Spiking Transformer with Experts Mixture
Authors: Zhaokun Zhou
Yijie Lu
Yanhao Jia
Kaiwei Che
Jun Niu
Liwei Huang
Xinyu Shi
Yuesheng Zhu
Guoqi Li
Zhaofei Yu
Li Yuan
Venue: NeurIPS 2024 poster (2024)
Keywords: Spiking Neural Networks; Transformer; Mixture of Experts; Neuromorphic Computing; Spike-driven;
Primary area: machine_learning_for_other_sciences_and_fields
TL;DR: A Spike-driven Mixture of Experts Mechanism for the Spiking Transformer.

Abstract: Spiking Neural Networks (SNNs) provide a sparse spike-driven mechanism which is believed to be critical for energy-efficient deep learning. 
Mixture-of-Experts (MoE), on the other side, aligns with the brain mechanism of distributed and sparse processing, resulting in an efficient way of enhancing model capacity and conditional computation. 
In this work, we consider how to incorporate SNNs’ spike-driven and MoE’s conditional computation into a unified framework. 
However, MoE uses softmax to get the dense conditional weights for each expert and TopK to hard-sparsify the network, which does not fit the properties of SNNs. 
To address this issue, we reformulate MoE in SNNs and introduce the Spiking Experts Mixture Mechanism (SEMM) from the perspective of sparse spiking activation. 
Both the experts and the router output spiking sequences, and their element-wise operation makes SEMM computation spike-driven and dynamic sparse-conditional. 
By developing SEMM into Spiking Transformer, the Experts Mixture Spiking Attention (EMSA) and the Experts Mixture Spiking Perceptron (EMSP) are proposed, which performs routing allocation for head-wise and channel-wise spiking experts, respectively. Experiments show that SEMM realizes sparse conditional computation and obtains a stable improvement on neuromorphic and static datasets with approximate computational overhead based on the Spiking Transformer baselines.

Spiking Transformer with Experts Mixture
Zhaokun Zhou Yijie Lu!,
Yanhao Jia
Kaiwei Chell2, Jun Niu!, Liwei
Huang*2 4.2 Xinyu Shis.4,
Yuesheng Zhu',
Guoqi Li
Zhaofei Yu
Li Yuan
School of Electronic and Computer Engineering, Shenzhen Graduate School, Peking University
*Peng Cheng Laboratory
College of Computing and Data Science, Nanyang Technological University
"Scoool of Computer Science, Peking University
Inntiitte for Artificial Intelligence, Peking University IInttittte of Automation, Chinese Academy of Sciences Deep NeuroCognition Lab, 12R and CFAR, Agency for Science, Technology and Research
Abstract
Spiking Neural Networks (SNNs) provide a sparse spike-driven mechanism which is believed to be critical for energy-efficient deep learning. Mixture-of-Experts (MoE), on the other side, aligns with the brain mechanism of distributed and sparse processing, resulting in an efficient way of enhancing model capacity and conditional computation. In this work, we consider how to incorporate SNNs spike-driven and MoE's conditional computation into a unified framework. How- ever, MoE uses softmax to get the dense conditional weights for each expert and TopK to hard-sparsify the network, which does not fit the propertics of SNNs. To address this issue, reformulate MoE in SNNs and introduce the Spiking Experts Mixture Mechanism (SEMM) from the perspective of sparse spiking activation. Both the experts and the router output spiking sequences, and their element wise operation makes SEMM computation spike-driven and dynamic sparse-conditional. By developing SEMM into Spiking Transformer, the Experts Mixture Spiking At- tention (EMSA) and the Experts Mixture Spiking Perceptron (EMSP) are proposed, which performs routing allocation for head-wise and channel-wise spiking experts, respectively. Experiments show that SEMM realizes sparse conditional computa- tion and obtains stable improvement on neuromorphic and static datasets with approximate computational overhead based on the Spiking Transformer baselines.
Introduction
The spiking neural networks (SNNs) are regarded as the third generation of neural networks M. distinguished by biological plausibility (2). spike-driven characteristic, and low power consumption. SNNs emulate the dynamics of biological neurons at a microscopic level, utilizing asynchronous binary spikes for information transmission. The membrane potential of spiking neurons in SNNs is only updated upon the arrival of spikes, avoiding calculations of zero values. The inherent features make SNNs promising candidates for low-energy consumption neuromorphic hardware, such as TrueNorth 3 and Loihi [4. There are lots of architectures in SNNs include Spiking Recurrent Neural Networks (5) ResNet-like SNNs 9 Spiking Graph Neural Networks [10,, and Spiking
Transformers [1113]] Spiking Transformers stands at the forefront. Spikformer [1]] introduces Spiking Self-Attention (SSA). The Spike-Driven Driven Transformer [12 introduces Spike-driven Self-
'Equal *Coresponding author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Attention. Other works explore the Spiking Transformer in termss of structural improvements 14 17
training methods [8], and different tasks 19] respectively.
Mixture-of-Expere (MoE) [20 21 is known for allowing each expert to learn specific tasks or features, showing better performance, conditional computing and dynamic adaptability, which are crucial features in the brain mechanism [22] 23). In this work, we are committed to exploring the effective integration of MoE and Spiking Transformer. As shown in Fig a). MoE introduces large number of parameters based on the original Transformer, and the conditional computation is achieved by calculating the routing probability of each token on each expert through the softmax function. Selecting Top-K experts based on the routing probability, MoE achieves hard sparsification. However, SNN calculations need to avoid multiplication and cannot use complex softmax functions. The features in SNNs are dynamically sparse and do not require additional TopK. All the parameters of the expert must loaded, which aggravates the difficulty of neuromorphic chip deployment. These factors make porting MoE to SNNs non-trivial. To tackle this problem, we develop the Spiking Experts Mixture Mechanism (SEMM), as shown Fig universal SNN-MoE paradigm with the following three main features. 1) The SEMM is spike-driven. The outputs of the expert and the router are spike sequences, and the glement-wiss operation between them conforms to the SNN char- acteristics, i.e., avoiding multiplication. 2) SEMM leverages the sparse spiking activation of SNNs to achieve dynamic conditional computation of MoEs, which more flexible than the fixed hard sparsi- fication of Artificial Neural Network MoE (ANN-MoE). 3) With reasonable parameter count settings, SEMM enables Spiking Transformers to achieve stable performance gains with negligible overhead.
Fig[a),
Based on SEMM, we modify the Spik- ing Self-attention (SSA) and Multi-layer Perceptron (MLP) of Spiking Transform- ers to obtain the Experts Mixture Spiking Attention (EMSA) and the Experts Mix- ture Spiking Perceptron (EMSP). EMSA treats each head of SSA as an expert, com- putes respective attention, and employs temporal-aware router to integrate atten- tion. As for Multi-layer Perceptron (MLP), is common to substitute the entire ML.P with MoE (24). which leads better per- formance but significant increase in the overall parameter number. EMSP imple- ments a channel-wise MoE within MLP to overcome the shortcoming. EMSA and EMSP can be inserted directly and seamlessly into existing Spiking Transformer variants
R outer
S pikkng Input
Saftmax
Spiking
Saking aaking
Rotter
H4M
Equett
Expen2
Expen
4xpert
Spiking
Spiking
lal ANN McE
[b) SEMN
Figure 1: ANN-MoE and Spiking Experts Mixture Mccha- nism (SEMM). N denotes the length of image patches.
[ 4 [2].
Our work contributes in three main aspects:
We introduce the Spiking Experts Mixture Mechanism (SEMM), universal SNN-MoE paradigm. SEMM spike-driven, capable of efficient dynamic sparse conditional computation.
Based SEMM, develop the Experts Mixture Spiking Attention (EMSA), whose information from all head-wise experts is selectively integrated through a temporal-aware router. We restructure MLP by channel-level spiking-sparse SEMM, named the Experts Mixture Spiking Perceptron (EMSP). They can seamlessly replace self-attention and MLP in Spiking Transformers.
Extensive experiments demonstrate the stable performance improvement of SEMM on both static and neuromorphic datasets. Notably, Spike-driven Transformer-8-55 with SEMM achieves remarkable 76.62% accuracy on ImageNet with 4 time steps, surpassing the baseline (74.57%).
2 Related Work
Deep Spiking Neural Networks and Spiking Transformers. Spatio-temporal backpropaga- tion(STBP) (25] directly trains SNNs by performing backpropagation on both spatial and temporal domains. Temporal backpropagation (26] computes the gradients of the timings of existing spikes for the membrane potential at the spike timing. Treshold-dependent batch normalization (idBN) g used to extend the network depth. Sew-ResNet 7 proposed the spiking element- wise residual for SNNs. Spikformer [I] firstly converts all the components of Vision Transformer (ViT) into

