Training High Performance Spiking Neural Network  by Temporal Model Calibration
Authors: Jiaqi Yan
Changping Wang
De Ma
Huajin Tang
Qian Zheng
Gang Pan
Venue: ICML 2025 poster (2025)
Keywords: Spiking Neural Networks
Direct Training
Temporal Heterogeneity
Temporal Model Calibration
Primary area: deep_learning->sequential_models_time_series

Abstract: Spiking Neural Networks (SNNs) are considered promising energy-efficient models due to their dynamic capability to process spatial-temporal spike information. Existing work has demonstrated that SNNs exhibit temporal heterogeneity, which leads to diverse outputs of SNNs at different time steps and has the potential to enhance their performance. Although SNNs obtained by direct training methods achieve state-of-the-art performance, current methods introduce limited temporal heterogeneity through the dynamics of spiking neurons or network structures. They lack the improvement of temporal heterogeneity through the lens of the gradient. In this paper, we first conclude that the diversity of the temporal logit gradients in current methods is limited. This leads to insufficient temporal heterogeneity and results in temporally miscalibrated SNNs with degraded performance. Based on the above analysis, we propose a Temporal Model Calibration (TMC) method, which can be seen as a logit gradient rescaling mechanism across time steps. Experimental results show that our method can improve the temporal logit gradient diversity and generate temporally calibrated SNNs with enhanced performance. In particular, our method achieves state-of-the-art accuracy on ImageNet, DVSCIFAR10, and N-Caltech101. Codes are available at https://github.com/zju-bmi-lab/TMC.

Training High Performance Spiking Neural Network by Temporal Model Calibration
Jiaqi Yan 12
Changping Wang 12
De Ma
Huajin Tang' Qian Zheng'
12 Gang Pan" 12
Abstract
iors of biological neurons and have spatio-temporal informa- tion process capability via spike transmission (Maass, 1997; Hu et al., 2021; Li et al., 2024a). The rise of neuromorphic computing (Ma et al., 2024; Davies et al., 2018) has allowed SNNs to achieve enhanced performance and greater energy efficiency, garnering extensive attention (Yan al., 2024; Liu et al., 2024; Hu al., 2023) and driving widespread applications (Liao al., 2024; Zhang et al., 2024; Yan et al., 2025). Previous works have demonstrated that neuronal dynamics lead to temporal heterogeneity in SNNs (Perez- Nieves et al., 2021; Chakraborty & Mukhopadhyay, 2024; Wang et al., 2025; Chen et al., 2025). This characteristic enables SNNs to extract temporal dynamic features and produce temporally extended diverse outputs that are trans- formations of those features (Wu et al., 2025; Gast et al., 2024; She et al., 2021b; Savin & Denève, 2014). By integrat- ing the diverse outputs to make decisions, the performance of SNNs can be further improved, especially in tasks with complex temporal structures, such as neuromorphic datasets (Gast et al., 2024; Zheng et al., 2024; She et al., 2021b; Chakraborty & Mukhopadhyay, 2023).
Spiking Neural Networks (SNNs) are considered promising energy-efficient models due to their dy- namic capability to process spatial-tempone spike information. Existing work has demonstrated that SNNs exhibit temporal heterogeneity, which leads to diverse outputs of SNNs at different time steps and has the potential to enhance their performance. A Although SNNs obtained by direct training meth- ods achieve state-of-the trr performance, current methods introduce limited temporal heterogene- ity through the dynamics of spiking neurons or network structures. They lack the improvement of temporal heterogeneity through the lens of the gradient. In this paper, we first conclude that the diversity of the temporal logit gradients in cur- rent methods is limited. This leads to insufficient temporal heterogeneity and results in temporally miscalibrated SNNs with degraded performance. Based on the above analysis, we propose a Tempo- ral Model Calibration (TMC) method, which can be seen as a logit gradient rescaling mechanism across time steps. Experimental results show that our method
The rate-coding based SNNs trained using direct training methods based on BackPropagation Through Time (BPTT) and surrogate gradient (Wu et al., 2018; Deng et al.) have achieved state-of-the-ee performance (Zhou et al., 2024; Yao et al.). To further explore the potential of SNNs, re- searchers focus on enhancing the temporal heterogeneity of the rate-coded SNNs during direct training from two per- spectives. The first category (Rathi & Roy, 2021; Fang et al., 202 lb; Yao al., 2022) increases the heterogeneity of neu- rons by proposing parametric spiking neurons. They set one or more membrane-related parameters learnable to enlarge the representation space and improve the performance of SNNs. The second category (She et al., 2021b:a) investi- gates the relationship between heterogeneity and network structure and demonstrates that recurrent connections beneficial to capture different features of temporal patterns.
improve the temporal logit gradi- ent diversity and generate temporally calibrated SNNs with enhanced performance. In particu- lar, our method achieves state-of-the-art accuracy on ImageNet, DVSCIFAR1O, and N-Caltech1O 101. Codes are available https://ittuu..com/jjuu lab/TMC.
1. Introduction
Spiking Neural Networks (SNNs), unlike traditional Artifi- cial Neural Networks (ANNs), mimic the dynamic behay-
'The State Key Lab of Brain-Machine Intelligence, Zhe- jiang University *College of Computer Science and Technol- ogy, Zhejiang University. Correspondence to: Qian Zheng <qianzheng @ zju.edu.cn.>, Gang Pan <gpan www.eiw...............
However, gradient descent in the temporal dimension de- termines the learning of temporal features (Wu et al., 201 8; Huh Sejnowski, 2018), and how to enhance the temporal heterogeneity of SNNs through the lens of gradient has not been fully studied. Following existing works (Huh & Se-
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Training High Performance Spiking Neural Network by Temporal Model Calibration
jnowski, 2018; Perez-Nieves & Goodman, 2021), the gradi- ents in direct training methods of SNNs can be decomposed into two components: the temporal gradients propagated backward through hidden layers (temporal hidden layer gra- dients) and the temporal gradients of the loss function with respect to the logits (temporal logit gradients). The temporal hidden layer gradients, determined by the spiking neuron behaviors, have been shown to exhibit temporal heterogene- ity (Perez-Nieves & Goodman, 2021). Nevertheless, the temporal logit gradients, which are derived from the loss function and calculated as the error between model outputs and the optimization target across time steps, have not been sufficiently investigated.
in tasks with complex temporal structures, such as neuro- morphic datasets (Gast et al., 2024; Zheng et al., 2024; She et al., 2021b; Chakraborty & Mukhopadhyay, 2023). Fur- thermore, the relationship between temporal heterogeneity and the performance of SNNs has been widely studied. Re- cent work demonstrates that temporal heterogeneity enables SNNs to extract dynamic features and produce temporally diverse outputs that are transformations of those features (Wu et al., 2025; Gast et al., 2024; She et al., 2021b; Savin & Denève, 2014). Based on rate coding, making decisions by integrating the diverse outputs improves the performance of SNNs.
Direct Training Strategies for SNNs. To enhance the performance of SNNs, existing direct training methods in- troduce training strategies from both forward and backward perspectives. On the one hand, to enlarge the representation space of SNNs, current studies (Rathi & Roy, 2021; Fang et al., 2021b; Yao et al., 2022) propose different parametric spiking neurons, which set one or more membrane-related parameters to be learnable during training. In addition, researchers (She et al., 2021 introduce recurrent connec- tions to approximate temporal mapping functions. On the other hand, current works propose different loss functions to determine the temporal optimization of SNNs. Specif- ically, Standard Direct Training method (Wu et al., 2018) uses the average membrane potential of the last layer as the final output and optimizes it towards the one-hot label. The Temporal Efficient Training (Deng et al.) method assigns the true label as the optimization target for each time step. Recently, new loss functions have been proposed (Dong et al., 2024; Zhao et al., 2025; Qiu et al., 2024; Zuo et al., 2024), which introduce additional optimization constraints of outputs based on SDT.
In this work, we investigate the temporal logit gradients under existing loss functions and conclude that they have insufficient diversity in the temporal dimension. We demon- strate that this is not beneficial to capture temporal dynamic features across time steps, limits the temporal heterogeneity, and results in temporally miscalibrated SNNs with degraded performance. To address this, we introduce the concept of model calibration in ANNs (Mukhoti et al., 2020; Ross & Dollár, 2017; Tang et al., 2020), and propose a Temporal Model Calibration (TMC) method. TMC can be seen as temporal gradient rescaling mechanism to generate di- verse logit gradients in the temporal dimension. Our main contributions can be summarized as follows:
We analyze the temporal logit gradients calculated in existing direct training methods and conclude that they have insufficient diversity in the temporal dimension. This restricts temporal heterogeneity and leads to mis- calibrated SNNs with degraded performance.
Inspired by the concept of model calibration in ANNs, we propose the TMC method, which can be seen temporal logit gradient rescaling mechanism.
Model Calibration in Deep Learning. Recent evidence has shown that deep networks are prone to make overcon- fident predictions due to miscalibrated output probabilities (Mukhoti et al., 2020; Müller et al., 2019). To mitigate the issue, two main families of approaches have emerged re- cently: post- post-processing methods (Tomani et al., 2021; Ma & Blaschko, 2021) and cearning-based methods (Ovadia et al., 2019; Szegedy et al., 2016; Mukhoti et al., 2020). Never- theless, post-processing methods do not work well under data distribution shifts (Ovadia et al., 2019). Thus, learning- based calibration methods have become more powerful choice. Muller et al. (Szegedy al., 2016) propose Label Smoothing on soft targets, which aids in improving cali- bration. Mukhoti et al. (Mukhoti et al., 2020) showed that focal loss (Ross& & Dollár, 2017) can implicitly calibrate models by reducing the KL-divergence between predicted and target distribution. Recently, common learning- based calibration methods use additional loss terms as a regular- ization term (Chen et al., 2024; Pereyra et al., 2017), which has achieved ofth-oofthhear performance. It is notable that
Extensive experimental results demonstrate that the proposed method can improve the temporal logit gradi- ent diversity and generate temporally calibrated SNNs that exhibit enhanced performance. It is worth noting that our method achieves state-of-the-art performance ImageNet and neuromorphic datasets.
2. Related Work
Temporal Heterogeneity of SNNs. As the basic informa- tion processing units, spiking neurons exhibit rich dynamic behaviors and determine the intrinsic properties of SNNs, which are different from ANNs (Wu et al., 2018; Roy et al., 2019). Existing work has shown that neuronal dynamics lead to temporal heterogeneity in SNNs (Perez-Nieves et al., 2021; Chakraborty& & Mukhopadhyay, 2024; Wang et al., 2025). Experimental evidence shows that temporal hetero-
geneity enhances the performance of SNNs, particularly
2

