relationship between (t) and (t)r(t) is equivalent to (t) = H (t)) 8¹), where (-) denotes Heaviside step function. As Heaviside function is non-differentiable, researchers consider using
a surrogate function, which is approximate to Heaviside function but differentiable, to handle the
term (m) in the back-propagation chain. For example, Bemiy (t) = sign (Imirr (1) 8"| s g')
0e(ue())
0
2
describes the well-known rectangular surrogate function.
Quantized ANNs. The quantized ANN model is widely used structure in the field of ANN-SNN Conversion. Compared to traditional ANNs, quantized ANNs usually use the following Quantization- as their activation function:
Clip-Floor-She -hift (QCFS) function
28
2
g
W l''a ay'a'-170 TT, + s¹
a'
T clip
,0.T.)
(4)
Ol
Here a' and sp represent the activation output and shift factor, while T,, and oe denote the quantization level and learnable scaling factor. If we set T. T, g 0¹, a' Σr= (t)( /T, v/) ف! one can find that the so-called QCFS function actually simulate the average spike firing rate of the IF model (we set (Tg) = Ei= (1)0') /T₂,) under the condition of the uniform input current and soft-reset mechanism. This conclusion suggests that SNNs have the potential to maintain the same level of performance ANNs under specific conditions.
3

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H Giet
Time-tepp
mn 00
0
Time-slap [0-22
Time-sop
III
mo
(c)
unifomm resion
Hoavisde gradiem rigton
Sunuyale gradieni
ftring Tigion
unifom finig migion
IOgDOn
Figure I: Forward and backward propagation of the M-HT model. (a)-(c): mathematical relationship between the M-HT model and vanilla IF model. (d)-(e); surrogate gradient calculation for the M-HT model.
4 Methodology
4.1 The Multi-hierarchical Threshold (M-HT) Model
In this section, we first introduce the M-HT model, which has equidistant multi-level thresholds and will select the threshold closest to current membrane potential at each time-step achieve the process of firing spikes and resetting potential. Eqs. M-HT model.
5 -) describe the dynamic equations of the
m (t) = - 1) ((),, (')) = (
(5)
L, (()) - 8(1)0'', 8'(1 = h kdt <k+)))( k L-1
m (1) 2 Lo
(6)
0,
otherwise
Here L denotes the number of level for the firing threshold. Regarding the surrogate gradient 0s'(t)
calculation of the M-HT model, similar to the vanilla spiking models, we propose 9m'(
sign (Lo² S mt) S (L + 1)0t), which covers wider range of the membrane potential, as shown in Fig (d)- As the M-HT model has L different firing options at each time-step, the information transmitted by the M-HT model within one time-step as an information integration of the vanilla model for L time-steps. Therefore, we attempt to bridge mathematical equivalent relationship between the M-HT and IF model:
consider
Lemma 4.1. Vt E [1, if 1) E [0,0), the effect of inputting current I'(t) into a M-HT model with L-level threshold at the t-th time-step. equivalent to continuously in- putting uniform current (1(1)/ for L time-steps into IF model with (0) o0(1 - 1), i.e.
s' (1) = -clip ( 10 ...L) -/ 8).()).
(1
Lemma 4.T indicates that the M-HT model under a single time-step can be used to simulate the total number of spikes emitted by the IF model under uniform input current within L consecutive time-steps. In addition, note that s' st))) which is equivalent the QCFS function mentioned before in quantized ANNs. The above conclusion preliminarily demonstrates that the M-HT model can achieve the same-level performance as pre- trained ANNs with L-level quantization under single-step condition.
Lemma 4.1 can also calculated through clip (L.J. :))
4.2
The Representation Ability of the M-HT Model on Multiple Time-steps
Based on Lemma
4.1 further consider the information representation of the M-HT model on multiple time-steps:
Theorem 4.2. When = 1, (0) E [o, ), for M-HT model with I-level threshold, after T sime-steps, we will derive the following conclusions:

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I
local inforsaiinn
glebal infecmation
traepora attrlbuta
rale albibule
R
I
A2.0
V8.m 10
Inttaalized
QCFSANN
F
1500
6))
?
MH
Mais st
0005
01000000
(i)
LMHH Cul
0CFS Cane
Figure 2: The STBP learning framework based on the LM-HT model. (a): vanilla STBP training. (b): STBP training with the LM-HT model. (c): direct training of quantized ANNs. (d): hybrid training with the LM-HT model, here R-I Curve denotes Rate-Input Curve.
(i) If we further assume Vt [1, E [0. ). we will have: Vt s.. LT
E E)=L(-11)+1 (i), v' = (Lt), Ei= s'(t) = Ef= s)r(
(ii) If we further assume (1) = = (T), we will have: E
=
clip ([ ('10)+ I'()) 0, LT).
8
Here the IF model has uniform input currents 1' (1)/L, I' (T))) respectively within every L steps and satisfies (0) (0).
The proofs of Lemma and Theorem 4.2 have been provided in the Appendix. From Theorem i) and Fig [ (c), one can find that the M-HT model is actually equivalent to dividing the spike firing sequence of the IF model on consecutive LT' steps into T L-step time windows. Combining with the soft-reset mechanism, the M-HT model actually focuses on a specific time window of the vanilla IF model at each time-step and maintains an equal membrane potential with the IF model at the end of each time window (i.e. Vt [1, v' (t) = v/r(Lt)) Lt)). The M-H' model follows the assumption of uniform input current within each window, while maintaining the basic calculation properties of spiking neurons between different windows. When the input current follows complete uniform distribution, according to Theorem 4.2 ii), the M-HT model can further simulate the output of an ANN with LT level quantization.
4.3
The Learnable Multi-hierarchical Threshold (LM-HT) Model
The uniform and uneven firing regions in the M-HT model. For a specific spike firing rate, the
M-HT model can often provide multiple spike firing sequences. For example, [1. 1]. [0, 2), [2, 0J can
all represent the situation where spikes are emitted within time-steps, while only [1, 1] can be
viewed as a case of uniform firing situation. However, even when the input current is uniformly distributed, as the sum of spikes that cannot be divided by [0, LT) is unable to be represented by uniform spike output sequence, there are still uneven firing situations:
IT),,, for a M-HT model with L-level s (T) is only satisfied when (1( E [kO', kBl + 8¹ /T), vk = 0,..., I. or (1) E (-o0, U [LO +oo).
Corollary 4.3. If X 1, v (0) = O and (1)
threshold, (1)
The proof is provided in the Appendix. From Corollary we can divide the input current into uniform and uneven firing regions according to the corresponding intervals, as shown in Fig ab) Note that the uneven spike sequences emitted by the I-th layer may further cause the input current of the + 1-th layer to no longer follow the uniform distribution. That is to say, as the number of layers increases, the uneven firing cases will tend increase gradually without introducing extra regulation.
Learnable Temporal-Global Information Matrix and leaky parameters. Enhancing the uniform firing pattern can promote SNNs to achieve superior performance similar to quantized ANNs, while
5