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Linear BN
Linear BN
BN
AULA
#1
DWCC
PS
RPE
m
A1
A2
1 x S2000
SSA
i
I
MLP
piking R Router
Spiking Router
Linear BN
Element-wise Add
DWC
CH
Hadamard Product Depth-wise Conv
Spiking Transformer
(b) EMSA
(c) EMSP
Figure 2: (a) The overview of Spiking Transformer. (b) The Experts Mixture Spiking Attention (EMSA). (c)
The Experts Mixture Spiking Perceptron (EMSP). EMSA and EMSP can directly replace SSA and MLP (a).
spike-form and pioneers the field of SNN-Transformer. Spike-driven Transformer 12] goes further by introducing linear spike-driven self-attention. Spikingformer [4] proposes hardware-friendly spike-driven residual learning architecture. Besides, Masked Spiking Transformer [27 combines SNNs and Transformers from the perspective of the ANN-to-SNN conversion. However, the SNN Transformer architecture that can bring SNNs' superiority into full play is still ongoing research.
Mixture-of-Experee
Expertss... The Hixture-of-Expeer (MoE) 28 29 combines the predictions of multiple
specialized experts, which effective in handling high-dimensional data and complex problems.
Researchers explore diverse gating mechanisms (30
31
optimizing expert allocation strategies [32
33), and enhancing the scalability of MoE models [34 35 Sparsely-Gated Mixture-of-Experts (30
adopted MoE into architectures such as the Long Short-Term Memory (LSTM) [36), showcasing
effectiveness in Language Modeling. The Transformer also benefits from MoE with the substitution
of the Multi-layer Perceptron (MLP) [24] 37). Switch Transformer (35) has scaled the models with trillions of parameters. Currently, there is no existing work on MoE with Spiking Transformers.
Methodology
3.1 Preliminaries and Overall Architecure
Spiking neuron the basic unit of SNNs. For the dynamics the Leaky Integrate-and-Firr (LIF) neuron used in this work, the t - th-time-step membrane potential (tt) is equal to the sum of the state potential H exceeds the threshold ues, the neuron will fire a spike, otherwise, it remains inactive, Consequently, the output S[t] only contains binary values, either 1 or O. Hea(-) is Heaviside function that satisfies Hea. (x) = when 2 0, otherwise Hea (x) = 0. H[t) is the temporal state output, and Vreset denotes the reset potential after a spiking event. ,3 < 1 determines the rate of decay. the neuron remains inactive, the potential [(f] decays towards H [t] over time. LIF can be described as:
at the previous time step and the input [t). When membrane potential
U[t] = H - 1] + [t),
(1)
S[t] Hea (U upk)
(2)
H[4 Vreser
(BU(4) (1 S(())
Spiking Transformer [I] 14 12] baselines contain Patch Splitting (PS), Relative Position Em- bedding (RPE), Spiking Self Attention (SSA) (e.g., Spiking Self-Attention [1]] and Spike-driven Self-Attentin MLP and linear classification head, as shown in Fig. [[.. Given a image sequence I E x where T, C, H. and W denote time-step, channel, height and width, the PS splits it into a sequence N flattened spike patches x with D dimensional channel. Convolution-Be
BatchNorm-Lee Convolution-BatchNooee
LIF block generates Relative Position Embedding (RPE):

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X, PS (I) RPE,
(4)
X; SSA(X,-1) X1-1,
(5)
X, MLP((X) + +X,,
Y CH(GAP(X:)),
where the Xo is fed into the T-blocks and each block consists of SSA and a MLP. X,, X, are spike sequences and I = I...I is layer. A Global Everage-pooling (GAP) is utilized on the X and the linear Classification Head (CH) to output the prediction Y. See Appendix, for more details.
3.2 Spiking Experts Mixture Mechanism
Before exploring the adaptation of spiking MoE in SNNs, we first review MoE in ANNs. An ANN- MoE layer typically comprises a set of m experts EA {E,, E2...... .... Em along with router RA for selecting the corresponding experts. Given an input sequence XA, the resulting output can be expressed as the sum of the Top-K selected experts from m candidates using router:
y
(x( Ex(XA),
(8)
1=
R(x) =
XA W We, K)),
(9)
TopK(v, K) = v if v is in the top K elements
(10)
otherwise
where WR is the router weight matrix. The TopK(-; K) together with the softmax(.) sets all elements of the routing vector zero except the largest top K values. The sparse conditional computing is obtained from the selecting of Topk and the different routing weights of the softmax, while K is usually taken 0.5 m. We argue that the ANN-MoE is not suitable for SNN for two main reasons. First, the float-point routing-expert and the softmax which involve exponentiation and division, do adhere to the computation principles of SNNs. Second, SNN experts are inherently highly sparse, so the hard sparsification approach of additional TopK is unnecessary. An event- triggered spike-based sparse conditional computation on asynchronous neuromorphic chips needed more than TopK. To bridge these gaps, we introduce a generalized representation of the Spiking Experts Mixture Mechanism (SEMM), which is as follows.
SEMM(E, R, F(-)) = F(E, R).
(11)
where (E, E2, .... Em represents the spiking sequence of m spiking experts in the Spiking Transformer, and R E (0, 1 xm represents the spiking router for allocating computation. F(-) denotes the element-wise form of the operation between the router and the expert output spiking sequence, i.c., Hadamard product and addition.
As MoE mechanism specifically designed for SNNs, SEMM has the following three significant advantages. i) Spike-driven. The SEMM is spike-driven, which important for SNNs. Due to the spike-driven computation mode of experts, i.e., Spiking Self-attention and Spiking-MLL SEMM computations are triggered by sparse spikes of experts and require only synaptic manipulation. For example, the Hadamard produet between the spiking signals R and E is equivalent to mask. ii) Sparse-spiking conditional computation. The SEMM subtly utilizes the sparse activation of the spiking routers for the conditional computation of MoE. SEMM has variable sparsity when dealing with different data. Additionally, SEMM does not suffer from the load imbalance problem in ANN- MoE, i.e., TopK selects fixed number of experts. The sparse conditional computation is distributed to each expert. iii) Eflicient computation. Unlike loading with multiple heavy expert modules of ANN-MOE, the SEMM has comparable parameters and operations the previous SSA and MLP of Spiking Transformers. These are further discussed in Sec. 3.5 Without loss of generality, we use the mainstream architecture of Spiking Transformer for SEMM embedding Sec. 3.3 and Sec. 3.4
3.3 Experts Mixture Spiking Attention
We begin by reviewing the processing of Spiking Self-Attention (SSA). Different from vanilla ANN- Transformers (38). it discards the softmax normalization for the attention map. The SSA mechanism

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can be described by the following equation:
= SN (=SNQLL-EN((XXI K SN (L-BNL-) V (L
(12)
SN s), for Spiking Self- Attention SN (SUMc (Q K)) V for Spike- Driven Self-Attentin
A = S.) =
(13)
where SN represents the spiking neuron and L.-BN represents that the features pass sequentially through Linear and BatchNorm. is the scaling factor and Q K. V {0, 1) 1)TxXxx are in spike- form. A E (0, 1)>xNxD is the spiking output of SSA. O is the Hadamard product, and (?M
denotes the sum of each column. As shown in Fig 2 b), the Experts Mixture Spiking Attention (EMSA) based on SEMM and SSA is formulated as:
Am SSA(Qm: m: K, V), E = {A1, A Amh
(14)
(15)
R = (N((N((Wx)) RR)
fri F.. Fm):
(16)
SEMM(E, R. F(-)) =
Σ + A,
(17)
EMSA = (B(NESSEM R. F())))
(18)
where Am E {0, (0,1]7xNxx is the output of each SSA experts, and WR E Dxm is the router weight matrix. R E {0, 1] is the routing sequence. We set d S D by default to avoid introducing many parameters. The operation F(-) between expert-router is computed by masking the expert using router pair-by-pain and then summing them up. The output of EMSA is obtained by performing matrix multiplication between SEMM(E, R, F(-)) and synaptic weight W, R the same as the last layer of SSA block in Spiking Transformers.
3.4 Experts Mixture Spiking Perceptron
As illustrate in Fig. reffigmmehodd, the Experts Mixture Spiking Perceptron (EMSP) launches the channel-wise SEMM within MLP. The architecture can be written as follows:
E SX(IDNCCSS
{0, 1) Nxm
(19)
R = (XWR)) E 1XxNNm
(20)
SEMM(E, R. F(-) E
(21)
FMSP SN SBB(BBBSEM R, F())W.) E (0.1)TXXXD,
(22)
where WI, WR E Rxmm is the weight matrix of first layer MLP and router, respectively. We integrate 3 x 3 Depth Wise Convolution layer DWC to capture local features each channel expert, which has fewer parameters and is computationally efficient compared to original convolution. W. m D is the wight matrix of output layer. The original Spiking Transformer's 's MLP would have a D to 4 D channel dimension increase after the first layer and the second layer would reduce the dimension back to D. In EMSP the original MLP. EMSP integrates sparse routing of multiple thannel-wise experts. It can also be viewed as the channel- wise gating, which is suitable for temporal information processing, allowing information to flow unimpeded through potentially many time steps 39 The gate (router) branch and expert branch can be also regarded as incorporating the Spike Element- -eine (SEW) 7 residual block into EMSP. Channel-wist convolution and element-wise (Hadamard) products only introduce a minor increase in computational cost. By appropriately configuring the number and dimensions of expert networks, our EMSP achieves more efficient computation compared to the original spiking MLP. More details of the computational overhead are given in the next sub-section.
set to (8//3) D to try match the parameter number of
3.5 Characteristics of SEMM
We explain each of the three advantages of SEMM, i.e., Spike-driven, Sparse-spiking conditional computation and efficient computation.
Spike-driven has the following formal definition, meaning that the gathering of input current is initiated by sparse spikes released from presynaptic neurons:
5