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Training High Performance Spiking Neural Network by Temporal Model Calibration
learning-based model calibration methods fundamentally have the rescaling effect to the logit gradient (Tang al., 2020; Lee et al., 2022).
et al., 2022). The second term, which represents the tem- poral input current, can be enriched through two primary approaches: Diversifying the input spike train of through re- current connections, which has been investigated in current studies (She et al., b;). Expanding the representation space of W. However, this approach is a promising yet under-researched approach. In the following section, we present a comprehensive analysis of the temporal training dynamics for W.
3. Method
3.1. Problem Definition
Temporal Heterogeneity Analysis. Here, we provide the analysis of the temporal heterogeneity within spiking neu- ron under both continuous and discrete time.
Temporal Gradient Analysis. In the direct training meth- ods, SNNs are updated by computing the gradient of the loss function with respect to the synaptic weights in each layer. Using Backpropagation Through Time (BPTT) with surrogate gradients ( (Wu et al., 2018), the weight gradients can be calculated as:
Continuous Time. The behavior of a spiking neuron over continuous time can be formulated as follows:
du: -(u - ureset) + Its ue < Veh: dt
(1)
Σö(t- ty
8(t tz), (t),
ÔLC
1 1 Σ az Y,) T
Oz. OW
Ot
(5)
aw
where 2n is the membrane potential, J is the time content, T is the input current, and Ve, is the threshold. 5() is the Dirac delta function. The spike train og is generated when u, reaches Vea at time 1y. and un is reset to the resting potential Ureast-
where T-EE represents the cross-entropy (CE) loss. At time step t, at is the optimization objective, which determined by the form of loss functions, and Y, is the optimization target. The synaptic weights W are used to process the given input and produce the logit Z,, 7 is the total number of time steps.
In Equation A.5, the change of u, is quantified by dus which is determined by two components - -wu, Ureset) ) and 4. The term -u, - Ureset) reflects the natural decay of the membrane potential back toward the resting potential Wreset- Meanwhile, I, captures the influence of the tempo- rally varying external current on the membrane potential and spike firing. Overall, | is likely to be greater than zero, indicating ongoing changes in the membrane potential and spike train. Notably, for complex temporal structures, such as those found in neuromorphic datasets, a higher degree of temporal heterogeneity is necessary to effectively capture dynamic features. More detailed analysis can be found in Appendix B.
In Equation (5), the update gradient of W at each time step can be decomposed into two terms. The first term aw the gradients propagated backward down to the input layer across time steps (temporal hidden layer gradient). It is associated with the dynamic behaviors in spiking neurons and have been proven to be heterogeneous in the temporal dimension (Perez-Nieves & Goodman, 2021). The second term (a:-Y,)) is the partial derivative of the loss function to the logit over time steps (temporal logit gradient). is cal- culated as the error between the model's outputs and the true labels, determining the optimization objective of SNNs. To effectively represent temporal dynamic features, (ar. Y,) should provide a reasonable optimization magnitude and direction of W at each time step while maintaining temporal diversity across time steps. However, as demonstrated in the following section, the temporal logit gradients cur- rent direct training methods don't exhibit these properties simultaneously.
az. is
Discrete Time. To establish the computational link along the spatial-temporal dimension, discrete-tine time spiking neuron model is utilized for constructing SNNs as follows:
u/f1 = (1 - (ut - Vihot) + Σ Wid
(3)
Standard Direct Training (SDT), as defined by Deng et al. (Deng et al.), calculates the cross-entropy loss between the average pre-synaptic input of the output layer and the true label. For SDT, the temporal logit gradient 7Z, at time step can be derived as:
of = H(u - Vin),
(4)
where W is the trainable synaptic weights, and H.) is the Heaviside step function.
In Equation 3, the change of w/+1 is determined by (1 (uf Vinof) and Σ, W The first term, that is, the neuron's 's intrinsic temporal membrane potential, can be enriched by incorporating one or more trainable parameters related to the membrane, which has been explored in cur- rent studies (Rathi & Roy, 2021; Fang et al., 2021b; Yao
VZADT = (P avg Y).
(6)
Formaaly, the temporal logit gradient is defined as the partial derivative of the loss function with respect to the logit output Z., at in the mathematical form. In the rest of the paper, we will or of use the symbol VZ, for its simplicity.
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Training High Performance Spiking Neural Network by Temporal Model Calibration
where Pa Pavg is the softmax probability of the average output logit and Y is the one-hot encoded training label. Given that V is the same across time steps, it constrains the gradient diversity within SDT. Recently, new loss functions have been proposed in (Dong et al., 2024; Zhao et al., 2025; Qiu et al., 2024; Zuo et al., 2024). They introduce additional loss terms as a regularization term, which can be regarded as variants of SDT. Their objective is to realize the consistent representation of SNNs, which is not compatible with the temporal heterogeneity of SNNs.
extend the definition to the temporal dimension for SNNs, which is defined as follows:
Definition 3.1. At time step t, an SNN is temporally per- fectly calibrated if and only if:
P. = P()))
fel.:......._1.......... 11.2....,
(8)
As our work focuses on the rate-coded SNNs that currently achieve sate-offthhearr performance, we incorporate spe- cialized analysis for these SNNs based on Definition 3.1. Specifically, a rate-coded SNN makes decisions according to the average accumulated temporal information. We use Z, - + Ei= Z, refer to the average accumulated logit at time step 4, where I E 2 T)- Then, the tem- poral confidence P. can be represented by the maximum value of Softmax(Z.). Notably, Li et al. (Li et al., 2024b) suggest that the accuracy of rate-coding SNNs should in- crease monotonically with time steps if temporal hetero- geneity exists. Based on this observation, the confidence of a temporally perfectly calibrated SNN should increase monotonically with time steps, and we have the following remark:
Temporal Efficient Training (TET) (Deng et al.) first calculates the cross-entropy loss between the pre-synaptic inputs of the output layer with the true labels at each time step. Then, it employs the mean of these losses as the final loss. For TET, the temporal logit gradient is expressed as:
V2 TET (Pr Y),
(7)
where P. is the softmax probability of the output logit at time step t. Although the logit gradients have different magnitudes across time steps due to varying Pt. they ho- mogenously guide the optimization of temporal outputs in the same direction toward the one-hot encoded training label Y. However, existing studies have presented that minimiz- ing the CE loss optimizes P. to infinitely match the training label Y. This results in temporally miscalibrated SNNs with overconfidence issues and limited performance (Mukhoti et al., 2020; Müller et al., 2019).
Remark 3.2. Given a rate-coded SNN with a T time steps, let P. denote the predicted probability confidence at time step I. If the SNN is temporally perfectly calibrated, then for any t, f = 1, ...,T 1, it satisfies P. < Pit1-
Temporal Logit Gradient Rescaling, To generate the tem- porally perfectly calibrated SNNs, we draw inspiration from the learning-based model calibration methods. These meth- ods introduce novel loss functions built upon the CE loss, demonstrating the most effective performance for model calibration in ANNs (Chen et al., 2024; Pereyra et al., 2017). We analyze the optimization effect of the loss functions in the sauring-bassee model calibration methods, and then ex- tend the effect to the temporal dimension for direct training of SNNs.
3.2. Gradient Rescaling in Model Calibration
Based on the above analysis, the temporal logit gradients should provide a reasonable optimization magnitude and direction of W at each time step while maintaining temporal diversity across time steps to generate temporally calibrated SNNs. In the following sections, we explicitly define the temporally calibrated SNNs and propose the temporal logit gradient rescaling mechanism that can generate the SNNs.
Temporally Perfectly Calibrated SNN. At each time step, consistently optimizing the predictive output P. towards the one-hot encoded training label Y, achieves impressive performance and is commonly used in the direct training of SNNs. However, based on the concept of model calibration, this approach leads to an overfitting issue and generates miscalibrated SNNs with limited performance. To address this issue, we first introduce the definition of a perfectly calibrated model in ANNs and then extend the definition to the temporal dimension for SNNs.
Specifically, the loss functions in current learning- based model calibration methods realize calibration during opti- mization by applying the rescaling effect to the logit gradient of the CE loss. This can be formulated as:
V2" Ø V = g()),
where 2 is the element-wise division operator, Z is the output logit, VZ is the logit gradient of the loss function in the model calibration method, and VZ is that of CE loss. P is the softmax probability of Z, and the highest value within P denotes the predicted probability confidence. g()) represents the rescaling function, which is influenced by the confidence and is adaptively diverse between differ- ent samples and training phases. The confidence-relatee rescaling effect on the logit gradient of C finally pro- duces the calbbaead logit gradient VZ". Thus, the
Specifiaclly, model is perfectly calibrated if and only if the predicted probability confidence of each sample equals the model's i accuracy. This can be mathematically expressed as P = Pj- = y)P), where P is the confidence, that is, the highest value within the predictive probability P. y denotes the predicted class, and y denotes the true label. We further
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Training High Performance Spiking Neural Network by Temporal Model Calibration
model calibration methods in ANNs can adaptively generate model-calibrated logit gradients and alleviate the overfitting issue where the predicted probability confidence is infinitely close to 1, ignoring the model accuracy.
Modification 1. To address the above problems, we replace P. with the output predictive probability of the target class, denoted as Be. In addition, we detach the historical logits in Equation 13 from the calculation graph, retaining only the logit at the current time step t. The modification can be defined as:
With further instantiation specific to the rate-coding SNNs, we extend the logit gradient rescaling effect to the tempo- ral dimension for direct training of SNNs. We denote the temporal rescaling factor as Ir and define as:
B. exp(C.)
Σj exp((r)
Definition 3.3. Let VZ; be the temporal logit gradient of the loss function in the temporal model calibration method. Similar to Equation 9, St is defined as:
(14)
Q = SG(Z,) + Z,]-
VZZ Ø V g(,, Zi, .... Ze).
(10)
where Q, is the average accumulated logits, k refers to the index of the target class, C' is the total number of class, and SG(.) indicates the stop-gradient operation. We need to highlight that, under the application of Bi, our loss func- tion's 's effect is to optimize the probabilities of the target class to serve the confidence over time steps during training. Thus, in the following content, the optimization objective "confidence" refers to the probability of the target class,
Notably, due to the rate coding mechanism of SNNs, Jt required to satisfy two properties: First, It should be dependent on the accumulated logits of the model up to the time step L. Second, Ir should optimize the rate-coded SNN with linearly increasing confidence over time steps as described in Remark 3.2.
The term 3, regularizes the logit value of the target class during training and is already able to alleviate overconfidence to some extent. However, we want ex- plicitly enhance the effect of regularization when 3e is close to I, that is, the model faces a high risk of overconfidence. Thus, we replace Br with tç:
Modification 2.
3.3. Temporal Model Calibration for SNNs
In this section, we propose a Temporal Model Calibration (TMC) method with a new loss function to generate the temporal gradient rescaling factor 9t as discussed above.
Confidence Regularization. Inspired by the effective learning-based model calibration methods that add regular- ization term to the CE loss (Chen al., 2024; Pereyra et al., 2017), at each time step, we introduce confidence regular- ization to the CE loss. The loss function can be expressed as:
3e 1 11-8 3t
exp(ex) ΣS ES exp(!!)
0.
(15)
where 0, is highly sensitive to the change of 3r. When 3e approaches 1, even a small change in B, can result in significant increase of 0,
L- = LeE(PP Y) + Pr.
(11)
p. argmax(softmax (Z:)),
(12) Linear Increasing Confidence Constraint. Notably, the use of SG(-) will not hinder our training objective of the lin- early increasing confidence over time steps. Specifically, we can prove that, under the guidance of the non-optimizable constant bias, if the logit of the target class Z at each time step can be optimized to increase monotonically, the accu- mulated logit value of the target class Z,₁ will also increase monotonically. Consequently, the temporal confidence can achieve a linear increase over time steps. To formalize this, we have the following proposition (a detailed proof can be found in Appendix C):
1 Σ Zi,
Ze = 7 1=1
(13)
where P. is the temporal confidence of a rate-coded SNN as defined in Section 3.2.
However, directly using P, as regularization term is likely to cause suboptimal solutions and could hinder the gradient analysis due to two issues: First, the class index correspond- ing to the confidence P. may vary during training. While our goal is to optimize P. only when its class index corresponds to the target class. In cases where this is not true, we aim to optimize the probability of the target class to be the highest value in the probability distribution. Second, in Equation 13, P, is generated by the model's temporal outputs up to the current time step f. This leads to a complex calculation graph during gradient backpropagation and complicates the formulation of gt-
Proposition 3.4. Let Z be the logits at single time step t, and Z be the average logits of the cumulative time step. Given a perfectly calibrated SNN, ifr any I, t = 1,.... T 1, z < Zit it satisfies Z < <Zi+z
This implies that the effect of 0, should decrease as t in- creases. Given that minor changes in Br can lead to sig- nificant differences in lr it is necessary to introduce an
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Training High Performance Spiking Neural Network by Temporal Model Calibration
exponential term; to Og to control its effect. Moreover, we define A - which linearly decreases with time steps, to ensure that the effect of Or can decrease across time steps. Overall, the final version of the loss function in our method is presented as follows:
4. Experiments
We primarily validate our proposed method TMC on the im- age classification tasks, where static datasets (CIFARIO, CI- FAR100, and ImageNet) and neuromorphic datasets (DVS- CIFARIO, N-Caltech101, DVS-Gesture, and SL-Animals- DVS) are used. The network architectures in this paper include ResNet-19, VGG-SNN, Meta-SpiikFormer, and Hierarchical Spiking Transforrre More details of the exper- imental configurations can be found in Appendix A.1. The results on the DVS-Gesture and SL--Animal--DY are re- ported in Appendix F.2. Additionally, to further demonstrate the effectiveness of our method, we conduct experiments the text classification task, where the Quora Question Pair (QQP) dataset and the SpikingBERT architecture (Bal & Sengupta, 2024) are used. The results of this task are reported in Appendix F.1.
L = (er Y) 02
(16)
Gradient Rescaling Factor Analysis. After designing the loss function of our method, we use it to instantiate the gra- dient rescaling factor 9: and see if it can match the desired property discussed in Section 3.2.
We first apply the temporal gradient calculation analysis framework in Section 3.1 to our method and derive the expression of the partial derivative of our loss function to the logits VZ,MC as follows:
V.TTMc. P y* +
1
t Ae0
4.1. Analysis of Temporal Gradient Rescaling Effect
1
P
(17)
VZ TMC,m-
Plr -m 7 >100
We decompose It into the temporal logit gradient rescaling factor on the target class g and the factor on non-target classes g.th 1 C-1 Eemth g." where C' is the total class number. Then, in Figure 1, we visualize the relationship be- tween confidence and gr as well as the relationship between (1 (I-confidence) and g. & of 500 samples at the end of training on DVSCIFARIO. There are two important observations: Firstly, at each time step, we can observe that gr and gt" show the symmetric distribution pattern. This implies that despite of and g." having slightly different formulations in Equation (18), the rescaling factor across different classes may have close values with little variation. In such way, the rescaling mechanism will not bring much variation to the direction of temporal gradient. This property is similar to the rescaling factor of some existing model calibration methods, the expectation of which has been proven to be consistent across different classes (Tang et al., 2020). In addition, when the network is under-confident, of and g, will be closer to 1. Similarly, they will likely be pushed toward 0 or become negative numbers when the network is overconfident. Secondly, over time steps, the distributions of confidence and rescaling factors increasingly converge towards 1. This is compatible with what we discussed in Section 3.3 of Gradient Rescaling Factor Analysis. Overall, Ia exhibits diversity distribution in the temporal dimension and at the data instance level. In addition, we provide a evaluation of the training stability of gi in Appendix F.3.
Here P:- = softmax Q2). l and m refer to the index of the target class and the index of the non-target classes, respec- tively. Then, the temporal logit gradient rescaling vector Jt of our method can be written as
CE,K =1 1 X.
9
VZ
11 P..
(18)
VZ, TCM,m
1 1x prt
=1
VZ, CE
t 1 p pm
The detailed derivations of Equation 17 and Equation 18 can be found in Appendix D.
In Equation 18, as time step increases, the terms 1 and d & 1-Pe and 1 1.00²0 pm 1 1-Pp pru in of and g,",
weaken the effect of 1 AM
respectively. This means the logit gradient effect of the CE loss will take over, making the later time steps tend to be more confident. Thus, the linearly increasing temporal confi- dence of the trained SNNs will align with the monotonically increasing accuracies to satisfy the Definition 3.1. More detailed theoretically analysis of the temporal confidence of a SNN trained with TMC could converge to its accuracy can be found in Appendix E.
We further analyze the diverse distribution of ge across time steps. From Equation 18, we can infer that the distribution of will be affected by the confidence, that is, 9t has a negative correlation with pt and P Since the temporal confidence monotonically increases with time steps, this will introduce diversity to the distribution of 9. across time steps. We carried out experiments in Section 4. I to demonstrate the detailed correlation between 9t and confidence, as well as their diversity of 9: over time steps.
4.2. Evaluation of Model Performance
Classification and Calibration Performances. To verify the temporal gradient rescaling effect on model classifica- tion and calibration performances, we evaluate the calibra- tion errors and accuracies of our method at each time step DVSCIFAR 10 and compare them with SDT and TET.
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first train ResNet-19 on Training High Performance Spiking Neural Network by Temporal Model Calibration
Figure 1. Visualization of the temporal relationship between predicted probabilitics and gradient rescaling effect.
Table I. Classification and calibration performances for different methods across time steps DVSCIFAR1O
Methods
T=l
T=2
T-3
T=4
T=5
Acci
ECEE
AdaECCE
Acc
ECEL
AdaECE
Acc
ECET
AdaFCE
Acc
FCEL
AdaECT
Aue
ECEL
AdaECIET
SDT
19.24
0.64
0.69
48.42
0.36
0.42
62.00
0.27
0.24
19.24
0.22
0.20
67.28
0.21
0.15
TET
63.64
0.26
0.27
73.22
0.22
0.22
77.26
0.18
0.18
78.45
0.18
0.17
79.64
0.18
0.17
63.65
0.25
75.56
0.13
T=6
T=7
T=8
T=9
T=10
Methods
Accf
ECET
AdaECEL
Ace
ECEE
AdaECE
Acef
ECET
AdaECEL
Accf
EECEL
AdaECE,
Avc7
ECCE AdaECET
SDT
0.25
0.18
71.25
0.23
0.20
70.06
0.25
0.22
69.63
0.25
0.20
70.05
0.26
0.18
TET
79.22
0.18
0.17
67
0.17
0.17
78 45
0.17
0.17
78.04
0.17
0.17
81.22
0.18
0.16
81.91
83.37
83.08
83.94
0.05
Following previous studies, we use three standard metrics for network calibration performance: Expected Calibration Error (ECE) (Naeini et al., 2015), Adaptive ECE (AdaECE) (Mukhoti et al., 2020), and Classwise- (CECE) (Kull al., 2019). The specific introduction and calculation for- mulas for these three metrics can be found in Appendix A.2. We report the main results in Table 1 and the calibration rors in CECE metric reported in Table A.5 in Appendix F.5. Firstly, compared to SDT and TET, over time steps, our method exhibits very competitive classification accura- cies. Compared to TET, our method achieves the highest accuracy improvement of 5.83% at time step Apart from slight fluctuation at time step 8, overall, the accuracy of method increase progressively with the time step, while SDT and TET don't have this property. Secondly, the calibration errors of our method drop significantly in both ECE and AdaECE metrics. In particular, the calibration errors are lower in the later time steps since our proposed method has more advantages in mitigating overconfidence in this phase. In Table A.1, We further show the overall classification and calibration performances by evaluating the average predicted probability across time steps of SNNs. Note that we chose @Krrrmer architecture to evaluate the performance on ImageNet since it is the perf-maeformance architecture. Noticeably, our method achieves the best accu- racy and lowest calibration errors on various datasets.
Temporal Scalability. The temporal heterogeneity enables SNNs to extract dynamic features and produce temporally extended diverse outputs. This implies that, with increasing time steps, the performance of SNNs can be improved by incorporating diverse temporal outputs. Here, we verify the temporal scalability of SNNs trained with our proposed loss function. Specifically, CIFAR 100 with small time step length of 6 on CIFAR100. Then, we evaluate the trained SNN directly changing the time step length from to 10 without fine-tuning on the test set. Figure displays the results under different time step lengths of our method, SDT and TET. We can see that just under the time step length of our method has surpassed the accuracy of SDT and TET under the time step length of 10. When increase the test time step length, the accuracy of our method exhibits a more explicit increase, while the accuracy of SDT has a limited increase trend and that of TET remains stable. This implies that the SNNs trained with our method can generate diverse temporal outputs and further introduce significant performance improvement as time step increases.
4.3. Ablation Studies
The proposed regularization term consists of two compo- nents: the base 0, and the exponent A, In this section, we verify the impact of these two components on the proposed regularization term, respectively.