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Cocy BN
Cony & BN
weighs host
T-GIM
T-GiI
Regarameleriation
BN
T-GIM
ER
M-T IF
75
Core BN
Conv & FIN
I-GIM
T-GIM
Regerameteriation
4
Coca BN
Corv & BNN
1
F-GIM
T-GIM
(a) SNN SoNiiiniiggSSag
(b) SNN Infereence
Figure 3: Reparameterization procedure of the LM-HT model.
uneven spike sequences retain more temporal and biological characteristics. Therefore, how to comprehensively utilize these two spike firing patterns becomes a critical problem. To address this issue, we first introduce the concept of cemporal-Global Information Matrix (T-GIM):
Vt [1, I'
Σ w' (2)0" j=1
Here w/, is the element at row and column j of the T-GIM 2 Ω E BTxT, As shown in Eq. and Fig this brand-new input current adopts multi-step current weighting form, allowing model to simultaneously focus on the global information along the time dimension. Note that the new input current will follow a uniform distribution when Vi,j [1. T. wly = 7 and degrade
vanilla input current when 52' = diag(1,.... 1). For the first case mentioned above. if we further add the condition X' = 1, according to Theorem 4.2 ii). one can find that the output of the model will be consistent with the activation output of LT level quantized ANN layer by layer, as shown in (b) For the second case, when I, = 1, the model will degenerate into vanilla LIF model, as shown in aa-
To enable the model to dynamically adjust the above calculation process, we set both Ω and X as learnable parameters. The initial values of S2' and X' are set 1/T and 1, respectively, During the training process, we choose the Sigmoid function o--) to control the parameters for fulfilling smooth gradient updates within bounded learning range. We call this novel model as Learnable multi-hierarchical Threshold (LM-HT) Model, which combines T-GIM and learnable attributes. We think the LM-HT model can regulate its spike firing pattern more flexibly and reasonably.
Since we can regulate the computational relationships between different stepssteps through learnable S and X in the LM-HT model, during the back-propagation process, unlike Eq. we detach 3mm 0m)((t) 0e'(L-1)
the term (t) @v'(t-1) (t-1 from the gradient calculation graph, thereby reducing redundant calculations and completely leaving the gradient propagation between different time-ster steps Q and X for control. The pack-propagation calculation chains for the LM-HT model have been described as follow. Here @ denotes the Hadamard product.
dml (t) (s((t) 0m)(t)' 0m'(t) sign (20r s m ml) L+ 1 2))8)
Oc OC (s)(1) as
(8)
Oc dL ax (t)
0c Oc
① v'(tt 1), awl, dml(1) (W'ss-1()00-
(9)
4.4 Hybrid Training based the LM-HT Model
Although the traditional ANN-SNN Conversion frameworks have much lower computational overhead than STBP training algorithm, serious performance degradation phenomenon often exists on the

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Table 1: Ablation study for the LM-HT model on a
Table Validation for the reparame-
subset of ImageNet-Ik lk.
terization procedure.
Model
T-GIM
Arch.
Acc.(%)
SOPs(G)
F.(mJ)
Arch. Acc.(%
SOPs(G)
E.mJ)
1=1,1=4
w/o
76.22
1.60
1.44
Before reparametenzaatinn (I=2, T=2)
L=2,T=2
w/o
ResN-18
80.52
1.08
0.97
VGG-13
61,64
0.26
0.23
L=2,T=2
80.56
0.73
0.66
ResN-18
64.44
0.52
T=1,1=4
65.62
3.20
2.88
After reparamcterization (L=1. T=4)
L=2.T=2
W/o
ResN-34
82.18
2.42
2.18
VGG-13
61.66
0.26
0.23
1=2,T==2
W/
82.72
1.72
1.55
ResN-18
64.50
0.52
0.46
converted SNNs under low time latency 17
address this problem, previous researchers [9
considered adopting STBP training for few epochs on the pre-trained ANN models to enhance the performance of the converted SNNs under fewer time-steps, which is called as hybrid training. In this work, we propose brand-new hybrid training framework based on the LM-HT model.
We firstly choose QCFS function train the quantized ANN models and then replace the QCFS function modules layer by layer with the LM-HT models under specific initialization (Vi, E
u,, T 2 = 1, 0 0¹ (0) %¹) shown in Fig 2).. Combining with the conclusion pointed out by 2 one can note that the initialized LM-HT model and the QCFS function before substitution have equivalence in terms of mathematical expectation, which has been described as the following theorem:
Theorem 4.4. When ΣT ['t/LLT W'T (T,) and ET,1'' [0, LTO"), if Vi,j
[1,], aut a (130'
to and X2 1, -,, (0) 2. for L, T.T.Ta with arbitrary values, we have: (1 W 1''''-'TTTN + 0, ..T.T))
g' T. clip
Theorem 4.4
indicates that regardless of whether the time-steps choose during the STBP training phase is equal to the inference steps simulated in ANN-SNN Conversion, the average spike firing rate of the LM-HT models under the initial state of STBP training maintains mathematical equivalence with that simulated by the QCFS function modules in the previous stage. Therefore, under this new training framework, can adopt STBP algorithm to optimize the inference performance of SNN under any degree of time latency. The detailed pseudo-code has been provided in the Appendix.
4.5 Reparameterize the LM-HT model to vanilla LIF model
As discussed in Section 4.2 the mathematical essence of the LM-HT model is simulate the spike firing situation of vanilla LIF neurons within each time window. Considering that the current neuromorphic hardware mainly supports single threshold models, we propose reparameterization scheme that can transform the LM-HT model obtained during the training stage into vanilla LIF model, which can further be deployed on hardware for inference.
As shown in Fig 3, for a L-level LM-HT model within T steps, we expand it into a vanilla LIF model within LT steps, where the membrane leakage factor between different time windows set to X'. In addition, T-GIM will be extended from T to F. I.TTTTT and the parameters are averaged within each L sub-region, ensuring that the input current meets the precondition Theorem 4.2 We also rectify the bias terms in synaptic layers, which involve addition operations at each time-step. By performing layer-by-layee reparameterization the above manner, we will obtain single threshold SNN model with theoretically lossless accuracy.
5 Experiments
To validate the effectiveness of our proposed STBP and hybrid training frameworks based on the LM-HT model, we consider multiple static and neuromorphic datasets with different data scale, including CIFAR-10(100) [23] ImageNet-200( 1k) [6] and CIFARIO-DVS (26). Consistent with the previous works, we also choose VGG 44 and ResNet [9] as the basic network architecture We evaluate the computational overhead of SNNs based on the number of synaptic operations (SOPs) and the calculation standard for related energy consumption refers to (55). In addition, as the information
transmitted by L-level LM-HT model within T time-steps remains the same level that of the
7