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Definition 1. In SNN, the operation is spike-driven if the input currents satisfy the following form,
Σ W s
Σ
Wij.
(23)
where I&( is the input current of the i-th postsynaptic neuron time step 8,14 E {0, is the spike oulpul of the j-th pre-synaptic neuron, Waj is the weight of the synaptic connection from j to i.
For EMSA, the input current for the LIF in Eq. 18 at specific time step t is given by:
[[4] = SEMM R(),
r:l4
+ A
(24)
1=1
m
1.44 = Σ rl Σ atppa4/uur
Σ I
W1,4
(25)
where [[t] has a dimension of Px Q, with and q represent the p-th row and g-th column, respectively, EMSA is essentially spike element-wise addition after masking operation on SSA, consistent with the characteristics of spike-driven. For EMSP, the operation between two spiking sequences E and R which also conforms to the spike driven,
corresponds to the logical AND function 7
I[t] SEMM R(t)
(Elt) O R.
(26)
1..04 = Σ
ep em.10mmm4400
Σ
W1.a
(27)
(ogaltinrpp 20) (0)
Sparse-spiking conditional computation means using spiking routers to dynamically allocate the computation in temporal and spatial dimension. More analysis is detailed in experiments.
CIFAR100
CIFAR10-DVS
12.5
Efficient computations means that SEMM approximates SSA and MLP in terms of number of parameters and the- oretical synaptic operations. Tab. [ demonstrates the computation load for EMSA, EMSP versus SSA and MLP. In terms of parameter number, despite having routing layer mD EMSA has a smaller number of parameters than SSA when the number of experts is within reason- able range (2 2). EMSP additionally introduces a 3 x 3 depthwise convolution, and the number is slighter higher than MLP. Due to the similarity of the actual spiking rates, EMSA is smaller than SSA on the TN D term, while the calculation of the additional introduced by SEMM is on the TND term (much smaller than D22 and there- fore can be ignored. The situation is similar on EMSP, with depth-wise convolution adding a slight computational overhead. The design of EMSP is different from ANN-MoE, which selects multiple heavy MLPs as experts. We demonstrate it's validity Fig. i..., it performs better while the number of parameters is much smaller than ANN-MoF.
70.5
51.0
10.0
71.5
10.5
Splurer Spiingbrra
15
15
70
5
Permmn (M]
Pwamre (4)
Figure 3: Comparison of parameters- accuracy for different MoEs. Methods of each line from left to right correspond to: 1. the baseline, 2. EMSP, 3. ANN router with four heavy MLP experts, 4. Spiking-route with four heavy MLP experts. respectively. The last two softmax and TopK (K 2).
4 Experiments
4.1 Sparse Conditional Computation Analysis
We analyze the average spiking rate (ASR) of routers for EMSA and EMSP on the ImageNet validation set, which is shown in Tab. 2 The 10-TTrnsformer-85511 is used for the analysis. The ASR of EMSA is around 0.5, which comparable to the regular TopK setting of ANN-MoE. The ASR of EMSP is low and gradually decreases the block deepens, compared to the fixed TopK hard sparse in ANN-MoF, SEMM fully reflects the advantage of SNN dynamic sparse conditional computation. To further verify the spiking router, we ablate it, i.c., cancel it in EMSA and EMSP, as

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Table I: Number parameters and theoretical synaptic operations. R. R, R and R denote the average spike firing rates (the proportion of non-zero elements in the spike matrix) in various spike matrices. T, N, and D are the time step, sequence length, and channel dimension of the input features, respectively. d is the channel dimension of Am and m is the number of experts. The details are provided in the Appendix. B
Param
OPs
SSA
4
ARTN 2R
EMSA
(1+ D2 + (2d + m)D (1+ RTND (2d m) BNRTND (D md) RTN
MLP
8D2
EMSP
8D 24D
SRNND2 24RTND
CIFAR10
CIFARIOO
96.2
80.4
96.0
75.8
EHSP Router
wio
05.0
Acaracy (88
74.2
05.8
70.6
95.4
78.0
95.2
17.4
95.0
76.0
Spiclormet
Spike-drioen Tatuformer
Spitinglorme
Spiiformer
Spikedriven Transdfrmer Spikingformr
Figure 4: Ablation study on the EMSA and EMSP router.
shown in Fig. 4 the accuracy decreases significantly after canceling the router, and even lower than the baseline model on the Spikingformer. Directly analyzing the conditional computation of SEMM is challenging, we therefore use visualizations to illustrate this intuitively. As show in Fig. for the same image, each expert's router assigns a different computational region, e.g., the third router filters the background the expert's features, while the second router assigns the computation to the foreground target. The computation allocation of the spiking router to the irregular object "snake" " can be seen that the router is highly dynamic and effective. See Appendix. [ for more samples. In addition, shown in Fig. 6 we also report the ASR of spatial-temporal locations of routers in different kinds of images. As can be seen by the difference in average firing rates, spiking router has dynamic adjustment of ASR processing different kinds of images, further illustrating its data-dependent conditional computation property.
Table 2: The average spike rate of EMSA and EMSP router in 8 blocks testing the ImageNet.
Block0
Block1
Block2
Block3
Block4
Block5
Block6
Block7
EMSA
0.64
0.68
0.49
0.52
0.47
0.43
0.51
0.50
EMSP
0.25
0.30
0.33
0.27
0.20
0.10
0.05
0.02
4.2 Results on various Datasets
We conduct experiments on static datasets, i.e., Im- ageNet 400 and CIFAR ET), and neuromorphic datasets, i.e., SIFARIO-DVS 42), DVS128 Ges- ture (43 verify the effectiveness of SEMM. See Appendix. C for more details. ImageNet results are shown in Tab. which mainly compares SEMM with the Spiking Transformer baselines. At slightly lower model parameter counts, SEMM steadily su- perior to baselines. For instance, Spikformer-8-511 with SEMM is 2.55% higher than the baseline with 28.22M parameters, Spike-driven Transformer-8-366 with SEMM obtain 2.05% improvement. SEMM on Spikingformer presents similar findings.
662
CIAR10
CHARR100
72.0
Acarrccy 15.
73.6
34.0
37.0
1ptiina
368
Expert Num
Expert Num.
Figure 8: Accuracy with different experts number.

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Figure Visualization routers as masks. mask position (black) indicates router of here background image is the same for each subplot. We show the dynamic sparsity of spiking router for different experts (horizontal direction) and time steps (vertical direction).
(a)
(b)
Figure Average spiking rate of different kinds images in the ImageNct validation in dimension. The height is time step. (a) Japanese spaniel. (b) Volcano.
CIFAR10
CIFAR100
$2.0
1CEMM
+SCOM
99.0
Recuncy
$2.4
552
Spkiformer
Spitn-driven Transfomer
Spikingfamer
Spiifororr
Spike driven Transformer
Spikingfomer
Figure 7: Ablation study of EMSA and EMSP module.
The experimental results on CIFAR, CIFARIO-DVS, DVS128 Gesture are shown Tab.
These four datasets are relatively small. We basically keep the experimental setup in [11 12 including the network structure, training settings, etc., and details are given the Appendix. SEMM has demonstrated stable performance improvements across various datasets when integrated