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Training High Performance Spiking Neural Network by Temporal Model Calibration
4.4. Comparison to Existing Works
Table 2. Overall classification and calibration performances.
Dataset
Architecture Time Step Methed Accuracyt ECEJ AdaBCEL
In this section, we conduct experiments on static datasets (CIFARIO, CIFAR 100, and ImageNet) and neuromorphic datasets (DVSCIFARIO 10 and N-Caltech101) to evaluate the performance of our method and compare the experimental results with previous works. All of the experiment results are summarized in Table 4. We specify all the training details in Appendix A.I.
SDT 21.12
0.22
0.20
CIFARI00
ResNet-19
78.05
0.05
0.04
SDT
85.62
0.05
0.05
ImageNet
QKformer
TET
85.66
0.05
0.04
Ours
85.83
0.02
0.02
DYSCIFARIO VGG-SNN
10
SDT
73.86
83.11
0.20
0.19
6.12
0.11
SDT
79.86
0.14
0.14
N-Caltech101 VGG-SNN
TET
81.73
0.11
0.10
CIFARIO/100. We apply our method on CIFARIO and CI and report the mean and standard deviation of 3 runs under different random seeds. The results of our method and existing methods on CIFAR100 are shown in Table 4, and the results on CIFAR1O 10 can be found in Table A.4 in Appendix F.4. Our method achieves competitive performance on CIFARIO I0 and CIFARI00 compared with other methods. The accuracies of our method on these two datasets are slightly lower than TEBN and RMP-LLss since they apply extra membrane potential regularizing operations. Our method outperforms the finite difference surrogate gra- dient descent method DSpike and spiking neurons paramet- ric method GLIF on CIFARIO and CIFAR100. compared to TET, TKS, TCL, ETC, and TSSD, our method doesn't 't achieve the best performance since the superiority of our method can not be fully exhibited on the static datasets.
Ours
86.03
0.12
0.10
78-
76-
74-
72
SDT
70-
TCT
5
10
Time Stpps
Figure 2. Time scalability of 19 on CIFAR100.
ImageNet. Compared to the convolution-based SNNs, transformer-based SNNs have achieved superior perfor- mance on ImageNet. We apply our loss function to the two best pessoperforming transformer- based SNNs: Spike- Spike-Driven Transformer V2 and QKFormer. We train Spike-Driven Transformer V2 from scratch. Note that considering the high training cost required to train QKFormer from scratch, we load the trained model of the initial QKformer and fine-tune it using our proposed loss function for 10 epochs. As shown in Table 4, our method further improves the performance of these two architectures by 0.80% and 0.18%, respec- tively. Overall, our method achieves SOTA performance on ImageNet and exhibits significant accuracy advantages compared to the convolution-basee SNNs.
Effect of 0, To validate the regularization effect introduced by 6r, we first create a variant of our loss function by re- placing O2 with 3r, denoted as Ours". Then, we compare the classification and calibration performances of the SNNs trained using our loss function and the variant loss func- tion on the DVSCIFARIO dataset. In Table 3, we report the calibration errors in ECE and AdaECE metrics and test accuracies at each time step on DVSCIFARIO. 10. We can see that the calibration errors of the variant loss function are higher than those of our proposed loss function at the later time steps. In addition, the classification performance at the later time steps is limited. This indicates that 0, in our loss function exhibits more efficient effect for the later time steps mitigate overconfidence, and further improve the performance of SNNs.
Neuromorphic Datasets. DVSCIFAR10 and N-Caltech101 are the most challenging mainstream neuromorphic datasets. We apply our method on these two datasets and report the mean and standard deviation of 3 runs under differ- ent random seeds. As shown in Table 4, on DVSCIFARIO 10, our method achieves an accuracy of 87.63% top-1 accu- racy, outperforming the previous SOTA method. On N- 1altech101, the proposed method achieves 88.24 top-1 ac- curacy. The results show the potential of SNNs in handling neuromorphic data. Compared to static datasets, neuro- morphic datasets exhibit richer spatiotemporal components through their interaction with both spatial and temporal in- formation. TMC leverages temporal heterogeneous training, thereby capturing a richer set of task-relevant spatiotempo-
Effect of A₁ We conduct experiments to evaluate the tem- poral constraint of X on the strength of the proposed regu- larization term. First, We create a variant of our loss func- tion by removing de. denoted as Ours?. Then, we compare the classification and calibration performances on DVSCI- FARIO and the results reported in Table 3. Except for the first time step, the calibration errors of the variant loss func- tion are significantly higher than those of our proposed loss function. Moreover, the accuracy does not increase with time step. This means that without the constraint of > 0, can not have a desirable regularization effect to make temporal confidence increase progressively.
8