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Table 3: Comparison with previous state-of-the-aae works.
Dataset
Method
Type
Architecture
Time-steps
Accuracy( %
STBP-ddB [54]
Direct Training
ResNet-19
4
92.92
Dspike
29
Direct Training
ResNer-18
93.66
TET
Direct Training
ResNet-19
94.44
SLTT
Online Training
ResNet-18
94.44
CIFAR 10
ResNet-18
6.
94.15, 94.67, 94.88
GLIF
51
Direct Truining
Re
4,6
94.44, 94.85, 95.03
ResNet-18
2
96.25
LM-HT (L=2)
Direct Training
ResNet-19
2
96.89
Dspike 29
Direct Training
ResNet-18
73.35
TET
Direct Training
ResNet-19
4
74.47
SLTT
Online Training
ResNet-18
6
74.38
ResNet-18
2, 4,
74.60, 76. 42, 77.28
CIFAR-100
GL.F S1
Direct Training
ResNer-19
4.
75.48,
77.05, 77.35
RMP-Loss
Direct Truining
ResNet-19
2. 4.
74.66.
78 78.28, 78.98
LM-HT (L=2)
Direct Training
ResNet-18
2
79.33
ResNet-19
2
81.76
DCT [B]
Hybrid Training
VGG-13
125
56,90
Online-LTL 48
16
54.82
Hybrid Training
VGG-13
Offline-LTI 48
16
55.37
ImageNct-200
ASGLL 44
Direct Training
VGG-13
4,8
56.57, 56.81
LM-HT (L=2)
2,
61.09, 61.75
Direct Training
VGG-13
LM-HT (L=4)
2
62.05
STBP-tdBN [54
Direct Training
ResNet-34
6
63.72
TET [
Direct Training
ResNet-33
6
64,79
MBPN L5
Direct Training
ResNet-34
64.71
ImageNet-I6
RMP-Loss 14
Direct Training
ResNet-34
SEW ResNet
Direct Training
ResNet-34
67.04 04
GLIF [11
Direct Training
ResNet-34
67.52
LM-HT (L=2)
Direct Training
ResNet-34
70.90
STBP-AdBN 64
Direct Training
Re
I0
67.80
Dspike
Direct Training
ResNet-18
10
75.40
MBPN 15
Direct Training
ResNet-19
10
74.40
CIFARIO-DVS
RMP-Loss 14
Direct Training
ResNet-19
10
76.20
LM-HT (L=2)
2,4
80.70, 81.00
Direct Training
ResNet-18
LM-H' (I.=4)
2
81.90
vanilla LIF model within LT" time-steps, to make fair evaluation, we will compare the performance of the L-level LM-HT model within T' steps with that of the previous works within LT steps.
5.1 Ablation & Validation Studies for the LM-HT Model
As shown in Tab we investigate the impact of threshold levels and T-GIM for our proposed model. One note that vanilla IF neuron (L = 1, 4) is not well suited for deep networks (e.g. ResNet- 34) and causes relatively high energy consumption, while the M-HT series models (L = effectively overcome the performance degradation problem on deep networks. When we further utilize T-GIM to regulate global information on the time dimension, the learning ability of our model enhanced and the computational overhead in synaptic layers is significantly reduced.
2)
We also validate the feasibility about the reparameterization procedure mentioned above. As shown in Tab and Fig by copying and reparameterizing the parameters of synapses, T-GIM and LM- HT neurons layer by layer, we obtain a single threshold model that maintained almost the same performance and power consumption as the original LM-H'T model. This convertible property enables the LM-HT model to be more flexibly deployed on neuromorphic hardware.
5.2 Comparison with Previous SoTA Works
We first investigate the competitiveness of our proposed model in the domain of STBP learning.
As shown in Tab our comparative works incorporate previous state-of-the-art (SoTA) methods in various sub-domains of STBP training, including batchnorm layer optimization [54 15 improved
surrogate gradients [29]. learning function design g 14, energy-efficient training
and
advanced neuron models [51 44]
8

---

Table 4: The performance of hybrid training based the LM-HT model for CIFAR-100 dataset.
VGG-16
ResNet-20
Method
Time-steps
ANN Acc. (%)
SNN Acc.(%)
ANN Acc.
SNN Acc.
RMP 16
32, 64, 128
71.22
63.76
68.72
27.64. 46.91. 57.69
SNM 43
32, 64, 128
74.13
71.80, 73.69, 73.95
SRP 17
5. 6, 8
76.28
71,52, 74.31, 75.42
69.94
46.48. 53,96, 59.34
QCFS (7, [4) 2
2, 4,
76.11
63.33, 69.70, 74.12
63.90
38.04, 52.28, 61.77
75.97 (+6.27)
63.55 (+11.27)
LM-HT (L=2)
76.49 (+2.37)
64.87 (+3.10)
LM-HT (L=4)
76.38 (+2.26)
63,43 (+1.66)
QCFS (7)=8) 2
2, 4,8
77.31
64,85, 70,50, 74.63
69.56
19.76, 34.17, 55.50
76.31 (+5,81)
67.08 (+32.91)
LM-II (L=2)
76.79 (+2.16)
69.00 (+13.50)
LM-HT (I=4)
76.08 (+1.45)
67.21 (+11.71)
CIFAR-10 & CIFAR-100. For conventional static datasets, one can find that our solution demon- strates significant performance advantages. For ResNet-18 structure, we achieve the top-1 accuracies of 96.25% and 79.33% with merely 2 time-steps on CIFAR-IO and CIFAR-I00 datasets, respectively, For ResNet-19 network with larger parameter scale, our method fulfills the precisions of 96.89% and 81.76% within time-steps, which at least outperforms other corresponding works with 2.04% and 3.48% under the same time latency. In addition, it is worth noting that above results have even exceeded the performance of other works with more time-steps (e.g. 6 steps).
ImageNet-200 & ImageNet- For large-scale datasets, we also confirm the superiority of the LM-HT model. For the two-level LM-HT model, we respectively reach the top-1 accuracies of 61. 61.09% and 70.90% within 2 time-steps
ImageNet-200) and ImageNet- 1k datasets, which is 4.52% higher than ASGL (4 steps) and 3.38% higher than GLIF (4 steps) under the same-level time overhead. For larger training time-step, one can note that our method will also demonstrate significant advantage. For example, the two-level LM-HT model reaches the precision 61.75% with 4 stees, which has surpassed ASGL (8 steps) with 4.94%.
We also evaluate the effectiveness of our approach on neuromorphic datasets. Compared to other previous methods, our proposed model can achieve better results on shallower networks with fewer time-steps. For instance, the two-level LM-H'T model can achieve the accuracy of 80.70% after merely 2 time-steps.
CIFARIO-DVS.
5.3 Performance Analysis of Hybrid Training
In our hybrid training framework, we first choose [2] as the hackhone for our ANN-SNN Conversion stage. Subsequently, we replace the QCFS function layer by layer with the initialized LM-HT model and conduct STBP training for merely 30 epochs. Furthermore, we also consider other advanced and multi-stage error correction method [17] as our comparative works.
conversion methods [16 43
As shown in Tab after conducting the STBP fine-tuning optimization with relatively low computa- tional overhead, note that the performance of the converted SNNs under different quantization levels has been significantly improved and surpass other previous methods, especially under low time latency. For instance, compared the ResNet-20 network after eight-level quantization (i.e. T,==),, the two-level LM-H'T model has achieved a performance improvement of 32.91 % and 13.50% with and 4 time-steps, respectively.
Conclusions
In this paper, first investigate the mathematical equivalence among the multi-threshold model, vanilla spiking model and quantized ANNs, then propose an advanced STBP training method based on the LM-HT model, which has been proven cover the representation range of vanilla STBP and quantized ANNs training frameworks, thereby promoting SNNs to achieve superior performance at the same level as quantized ANNs. Furthermore, the LM-HT model can achieve lossless transformation towards single threshold models or quantized ANNs under specific parameter configuration. Numerous experimental results have verified the effectiveness our method. We believe that our work will further promote in-depth research on advanced spiking neural model.

