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_cc_2024_Conference
Published: 2024-04-23 04:26:23

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.

Main paper: 
 introduction: 
 Recognized as 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 on neuromorphic hardware [35, 5, 37] . Currently, SNNs have further fulfilled a role in multiple application scenarios including object detection [22] , natural language processing [32] , and 3D recognition [24] . Spatial-Temporal back-propagation (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 gradient adjustment [29, 8, 14] and structural improvement [54, 51, 44, 50] , there is still a certain performance gap between ANNs and SNNs.Recently, the STBP learning algorithm based on multi-threshold models [41, 42] is considered as another potential way to 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 multi-hierarchical and equidistant threshold model based on global input 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 a vanilla single threshold model through a layer-by-layer 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:• 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 a 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 a vanilla single threshold model losslessly during the inference stage. By adopting different parameter initialization schemes, the LM-HT model can further establish a bridge between the vanilla STBP and quantized ANNs training. • We further design a 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-art 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 et al. [8] and Guo et al. [14] respectively proposed brand-new target loss functions by analyzing the temporal distribution of the spike sequence and membrane potential. Furthermore, various temporal-dependent batch normalization layers [54, 10, 15] and advanced spiking neuron models [51, 44] have been pointed out, which enhances the capability and stability of SNN learning.The researchers also designed a variety of residual blocks [11, 20] 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-First-Spike (TTFS) [21] algorithm conduct one-time back-propagation based on the specific firing moment. Meng et al. [34] 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.ANN-SNN Conversion. ANN-SNN Conversion is another widely used method for obtaining highperformance 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 a 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 et al. [16] and Li et al. [28] classified and summarized the relevant errors existed in the conversion process. Deng et al. [7] and Bu et al. [2] 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 sequences, multiple optimization strategies have been proposed successively, including memorizing the residual membrane potential [17] , firing negative spikes [43, 25] , calibrating offset spikes [18] and hybrid finetuning training [45] . Currently, ANN-SNN Conversion has been further applied to the training of large-scale visual and language models [46, 32] .Spiking neural models with multi-threshold. 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 multi-threshold 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. 
limitations: 
 Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [NA] . Justification: We find no limitation which needs to be emphasized here.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
preliminaries: 
 The spiking neuron models for SNNs. The Leaky-Integrate-and-Fire (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:EQUATIONEQUATIONEq.( 1) describes the charging process: ∀t ∈ [1, T ], m l LIF (t) and v l LIF (t -1) respectively represent the membrane potential before and after the charging at the t-th time-step. I l (t) denotes the input current and λ l LIF characterizes the leakage degree of the membrane potential. When λ l LIF = 1, the LIF model will degenerate into a more specialized model called the IF model. Eq.( 2) depicts the reset and firing process: s l LIF (t) indicates the spike emitting situation and θ l is the firing threshold. Here we adopt the soft-reset mechanism, which means that the reset amplitude of the membrane potential is equal to the value of θ l . STBP learning algorithm for SNNs. The gradient calculation mode of STBP is inspired by the back-propagation Through Time (BPTT) algorithm in Recurrent Neural Network (RNN), which will propagate along the spatial and temporal dimensions of SNNs simultaneously. Following equations have described the specific propagation process:EQUATIONHere L denotes the target loss function. From Eq.