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Training High Performance Spiking Neural Network by Temporal Model Calibration
Table 3. Ablation study DVSCIFARIO.
Methods
ACCT
FCEL
AdaFCE.
ACCT
ECEJ AdaFCE.
ACC
PCEL
AdaECE
ACCT
FCE.
AdaFCE.
ACCT
ECEJ AdaECES
Ours
65.00
0.25
18.56
0.17
78.09
0.17
19.26
0.16
0.16
0.16
77.05
Ours
63.65
0.25
0.24
75,56
0.11
0.17
78.97
0.11
0.16
81.76
0.12
0.15
81.64
0.12
0.13
Methods
Tw6
T=7
T-8
Tw9
Tw10
4CCTBCCK
AdaBCE,
ACCT ECE, AdaECE
ACCT
ECET AdaECE,
ACCT
AdaECEL
ACCT AdaECET
Ouss
80.85
0.16
0.17
81.08
0.16
0.16
$1.38
U.17
0.16
81.35
0.17
0.35
B1.82
0.16
0.16
Ours!
81.20
0.17
0.17
80.40
0.17
0.16
81.02
0.17
0.17
80,80
0.18
0.18
81.32
0.17
0.16
0.13
83.37
0.08
83.08
0.06
0.05
83.94
0.08
Table 4. Performance comparison with sfate-of-the-eee methods.
Dataset
Model
Architecture
Time Step
Accuracy
DSpike(Li et al., 2021)
ResNet-18
24
GL.FF(Yan et al., 2022)
ResNet-19
6
77.35 77.35
TEBN(Duan 2022)
ResNet-19
6
78.76
RMP-Losst(uu et 2023)
ResNet-11
6
78.98
TET(Deng et al.)
ResNet-19
6
74.72
TKS(Dong al., 2024)
ResNet-19
4 76.20
CIFARI0O
ETCrZhao et al., 2025)
[CL(Q et al., 2024)
ResNet-19
ResNet-19
4
79.47
79.73
TSSDiZuo 2024)
VGG-9
74.69
6
78.05-00.18
Ours
ResNet-19
4
10.13
76.35 1 0.15
DSpike(Li et al., 2021)
VGG-16
5
71.24
GLIF( Yao et al., 2022)
ResNeet-3
6
69.09 09
DEan al., 2022)
SEW ResNet-34
4
68.28
RMP-Loss((Guo et al., 2023)
ResNeet-34
65.17
Deng et at.)
Spiking-Ress Restge--
6
64.79
Dong
2024)
SEW-ResNet-34
4
69.60
ImageNet
ETCiZhao et al., 2025)
Spiking- Resng----
6
69.64 64
TSSD 2024)
ResNet-34
2
66.13
Spike-Drivee Transformer Yao et al.)
Meta-Spike Sprmeeormerrrr
80.00
QKFomeenZhou al., 2024)
Hierarchical TaikiingTransforo
4
85.65
Ours
Hierarchical Spiking Transformer
Meta- Aeta-Sppikeoomer
4
4
85.83
80.80
DSpike(Li et al., 2021)
ResNet- 18
10
75.40
GLIF Yao et al., 2022)
Wide Net
16
78.10
TEBN(Duan ct al., 2022)
VGG-SNN
10
84.90
RMP-Lossi(inn Guo et 2012))
ResNet- 19
10
76.20
TET(Deng al))
VGG-SM
10
83.17
DYSCIFARIO
TKS/Dong et al., 2024)
VGG-SNN
10
85.30
TCL(Quu al., 2024)
VGG-SNN
4
10
ETCrZhao et al., 2025)
VGG-SNN
10
85.95
TSSD al., 2024)
VGG-9
16
84.37
Ours
VGG-SNN
10
87.63
Ramesh al., 2019)
N/A
N/A
66.80
TCIA- ANNA-S et al., 2024)
VGG-SNN
14
78.50
STCA-SNN SNN(Wu et al., 2023)
N/A
14
80.90
TET al.)
VGG-SNN
10
81.72
N-Caltech101
ETC et al., 2025)
VGG-SNN
10
85.53
DDon al., 2024)
VGG-SNN
10
84.10
Ours
VGG-SNN
VGG-SNN
10
16
20.24
86.03-LO.J
ral dynamics features. This enables TMC to outperform existing methods that rely on temporal homogeneous train- ing.
limit the temporal heterogeneity and the performance of SNNs. Inspired by model calibration methods, propose TMC, which can be seen as temporal gradient rescaling mechanism to generate diverse logit gradients in the tempo- ral dimension. Experimental results show that our method can improve the temporal hetcrogeneity and performance of SNNs. In particular, our method achieves state-of-the-art accuracy ImageNet, DVSCIFAR 10, and N-Caltech101.
5. Conclusion
In this work, we analyze the temporal logit gradient in exist- ing direct training methods and conclude that the insufficient diversity of the logit gradient in the temporal dimension will