---

Acknowledgements
This work supported by STI 2030-Major Projects 20212I00200300, the National Natural Science Foundation of China under Grant No. 62176003 and No. 62088108 and by Beijing Nova Program under Grant No. 20230484362.
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A Appendix
A.1 Proof of Theorem
A.1.1 Proof of Lemma 4.1 & Theorem 4.2
Before the proof of Theorem we first need to introduce Lemma A.
Lemma A.1. Assume a continuous T-step input current (1),, (T)), for a LM-IIT model with L-level threshold, when Vt [1. (')) E [0, L8') and (0) E [0, 00 X 1. we will have v' (T) E (),
((()))=((t) Proof. Vt E (0,T), (') E 10,0), as m'(t + = v'(t) + (t). have m + 1) [0. +I++++1 1)))). Therefore, after the firing process 11) 1)) (t)00', one can note that 1' E [0, 0¹). According to the idea of mathematical induction, if we directly set v' (0) E [0, 0'). we can have o' (T) E (0,
Theorem 4.2. When X' 00() E [0, 0¹), for M-IIT model with L-level threshold, after T
time-steps, we will derive the following conclusions:
(i) If we further assume E [1, (1) E [0. LO we will have: Vt E [1,T], s'
L 8)r( = (Lt), Ex= s(()) = ES= sir(
(ii) If we further assume (1)
(T)). we will have: Ets
(1" (1' r't)) ,,0, IT).
clip
Here the IF model has uniform input currents I' (T)/ respectively within every L steps and satisfies (0) (0).
Proof. (i) If we consider the pre-condition in Theorem and combine Eq. with Eq. 2 VI [1, I.T), we will have:
(t) - 1/) ( /L -- (ip())
(S1)
Similarly, if we set X' = and incorporate 55. Vt E []. we will have:
a' (t) na (t - (t) - s' (t)e².
(S2)
Then we accumulate Eq. ST) along the time dimension and obtain the following equation:
(Lt) e/r (L(t-1) - 1))
(t)
Σ
(S3)
As (1) E (0, T,O'), according to Lemma A.1 when o' (1- 1) = t/r (I(ll 1)) Ao (0,
we will have o' (t) [0,00] 0¹) and (Lt) E [0, )'). Considering st),
8)r E N.
s1) + Σ L (f), one can note that 1 [)=1(0-1)+1 ( - - ( I(0
o! ot 1)) u ff (Lt) d/p (I((t - 1))) |w'(t) (lp(((t))) l which will violate the conclusion Lemma A.1 Therefore, we can finally deduce that s² ( LLe 2== ()) Then we can
further have v (1) (LL) and Σ s0(1)= -/ (J).
(ii) If we accumulate Eq. S22 along the time dimension and divide Ø on both sides, we will have the following equation:
('(T) v0())
ΣT I'tt
Σ (t).
(S4)
01
0¹
((1) < or o +' 1'())
(1) 2 LO', is obvious that we will have Et
or Σx s1) = -clip (1
o'(0)+> ('(() 0, LT
clip
0, LT)
14

---

s' N. based on Eq. (S4. we can finally deduce that Ei= (l) x'(0)+2 F()) (T)
Ei= If (1) E [0, L8"), according to Lemma A.1 we will have (T) E 10,0') As
0
+1 1'
-lip (L o'(0)+ 11 0, T.T)
One can note that Lemma 4.1 is actually a special case of Theorem 4.2 under the condition of T therefore Lemma 4.1 is also proven.
A.1.2 Proof of Corollary 4.3
Corollary 4.3. If X' 1. rz (0) 0 and (1) =..= =
(T), for M-HT model with L-level
(T) is only satisfied when (1) E [k-), k0' + 0 /T), vk = 0,..., L 1 or (1) E (-00, 0) U [LO', +
threshold, s1)
Proof. If (1( < or (1) 2 LO it is obvious that we will have s1(1)
(T) = 0 or (1) = +T) = L. Otherwise, based on the conclusion Σ s'(t)
i)), when ((1) E [k0', ko DT/T), Vk O,.... L. - 1, we will have Ei= s'(t) = KT, Ak = O.... - 1. Note that VT E [1,T], we KT', 4k O,.... 1. Therefore. it can be concluded that s1(1 = ... s (T) k. Instead, if (1) E L0') AI $ iK0', k0' + OTT) Vk 0, ..., -- we will have Σi= s' (t) * KT, vk = O,... L 1. Therefore, s'(1) = = s' (T) does not hold true.
clip (L +' (()
LT in Theorem i.2
can further have Σ st(t)
A.1.3 Proof of Theorem 4.4
Theorem 4.4. When Σt I't/ W'T and e 10, LT, if ji,
[1, [17]], wt s' (136"
$ and X2 1, ,4, 00)' = g' 7. for L,, T.T.Ty with arbitrary values, we have:
o² clip (1
w ' (T, + 2 0. O.T,)) =0.
LT T,
Proof. If Vi,j E [1,
7. == 1,0 or and v'(0)
o²
2. com-
bining with the conclusion mentioned in Theorem ii) we will have ΣT of (1)0
LT
of clip (1 [ET that E clip (1 LT + ,,TTTT T.
I'm + (Et,/9
.0, .O.TT. According to the conclusion pointed out in (2], we have known
cllpp
(1=t) - .0.TT) = O, here at E [0,0]]
Therefore, we directly set x = Σt= Ltt = ''r (IT) and then we will draw the fi- nal conclusion.
A.1.4 Computational Equivalence about the Reparameteriizttoon Process
Theorem A.2. Vi,i E [1, Vj e 1) Vk [L(i - 11) + 1, Lū), when s - Lt E)= T (f), 03 b.LL,w,,, = wls/L, we will have F'(t) = Le Et= [1).)
Here b² wt denote the rectified bias term and T- T-GIM layer after the reparametericzation process.
J-1 Proof. Firstly, is obvious that (1(),, (3) can be rewritten ('(1)= = - (W's''
and ()) E a3 (W' S (i) + b)). Considering the precondition b) /L/L,
w = w/// L.
15

---

we will have:
Li
[1)
Σ
Σ
( (F (k) b))
11-11)+11
T
(W'S + 4
(S5)
M(i 1)+1
Then further have:
Lt
L1 Σ
La E
(W's + ")
Σ
[))(1)
1=
j= 1)+1 i=1
k=
T
Lt
Li
Σ W
Σ
(wi
Σ
( + 0))
j=
k=
Lt
Σ
Σ
j=L(1-1)+1
T
Σ
WW'' - (1) +b)
(t).
(S6)
Due to the fact that the calculation process of the spike sequences passing through Conv & BN and T-GIM layers can be abstractly described by Theorem A.2 we can conclude that the sum of the input currents within the corresponding time windows before and after reparameterization remains unchanged. The spike sequences obtained by passing the imput currents through the spiking neuron
layer will also satisfy the precondition Theorem A.2 (t) = Σ (i)). Therefore, we can prove the computational equivalence before and after the reparameterization process.
A.2 Comparison with Other Advanced Network Backbones
As shown in ST we have made comparison with related advanced works [20 55 49 38) on CIFAR datasets. One can find that our LM-HT model has superior scalability and can demonstrate its effectiveness on multiple different backbones. For example, for CIFAR-100 dataset, compared to GAC-SNN (38), achieve accuracy improvement of 1.35% on MS-ResNet-1D 18. For Transformer- 4-384 architecture, our method also outperforms Spikformer 55 and Spike-driven Transformer 49 in terms of performance.
Table SI: Comparison with previous methods based advanced backbones and attention mechanism.
Dataset
Method
Architecture
Time-steps
Accuracy( %)
MS- MetResNet
20
MS MSS-ResNet-13 18
6
94,92
GAC-SNN
38
MS-ResNet-18
4
96.24
Spikformer
55
Transformer-4-384
4
95,51
CIFAR-10
Spike-driven Transformer
49
Transformer-4-384
4
95.6
MS-ResNet-118
4
96.38
Ours
Transformer-4-384
95.82
NetResNet
20
MS-ResNet-188
6
76.41
GAC-SNN
38
MS-ResNet-188
4
79.83
Spikformer
55
Transformer-43344
4
78.21
CIFAR-100
Spike-driven Transformer
49
Transformer-4-384
4
78.4
MS-ResNet-118
81.18
Ours
Transformer-4-384
4
79.03
16