( 2) one can note that the mathematical relationship between s l LIF (t) and m l LIF (t) is equivalent tos l LIF (t) = H(m l LIF (t) -θ l ), where H(•) 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 ∂s l LIF (t) ∂m l LIF (t) in the back-propagation chain. For example,∂s l LIF (t) ∂m l LIF (t) = sign m l LIF (t) -θ l ≤ θ l 2describes the well-known rectangular surrogate function.Quantized ANNs. The quantized ANN model is a widely used structure in the field of ANN-SNN Conversion. Compared to traditional ANNs, quantized ANNs usually use the following Quantization-Clip-Floor-Shift (QCFS) function [28, 2] as their activation function:EQUATION)Here a l and φ l represent the activation output and shift factor, while T q and ϑ l denote the quantization level and learnable scaling factor. If we set T q = T, ϑ l = θ l , a l = T t=1 s l IF (t)θ l /T, v l IF (0) = φ l , one can find that the so-called QCFS function actually simulate the average spike firing rate of the IF model (we set r l IF (T q ) = Tq t=1 s l IF (t)θ l /T q ) 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 as ANNs under specific conditions. In this section, we first introduce the M-HT model, which has equidistant multi-level thresholds and will select the threshold closest to its current membrane potential at each time-step to achieve the process of firing spikes and resetting potential. Eqs. ( 5)-( 6) describe the dynamic equations of the M-HT model.m l (t) = λ l v l (t -1) + I l (t), I l (t) = W l s l-1 (t)θ l-1 .(5)EQUATIONHere L denotes the number of level for the firing threshold. Regarding the surrogate gradient calculation of the M-HT model, similar to the vanilla spiking models, we propose s l (t) m l (t) = sign 1 2 θ l ≤ m l (t) ≤ (L + 1 2 )θ l , which covers a wider range of the membrane potential, as shown in Fig. 1(d)-(e) . As the M-HT model has L different firing options at each time-step, we can consider 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 a mathematical equivalent relationship between the M-HT and IF model:Lemma 4.1. ∀t ∈ [1, T ], if v l (t -1) ∈ [0, θ l ), the effect of inputting current I l (t) into a M-HT model with L-level threshold at the t-th time-step, is equivalent to continuously inputting uniform currentI l (t)/L for L time-steps into a IF model with v l IF (0) = v l (t -1), i.e. s l (t) = clip v l (t-1)+I l (t) θ l , 0, L = L j=1 s l IF (j).Lemma 4.1 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 l (t) in Lemma 4.1 can also be calculated through clip (⌊•⌋ , •, •), which is equivalent to 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 pretrained ANNs with L-level quantization under single-step condition.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [Yes] . Justification: We provide the corresponding assumptions and proofs in the Appendix section. Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.• Theorems and Lemmas that the proof relies upon should be properly referenced. 
the representation ability of the m-ht model on multiple time-steps: 
 Based on Lemma 4.1, we further consider the information representation of the M-HT model on multiple time-steps: Theorem 4.2. When λ l = 1, v l (0) ∈ [0, θ l ), for a M-HT model with L-level threshold, after T time-steps, we will derive the following conclusions: (i) If we further assume ∀t ∈ [1, T ], I l (t) ∈ [0, Lθ l ), we will have:∀t ∈ [1, T ], s l (t) = Lt j=L(t-1)+1 s l IF (j), v l (t) = v l IF (Lt), T t=1 s l (t) = LT j=1 s l IF (j).(ii) If we further assume I l (1) = ... = I l (T ), we will have:T t=1 s l (t) = clip v l (0)+ T t=1 I l (t) θ l , 0, LT .Here the IF model has uniform input currents I l (1)/L, ..., I l (T )/L respectively within every L steps and satisfies v lIF (0) = v l (0).The proofs of Lemma 4. ], v l (t) = v l IF (Lt)).The M-HT 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 a 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.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, 0] can all represent the situation where 2 spikes are emitted within 2 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 L in [0, LT ] is unable to be represented by a uniform spike output sequence, there are still uneven firing situations:Corollary 4.3. If λ l = 1, v l (0) = 0 and I l (1) = ... = I l (T ), for a M-HT model with L-level threshold, s l (1) = ... = s l (T ) is only satisfied when I l (1) ∈ [kθ l , kθ l + θ l /T ), ∀k = 0, ..., L -1 or I l (1) ∈ (-∞, 0) ∪ [Lθ l , +∞).The proof is provided in the Appendix. From Corollary 4.3, we can divide the input current into uniform and uneven firing regions according to the corresponding intervals, as shown in Fig. 1(a)-(b) . Note that the uneven spike sequences emitted by the l-th layer may further cause the input current of the l + 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 to increase gradually without introducing extra regulation. 