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Training High Performance Spiking Neural Network by Temporal Model Calibration
Acknowledgements
Davies, M., Srinivasa, N., Lin, T.-H., Chinya, G., Cao, Y., Choday, S. H., Dimou, G., Joshi, P., Imam, N., Jain, S., et al. Loihi: A neuromorphic manycore processor with on-chip learning. IEEE Micro, 38(1):82 294882999999 2018.
This work was supported in part by the National Key Research and Development Program of China (No. 2020AAA0109002), in part by the National Natural Science Foundation of China (62376247, 62334014, and 62436008), and in part by the grant from Key R&D Program of Zhejiang (2022C01048).
Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L. Imagenet: A large-scale hierarchical image database. In Proceedings of the IEEE Conference on Computer Vision and Puttern Recognition, pp. 248-255, 2009.
Impact Statement
Deng, S., Li, Y., Zhang, S., and Gu, S. Temporal effi- cient training of spiking neural network via gradient re- weighting. In International Conference on Learning Rep- rexentations.
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be
specifically highlighted here.
Dong. Y., Zhao, D., and Zeng, Y. Temporal knowledge sharing enable spiking neural network learning from past and future. IEEE Transactions on Artificial Intelligence, 2024.
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Zhu, R.-J., Zhang, M., Zhao, Q. Deng, H., Duan, Y., and Deng, L.-J. TCJA-SNN: Temporal-che channell joint atten- tion for spiking neural networks. IEEE Transactions on Neural Networks and Learning Systems, 2024.
Zuo, L., Ding, Y., Jing, M., Yang, K., and Yu, Self- distillation learning based temporal-spatial consis- tency for spiking neural networks. arXiv preprint arXiv:2406.07862, 2024.
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Training High Performance Spiking Neural Network by Temporal Model Calibration
A. Details of Experiments
A.1. Datasets and Training Details
CIFAR10/100. CIFARIO/IO (Krizhevsky et al., 2009) contains 10/100 classes and consists of 50k training images and 10k testing images with the size of 32 x 32. The original random horizontal flip and crop are applied to the training image augmentation. We train ResNet-19 architecture on these two datasets under time step lengths of 2, 4, and for 300 epochs, respectively. We use Adam optimizer with learning rate of 0.01 and apply cosine decay, gradually reducing the learning rate to O.
ImageNet. ImageNet (Deng et al., 2009) is the challenging image recognition dataset with 1.28 million training images and 50k test images from 1000 object classes. We crop the images to 224x224 and use the standard augmentation for the training data. The detailed experimental settings are the same as Spike-Driven Transformer V2 (Yao et al.) and QKformer (Zhou et al., 2024).
DVSCIFAR10. DVSCIFARIO (Li et al., 2017) is converted from CIFAR10. It has 10k images with size of 128x128. We apply a 9: 1 train-test split (i.e. 9k training images and Ik test images) to DVSCIFAR 10. In our training, we integrate the event data into 10 frames and resize the resolution to 48 x 48. Random horizontal flip and random roll within 5 pixels are taken as augmentation (Li et al., 2022). We train VGG-SNN architecture on DVSCIFAR 10 under time step lengths of 10 for 300 epochs and use an Adam optimizer with a learning rate of 0.01 and cosine decay to 0.
N-Caltech101 (Orehard et al., 2015) a neuromorphic version of the Caltech101 dataset, encompassing 101 categories. We train VGG-SNN architecture on DVSCIFARIO under time step lengths of 10 and 14 for 300 epochs, respectively, In addition, we use Adam optimizer with learning rate of 0.01 and cosine decay to 0.
N-Caltech101
DVS-Gesture. DVS-Gesture (Amir et al., 2017) is neuromorphic dataset for gesture recognition. DVS-Gesture contains a total of 11 event stream samples of gestures, 1176 for training and 288 for testing, with spatial size of 128x128 for each sample. For WVS-Gesture data, the event stream is integrated into frames in 30ms units and downsampled to 32 x 32. We train VGG-SNN architecture on VVS-Gesture under time step lengths of 10 for 300 epochs. In addition, we use an Adam optimizer with a learning rate of 0.01 and cosine decay to
SL- SL-Annmals-DDV. SL-AAnmals-DVS (Vasudevan et al., 2022) is a event-based Spanish sign language dataset. It composed of more than 1100 samples of 59 subjects performing 19 signs in isolation corresponding to animals. We train VGG-SNI architecture on SL-Animals-DVS under time step lengths of 16 for 300 epochs and use an Adam optimizer with learning rate of 0.01 and cosine decay to 0.
Quora Question Pair (QQP). The QQP dataset (Sharma et al., 2019) consists of a training set of 404,290 question pairs, and test set of 2,345,795 question pairs and is provided as part of a Kaggle competition. The detailed experimental settings are the same as SpikingBERT (Bal & Sengupta, 2024).
A.2. Calibration Metrics
ECE, The ECE of SNN at time step can be defined as:
ECE, = Eyi IP66 y,,,) - -Pel--
(A.1)
Since we only have finite samples, an approximate estimation is used to calculate ECE, given a finite sample size of N. Specifically, we group the probability predictions at time step t into M equispaced bins. Let Bm, denote the set of samples with predicted confidence belonging to the my bin, where the interval is [i-1 M M. Then, P(5, = y,le,) in Equation (A.1) can be denoted as the accuracy of Bm Am, T....l EiEB 16 y)), where is the indicator function. Similarly, Pe in Equation (A.I) can be represented as the mean confidence of Bm,,, that is the average confidence of all samples in the bin: Cm,t= [mmaal EieBm, p/. Then, ECE, can be approximated as a weighted average of the absolute difference between the accuracy and confidence of each bin:
M Σ
Bmmel Am,, Cm,el-
ECE,
(A.2)
AdaECE.
One disadvantage of ECE is the uniform bin width. For a trained model, most of the samples lie within the highest confidence bins, and hence these bins dominate the value of the ECE. We thus also consider another metric, AdaECE, for
14