---

A.3 Experimental Configuration
For static datasets, we attempt to suppress the possible overfitting phenomenon by utilizing data augmentation techniques including AutoAugment [4] and Cutout 99. For CIFARIO-DVS dataset, we resize each image to 48 48 pixels and split into 10 frames. For ImageNet-Ik dataset, we further consider MS-ResNet architecture
[20] and Mixup technique [53] to strengthen the generalization ability of our network. We respectively try to use SGD [T] and AdamW [30] as our optimizers. The corresponding initial learning rate and weight decay set to 0.025, x 10 (0-4 for SGD on CIFAR- 10(100), 0.0125,5 5 x 10-4 for SGD on ImageNet-2000 and 0.02, 0.01 for AdamW on CIFARIO-DVS. For ImageNet- Ik dataset, Furthermore, in the hybrid training framework, our initial learning rate and weight decay are both set to 5 10 The QCFS pretrained models are selected from related open-source source checkpoints and self- self-immlemennatioon For all experimental cases, we choose the Cosine Annealing scheduler [i] to dynamically regulate the learning rate. Our experiments implemented NVIDIA RTX A5000 and 4090.
SGD our optimizer and set the corresponding weight decay O.
A.4 The Pseudo-Code of Hybrid Training Algorithm
Algorithm/ Hybrid training framework based on the LM-HT model.
Require: Pretrained QCFS ANN model fANN (W, Tqp 0) with LN layers; Dataset D; Number of time-steps choosed for STBP training T.
Ensure: SNN model fSNN W,S, S2, L, X,
Convert ANN to SNN
2: for 1 = 1 to LN do
3:
fSNN. W SANN. W
4:
JSNN.0 0' =JNNN
5:
JSNN. St' =T
6:
JSNN =
7: fSNN. (0) fsNN.0' .'
8: end for
9: # STBP training based the LM-H model
/SNN. 11: for Label) in D do
10: # Set JS' / as learnable parameters and /sNv.0' as scalars
1=110 12: for 1or1110 to LN do
Is the first layer then 14: for t = 1 to T do
(t) (t) X
16:
end for
end if
18: LM-HT model performs forward propagation based on Eqs. 5 66 and Eq.
19:
LM-HT model performs pack-propagation based on Eqs.
20:
end for
21:
end for
22:
return fSNN W. S2 0,
NeurIPS Paper Checklist
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Reviews:
[{"review_id": "l9IeTsU56V", "reviewer": "Reviewer_zyi4", "summary": "This paper introduces the Learnable Multi-hierarchical Threshold (LM-HT) model, a novel approach to enhance the performance of SNNs to match that of ANNs. The LM-HT model dynamically adjusts the global input current and membrane potential leakage, and can be reparameterized into a standard single-threshold model for flexible deployment. The proposed model can integrate with ANN-SNN conversion framework to enhance the performance of converted SNNs under low time latency.", "questions": "According to Section 4.5 and Figure 3, it seems that the reparameterization procedure is simultaneously required for Conv&BN, T-GIM and neuron layers. Can the authors make a more detailed explanation about the parameter rectification for these three types of modules?", "limitations": "N/A", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1. The LM-HT model provides a mathematical bridge between multi-threshold SNNs and quantized ANNs. The proposed reparameterization scheme allows for efficient hardware deployment.\n\n2. The hybrid training framework based on the LM-HT model effectively addresses performance degradation issues in traditional ANN-SNN conversion methods.\n\n3. Extensive experimental results demonstrate the superior performance across multiple datasets.", "weaknesses": "1. The author need to clarify the benefits of using the LM-HT model during the training phase and reparameterizing it to a vanilla single threshold model during the inference phase. For example, what are the advantages of this approach compared to directly training a single threshold model based on T-GIM modules?\n\n2. Theorem 4.2 and Theorem 4.4 have demonstrated the equivalence of layer-by-layer output between the LM-HT model and quantized ANN during forward propagation, but the authors still need to analyze the equivalence of gradient calculation between the LM-HT model and quantized ANN from the perspective of back-propagation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## To Reviewer zyi4\nThanks for your constructive and valuable feedback. We are encouraged that you found our proposed method \"effctive\", \"superior\" and \"provides a mathematical bridge between multi-threshold SNNs and quantized ANNs\". We would like to address your concerns and questions in the following.\n\n> 1. What are the advantages of this approach compared to directly training a single threshold model based on T-GIM modules?\n\nThanks for the question. Firstly, there is an advantage in terms of computational cost. During the training phase, compared to single-threshold models, using LM-HT models saves $L \\times$ in both time and memory overhead ($O(LT)$ v.s. $O(T)$). Next is the enhancement of performance. The LM-HT model enhances the uniformity of spike sequences within each time window, combined with the gradient calculation mode of Eqs.(8)-(9), making LM-HT SNN have an equivalent training mode to quantized ANN, significantly improving its learning ability. Ultimately, our method can also reparameterize LM-HT SNN to vanilla SNN at no cost during the inference stage.\n\n> 2. The authors still need to analyze the equivalence of gradient calculation between the LM-HT model and quantized ANN from the perspective of back-propagation.\n\nThanks for your comment. According to Eqs.(8)-(9), we can have\n\n$\\frac{\\partial \\mathcal{L}}{\\partial m^l(t)} = \\sum_{j=1}^T \\frac{\\partial \\mathcal{L}}{\\partial m^{l+1}(j)} \\omega^{l+1}_{jt} \\theta^l {W^{l+1}}^{T} \\frac{\\partial s^l(t)}{\\partial m^l(t)}$,\n\nand $\\frac{\\partial \\mathcal{L}}{\\partial W^l} = \\sum_{t=1}^T \\frac{\\partial \\mathcal{L}}{\\partial m^l(t)} \\sum_{j=1}^T \\omega^l_{tj} s^{l-1}(j)\\theta^{l-1}$. \n\nWhen $\\forall i,j\\in T, \\omega^l_{ij}=\\frac{1}{T}$ and $\\forall t\\in T, m^l(t)\\in[\\frac{\\theta^l}{2}, (L+\\frac{1}{2})\\theta^l ]$, as $\\frac{\\partial s^l(t)}{\\partial m^l(t)} = \\text{sign}(\\frac{\\theta^l}{2} \\leq m^l(t) \\leq (L+\\frac{1}{2})\\theta^l )$, we will find that\n\n$\\forall t\\in [1,T], \\frac{\\partial \\mathcal{L}}{\\partial m^l(t)} = \\frac{\\theta^l }{T} \\sum_{j=1}^T \\frac{\\partial \\mathcal{L}}{\\partial m^{l+1}(j)} {W^{l+1}}^{T}$,\n\nwhich means that $\\frac{\\partial \\mathcal{L}}{\\partial m^l(1)} = ... = \\frac{\\partial \\mathcal{L}}{\\partial m^l(T)}$.\n\nIn additon, we can derive that $\\frac{\\partial \\mathcal{L}}{\\partial W^l} = \\sum_{t=1}^T \\frac{\\partial \\mathcal{L}}{\\partial m^l(t)} r^{l-1}, r^{l-1} = \\frac{1}{T} \\sum_{j=1}^T s^{l-1}(j)\\theta^{l-1}$, which is equivalent to the back-propagation mode of quantized ANN.\n\n> 3. Can the authors make a more detailed explanation about the parameter rectification for Conv&BN, T-GIM and neuron layers?\n\nThanks for the question. For the bias terms in Conv&BN layers, due to the $L\\times$ time-steps expansion in the inference stage, the addition operation at each time-step needs to be divided by $L\\times$ (to ensure $I^l(t) + b^l_t = \\sum_{j=L(t-1)+1}^{Lt} I^l_{IF}(j) + \\hat{b^l_j}$, we set $\\forall j\\in [L(t-1)+1, Lt],  \\hat{b^l_j} = b^l_t / L$). \n\nFor T-GIM layers, due to the precondition that the multi-threshold model in the training stage is based on the uniform distribution of the input current within each time window, the values at each position within the $L\\times L$ windows need to be divided by $L\\times$ while expanding the shape of T-GIM in the inference stage (to ensure $\\sum_{i=1}^T \\omega^l_{ti} I^l(i) = \\sum_{j=L(t-1)+1}^{Lt} \\sum_{i=1}^{LT} \\hat{\\omega^l_{ji}} I^l_{IF}(i)$, we set $\\forall i\\in[1,T], \\forall j\\in [L(t-1)+1, Lt], \\forall k\\in [L(i-1)+1, Li], \\hat{\\omega^l_{jk}} = \\omega^l_{ti} / L$). \n\nFor spiking neural layers, the multi-threshold model during the training phase has no membrane potential leakage within each time window, and the leakage degree between windows can be directly copied from the leakage parameters $\\lambda^l$ during the training phase ($\\forall t \\in [1,T], \\forall j\\in [L(t-1)+1, Lt), \\hat{\\lambda^l_j} = 1; \\forall j=L,...LT,  \\hat{\\lambda^l_j} = \\lambda^l_{j/L}$).", "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": "", "extra_scores": {}}, {"review_id": "5UUoiPlLMe", "reviewer": "Reviewer_JrEi", "summary": "This paper proposes a learnable multi-hierarchical threshold model for SNNs and call it LM-HT. The authors theoretically analyzes the equivalence between LM-HT, vanilla spiking models, and quantized ANNs, and demonstrate that LM-HT can achieve comparable performance comparable as quantized ANNs under a two-stage training-inference framework. The authors also introduce a hybrid training framework that improves the performance of converted SNNs within fewer time-steps.", "questions": "* From Fig.1(d)-(e), it seems that the surrogate function used in this work is the Rectangle Surrogate Function. How about other types of surrogate functions?", "limitations": "This work can benefit from further discussion on the details of model parameters and method implementation.