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 Temporal-Global Information Matrix (T-GIM):Conv & BN T-GIM Conv & BN T-GIM Conv & BN T-GIM Conv & BN T-GIM Conv & BN T-GIM Conv & BN T-GIM Conv & BN T-GIM Conv & BN T-GIMEQUATIONHere ω l tj is the element at row t and column j of the T-GIM Ω l , Ω l ∈ R T ×T . As shown in Eq.( 7) and Fig. 2 (b), this brand-new input current adopts a multi-step current weighting form, allowing the model to simultaneously focus on the global information along the time dimension. Note that the new input current will follow a uniform distribution when ∀i, j ∈ [1, T ], ω l ij = 1 T and degrade to the vanilla input current when Ω l = diag(1, ..., 1). For the first case mentioned above, if we further add the condition λ l = 1, according to Theorem 4.2(ii), one can find that the output of the model will be consistent with the activation output of a LT -level quantized ANN layer by layer, as shown in Fig. 2(b)-(c ). For the second case, when L = 1, the model will degenerate into vanilla LIF model, as shown in Fig. 2(a)-(b) .To enable the model to dynamically adjust the above calculation process, we set both Ω l and λ l as learnable parameters. The initial values of Ω l and λ l are set to 1/T and 1, respectively. During the training process, we choose the Sigmoid function σ(•) to control the parameters for fulfilling smooth gradient updates within a 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 time-steps through learnable Ω l and λ l in the LM-HT model, during the back-propagation process, unlike Eq.( 3), we detach the term ∂L ∂m l (t)∂m l (t) ∂v l (t-1) ∂v l (t-1)∂m l (t-1) from the gradient calculation graph, thereby reducing redundant calculations and completely leaving the gradient propagation between different time-steps to Ω l and λ l for control. The back-propagation calculation chains for the LM-HT model have been described as follow. Here ⊙ denotes the Hadamard product.EQUATIONEQUATIONAlthough the traditional ANN-SNN Conversion frameworks have much lower computational overhead than STBP training algorithm, a serious performance degradation phenomenon often exists on the We firstly choose QCFS function to train the quantized ANN models and then replace the QCFS function modules layer by layer with the LM-HT models under specific initialization (∀i, j ∈[1, T ], ω l ij = 1 T ; λ l = 1, θ l = ϑ l , v l (0) = θ l2 ), as shown in Fig. 2(d) . Combining with the conclusion pointed out by [2] , one can note that the initialized LM-HT model and the QCFS function before substitution have an equivalence in terms of mathematical expectation, which has been described as the following theorem:Theorem 4.4. When T t=1 I l (t)/LT = W l r l-1 IF (T q ) and T t=1 I l (t) ∈ [0, LT θ l ], if ∀i, j ∈ [1, T ], ω l ij = 1 T and λ l = 1, θ l = ϑ l , v l (0) = θ l 2 ,for L, T, T q with arbitrary values, we have: E T t=1 s l (t)θ l LT -ϑ l Tq clip W l r l-1 IF (Tq)Tq ϑ l + 1 2 , 0, T q = 0.As discussed in Section 4.2, the mathematical essence of the LM-HT model is to 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 a reparameterization scheme that can transform the LM-HT model obtained during the training stage into a 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 is set to λ l . In addition, T-GIM will be extended from R T ×T to R LT ×LT and the parameters are averaged within each L × L sub-region, ensuring that the input current meets the precondition in Theorem 4.2. We also rectify the bias terms in synaptic layers, which involve addition operations at each time-step. By performing layer-by-layer reparameterization in the above manner, we will obtain a single threshold SNN model with theoretically lossless accuracy.Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? Answer: [Yes] . Justification: We provide the detailed experimental configuration in the Appendix section. 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In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. , with an open-source dataset or instructions for how to construct the dataset). 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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 CIFAR10-DVS [26] . Consistent with the previous works, we also choose VGG [40] and ResNet [19] 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 our L-level LM-HT model within T time-steps remains at the same level as that of the vanilla LIF model within LT time-steps, to make a 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.As shown in Tab.1, we investigate the impact of threshold levels and T-GIM for our proposed model. One can note that vanilla IF neuron (L = 1, T = 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 = 2, T = 2) can 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 is enhanced and the computational overhead in synaptic layers is significantly reduced.We also validate the feasibility about the reparameterization procedure mentioned above. As shown in Tab.2 and Fig. 3 , 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-HT model. This convertible property enables the LM-HT model to be more flexibly deployed on neuromorphic hardware.We first investigate the competitiveness of our proposed model in the domain of STBP learning.As shown in Tab.3, 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 [8, 14] , energy-efficient training [13, 48, 34] and advanced neuron models [51, 44] . ImageNet-200 & ImageNet-1k. 