---

Training High Performance Spiking Neural Network by Temporal Model Calibration
which bin sizes are calculated evenly distribute samples between bins (similar the adaptive binning procedure in (Nguyen & O'Connor, 2015)). The AdaECE, of SNN at time step & can be calculated as:
AdaECE, Σ [Bm.e 14m,t C.,,l S.L.i,j |Bm,el L.Vi,,-IBB..ll
B3,rl-
(A.3)
CECE. The ECE metric only considers the probability of the predicted class, without considering the other scores in the softmax distribution. A stronger definition of calibration would require the probabilities of all the classes in the softmax distribution to calibrated. This can be achieved with a simple classwise extension of the ECE metric. The CECE, of SNN at time step can be calculated as:
M
K
B N
CECE =
A - C Cm,ele
(A.4)
m=
where K the number of classes, B denotes the set of samples from the j-th class in the m-th bin at time step t,
A I EKEB 1(j = yt) and C [B E Pt k.
Note that we the number of bins for ECE and CECE 15 in our experiments.
B. Detailed Analysis of the Temporal Heterogeneity in SNNs
B.1. Neuronal Behavior Definition
The behavior of a spiking neuron over continuous time can be formulated as follows:
duc -(ue ureset) + Ie. u < Vih
T dt
(A.5)
o-= - 8( 4
(A.6)
where u, is the membrane potential, is the time content, J, is the input current, and Ven is the threshold. 8(.) is the Dirac delta function. The spike train oe is generated when u, reaches Vo, at time ty, and u, is reset to the resting potential treeet
B.2. Neuronal Temporal Heterogeneity Analysis
Firstly, the change of u, is quantified by du, di, which is determined by two components:
1) Decay of Membrane Potential - (w+ (reset). This term indicates that the membrane potential naturally decays towards the resting potential Wreset- If u, > Ureset, this term is negative, indicating a decrease in membrane potential. If uz < Uresets this term is positive, indicating an increase in membrane potential.
2) Input Current 7. represents the effect of the external current on the membrane potential. If T is positive, it drives the membrane potential up. If I is negative, pushes the membrane potential down.
Secondly, investigate the temporal dynamic changes of duz. de
du. > 0, means the membrane potential u, is increasing. This could be due to an input current I that is 1) When di sufficiently large overcome the natural decay of the membrane potential, because the effect of the natural decay is relatively small at the current moment.
2) When dus < 0, the membrane potential u, is decreasing. This could be because the input current I, is small or negative, de
not enough to counteract the decay of the membrane potential, or the effect of the natural decay is relatively large at the current moment.
3) Note that dus dr
0 is highly unlikely to occur unless specific conditions are met.
Thus, du: > 0 indicates that the membrane potential is continuously changing, either increasing or decreasing. This change a direct reflection of the neuron's 's response to temporal dependency and input signals and is key evidence of temporal heterogeneity.
15

---

Training High Performance Spiking Neural Network by Temporal Model Calibration
C. Proof of Proposition 3.4
Proof. The cumulative logits of the target class, denoted as Z can be written as Ei= Z Thus get
z
Z++1
H+1
Σ
z
t+1
im1
1+1
H+1
Z
(A.7)
F=1
t
x
zit1
1=1
1=1
Σ
z
<
Z4+1
By subtracting Zü on both sides of Equation A.7, get
z 7
<
x
(A.8)
for any 1, = l,...,? T 1, 1 < 2 it holds that
1
z -
50<
(A.9)
Hence satisfies Z z Z
D. Derivations in Section 3.3
D.1. Derivation of Equation 17
The loss function of our TMC method is:
L TCM LCR(Or. 8) + 02.
(A.10)
According to the chain rule, the partial derivative of this loss function with respect to the logit vector at time step is:
0LCr
80,--
VZTCM.
82;
az;
(A.11)
0Lcg
a0,¹
00, a ag 00,- 00, Op ag
a2;
00, drt
0o oz;
00, dr""" dg o.,
16