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "* **Theoretical Analysis**: This paper provides a rigorous mathematical analysis of the equivalence between LM-HT and other models, offering a new perspective of SNNs and their learning capabilities.\n\n* **Performance Improvement**: Experimental results demonstrate that LM-HT significantly outperforms previous SOTA methods on various datasets.", "weaknesses": "- **Limited Analysis of T-GIM**: While the paper introduces the Temporal-Global Information Matrix (T-GIM), it is unclear how it affects the learning process and performance of the LM-HT model.\n\n- **Advantages of the LM-HT model**: While it is interesting to combine SNNs with quatization, it is unclear what advantages the proposed models have over traditional SNNs and quantized ANNs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## To Reviewer JrEi\nThanks for your insightful and valuable feedback. We are glad that you found our work \"provides a rigorous mathematical analysis\", \"offering a new perspective of SNNs\" and \"significantly outperforms previous SOTA methods\". We would like to address your concerns and questions in the following.\n\n> 1. While the paper introduces the Temporal-Global Information Matrix (T-GIM), it is unclear how it affects the learning process and performance of the LM-HT model.\n\nThanks for the question. From the perspective of forward propagation, as shown in Fig.2, we can observe that the T-GIM layer can encompass two training modes: vanilla STBP training (when $\\Omega^l = \\text{diag}(1,...,1)$) and quantized ANN training (when $\\forall i,j\\in[1,T], \\omega_{ij}^l=\\frac{1}{T}$). In the training stage of LM-HT SNN, we set the T-GIM layer parameters to be learnable, while combining the learnable leakage parameters $\\lambda^l$ of the multi-threshold model, so that the LM-HT model can be dynamically optimized to any parameter state between vanilla STBP training and quantized ANN training.\n\nFrom the perspective of back-propagation, when we want to enhance the learning properties of LM-HT SNN simliar to that of quantized ANN, we can detach the temporal gradient term $\\frac{\\partial m^l(t)}{\\partial m^l(t-1)}$, then we will have $\\frac{\\partial \\mathcal{L}}{\\partial W^l} = \\sum_{t=1}^T \\frac{\\partial \\mathcal{L}}{\\partial m^l(t)} \\sum_{j=1}^T \\omega^l_{tj} s^{l-1}(j)\\theta^{l-1}$.\n\nHere $\\sum_{j=1}^T \\omega^l_{tj} s^{l-1}(j)\\theta^{l-1}$ denotes the comprehensive global information that extracted from different time windows; while $\\frac{\\partial \\mathcal{L}}{\\partial m^l(t)} = \\sum_{j=1}^T \\frac{\\partial \\mathcal{L}}{\\partial m^{l+1}(j)} \\omega^{l+1}_{jt} \\theta^l {W^{l+1}}^{T} \\frac{\\partial s^l(t)}{\\partial m^l(t)}$, which means that the gradients at each time-step is the comprehensive weighted result of the gradients at each time-step in the post-synaptic layer. At this point, we can observe that LM-HT SNN maintains computational equivalence with quantized ANN in both forward and backward propagation, while T-GIM layer plays an important regulatory role in it.\n\n> 2. While it is interesting to combine SNNs with quantization, it is unclear what advantages the proposed models have over traditional SNNs and quantized ANNs.\n\nThanks for the question. The essence of the LM-HT model is to extract information from spike sequences in units of time windows with specific length, while the introduction of the T-GIM layer ensures that the LM-HT model can simultaneously focus on global information on the time dimension. In addition, the precondition that the input current is uniformly distributed within the time window has enabled the performance of LM-HT SNN to reach the same level as quantized ANNs.\n\nBy comparison, vanilla spiking models can only extract local temporal information unidirectionally and have a certain degree of loss in learning accuracies, while quantized ANNs can only learn global average information on the time dimension. As shown in  Fig.2, we can note that the LM-HT model has the advantages of these two training modes and can effectively avoid the related disadvantages of them.\n\n> 3. From Fig.1(d)-(e), it seems that the surrogate function used in this work is the Rectangle Surrogate Function. How about other types of surrogate functions?\n\nThanks for pointing it out. We have tried other types of surrogate functions before, but we tend to think that Rectangle Function is the most ideal choice. As the membrane potential of multi-threshold models before firing spikes actually represents comprehensive information within a time window, therefore, its surrogate gradients cannot be directly calculated by fitting Heaviside Function. \n\nFor example, when $m^l(t)\\in(k \\theta^l, (k+1)\\theta^l), k=0,...,L-1$, for vanilla surrogate functions (e.g. Sigmoid Function, Triangle Function) which aim to fit Heaviside Function, $\\frac{\\partial s^l(t)}{\\partial m^l(t)} \\rightarrow 0$, thereby leading to the gradient vanishing problem, although they can more richly reflect membrane potential information at each time-step. Therefore, we tend to think that this series of surrogate functions might be more suitable for vanilla single-threshold models.", "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": "", "extra_scores": {}}, {"review_id": "imairlicz0", "reviewer": "Reviewer_wxgA", "summary": "Traditional SNNs adopt binary spike communications, resulting in relatively poor performance compared to their ANN counterparts. In this work, the authors propose an advanced Multi-hierarchical Threshold (LM-HT) Model. In my view, LM-HT models introduce several low-bit representation precisions to model the spiking number during short time periods. This perspective not only brings a nice mathematical relationship with quantized ANN training but also enhances SNN performance on several image classification tasks and certain types of neuromorphic datasets.", "questions": "The authors lack sufficient disucssions for its limiations. The proposed methods seem to be hard to extend to a broader range of sequential learning tasks.", "limitations": "-\tIn Table 3, the types of learning methods should be well explained in the caption. What is the meaning of efficient training?\n\n- It is not straightforward to understand the time-steps in CIFAR10-DVS, as it is normally assumed to classify a gesture sequence. Given such high performance with only two time steps, have the authors tested the results with a single time step?", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The LM-HT offers a nice perspective on how to further improve SNN performance with solid theoretical analysis. The overall idea makes sense to me, and the writing is generally good.", "weaknesses": "- It is less significant to only report the best accuracy. The authors are encouraged to provide mean accuracy and standard deviation over at least three different seeds for statistical significance.\n- The use of multi-precision spikes in SNNs has been reported in several previous studies. To cater to a broader background of readers, the authors should better clarify their motivations, either from the biological rationale or performance perspective.\n\nMinor issues:\n- The authors should correct the name of DVSCIFAR-10 and several typos, such as “with larger parameter scale” to “with a larger parameter scale.”", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## To Reviewer wxgA\nThanks for your insightful and valuable feedback. We are glad that you found our work \"offers a nice perspective\", \"has solid theoretical analysis\" and \"the writing is generally good\". We would like to address your concerns and questions in the following.\n\n> 1. The authors are encouraged to provide mean accuracy and standard deviation over at least three different seeds for statistical significance.\n\nThanks for the question. Due to time constraints, we conducted 5 rounds of validation using different random seeds on CIFAR-10/ResNet-18 and obtained an statistical accuracy result of $96.20$% ± $0.10$%, which can verify the superiority of our method.\n\n> 2. The use of multi-precision spikes in SNNs has been reported in several previous studies. To cater to a broader background of readers, the authors should better clarify their motivations.\n\nThanks for your suggestion. We think that the core motivation and contribution of this paper lies in revealing the relationship between multi-threshold models, vanilla spiking neurons and quantized ANNs, thus establishing a bridge between ANN-SNN Conversion and STBP training, which are two core training algorithms in the SNN community, while previous works focused more on using multi-precision spikes from the perspective of enhancing the richness of information transmission or reducing inference errors in ANN-SNN Conversion learning. We think that this point might be overlooked in previous works based on multi-precision spikes.\n\n> 3. The authors should correct the name of DVSCIFAR-10 and several typos.\n\nThanks for pointing it out. We will revise them in the final submitted version.\n\n> 4. The proposed methods seem to be hard to extend to a broader range of sequential learning tasks.\n\nThanks for the question. As shown in Eqs.