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Table 3: Results on ImageNet-1k, lk. Model-L--D represents model with L encoder blocks and D channels.
Methods
Architecture
Param (M)
Time Step
Top-1 Ace (%)
SEW-ResNet-34
21.79
67.04
SEW ResNet f
SEW-ResNct-101
44.55
68.76
SEW-ResNet-152
60.19
69.26
MS-ResNet-18
11.69
6
63.10
MS-ResNet
MS-ResNet-34
21.80
69.42
MS MS-ResNet-104"
77.28
76.02
Spikformer-8-365
16.81
70.24
Spikformer
Spikformer-8-5/1
29.68
73.38
Spikformer-8-23 8-3434
16.05
72.86
Spikformer SEMM
Spikformer-8-55
28.22
75.93
SD-Transformer--8B33
16.81
72.28
Spike-driven Transformer
SD-Transformer-8855
29.68
74.57
SD-Transforerr-8-
16.05
73.93
SD Transffforeer SEMM
SD-Transfomer--8-1
28.22
76.62
Spikingformer-
8-344
16.81
72.45
Spikingformer 44
Spikingformer-8-511
29.68
74.79
Spikingformer-8855 8-3-8-84
16.05
73.58
Spikingformer SEMM
Spikingformer-85512
28.22
76.03
Table 4: Results on CIPAR1O-DVS, DVS128 Gesture, and CIFAR.
CIFARIO-DVS
DVS128 Gesture
CIFAR-I0
CIFAR-100
Methods
T
Acc
T
Acc
T
Acc
T
Acc
ldBN S
10
67.80
40
96.90
6
93.20
PLIF 44
20
74.80
20
97.60
8
93.50
DIET-SNN 45
5
92.70
5
69.70
Dspike
46
10
75.40
6
94.30
6
74.20
DSR
47
10
77.30
20
95.40
20
78.50
10
78.90
10
96.90
Spikformer
4
95.19
77.86
16
80.90
16
98.30
82.32
10
97.56
Spikformer SEMM
4
95.78
4
79.04
82.90
16
98.63
78.90
10
96.90
Spike- Spike-Driven Transformer
12
4
95.60
4
78.40
80.00
16
99.30
10
81.10
10
97.56
Spike-Driven Transformer SEMM
96.12
80.26
16
82.42
16
99.30
79.90
10
96.20
Spikingformer
4
95.81
4
79.21
16
81.30
16
98.30
10
80.70
10
96.88
Spikingformer SEMM
96.16
80.24
16
82.10
16
98.56
into different Spiking Transformer baselines. Specifically, SEMM achieves SOTA on CIFAR-10 (96.16%) 16%), CIFAR-100 (80.26%), CIFARIO-DVS (82.9%) and DVS 128 Gesture (99.3%).
4.3 Ablation Study and Hyperparameter Sensitivity
Module ablation. EMSA and EMSP together improve the performance of the baseline, shown in Fig. The use both EMSA and EMSP alone is better than baseline, illustrating their respective superiority.
Experts number. We examine the utility brought by different numbers experts of EMSA. The results of Spiking Transformer baselines CIFAR are presented in Fig. 8 It indicates that within certain range of expert numbers, the results can still be competitive and robust. Among these, we select experts the parameter setting for our final application.

---

5 Conclusion
In this work, we explored the feasibility of adapting MoE in Spiking Neural Networks and formulated an SNN-MoE paradigm named the Spiking Experts Mixture Mechanism. Unlike the vanilla MoE, which uses softmax and TopK hard sparse, SEMM implements dynamic conditional computation from the viewpoint of spiking sparse activation. With redesigned Router-Expert pairs and element-wise spike-driven operations, SEMM is computation-effieenn and SNN-compatbll. Embedded in Spiking Transformers, SEMM-based EMSA and EMSP can bring stable performance improvement on static and neuromorphic datasets. SEMM paradigm can inspire future exploration of high perffrrrrrmm high-capacity Spiking Transformers. We hope SEMM can bring vitality to dynamic conditional computation and the design of next-generation architecture for SNNs. Future work will explore SEMM implementation in a wider range of tasks and larger SNN models.
6 Acknowledgments and Disclosure of Funding
We sincerely thank Dr. Wei Fang for his assistance with this work. This work is sup- ported by grants from the National Natural Science Foundation of China (62088 102, 62202014, 62332002, 62425101, 62236009, U22A20103, 62441606), Shenzhen Basic Research Program the major key project of the Pengcheng Laboratory (PCL2021 A 13) and National Science Foundation for Distinguished Young Scholars (62325603). Computing support was provided by Pengcheng Cloud Brain.
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A Overall architecture of Spike-driven Transformer/Sppkingfrmer with SEMM
We provide detailed description of the overall structure of Spike-driven Transformer with SEMM, and Spikingformer with SEMM. Given 2D image sequence I TR IRTXXxXx* the Patch Splitting Module (PSM), consisted of three Convolution-Banl Convolution-Batee Normalization-Man Maxxpoolinn layer, used to projects and splits I into a sequence of N flattened spike patches with D channels. The Relative Position Embedding (RPE) module, Dnvolution-BatchNone
Normalization-S5 SSN L----Mppooling layers and one
layer, is used to generate position embedding to get Ug. S.((.) means the spike neuron layer. Then the Uo is passed to the L-blocks encoder. The encoder block consists of an Experts Mixture Spiking Attention (EMSA) and an Experts Mixture Spiking Perceptron (EMSP) block. Membrane-shortee
residual connections are applied in both the EMSA and EMSP blocks. global average-pooling (GAP) is utilized on the processed feature from the encoder and outputs the D-dimension feature which will be sent the mully-connected-laye classification head (CH) to output the prediction Y. The architecture can be written as follows:
PSM (T):
I RT 3 .TERTXNDD
(28)
(N),
SERTXNDD
(29)
RPE BB((Con2d(s)),
RPE E RXNxD
(30)
U, = u R.E.
U, E TRNND
(31)
S. SN,,
So E RTxNxD
(32)
U, EMSA(S/-1) + U--1,
U RRRNNND 1...L
(33)
S/ SN(U)),
S/ E 1
(34)
S SNVEMSP(S) U,),,
S E RTXNDD 1=
(35)
Y = CH(GAP(SL))
(36)
Following the Spiking Experts Mixture Mechanism and Spike-driven Self-Attention (SDSA) used in Spike-driver Transformer, the EMSA can be written as:
Q SNQ(L-BNg (X)) K
()), V SNv(L-BNv(X)), A = (Q, K, V) = SN (SUM, (Q @ K)) @ V.
(37)
(38)
The rest operate according to standard EMSA 18| The EMSP in Spike Driven Transformer is also much the same as the standard 22 except that no spiking neuron layer in the output layer.
B
Computation Overhead Details of SEMM
B.1
EMSA
The linear layers that generates m Query Qm have ml (2 = D2 parameters. The linear layer for m
Key K has D parameters. Generating Value V requires d.D parameters. The output linear has dD parameters. The router linear layer transforms to expert number m, and has mD parameters. The number of total parameters is (1 + 1/m) (2d + m m) For operations, obtaining Query and Key needs (1 + 1 //m) RTN 2 Computing Value or the output of EMSA requires d dRT D.. The numher of operations between Qm, K, V is (D + md) Rgg 2 The number router matrix computation operations m/iT D. Due to the sparsity of SEMM and the smaller data dimensions, the number of element- wise operations for the router and expert is negligible.
B.2 EMSP
The parameter number of W1, WR, W. are all (8//3)D2. The number of parameters for 3 x 3 depthwise convolution is 3 x (8//3)D 24D. The all parameter number is 8D2 24D. The operation number of W1, W W. are all (8//3)TND?. The number of operations for depthwise convolution is 24RTND
14

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C Experiment Details
C.1 Experimental Setup
Our experimental framework closely aligns with the methodology outlined in m. All of Baseline's code is publicly available, and we abide by their credentials. For the ImageNet- IK, we utilize fixed timestep count of T = 4. The optimizer is the AdamW, with a batch size of 128 or 256 over the course of 310 training epochs. The learning rate is governed by a cosine-decay schedule, starting from an initial value of 0.0004. We incorporate suite of standard data augmentation techniques, including random augmentation, mixup, and cutmix, into our training. For four small datasets, we adapt the SEMM to variety of baseline models, following the precedents set by [ 12. For CIFAR, we maintain a timestep count of 7 4. For the neuromorphic datasets, we increase this to 7 and
T = 16, respectively. Our experimental setup is consistent with each Spiking Transformer baselines, as detailed below.
Spikformer with SEMM. The training epoch is set to 310 for CIFAR, 200 for DVS128 Gesture, and 106 for CIFARIO-DVS, The batch size is 128 for CIFAR, 16 for DVS128 Gesture and CIFARIO-DVS. The learning rate is initialized to 0.0005 for CIFAR10/100, 0.001 for DVS128 Gesture and CIFAR1O- DVS. All of them are reduced with cosine decay. We follow [I] to apply data augmentation on DVS128 Gesture and CIFAR 10-DVS. In addition, the network structures used in CIFAR-10, CIFARIO-DVS, and DVS128 Gesture are Spiking Transformee-4-384, Spiking 2-5556 and Spiking 2-5556.
Spike Driven Transformer with SEMM. The training epoch is set 210 for CIFARI0/100 datasets, 200 for DVS128 Gesture, and 106 for CIFARIO-DVA The batch size is 32 for CIFAR 10/100, 16 for DVS128 Gesture and CIFAR 10-DVS. The learning rate initialized to 0.0005 for CIFAR10/100 0.0003 for DVS128 Gesture and 0.01 for CIFARIO-DVS, The rest is consistent with Spikformer.
Spikingformer with SEMM The training epoch is set 410 for CIFAR10/100 datasets, 200 for DVS128 Gesture, and 106 for CIFAR 10-DVS. The batch size is 64 for CIFAR 10/100, 16 for DVSI 128 Gesture and CIFAR OODDVS. The learning rate is initialized to 0.0005 for CIFAR10/100 0.1 for DVS128 Gesture and CIFARIO-DVS. The rest consistent with Spikformer.
C.2 Model Details
In our experiments, use 8 NVIDIA-4099 GPUs for ImageNet, and NVIDIA-4090 GPU for other datasets. We adjust the value of membrane time constant in spike neuron when training models on DVS datasets. In direct training SNN models with surrogate function,
Sigmoid(x) 1 + exp(-ax)
(39)
We select the Sigmoid function as the surrogate function with a = 4 in all experiments.
C.3 Additional Experiments
Table 5: Ablation study results on the time step.
Datasets
Models
Time-Step Topl-Aco(%k
94. 94.11/74.67
Spikformer SEMM
2
84.87778.49
4
95. 05.78/79.04
95.99/79.29 6
|
94.97/76.77
CIFARI0/100
Spike Driven Transformer SEMM
2
95 58/78/77.. .9
4
96. 06.02/00.222 .26
46.48/80.87 6
|
94 04.54/77.11 16
Spikingfonner SEMM
2
95. 95.42/78.5 85
6
96.67/81.22
The accuracy regarding different simulation time steps of the spiking neuron is shown in Tab. [5] At all the time steps, our Spiking Transformer with SEMM shows competitive performance. When the
15