---

Training High Performance Spiking Neural Network by Temporal Model Calibration
where superscript i is used separate VZt by different classes. Superscript k and refer to the index of the target class and the index of non-target classes, respectively. For the element of the target class, have:
OLCE 02,
Pt pk
00p-
1,0%
00,
00,
apt 1 p.
00,
p
OP.
(1 P2)2'*
(A.12)
OP,
exp(e,)
exp(O.)
= - -P!))
exp(e()
E exp((?))
exp(e)) Ej Ej exp(C!)
OP."
= PP (1 Pt))
0
ES exp(C!)
0 1
az,
E'
In Equation A. C' is the total number of classes. By substituting Equation A.12 into Equation A.11, we have:
pd.
T.T.MGK ph yk + 1100
1
[Pt
(p])2]-
Pt (1 p?))
1 ph
(1
p/12
ph
p
ph- yh + 2. -1
(1- - P!)) 1 + Ne
p 1
(A.13)
1
pk
t
1 Pt
t.
1
p+ - yh + Xx.0eee
Finally, we have:
VZ.
TMC,k
Pp - 1h + Negpr
(A.14)
In the same way, we can have:
P"m
VZ Pm
TMC m
pm
(A.15)
f
p
D.2. Derivation of Equation 18
The gradient rescaling vector ge of our method is defined as:
VZ,MC
VZ TET 9t-
(A.16)
Since VZ,TC has different expressions for the element of the target class and the other elements, should also be derived separately for those two categories. We denote ge as the gradient rescaling factor of the target class, and g as the gradient rescaling factor all the other classes.
For the target class, we have:
V.Z.MCC
pk pk + tx.07-
1 +
(A.17)
VZ TET
pk- pk
p yt
The denominator p - 4t in Equation A.17 can also be written as -(1 ph)) ) Equation A.17 can be re-written
VZ
TMC,
9
(A.18)
VZ TET
1 pt
17

---

Training High Performance Spiking Neural Network by Temporal Model Calibration
Similarly, for the non-target classes Dt", have:
1 m
P ym 1 700
pm p.
x20 p."
(A.19)
VZ m
p
pm
t 1 P pr
because y = 0.
E. Convergence Analysis of TMC
E.1. Temporally Perfectly Clibrated SNN
Definition 3.1 requires a rate-coding SNN satisfy the following two properties:
1) Property I: P. PP yl:t), (1, ...,, T).
2) Property 2: P, Pi+1-
E.2. Optimization Objective
Convert the realization of these two properties into the optimization objective of TMC:
1) Note: Since the predicted outputs of trained SNN typically have the highest probability for the target class across time steps, P. be expressed P. Ph. Here, Pt the probability of the target class l: in the distribution of softmax(Z.).
2) Objective 1: During training, the realization of Property 1 can be converted to optimize IP. IP( y|p:)
This can
be achieved by introducing confidence regularization term, 6, to penalize the under-confidence issue P. < P(j = yP, and, especially, the over-confidence issue P. > P( y(Pr) with high sensitivity.
3) Objective 2: During training, the realization of Property 2, P. Pi+1. can be converted to P. < P.+1- This can be achieved by introducing a linearly decreasing exponent, Xe, into 0g to optimize z Z441 as described Proposition 3.4.
E.3. Convergence Analysis
With the loss function of TMC, the rescaling factor for the target class is generated to optimize confidence. Specifically:
Ae CE,K = =1=f(t) h(t), f(t) = t h(t) 1 Ph
(A.20)
VZ
Here, f(t) decreases with time steps.
the initial training phase, (tt) follows a random uniform distribution, and of increases with time steps to optimize z 2 thereby meeting Objective 2.
2) During training, at time step t, if 2t is high, the probability of the target class in the distribution of softmax (Z.). denoted as Pt may lead to overconfidence. In this situation, h(t) increases, causing of to decrease potentially even to negative value, penalize the overconfidence issue. Conversely, underconfidence occurs when Z is low, and of increases to address this issue, thereby meeting Objective 1.
3) At the end of the training, o. for different samples will converge to an interval. We have visualized the distribution of of values for 500 samples of a trained SNN in Figure Notably, for most samples with reasonable confidence, their g, values are centered within an interval that shifts closer over time. This indicates the achievement of Objective 2. It can be seen that across time step, some samples' of values are close to I negative numbers. This is consistent with Objective 1, which aims penalize particularly underconfident and overconfident samples. Overall, TMC has the effect of realizing these two optimization objectives and generates temporally calibrated SNNs.
F. Experimental Results
F.1. Performance Comparison on QQP
We apply our method in the direct training phase of SpikingBERT (Bal & Sengupta, 2024) model proposed in () on the widely used dataset QQP. We compare the classification performance of TMC with SDT (as used the original paper) and
18

---

Training High Performance Spiking Neural Network by Temporal Model Calibration
Table A.I. Performance comparison for different methods QQP.
Dataset
Architecture
Time Step
Method
Accuracy
SDT TET 42.72 Ours
71.12
QQP
SpikingBERT
80
78.05
TET. TMC achieves a higher accuracy of 87.86% than SDT((6.82%)) and TET(87.03%)
E2. Performance Comparison on Neuromorphic Datasets
We train VGGSNN with T=10 on DVS-Gesture and compare the performance of TMC with current works in Table A.2. TMC achieves SOTA performance with an accuracy of 12%.
Table A.2, Performance comparison with stale-of the-art methods on DVS-Gesture
Dataset
Model
Architecture
Time Step
Accuracy
STBP-idBN(Wi et al., 2018)
ResNet-17
96.87
(Fan el al., 2021b)
5Cunv, 2FC
20
97.57
SEW ResNet
2021a)
7B-Net
97.92
TA-SNN Yau el al., 2021)
TA-SNN
20
98.61
DVS-Gesture
TZuu al., 2024)
5Conv, 2FC
20
99.00
STSC Yu el al., 2022)
5Conv. 2FC
20
98.96
SLT (Anuaas
2024)
VGG-SNN
10
98.43
Spikeformer(Li et al., 2024c)
Spikeformer-775 x
98.96
Ours
VGG-SNN
99.12
We train VGGSNN with T=16 using TMC on SL-Animals-DD DDS and compare the performance with SDT and TET Table
A.3. TMC achieves a higher accuracy of 70.05% against SDT (66.75%) and TET (68.34%), demonstrating the superiority of TMC.
Table A.3. Performance comparison for different methods on SL-Animals-DYY
Dataset
Architecture
Time Step
Method
Accuracy
SDT TET Ours
66.75
SL-Animals-DVS VGG-SNN
16
68.34 70.05
F.3. Training Stability Evaluation TMC
We compare the trends of test loss variation during the training process among SDT, TET, and TMC on DVSCIFARIO in Figure A1. The results demonstrate TMC's stable training, driving loss to minimal values, unlike SDT and TET, which suffer from overconffidence-nnduedd oscillations.
2.50
TMC
2.25
2.00
1.75
1.50
1.25
1.00
50
100
150 Epochs
200
250
300
Figure A1. Training Stability Evaluation DVSCIFAR10.
19

---

Training High Performance Spiking Neural Network by Temporal Model Calibration
F.4. Performance Comparison CIFARIO
Table A.4. Performance comparison with tate-of-the-arr methods on CIFARIO.
Dataset
Model
Architecture
Time Step
Accuracy
DSpike(Li et al., 2021)
ResNet-18
94,25
GLIF( Yao et 2022)
ResNet-19 19
95.03
TEuN al., 2022)
ResNet-19
95.60
RMP-Loss et el., 2023)
ResNet-19
96.10
TET(Deng et al.)
ResNet-19
94.50
TKS(Dong et al., al. 2024)
ResNet-19
95.30
CIFAR1O
(Cii et al., 2024)
ResNet-19
95.03
ETC(Zhao al., 2025)
ResNet-19
95.87
TSSD(Zuo et al., 2024)
VGG-9
2
94.41
95.23
Ours
ResNet-19
95.08-10.10
94.8710.08
E.5. Calibration Error in CEC Metric DVSCIFARIO
Table A.5. Calibration error CECE metric for different methods across time steps on DVSCIFARIO.
Methods
T=1
T=2
T=3
T=4
T=5
T=6
T=7
T=8
T=9
T=10
SDT
0.13
0.08
0.07
0.06
0.05
0.05
0.05
0.05
0.05
0.05
TET
0.07
0.05
0.05
0.04
0.04
0.04
0.04
0.04
0.04
0.04
Ours
0.06
0.05
0.05
0.04
0.04
0.04
0.03
0.03
0.02
0.03
Compared to SDT and TET, our method achieves lower calibration errors. In particular, the calibration errors of our method drop significantly in the later time steps. This observation further demonstrates that our method can effectively alleviate the overfitting issue.
20