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 61.09% and 70.90% within 2 time-steps on 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 a larger training time-step, one can note that our method will also demonstrate a significant advantage. For example, the two-level LM-HT model reaches the precision of 61.75% with 4 time-steps, 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-HT model can achieve the accuracy of 80.70% after merely 2 time-steps.In our hybrid training framework, we first choose [2] as the backbone 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 conversion methods [16, 43] and multi-stage error correction method [17] as our comparative works.As shown in Tab.4, after conducting the STBP fine-tuning optimization with relatively low computational overhead, we 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 to the ResNet-20 network after eight-level quantization (i.e. T q =8), the two-level LM-HT model has achieved a performance improvement of 32.91% and 13.50% with 2 and 4 time-steps, respectively. 
conclusions: 
 In this paper, we 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 to 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 of our method. We believe that our work will further promote in-depth research on advanced spiking neural model. 
experiment statistical significance: 
 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [No] .Justification: The experiments choose a shared random seed to ensure fairness and reproducibility.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors). • It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
experiments compute resources: 
 Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?Answer: [Yes] .Justification: We make description about the computation resources in the Appendix section.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. • The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. • The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). 
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broader impacts: 
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 Appendix: A Proof. ∀t ∈ [0, T ), if v l (t) ∈ [0, θ l ), as m l (t + 1) = v l (t) + I l (t), we have m l (t + 1) ∈ [0, (L + 1)θ l ). Therefore, after the firing process v l (t + 1) = m l (t + 1) -s l (t)θ l , one can note that v l (t + 1) ∈ [0, θ l ). According to the idea of mathematical induction, if we directly set v l (0) ∈ [0, θ l ), we can have v l (T ) ∈ [0, θ l ).Theorem 4.2. When λ l = 1, v l (0) ∈ [0, θ l ), for a M-HT model with L-level threshold, after T time-steps, we will derive the following conclusions: (i) If we further assume ∀t ∈ [1, T ], I l (t) ∈ [0, Lθ l ), we will have:∀t ∈ [1, T ], s l (t) = Lt j=L(t-1)+1 s l IF (j), v l (t) = v l IF (Lt), T t=1 s l (t) = LT j=1 s l IF (j).(ii) If we further assume I l (1) = ... = I l (T ), we will have:T t=1 s l (t) = clip v l (0)+ T t=1 I l (t) θ l , 0, LT .Here the IF model has uniform input currents I l (1)/L, ..., I l (T )/L respectively within every L steps and satisfies v l IF (0) = v l (0).Proof. (i) If we consider the pre-condition in Theorem 4.2 and combine Eq.( 1) with Eq.( 2), ∀t ∈ [1, LT ], we will have:v l IF (t) -v l IF (t -1) = I l t L /L -s l IF (t)θ l . (S1)Similarly, if we set λ l = 1 and incorporate Eq.( 5), ∀t ∈ [1, T ], we will have:v l (t) -v l (t -1) = I l (t) -s l (t)θ l . (S2)Then we accumulate Eq.(S1) along the time dimension and obtain the following equation: (ii) If we accumulate Eq.(S2) along the time dimension and divide θ l on both sides, we will have the following equation:v l IF (Lt) -v l IF (L(t -1)) = I l (t) - Lt j=L(t-1)+1 s l IF (j)θ l . (S3) As I l (t) ∈ [0, Lθ l ), according to Lemma A.1, when v l (t -1) = v l IF (L(t -1)) ∧ v l (t -1) ∈ [0, θ l ), we will have v l (t) ∈ [0, θ l ) and v l IF (Lt) ∈ [0, θ l ). Considering s l (t), Lt j=L(t-1)+1 s l IF (j) ∈ N, if s l (t) ̸ = Lt j=L(t-1)+1 s l IF (j), one can note that | Lt j=L(t-1)+1 s l IF (j)θ l -s l (t)θ l | = |(v l (t) - v l (t -1)) -(v l IF (Lt) -v l IF (L(t -1)))| = |v l (t) -v l IF (Lt)| ≥ θ l ,EQUATION)If I l (1) < 0 or I l (1) ≥ Lθ l , it is obvious that we will haveT t=1 s l (t) = clip v l (0)+ T t=1 I l (t) θ l , 0, LT = 0 or T t=1 s l (t) = clip v l (0)+ T t=1 I l (t) θ l , 0, LT = L.If I l (1) ∈ [0, Lθ l ), according to Lemma A.1, we will have v l (T ) ∈ [0, θ l ). As T t=1 s l (t) ∈ N, based on Eq.