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time step is 1, our Spikformer with SEMM is 1.7% lower than the network with T = 4 on CIFARIO, and our Spikformer with SEMM with time step still achieves 74.67% on CIFARI00. The above results show that Spikformer with SEMM is robust under fewer time steps.
In EMSP, the role of Deep Wise Convolution layer (DWC) is independently extract features the channel-level experts using 3x3 receptive field, thereby enhancing representational capacity. The ablation study about DWC in Tab. ddemonstrates its effectiveness.
Table Ablation on Deep Wise Convolution layer.
Model
Module
CIFAR100
CIFARIO-DVS
Spikformer
EMSP
78.53
82.32
Spikformer
EMSP w/o DWC
78.17
81.30
Spike-driven Transformer
EMSP
79.81
81.10
Spike-driven Transformer
EMSP w/o DWC
78.95
80.51
Spikingformer
EMSP
79.81
81.95
Spikingformer
EMSP w/o DWC
79.44
81.20
D Visualization
D.1 Image Samples Visualization
More dynamic allocation visualizations of the spiking router are given in Fig. 9
E Discussion on the Feasibility of Hardware Deployment for SEMM
SEMM can theoretically be deployed on chips that are synchronous within layers and asyn- chronous between layers, such TrucNorth [48]. Specifically, multiple spiking router matrices, iee. (21,r2, I' ). can be implemented by one block can be mapped to the core are considered synchronous. The spiking Hadamard product and addition that occur in SEMM analogous the AND and ADD operations in SEW ResNet [7]. These element-win spiking operations can be adapted on neuromorphics chips that support multiple branches and residuals, such Speck V2 49
thus of the neuromorphic chip for computation. Outputs from the same
16

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Figure 9: More visualization samples of routers as masks.
17