(8)-(9), we designed the the back-propagation calculation chain of the LM-HT model in a form similar to quantized ANNs (detach the term $\\frac{\\partial m^l(t)}{\\partial m^l(t-1)}$ and establish the correlation calculation of gradients at different time-steps solely through the T-GIM layer), which means that the LM-HT model is more inclined to learn global sequential information on the time dimension. \n\nWe think that if there is a need to learn towards long-term or local temporal information, one possible solution is to consider adjusting the back-propagation mode of the LM-HT model to the computational mode of vanilla LIF model in STBP training (consider the temporal back-propagation term $\\frac{\\partial m^l(t)}{\\partial m^l(t-1)}$), as shown in Eq.(3).\n\n> 5. In Table 3, the types of learning methods should be well explained in the caption. What is the meaning of efficient training?\n\nThanks for pointing it out. The term \"efficient training\" here actually refers to online training, which means that the backward gradient will be updated immediately after the forward propagation is completed at a specific time-step, thereby keeping the GPU memory overhead at a constant level. we will make relevant modification in the final version.\n\n> 6. It is not straightforward to understand the time-steps in CIFAR10-DVS, as it is normally assumed to classify a gesture sequence. Given such high performance with only two time steps, have the authors tested the results with a single time step?\n\nThanks for the question. As we mentioned in Appendix A.2, we will still divide CIFAR10-DVS into 10 frames, while the T-GIM layer located before the first LM-HT neural layer will compress these 10 frames into the required time-steps $T'$ ($\\Omega^1\\in \\mathbb{R}^{T'\\times T}$). The experimental results in Table 3 indicate that good performance can be achieved on CIFAR10-DVS with just such a simple treatment.\n\nWe find through subsequent experiments that when $T'=1$, LM-HT SNN can still maintain an accuracy of $81.0$%. We think that the above phenomenon might also hint that some sequence classification tasks can be effectively completed through global information on the time dimension.", "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": "", "extra_scores": {}}, {"review_id": "BW4sgnpvv7", "reviewer": "Reviewer_59n9", "summary": "This paper introduces a novel learning model for SNNs that dynamically adjusts input current and membrane potential leakage, enhancing SNN performance to match ANNs. The proposed LM-HT model can be seamlessly integrated with ANN-SNN Conversion frameworks, effectively improving the performance of converted SNNs under low time latency. Extensive experiments demonstrate the model's superior performance on various datasets, achieving state-of-the-art results.", "questions": "See  Weaknesses.", "limitations": "The author has discussed the limitations.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper's strengths include the introduction of the LM-HT model, which significantly enhances SNN performance through dynamic regulation of input current and membrane potential leakage. It also presents a seamless integration with ANN-SNN Conversion frameworks, leading to improved performance under low time latency. Additionally, the extensive experimental validation demonstrates the model's state-of-the-art performance across multiple datasets.", "weaknesses": "1. Checklist should be placed after appendix.\n2. The authors can also compare with paper [1][2] which obtains really good results by MS-ResNet-34 backbone on large imageNet datasets.\n3. The paper might not address the scalability of the LM-HT model to different SNN backbone. The authors can conduct experiments on more backbones (MS-ResNet [3] and Spikformer [4][5]) to demonstrate the effectiveness of the proposed LM-HT SNN.\n4. While the LM-HT model can be reparameterized to a single-threshold model for inference, the paper may not fully explore the implications of this transformation for deployment on neuromorphic hardware.\n\n\n\n\n[1] Yao M, Zhao G, Zhang H, et al. Attention spiking neural networks[J]. IEEE transactions on pattern analysis and machine intelligence, 2023.\n\n[2] Qiu X, Zhu R J, Chou Y, et al. Gated attention coding for training high-performance and efficient spiking neural networks[C]. Proceedings of the AAAI Conference on Artificial Intelligence. 2024, 38(1): 601-610.\n\n[3]Hu Y, Deng L, Wu Y, et al. Advancing Spiking Neural Networks Toward Deep Residual Learning[J]. IEEE Transactions on Neural Networks and Learning Systems, 2024.\n\n[4] Yao M, Hu J, Zhou Z, et al. Spike-driven transformer[J]. Advances in Neural Information Processing Systems, 2024, 36.\n\n[5] Zhou Z, Zhu Y, He C, et al. Spikformer: When Spiking Neural Network Meets Transformer[C]//The Eleventh International Conference on Learning Representations. 2022.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## To Reviewer 59n9\nThanks for your valuable and constructive feedback. We are encouraged that you found our paper \"significantly enhances SNN performance\", \"presents a seamless integration with ANN-SNN Conversion frameworks\" and \"has extensive experimental validation\". We would like to address your concerns and answer your questions in the following.\n\n> 1. Checklist should be placed after appendix.\n\nThanks for pointing it out. We will make relevant revision in the final version.\n\n> 2. The authors can conduct experiments on more backbones (MS-ResNet [3] and Spikformer [4, 5]) to demonstrate the effectiveness of the proposed LM-HT SNN. The authors can also compare with papers [1, 2] which obtain really good results by MS-ResNet-34 backbone on large imageNet datasets.\n\nThanks for your suggestions! As shown in Table R1-R2, we have made comparison with related advanced works [1, 2, 3, 4, 5] on CIFAR and ImageNet datasets. One can find that our LM-HT model has superior scalability and can demonstrate its effectiveness on multiple different backbones. \n\nFor example, for CIFAR-100 and ImageNet-1k datasets, compared to GAC-SNN [2], we achieve accuracy improvements of 1.35% and 1.13% on MS-ResNet-18/34, respectively. For Transformer-4-384 architecture, our method also outperforms Spikformer [5] and Spike-driven Transformer [4] in terms of performance.\n\n**Table R1: Comparison with MS-ResNet series works [1, 2, 3].**\n\n| Dataset | Arch. | time-steps | MS-ResNet [3] | Att-MS-ResNet [1] | GAC-SNN [2] | **Ours** |\n| --- | --- | --- | --- | --- | --- | --- |\n| CIFAR-10 | MS-ResNet-18 | 4 | 94.92 (6 steps) | - | 96.24 | **96.38** |\n| CIFAR-100 | MS-ResNet-18 | 4 | 76.41 (6 steps) | - | 79.83 | **81.18** |\n| ImageNet-1k | MS-ResNet-34 | 4 | 69.43 (6 steps) | 69.35 (6 steps) | 69.77 | **70.90** |\n\n**Table R2: Comparison with Spikformer series works [4, 5].**\n\n| Dataset | Arch. | time-steps | Spikformer [5] | Spike-driven Transformer [4] | **Ours** |\n| --- | --- | --- | --- | --- | --- | \n| CIFAR-10 | Transformer-4-384 | 4 | 95.51 | 95.6 | **95.82** |\n| CIFAR-100 | Transformer-4-384 | 4 | 78.21 | 78.4 | **79.03** |\n\n> 3. While the LM-HT model can be reparameterized to a single-threshold model for inference, the paper may not fully explore the implications of this transformation for deployment on neuromorphic hardware.\n\nThank for the question. As shown in Fig.3, the main differences before and after reparameterization are the adjustments to the bias terms of Conv&BN layers, the shape of T-GIM layers, and the membrane leakage parameters，which aims to keep the sum of the input currents within each time window during the inference phase consistent with the expected input currents at the corresponding time-step during the training phase, thus further ensuring that the sum of spike sequences emitted by each layer within each time window is consistent with the multi-bit spikes emitted at the corresponding time-step during the training phase.\n\nBesides these, there have been no changes in other parts of the network. Therefore, the reparameterized LM-HT SNN has consistent network architecture and firing mechanism with the SNN based on the vanilla LIF model, and we believe it can be effectively deployed on neuromorphic hardware.\nThe experimental results in Table 2 also confirm that after being reparameterized and converted to a vanilla single-threshold model, the LM-HT model can maintain consistent performance and SOPs with the multi-threshold model before the reparameterization procedure.\n\n\n[1] Yao M, Zhao G, Zhang H, et al. Attention spiking neural networks[J]. IEEE transactions on pattern analysis and machine intelligence, 2023.\n\n[2] Qiu X, Zhu R J, Chou Y, et al. Gated attention coding for training high-performance and efficient spiking neural networks[C]. Proceedings of the AAAI Conference on Artificial Intelligence. 2024.\n\n[3] Hu Y, Deng L, Wu Y, et al. Advancing Spiking Neural Networks Toward Deep Residual Learning[J]. IEEE Transactions on Neural Networks and Learning Systems, 2024.\n\n[4] Yao M, Hu J, Zhou Z, et al. Spike-driven transformer[J]. Advances in Neural Information Processing Systems, 2024.\n\n[5] Zhou Z, Zhu Y, He C, et al. Spikformer: When Spiking Neural Network Meets Transformer[C]. The Eleventh International Conference on Learning Representations, 2022.", "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": "", "extra_scores": {}}]