(S4), we can finally deduce thatT t=1 s l (t) = v l (0)+ T t=1 I l (t) θ l -v l (T ) θ l = v l (0)+ T t=1 I l (t) θ l = clip v l (0)+ T t=1 I l (t) θ l , 0, LT .One can note that Lemma 4.1 is actually a special case of Theorem 4.2 under the condition of T = 1, therefore Lemma 4.1 is also proven. I l (1) ∈ [kθ l , kθ l + θ l /T ), ∀k = 0, ..., L -1 or I l (1) ∈ (-∞, 0) ∪ [Lθ l , +∞).Proof. If I l (1) < 0 or I l (1) ≥ Lθ l , it is obvious that we will have s l (1) = ... = s l (T ) = 0 or s l (1) = ... = s l (T ) = L. Otherwise, based on the conclusionT t=1 s l (t) = clip v l (0)+ T t=1 I l (t) θ l, 0, LT in Theorem 4.2(ii), when I l (1) ∈ [kθ l , kθ l + θ l /T ), ∀k = 0, ..., L -1, we will haveT t=1 s l (t) = kT, ∀k = 0, ..., L -1. Note that ∀T ′ ∈ [1, T ], we can further have T ′ t=1 s l (t) = kT ′ , ∀k = 0, ..., L -1. Therefore, it can be concluded that s l (1) = ... = s l (T ) = k. Instead, if I l (1) ∈ [0, Lθ l ) ∧ I l (1) /∈ [kθ l , kθ l + θ l /T ), ∀k = 0, ..., L -1, we will have IF (T q ) andT t=1 I l (t) ∈ [0, LT θ l ], if ∀i, j ∈ [1, T ], ω l ij = 1T and λ l = 1, θ l = ϑ l , v l (0) = θ l 2 , for L, T, T q with arbitrary values, we have:EQUATIONT , λ l = 1, θ l = ϑ l and v l (0) = θ l 2 , combining with the conclusion mentioned in Theorem 4.2(ii), we will haveT t=1 s l (t)θ l LT = θ l LT clip T t=1 I l (t) θ l+ 1 2 , 0, LT . According to the conclusion pointed out in [2] , we have known thatE θ l LT clip x l LT θ l + 1 2 , 0, LT -ϑ l Tq clip x l Tq ϑ l + 1 2 , 0, T q = 0, here x l ∈ [0, θ l ].Therefore, we directly set x l = T t=1 I l (t)/LT = W l r l-1 IF (T q ) and then we will draw the final conclusion.Theorem A.2. ∀t, i ∈ [1, T ], ∀j ∈ [L(t -1) + 1, Lt], ∀k ∈ [L(i -1) + 1, Li], when s l-1 (t) = Lt j=L(t-1)+1 s l-1 IF (j), if bl j = b l t /L, ωl jk = ω l ti /L, we will have I l (t) = Lt j=L(t-1)+1 I l IF (j).Here bl , ωl denote the rectified bias term and T-GIM layer after the reparameterization process.Proof. Firstly, it is obvious that I l (t), I l IF (j) can be rewritten asI l (t) = T i=1 ω l ti (W l s l-1 (i) + b l i ) and I l IF (j) = LT i=1 ωl ji (W l s l-1 IF (i) + bl i ).Considering the precondition bl j = b l t /L, ωl jk = ω l ti /L, we will have:I l IF (j) = T i=1 Li k=L(i-1)+1 ωl jk (W l s l-1 IF (k) + bl k ) = T i=1 ω l ti L Li k=L(i-1)+1 (W l s l-1 IF (k) + b l i L ). (S5)Then we can further have:Lt j=L(t-1)+1 I l IF (j) = Lt j=L(t-1)+1 T i=1 ω l ti L Li k=L(i-1)+1 (W l s l-1 IF (k) + b l i L ) = T i=1 ω l ti L Lt j=L(t-1)+1 (W l Li k=L(i-1)+1 s l-1 IF (k) + b l i ) = T i=1 ω l ti L Lt j=L(t-1)+1 (W l s l-1 (i) + b l i ) = T i=1 ω l ti (W l s l-1 (i) + b l i ) = I l (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 input currents through the spiking neuron layer will also satisfy the precondition of Theorem A.2 (s l (t) = Lt j=L(t-1)+1 s l IF (j)). Therefore, we can prove the computational equivalence before and after the reparameterization process.As shown in Tab.S1, 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] , we achieve an accuracy improvement of 1.35% on MS-ResNet-18. For Transformer-4-384 architecture, our method also outperforms Spikformer [55] and Spike-driven Transformer [49] in terms of performance. For static datasets, we attempt to suppress the possible overfitting phenomenon by utilizing data augmentation techniques including AutoAugment [4] and Cutout [9] . For CIFAR10-DVS dataset, we resize each image to 48 × 48 pixels and split it into 10 frames. For ImageNet-1k 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 [1] and AdamW [30] as our optimizers. The corresponding initial learning rate and weight decay are set to 0.025, 5 × 10 -4 for SGD on CIFAR-10(100), 0.0125, 5 × 10 -4 for SGD on ImageNet-200 and 0.02, 0.01 for AdamW on CIFAR10-DVS. For ImageNet-1k dataset, we use SGD as our optimizer and set the corresponding weight decay as 0. Furthermore, in the hybrid training framework, our initial learning rate and weight decay are both set to 5 × 10 -4 . For all experimental cases, we choose the Cosine Annealing scheduler [31] to dynamically regulate the learning rate. Our experiments are implemented on NVIDIA RTX A5000 and 4090.A. (W , Ω, L, λ, θ ).1: # Convert ANN to SNN 2: for l = 1 to L N do 3: if Is the first layer then f SNN .W l = f ANN .W l 4: f SNN .θ l = f ANN .ϑ l 5: f SNN .Ω l = 1 T 6: f SNN .λ l = 1 7: f SNN .v l (0) = f SNN .θ l /• The answer NA means that the abstract and introduction do not include the claims made in the paper.• The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 