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Reviews:
[{"review_id": "CjhdUVxJJm", "reviewer": "Reviewer_dEFi", "summary": "This research introduces the Spiking Experts Mixture Mechanism (SEMM), a paradigm that combines Spiking Neural Networks (SNNs) with Mixture-of-Experts (MoE) to enhance the capabilities of Spiking Transformers. The proposed SEMM leverages the energy-efficient, sparse spiking activation characteristic of SNNs, resulting in an SNN-MoE framework that is both computation-efficient and SNN-compatible. By integrating SEMM into existing spiking Transformers, performance improvements are achieved. This work contributes to the exploration of high-performance and high-capacity Spiking Transformers.", "questions": "See weakness", "limitations": "See weakness", "rating": 3, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.\tThis study is the first attempt to utilize MoE on Spiking Transformers, leveraging the advantages of SNN and MoE for stable performance gains.\n2.\tThe experimental analysis on sparse conditional computation is thorough.\n3.\tThe writing is clear and accessible.", "weaknesses": "**Major:**\n\n1.\tAs mentioned in the first contribution (Line 68-69), the proposed SEMM is a universal SNN-MoE paradigm. Therefore the application of SEMM on the most state-of-the-art Spiking Transformers such as Spike-driven Transformer V2[R1] is crucial to establish the generalizability of SEMM. However, provided baselines do not include it, which weakens the persuasiveness of its generalizability. Moreover, SEMM-based Spiking Transformers underperform compared to [R1]'s peak results, suggesting MoE might not be essential for boosting performance. I would consider raising my rate if you can provide consistent performance improvements using SEMM on [1].\n2.\tCan you provide evidence or previous studies demonstrating the effectiveness of using DWConv after MLP.fc1? If such evidence exists, it should be cited; otherwise, conducting an ablation study on this layer is recommended. Without this clarification, the motivation behind adding this layer remains unclear, weakening the persuasiveness of SEMM's design applied to MLP.\n3.\tIs the ANN-MoE paradigm derived from prior ANN-MoE studies, or is it an original concept from your research? If it's based on earlier work, citations are necessary; otherwise, you should highlight the unique advantages of your proposed ANN-MoE paradigm compared to existing ones.\n\n**Minor:**\n\n1.\tThe description for Figure 3 is not clear enough. Given the explanation provided, I cannot fully understand the meaning of each line in the diagram.\n2.\tIn Figure 4, do the four graphs in each sub-figure relate to four images from the same class?\n\nReferences:\n\n[R1] Man Yao, JiaKui Hu, Tianxiang Hu, Yifan Xu, Zhaokun Zhou, Yonghong Tian, Bo XU and Guoqi Li.  Spike-driven transformer v2: Meta spiking neural network architecture inspiring the design of next-generation neuromorphic chips. In International Conference on Learning Representations (ICLR), 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed comments and suggestions for improvement. We would like to address your concerns and answer your questions below.\n\n> ***Weakness 1**: The application of SEMM on the most state-of-the-art Spiking Transformers such as Spike-driven Transformer V2.*\n\n**AW1**:  Following your suggestion, we report the Spike-driven Transformer V2 (SDT V2) with SEMM on CIFAR100 and ImageNet in Tab. R1-1. \n### Table R1-1\n| Model | CIFAR100（T=4） | ImageNet (T=1)|\n| -------- |:--------:|:--------:|\n| SDT-V2-15M  | 80.40    | 71.80    |\n| SDT-V2-15M with SEMM  | **81.04**      | **73.14**    |\n\nAfter configuring SEMM, SDT V2 demonstrates a consistent performance improvement, but the degree of improvement is not as large as that of SDT V1. This could be attributed to SDT V2 being a hybrid spiking CNN-Transformer framework, where the first two stages are based on spiking convolutions, with the third and fourth stages utilizing spiking transformers. Therefore, compared to SDT V1, the proportion of EMSA and EMSP blocks in SDT V2 is lower, resulting in a smaller gain. We will add these results to the manuscript appendix.\n\nFurthermore, we argue that the SEMM differs from the framework design of Spiking Transformers (Spikformer, Spike-driven Transformer V1/2, and Spikingformer, et al.). SEMM is a universal plug-and-play paradigm that can stably enhance performance. It provides a new direction for improving the Spiking Transformer.\n\n> ***Weakness 2**: Provide evidence demonstrating the effectiveness of using DWConv after MLP.fc1.*\n\n**AW2**: In EMSP, the role of DWConv is to independently extract features at the channel-level experts using a 3x3 receptive field, thereby enhancing representational capacity. \nThe ablation study about DWConv in Tab. R1-2 demonstrates its effectiveness:\n### Table R1-2\n| Model | Module |CIFAR100| CIFAR10-DVS|\n| -------- | -------- |:--------:|:--------:|\n| Spikformer  |EMSP  |  78.53   |  82.32  | \n| Spikformer  |EMSP w/o DWConv|  78.17 |  81.30  | \n| Spike-driven Transformer  |EMSP  | 79.81   | 81.10   | \n| Spike-driven Transformer  |EMSP w/o DWConv| 78.95  |  80.51   | \n| Spikingformer  |EMSP  | 79.81  | 81.95   | \n| Spikingformer  | EMSP w/o DWConv| 79.44  | 81.20  | \n\nThanks for your comments, we will include the ablation experiments in the manuscript appendix.\n\n> ***Weakness 3**: Is the ANN-MoE paradigm derived from prior ANN-MoE studies?*\n\n**AW3**: Thank you for your kind reminder. ANN-MoE represents a paradigm derived from previous works on MoE in ANNs [R1, R2],  integrating multiple experts through routing and hard sparsification. The referenced ANN MoE works in the manuscript adopt this paradigm, for example, __manuscript line 32 \"Mixture-of-Experts (MoE) [20; 21] is known for allowing each expert to learn specific tasks or features, showing better performance\" and line 89 \"The Mixture-of-Experts (MoE) [28; 29] combines the predictions of multiple specialized experts\"__. For clarity, we will append citations to these works following the term 'ANN-MoE' in the manuscript.\n\n> ***Weakness 4**: The description for Figure 3 is not clear enough.*\n\n**AW4**: Fig. 3 depicts the parameter number (X-axis) and performance (Y-axis) of various MoE strategies under multiple Spiking Transformer baselines. Each line consists of four points from left to right, representing: \n+ __1__. Baseline\n+ __2.__ EMSP\n+  __3.__ ANN-MoE paradigm with float-point router matrix (AM-F)\n+ __4.__ ANN-MoE paradigm with spiking router matrix (AM-S)\n\nFor clarity, we convert Fig. 3 into the following Tab. R1-3. It can be observed that our EMSP achieves better performance with fewer parameters, whereas the application of the ANN-MoE paradigm yields suboptimal results. We will revise Fig. 3 in the manuscript to enhance its readability.\n### Table R1-3\n| Model | MoE-Type|CIFAR100 Param(M)/ Acc| CIFAR10-DVS Param(M) / Acc|\n| -------- |:--------|:--------:|:--------:|\n| Spikformer  |Baseline | 9.32/77.86 |2.57/80.90\n| Spikformer  |EMSP | 9.39/78.53 |2.58/82.32|\n| Spikformer  |AM-F  | 23.60/77.35 |5.74/80.80|\n| Spikformer  |AM-S  | 23.60/76.79 |5.74/82.90|\n| Spike-driven Transformer  |Baseline | 10.28/78.40 |2.57/80.00|\n| Spike-driven Transformer  |EMSP | 10.30/79.81|2.57/81.10|\n| Spike-driven Transformer  |AM-F  | 22.91/79.64 |5.74/81.50|\n| Spike-driven Transformer  |AM-S  | 22.91/78.89 |5.74/81.09|\n| Spikingformer  |Baseline | 9.32/79.21 |2.57/81.30|\n| Spikingformer  |EMSP | 9.39/79.81|2.58/81.95|\n| Spikingformer  |AM-F  | 23.59/79.06 |5.74/81.00|\n| Spikingformer  |AM-S  | 23.59/78.81 |5.74/82.50|\n\n\n> ***Weakness 5**: In Figure 4, do the four graphs in each sub-figure relate to four images from the same class?*\n\n**AW5**: We speculate that you are referring to Fig. 5, not Fig. 4. The background image is the same for each subplot. We will add this explanation to the manuscript.\n\n[R1] Noam Shazeer, et al. \"Outrageously large neural networks: The sparsely-gated mixtureof-experts layer.\" International Conference on Learning Representations (2017).\n\n[R2] Yanqi Zhou, et al. \"Mixture-of-experts with expert choice routing.\" Advances in Neural Information Processing Systems (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": {}}, {"review_id": "AIP9HIPJ72", "reviewer": "Reviewer_RYQd", "summary": "The paper presents a novel integration of Spiking Neural Networks (SNNs) with Mixture-of-Experts (MoE) to form the Spiking Transformer, introducing the Spiking Experts Mixture Mechanism (SEMM). The SEMM enables dynamic sparse-conditional computation by having both experts and routers output spiking sequences, which aligns with the sparse and energy-efficient nature of SNNs. The proposed model incorporates two key components: Experts Mixture Spiking Attention (EMSA) for head-wise routing and Experts Mixture Spiking Perceptron (EMSP) for channel-wise allocation of spiking experts. Empirical evaluations demonstrate that SEMM improves performance on neuromorphic and static datasets with minimal additional computational overhead compared to traditional SNN-based transformers.", "questions": "1.Can the authors provide more empirical evidence demonstrating the effectiveness of SEMM on larger and more diverse datasets?\n2.How does the proposed SEMM adapt to varying levels of sparsity in the input data, and what is the impact on computational efficiency?", "limitations": "1.Generalizability: The experiments, though indicative of the SEMM's effectiveness, are limited to a specific subset of datasets. Extending the evaluation to include a wider variety of data, especially those with different characteristics and scales, would strengthen claims about the model's versatility and robustness.\n2. Implementation Details for Reproducibility: Although the paper outlines the conceptual framework of SEMM, more explicit details on the implementation and the source code, such as hyperparameter tuning strategies and specifics of the spiking sequences' generation and handling, would facilitate reproducibility and foster further advancements built upon this work.\n3. The novelty of the router mechanism for expert mixture is limited for the router mechanism seems have little difference with that in ANN MoE methods.", "rating": 3, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1.Originality: The fusion of SNNs with MoE principles is innovative and provides a fresh perspective on conditional computation.\n\n2.Quality: The paper offers a rigorous mathematical and conceptual framework for SEMM, demonstrating a thorough understanding of both SNNs and MoE.\n\n3.Clarity: While the concepts are complex, the authors have made efforts to explain them coherently, making the paper accessible to a wide audience.\n\n4.Significance: The work has the potential to impact energy-efficient AI, particularly in edge devices where power consumption is a primary concern.", "weaknesses": "1.The paper could benefit from more comprehensive empirical evaluations across diverse datasets to generalize the findings.\n\n2.The lack of theoretical guarantees or analysis regarding the convergence and stability of SEMM leaves room for skepticism about its robustness.\n\n3.A more detailed discussion on computational efficiency and scalability with respect to dataset size and complexity would be beneficial.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable comments. We will explain and discuss your concerns.\n\n> ***Weakness 1, Question 1, Limitation 1**: More experiments across diverse datasets to generalize the findings.*\n\n**A1**: __These datasets in experiments possess distinct characteristics and scales, which verify the effectiveness and generalization capability of SEMM.__ Specifically, the neuromorphic dataset exhibits temporal characteristics and sparse event features, validating SEMM's capability to handle sparse event data. The ImageNet-1K, with its 1,000 classes, verifies SEMM's ability to process complex large-scale data. \n__In terms of data scale,__ we apologize for not being able to provide results on larger datasets due to the rebuttal time and resource constraints.\n__In terms of data diversity,__ we add the validity and generalization experiments of SEMM on time series forecasting, which is a challenging task and aims to predict future values based on historical observations arranged chronologically. Addressing this task often involves modeling the temporal dynamics, resonating profoundly with the nature of neural coding and SNN. As shown in Tab. R2-1, SEMM demonstrates superior predictive capabilities over the baseline on four datasets, further illustrating its versatility and robustness.\n### Table R2-1\n| Model   |Horizon    | Metr-la [R1]   | Pems-bay [R1] |  Solar [R2]  | Electrictiy [R2]   |\n|---------|:---------:|---------|---------|---------|---------|\n|  |  | $\\text R^2$ / RSE | $\\text R^2$ / RSE | $\\text R^2$ / RSE  | $\\text R^2$ / RSE |\n| Spikformer |6 | 0.713/ 0.565 | 0.773/ 0.514 | 0.929/ 0.272| 0.959/ 0.373 |\n| Spikformer+SEMM |  6  | 0.721/ 0.557 | 0.776/ 0.510  | 0.930/ 0.270| 0.964/ 0.338 |\n| Spikformer |24 | 0.527/ 0.725 | 0.697/ 0.594 | 0.828/ 0.426| 0.955/ 0.371 |\n| Spikformer+SEMM |  24 | 0.534/ 0.719 | 0.707/ 0.584 | 0.835/ 0.423  | 0.968/ 0.363 |\n| Spikformer |48| 0.399/ 0.818 | 0.686/ 0.606 | 0.744/ 0.519| 0.955/ 0.379 |\n| Spikformer+SEMM | 48| 0.400/ 0.811 | 0.688/ 0.603 | 0.751/ 0.464 |0.968/ 0.320 |\n| Spikformer |96 | 0.267/ 0.903 | 0.667/ 0.621 | 0.674/ 0.586| 0.954/ 0.382 |\n| Spikformer+SEMM |  96 | 0.279/ 0.895 | 0.673/ 0.618 | 0.675/ 0.581 | 0.961/ 0.381 |\n\nHere, the horizon represents the prediction length. RSE stands for Root Relative Squared Error, where a smaller value indicates a lower error. $\\text R^2$ represents the coefficient of determination, with higher values indicating higher prediction accuracy.\n\n> ***Weakness 2**: The lack of theoretical guarantees or analysis regarding the convergence and stability of SEMM leaves room for skepticism about its robustness.*\n\n**AW2**: We regret that we are unable to provide a theoretical analysis of the convergence and robustness of SEMM. To the best of our knowledge, there are currently no works theoretically analyzing the convergence of deep SNNs. __However, to address your concern, we validate SEMM's convergence and robustness through experiments.__\n\n__Convergence: From an experimental perspective, SEMM does not affect the final convergence results of the spiking transformer.__ In Fig. R2-1 and  R2-2 of our submitted PDF rebuttal document, we compare the training loss and test accuracy curves between the Spikformer baseline and SEMM. The introduction of SEMM did not affect convergence.\n\n__Robustness: From an experimental perspective, SEMM does not compromise the robustness of the model.__ To show this, we add experiments about input noise on  CIFAR100. We add a Gaussian noise with mean 0 and variance $\\sigma^2$ on inputs $X$. The results are shown in Tab. R2-2. It can be found that at each noise level, introducing SEMM is still better than the baseline.\n### Table R2-2\n| Model |$\\sigma= 0$ (no noise) |$\\sigma= 0.1$|$\\sigma= 0.2$|$\\sigma= 0.3$|\n| -------- |:--------:|:--------:|:--------:|:--------:|\n| Spikfromer  | 77.86    | 72.62   | 70.37 | 68.39|\n| Spikformer with SEMM  | 79.04 | 73.47|71.05|69.54| \n\n> ***Weakness 3**: A more detailed discussion on computational efficiency and scalability with respect to dataset size and complexity would be beneficial.