Author rebuttal:
## To All Reviewers
Thanks for all your constructive and insightful feedbacks! Considering the concerns of some reviewers about detailed implementation of LM-HT SNN in the reparameterization process, we will provide an additional proof here regarding the computational equivalence before and after reparameterization:

### Computational Equivalence about the Reparameterization Process

**Theorem R.1** $\forall t, i\in[1,T], \forall j\in [L(t-1)+1, Lt], \forall k\in [L(i-1)+1, Li]$, when $s^{l-1}(t) = \sum_{j=L(t-1)+1}^{Lt} s_{IF}^{l-1}(j)$, if $\hat{b^l_j} = b^l_t / L, \hat{\omega^l_{jk}} = \omega^l_{ti} / L$, we will have $I^l(t) = \sum_{j=L(t-1)+1}^{Lt} I_{IF}^l(j)$. Here $\hat{b^l}, \hat{\Omega^l}$ denote the rectified bias term and T-GIM layer after the reparameterization process.

*proof* Firstly, it is obvious that $I^l(t), I_{IF}^l(j)$ can be rewritten as $I^l(t) = \sum_{i=1}^T \omega^l_{ti} (W^l s^{l-1}(i) + b^l_i)$ and $I_{IF}^l(j) = \sum_{i=1}^{LT} \hat{\omega^l_{ji}} (W^l s_{IF}^{l-1}(i) + \hat{b^l_i})$.

Considering the precondition $\hat{b^l_j} = b^l_t / L, \hat{\omega^l_{jk}} = \omega^l_{ti} / L$, we will have:

$I_{IF}^l(j) = \sum_{i=1}^{T} \sum_{k=L(i-1)+1}^{Li} \hat{\omega^l_{jk}} (W^l s_{IF}^{l-1}(k) + \hat{b^l_k}) $

$= \sum_{i=1}^{T} \frac{\omega^l_{ti}}{L} \sum_{k=L(i-1)+1}^{Li} (W^l s_{IF}^{l-1}(k) + \frac{b^l_i}{L})$.

Then we can further have:

$ \sum_{j=L(t-1)+1}^{Lt} I_{IF}^l(j) = \sum_{j=L(t-1)+1}^{Lt} \sum_{i=1}^{T} \frac{\omega^l_{ti}}{L} \sum_{k=L(i-1)+1}^{Li} (W^l s_{IF}^{l-1}(k) + \frac{b^l_i}{L}) $

$= \sum_{i=1}^{T} \frac{\omega^l_{ti}}{L} \sum_{j=L(t-1)+1}^{Lt} (W^l \sum_{k=L(i-1)+1}^{Li}  s_{IF}^{l-1}(k) + b^l_i) $

$= \sum_{i=1}^{T} \frac{\omega^l_{ti}}{L} \sum_{j=L(t-1)+1}^{Lt} W^l (s^{l-1}(i) + b^l_i) $

$= \sum_{i=1}^T \omega^l_{ti} (W^l s^{l-1}(i) + b^l_i)$

$=I^l(t)$.

Due to the fact that the calculation process of the spike sequences passing through Conv&BN and T-GIM layers can be abstractly described by Theorem R.1, we can conclude that the sum of the input currents within the corresponding time windows before and after reparameterization remains unchanged. The spike sequences obtained by passing the input currents through the spiking neuron layer will also satisfy the precondition of Theorem R1 ($s^{l}(t) = \sum_{j=L(t-1)+1}^{Lt} s_{IF}^{l}(j)$). Therefore, we can prove the computational equivalence before and after the reparameterization process.

Meta-review: The reviewers concurred that this work (1) has solid theoretical analysis and empirical validation, and (2) will be of interest to the NeurIPS community. The authors' reply to the reviewers comments helped strengthen the paper's position, both on the theory side (by showing additional derivations and providing justification of the method's usefulness) and the empirical side (by providing further experimental validation and better statistics).
Decision: Accept (poster)