Decision: 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).

Reviews: review: paper_summary: paper_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.
strength_weakness: strength_weakness: 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?

2. 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.
questions: 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: limitations: N/A
paper_summary: paper_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.
strength_weakness: strength_weakness: - **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.

- **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.
questions: 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: limitations: This work can benefit from further discussion on the details of model parameters and method implementation.
paper_summary: paper_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.
strength_weakness: strength_weakness: - 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.
- 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.

Minor issues:
- The authors should correct the name of DVSCIFAR-10 and several typos, such as “with larger parameter scale” to “with a larger parameter scale.”
questions: 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: limitations: -	In Table 3, the types of learning methods should be well explained in the caption. What is the meaning of efficient training?

- 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?
paper_summary: paper_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.
strength_weakness: strength_weakness: 1. Checklist should be placed after appendix.
2. The authors can also compare with paper [1][2] which obtains really good results by MS-ResNet-34 backbone on large imageNet datasets.
3. 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.
4. 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.




[1] Yao M, Zhao G, Zhang H, et al. Attention spiking neural networks[J]. IEEE transactions on pattern analysis and machine intelligence, 2023.

[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.

[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.

[4] Yao M, Hu J, Zhou Z, et al. Spike-driven transformer[J]. Advances in Neural Information Processing Systems, 2024, 36.

[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.
questions: questions: See  Weaknesses.
limitations: limitations: The author has discussed the limitations.

Comments: comment: Thanks for your rebuttal. I raise my score to 7.
I appreciate the author's response, which addresses my remaining concerns. I believe this is an interesting work for the neuromorphic community.
Thank you for your reply. I think this is a nice bit of discussion and could be added to the manuscript. In light of the additional discussion, I'd like to raise my score to a 7. This is an interesting piece of work and would be a nice addition to NeurIPS.