*\n\n**AW3**: Thanks for your comments. Following your suggestion, we show that SEMM exhibits higher computational efficiency compared to ANN-MoE paradigm on three diverse datasets, i.e., CIFAR100 (static images), CIFAR10-DVS (neuromorphic images), and  Solar (time series data). As shown in Tab.R2-3 (following), the number of parameters (Param) and computational operations (OPs) of SEMM on the three datasets are both lower than that of ANN-MoE, while also achieving the best performance. This demonstrates its computational efficiency and scalability for different datasets.\n### Table R2-3\n| Model | MoE-Type|CIFAR100 | CIFAR10-DVS | Solar |\n| -------- |:--------|:--------:|:--------:|:--------:|\n|   | | Param(M)/OPs (G)/ Acc (%) |Param(M)/OPs (G)/ Acc (%)|Param(M)/OPs (G)/ $\\text R^2$|\n| Spikformer  |Baseline | 9.32/0.83/77.86 |2.57/1.06/80.90|2.52/0.88/0.828\n| Spikformer  |SEMM | 8.98/0.94/79.04 |2.57/0.89/82.90|2.52/0.78/0.835\n| Spikformer  |AM-F  | 23.60/1.22/77.35 |5.74/1.45/80.80|5.69/1.16/0.821\n| Spikformer  |AM-S  | 23.60/1.34/76.79 |5.74/1.42/82.90|5.69/1.05/0.819\n\n> ***Question 2**: How does the proposed SEMM adapt to varying levels of sparsity in the input data, and what is the impact on computational efficiency?*\n\n**AQ2**: __Regarding question 2, could you further explain what you mean by data sparsity?__\n\n> ***Limitation 2**: Implementation Details for Reproducibility.*\n\n**AL2**: You can refer to the Appendix. C for the hyperparameters and find the source code of SEMM in the supplementary materials. \n\n[R1] Li, Yaguang, et al. \"Diffusion convolutional recurrent neural network: Data-driven traffic forecasting.\" arXiv preprint arXiv:1707.01926 (2017).\n\n[R2] Lai, Guokun, et al. \"Modeling long-and short-term temporal patterns with deep neural networks.\" The 41st international ACM SIGIR conference on research & development in information retrieval (2018).", "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": "IcHVx8hGsN", "reviewer": "Reviewer_rWD5", "summary": "The authors introduce MOE into SNN, propose the SEMM structure, and use this structure to enhance the attention and MLP modules in the Spikeformer-like architecture. They transform these elements into the EMSA and EMSP modules. The new network architecture achieved better performance under similar parameter settings compared to the original spiking transformer structures.", "questions": "1. The SEMM(E, R, F(·)) obtained from Eqn. 16 is described as containing only 0 or 1. But it should be a floating-point matrix.\n\n2. Line 217 What means “The ASR of EMSA is around 0.5, which is comparable to the regular TopK setting of ANN-MoE”? What's the relationship between the ASR in EMSA and the TopK rate in ANN-MoE?\n\n3. What is the purpose of Fig. 6? In my view, different inputs leading to different activations are natural.", "limitations": "NA", "rating": 7, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The SEMM proposed by the authors does not introduce floating-point matrix multiplication during the SNN forward process. Its SMSA structure effectively focuses on the relevant parts of the image. As a result, SEMM enhances the performance of various spiking transformers, achieving SOTA performance on different datasets.", "weaknesses": "1. The authors claim that SEMM's advantage lies in its spike-driven characteristics and suitability for asynchronous chips. However, the author did not run the SEMM structure on real asynchronous hardware. In my opinion, due to the spike-driven nature of an asynchronous chip lacking a hardware clock, it is difficult for multiple spike matrices in the router part (in SEMM) to arrive simultaneously and interact, resulting in most outputs being 0. The author needs to provide a more detailed explanation of the spike-driven advantage, especially on asynchronous hardware.\n\n2. Prior to Eqn. 14, the authors omitted the equation for partitioning the input X (how to get the $A_m$?).\n\n3. In Eqn. 17, $BN(SEMM(E, R, F(·)))W_o$ undoubtedly involves floating-point matrix operations; is there a writing error present here?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your detailed comments. We would like to address your concerns below.\n\n> ***Weakness 1**: The author needs to provide a more detailed explanation of the spike-driven advantage, especially on asynchronous hardware.*\n\n**AW1**: As you mentioned, deploying SEMM on a fully asynchronous neuromorphic chip is challenging. However, SEMM can theoretically be deployed on chips that are synchronous within layers and asynchronous between layers, such as TrueNorth [R1]. Specifically, multiple spiking router matrices, i.e. $\\{ r_1,r_2,...,r_m\\}$, can be implemented by one block (Linear-BatchNorm-SpikingNeuron) and thus can be mapped to the one core of the neuromorphic chip for computation. Outputs from the same core are considered synchronous. The spiking Hadamard product and addition that occur in SEMM are analogous to the AND and ADD operations in SEW ResNet [R2]. These element-wise spiking operations can be adapted on the neuromorphic chip that supports multiple branches and residuals, such as Speck V2 [R3].\n\n\n> ***Weakness 2**: Prior to Eqn. 14, the authors omitted the equation for partitioning the input X (how to get the $A_m$?).*\n\n**AW2**: Sorry for the confusion caused by ommitation. In Eqn. 14, Each $\\bf A_m$ is obtained as follows,\n$$\n{\\bf A\\_m} = \\text{SSA}(\\bf Q\\_m,K,V),\n$$\nwhere $\\bf Q_m$ is individual to each expert, and $\\bf K, V$ are shared among experts. We will revise it in the manuscript.\n\n\n> ***Weakness 3, Question 1**: In Eqn. 17, $BN(SEMM(E, R, F(·)))W_o$ undoubtedly involves floating-point matrix operations; is there a writing error present here? The SEMM(E, R, F(·)) obtained from Eqn. 16 is described as containing only 0 or 1. But it should be a floating-point matrix.*\n\n**AW3**: We apologize for our incorrect expression. The $\\text{F}(·)$ in Eqn. 16 involves spiking Hadamard products and element-wise addition. Since both $\\bf E$ and $\\mathcal{R}$ are in spiking form, the output of ${\\rm{SEMM}}({\\bf E}, \\mathcal{R}, {\\text F}(\\cdot))$ is an integer spiking form rather than a binary spiking form. $\\text{BN}({\\rm{SEMM}}({\\bf E}, \\mathcal{R}, {\\text F}(\\cdot)))\\bf W_o$  in Eqn. 17 can be decomposed into addition operations and multiplication is avoided, which is shown as\n$$ {\\rm{SEMM}}({\\bf E}, \\mathcal{R}, {\\text F}(\\cdot)) = \\sum_{i=1}^{m} {\\bf r}_i * {{\\bf A}_i} = \\bf s_1 + s_2 + \\cdots + s_m $$ \n$$ {\\rm{SEMM}}({\\bf E}, \\mathcal{R}, {\\text F}(\\cdot))\\bf W_o = ({\\bf s_1 + s_2 + \\cdots + s_m})\\bf W_o = \\bf s_1W_o + \\bf s_2W_o + \\cdots + \\bf{s_m}W_o$$ \nwhere $\\bf s_m$ is binary spiking form. We omit $\\text{BN}$ because its parameters can be integrated with the linear layer. We will revise it in the manuscript.\n\n>**Question 2**: Line 217 What means \"The ASR of EMSA is around 0.5, which is comparable to the regular TopK setting of ANN-MoE\"? What's the relationship between the ASR in EMSA and the TopK rate in ANN-MoE?\n\n**AQ2**:  In ANN-MoE [R4, R5], TopK typically selects 2 experts, which can be considered as 50% sparsity for a total of four experts in our settings. This is similar to the average spiking rate (sparsity) observed in EMSA.\n \n>**Question 3**: What is the purpose of Fig. 6? In my view, different inputs leading to different activations are natural.\n\n**AQ3**:  The purpose of Fig. 6 is to further demonstrate the dynamic condition computing characteristics of routers in SEMM. It illustrates the average spiking rate across spatial-temporal locations of routers across different classes of images (with 50 images per class), offering a more representative visualization compared to the single image in Fig. 5. Specifically, the spiking pattern of a router varies with time steps and spatial positions. Across different categories, the active routers differ; for example, in (a), routers 1 and 2 (top left and top right) are active, while in (b), routers 1 and 4 (top left and bottom right) exhibit high activation.\n\n[R1] Akopyan, Filipp, et al. \"Truenorth: Design and tool flow of a 65 mw 1 million neuron programmable neurosynaptic chip.\" IEEE transactions on computer-aided design of integrated circuits and systems 34.10 (2015): 1537-1557.\n\n[R2] Fang, Wei, et al. \"Deep residual learning in spiking neural networks.\" Advances in Neural Information Processing Systems 34 (2021): 21056-21069.\n\n[R3] Ole, Richter, et al. \"Speck: A Smart event-based Vision Sensor with a low latency 327K Neuron Convolutional Neuronal Network Processing Pipeline.\" IEEE International Symposium On Asynchronous Circuits and Systems (2023).\n\n[R4] Noam Shazeer, et al. \"Outrageously large neural networks: The sparsely-gated mixtureof-experts layer.\" International Conference on Learning Representations (2017).\n\n[R5] Yanqi Zhou, et al. \"Mixture-of-experts with expert choice routing.\" Advances in Neural Information Processing Systems (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": {}}, {"review_id": "1i4s3gWRnI", "reviewer": "Reviewer_eZ3Z", "summary": "The manuscript presents an innovative approach to integrating SNNs and MoE methodologies into a cohesive framework.  It introduces the Spiking Experts Mixture Mechanism (SEMM), which leverages the sparse spiking activations of SNNs and the conditional computation of MoE.  The proposed SEMM has been adapted into the Spiking Transformer architecture, resulting in two novel structures: the Experts Mixture Spiking Attention (EMSA) and the Experts Mixture Spiking Perceptron (EMSP). Experimental results are provided, showing SEMM's effectiveness in achieving stable improvements on neuromorphic and static datasets with manageable increases in computational overhead.", "questions": "1. Can you discuss the feasibility of adapting your proposed EMSA and EMSP modules to neuromorphic hardware? Are there any specific types of neuromorphic hardware or setups that are particularly well-suited or ill-suited for deploying SEMM? \n2. How does SEMM handle the trade-off between sparsity and accuracy?", "limitations": "While the authors indicate that the limitations are addressed in Section 5 (refer to Line 533 in the Checklist), it appears that this section does not currently include a discussion on the limitations.\nCan you include a discussion on the limitations of the work?", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Novel mechanisms. The introduction of SEMM, along with its derivatives EMSA and EMSP, is a novel contribution, particularly in how these mechanisms manage to perform conditional computation dynamically.\n2. The improvement of performance brought by SEMM is significant. The EMSA and EMSP modules can be adapted to any spiking self-attention and MLP modules, which may be beneficial to broader SNN research community.\n3. Experiments are comprehensive. Especially, Fig. 6 shows the average spiking rate of spatial-temporal locations of routers in different kinds of images, which proves the effectiveness of spiking routing functions.", "weaknesses": "1. Several Unclear Notations.\n(1) In Fig.1 (b) and Line 132, the authors use the notation of $N$, but what does the $N$ denote? I cannot find explanations on this issue. Does $N$ denote the length of image patches?\n(2) In Fig.2 (c), why does \"DWL\" stand for Depth-Wise Conv? I think it would be better to denote it as \"DWC\" or just explain it as Depth-Wise Linear.\n(3) In Equation 24 and 26, what’s the meaning of $p$ and $q$?\nPlease clarify these notations.\n2. The Section 4.3 is more like a sensitivity experiment of hyper-parameters rather than an ablation study. The real ablation study has been shown in Fig. 4. Please correct this issue to avoid misleading readers.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comments and suggestions for improvement. \n\n> ***Weakness 1**: Several Unclear Notations.*\n\n**AW1**: Sorry for the confusion caused by unclear notations. To clarify: __(1)__ $N$ denotes the length of image patches. __(2)__ Thanks for your advice. We will change the abbreviation representing the depth-wise convolutional layer from \"DWL\" to \"DWC\". __(3)__  The input current matrix (feature) $I[t]$ has a dimension of $P\\times Q$, where p and q represent the $p$-th row and $q$-th column, respectively. We will modify them in the manuscript.\n\n> ***Weakness 2**: The Section 4.3 is more like a sensitivity experiment of hyper-parameters rather than an ablation study.*\n\n**AW2**: We appreciate your suggestion and will revise the titles of the ablation study and the hyperparameter sensitivity experiment in the manuscript accordingly.\n\n> ***Question 1**: Can you discuss the feasibility of adapting your proposed EMSA and EMSP modules to neuromorphic hardware?*\n\n**AQ1**: \nSEMM can theoretically be deployed on chips that are synchronous within layers and asynchronous between layers, such as TrueNorth [R1]. The spike-driven operation of SEMM is similar to the AND and ADD residuals in SEW ResNet [R2]. Multiple branches and residuals are supported on certain neuromorphic hardware, such as Speck V2 [R3]. However, deploying SEMM on a fully asynchronous neuromorphic chip is challenging. \n\n\n> ***Question 2**: How does SEMM handle the trade-off between sparsity and accuracy?*\n\n**AQ2**: \nIn SEMM, we do not control sparsity using hyperparameters; instead, the model dynamically adjusts based on the input. Therefore, there is no trade-off between sparsity and performance. Higher sparsity does not necessarily imply lower accuracy. SEMM utilizes dynamic spiking sparsity to select feature representations within each expert, thereby enhancing accuracy.\n\n> ***Limitation 1**: While the authors indicate that the limitations are addressed in Section 5 (refer to Line 533 in the Checklist), it appears that this section does not currently include a discussion on the limitations. Can you include a discussion on the limitations of the work?*\n\n**AL1**: \nWe apologize for the confusion caused by our incomplete statement. In Sec. 5, we state that future work will focus on SEMM application on larger SNN models. In line with your suggestion, we further discuss the limitations of this work:\n\nSEMM has been validated on image datasets, demonstrating its effectiveness. However, no further verification has been conducted on other tasks and diverse datasets. Despite validating SEMM's effectiveness\nin time series forecasting tasks as suggested by reviewer RYQd, demonstrating its generalizability, there remain numerous tasks yet to be validated. Additionally, the effectiveness of SEMM on larger SNN models has yet to be validated. Future work will address these two limitations.\n\nWe will add this discussion to the appendix of the manuscript.\n\n[R1] Akopyan, Filipp, et al. \"Truenorth: Design and tool flow of a 65 mw 1 million neuron programmable neurosynaptic chip.\" IEEE transactions on computer-aided design of integrated circuits and systems 34.10 (2015): 1537-1557.\n\n[R2] Fang, Wei, et al. \"Deep residual learning in spiking neural networks.\" Advances in Neural Information Processing Systems 34 (2021): 21056-21069.\n\n[R3] Ole, Richter, et al. \"Speck: A Smart event-based Vision Sensor with a low latency 327K Neuron Convolutional Neuronal Network Processing Pipeline.\" IEEE International Symposium On Asynchronous Circuits and Systems (2023).", "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:
Dear ACs and Reviewers,

We sincerely appreciate the valuable time and feedback provided by each reviewer. We have responded to each of their comments individually.

In the rebuttal, we use 

> ***Weakness/Question/Limitation**: ...* 

to summarize the reviewers' comments. Responses start with '__AW, AQ, AL__'.

Best regards

Authors

Meta-review: The reviewers found that this paper presents a novel, interesting idea in a clear way. There was less consensus on the merits of the empirical validation presented, but after following the discussion I am convinced that the authors provided sufficient evidence to make the paper a useful contribution to the conference.

From the discussion among the reviewers, one minor point emerged that the authors should take care to fix in the final version of the paper. The explanation given in Appendix B for EMSA's parameter count in Table 1 seems incorrect. This is easily checked by comparison with the code given as supplementary material (for instance in `Supplementary Material/ImageNet/models_spikformer_v1_moe.py`, lines 68-124). In appendix B,
1. the authors say that the the output layer needs $N^2$ parameters. This is incorrect: the output layer maps a mixture of the output of the experts, which lives in dimension $d$, to an output of dimension $D$; therefore it has $dD=D^2/m$ parameters.
2. the authors say that $K$, $V$, and $Q$ all need $dD=D^2/m$ parameters each. This is correct only for $K$ and $V$ (which are shared among all experts) but incorrect for $Q$. Each experts has an independent $Q$, each of which has $dD$ parameters, so the total number of parameters associated with $Q$ is $mdD=D^2$.

In practice, the two errors cancel each other out in the formula reported in Table 1, so the formula is correct, but the explanation in Appendix B should be fixed, and the main text should highlight the fact that $K$ and $V$ are shared among experts.
Decision: Accept (poster)