Reviews:
[{"review_id": "o0MZlGnxii", "reviewer": "Reviewer_ksZk", "summary": "This work finds that the logit gradients have insufficient diversity in the temporal dimension during SNN training. The authors then rescale the gradient in each time step to improve diversity, resulting in SOTA performance for image classification tasks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Rebuttal Appendix:https://anonymous.4open.science/r/TMC-0262**\n\n**1.The relationship between temporal heterogeneity and the performance of SNNs remains unclear. It is not clear whether the so-called logit gradient should be diverse.**\n\n**1)Relationship between Temporal Heterogeneity and Performance**\n\nIn SNNs, the neuronal dynamics of the membrane potential $u_t$ can be formulated as:\n \\begin{equation} {\\tau\\frac{du_t}{dt}}=-(u_t-u_{reset})+I_t.\\tag{R1}\\end{equation}\nWhen $u_t$ reaches the threshold $V_{th}$, $u_t$ is set to $u_{rest}$ and the neuron fires a spike:\n\\begin{equation} s_t=\\sum_{t_f}\\delta(t-t_f).\\tag{R2}\\end{equation}\nThe temporal heterogeneity of SNNs implies $|\\frac{du_t}{dt}|>0$, yielding benefits as follows:\n+ **Reduction of Dead Neurons**: Ensuring neurons spike when necessary. \n+ **Sensitive Response** to Temporal Dependency and Input Information: Capturing spatio-temporal dynamics effectively.\n+ **Diverse Spike Firing Frequency**: Adjustable $\\Delta t$ in $s_t$ allows for diverse behaviors, such as burst spikes.\n\nOverall, accumulating temporal heterogeneous neuron responses to varied temporal inputs over time steps **enhances predictive confidence and performance**. \n\n**2)Temporal Heterogeneity Improvement**\n\nDirect training of SNNs involves the discrete model expression:\n \\begin{equation} {u_{t+1}^i}=\\lambda(u_t^i-V_{th}s_t^i)+\\sum_j\\mathbf{W_{ij}}s_{t+1}^j.\\tag{R3}\\end{equation}\nEnhancing temporal heterogeneity can be achieved by improving either the temporal dependency dynamics $\\lambda(u_t^i-V_{th}s_t^i)$ or the input mapping dynamics $\\sum_j\\mathbf{W_{ij}}s_{t+1}^j$ via $\\mathbf{W}$ training. **We concentrate on the latter, which is less explored.**\n\n**3)Logit Gradient Diversity Rationality**\n\nEnhancing linear separability, that is, increasing the rank of $\\mathbf{W}$ is crucial for dynamic input mapping. Increasing temporal gradient diversity is beneficial to explore the solution space and increase $\\mathbf{W}$ rank. However, existing methods lack exploration of logit gradient diversity.\n\n\n**2.The method is a minor modification of TET.This paper fails to show the significance of such marginal changes clearly.The regularization to improve diversity is not straightforward.**\n\nFrom the technological perspective, TMC is a regularization term modification based on TET. However, from the temporal gradient decent perspective, **TMC is a critical improvement of TET**. The significance of TMC is highlighted as follows:\n\n**1)Rethinking the Temporal Logit Gradient Calculation**\n\n+ TET focuses on increasing the momentum of temporal gradient descent but ignores appropriately assigning magnitude and direction to the momentum. This leads to an overconfidence issue. \n\n+ Introducing model calibration into SNNs, we rethink the temporal gradient calculation properties that SNNs should meet and propose a temporal gradient rescaling mechanism. This mechanism assigns appropriate magnitude and direction to the gradient descent momentum across time steps.\n\n**2)Effective and Adaptive Regularization Term**\n\n+ TET introduces the MSE regularization to prevent the occurrence of \"particular outliers\" but lacks deeper analysis on the issues. The regularization term appears to be an intuitive design. Moreover, the regularization term at each time step keeps same and fixed, relying on hyperparameters. \n+ During training, TMC with a new regularization term rescales the logit gradient effects of TET, adaptively responding to input data, training phases, and time steps without hyperparameter dependence. \n+ There exist straightforward model calibration techniques like label smoothing, but these methods have been found to be suboptimal relying on prior information and hyperparameters. Moreover, the regularization-based calibration techniques achieve SOTA performance.\n\n**3.Sequential tasks are more appropriate benchmarks to be considered.**\n\n+ **Text Sequential Classification Task.** TMC is applied during the direct training phase of the SpikingBert [R1] model on Quora Question Pair dataset. Compared to SDT(86.82%) and TET(87.03%), TMC achieves higher accuracy at 87.86%.\n+ **Dynamic Action Recognition Task.** We train VGGSNN with T=10 on DVS-Gesture and compare the performance of TMC with current works in Table R1 in the appendix. TMC achieves SOTA performance with an accuracy of 99.12%. \n\n**4.TMC works better on neuromorphic datasets than static images. Provide explanation and insights.**\n\n**TMC indeed exhibits more significant advantages on neuromorphic datasets.**\n+ Compared to static datasets, neuromorphic datasets have rich spatiotemporal components by interacting the spatial and temporal information.  \n+ TMC employs temporal heterogeneous training and has the advantage of capturing more task-relevant spatiotemporal dynamics features, outperforming existing methods, which employ temporal homogeneous training.\n\n[R1]Spikingbert: Distilling bert to train spiking language models using implicit differentiation.", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "The technical detail is limited:\n\n1. It is not clear whether the so-called \"logit gradient\" should be diverse across time steps. The surrogate gradient + firing rate-based loss framework (both SDT and TET) implicitly follows a rate coding scheme. Consequently, it might be that uniformity in behavior over time steps serves as a more reliable indicator of confidence. The relationship between temporal heterogeneity and the performance of SNNs remains unclear and needs a formal theoretical explanation.", "experimental_designs_or_analyses": "checked", "relation_to_broader_scientific_literature": "This work improves TET by increasing logit diversity in the temporal dimension.\n\nTET: Temporal efficient training of spiking neural network via gradient reweighting", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Strengths:\n1. The method is in a plug-and-play fashion and can be merged with most SOTA models and algorithms.\n2. It achieves SOTA performance on common image classification tasks.\n\nWeaknesses:\n1. Lack of novelty. The method can be regarded as a minor modification of TET: it is like a cumsum version of TET. Even the regularization term also exhibits a comparable effect to the $L_{MSE}$ regularization in TET.\n\n2. The logic is not convincing. While marginal modifications can indeed constitute valuable contributions, this paper fails to show the significance of such marginal changes clearly. It is not clear why diversity across time should be improved. And the regularization to improve diversity is not straightforward.", "other_comments_or_suggestions": "Suggestions:\n\n1. Given the focus on temporal heterogeneity, I believe that sequential tasks would serve as more appropriate benchmarks to be considered.  \n\n2. The importance of temporal heterogeneity should be clarified.", "questions_for_authors": "Interestingly, the proposed method works better on neuromorphic datasets than static images. Do the authors consider this a common phenomenon or not? Could the authors explain the possible reason? Are there fundamental insights within this observation?", "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": "9VQP8ujGnk", "reviewer": "Reviewer_GDhn", "summary": "this paper introduces a new training method for spiking neural networks called temporal model calibration . the goal is to improve the performance of snns by increasing their temporal heterogeneity, which is how much their outputs vary over time. the authors argue that existing training methods, like direct training using bptt, do not fully utilize temporal heterogeneity because the loss gradients remain too similar across time steps. their main idea is to rescale the loss gradients at each time step to encourage more diversity in the network’s responses over time. they do this by modifying the cross-entropy loss function with a new gradient scaling factor based on confidence calibration techniques used in deep learning. the method is tested on several datasets, including imagenet, dvscifar10, and n-caltech101, and achieves state-of-the-art accuracy in some cases. overall, the paper presents a novel way to enhance the learning dynamics of snns by focusing on how gradients evolve over time.\n\n## update after rebuttal\n\nThanks to the authors for a very detailed and thoughtful response. I appreciate the new experiments, the visualizations of temporal heterogeneity, and the extra analysis around computational cost and training stability — it’s clear a lot of effort went into the rebuttal. These additions definitely helped clarify several points I had raised.\n\nThat said, my overall opinion on the paper hasn’t changed much. While the new results are nice to see, they don’t fully address my bigger concerns about the lack of strong theoretical backing and the limited discussion on scaling to larger models and broader applications. The paper is solid and the method works well empirically, but I still feel it falls a bit short on the novelty and depth needed for acceptance. So I’m keeping my original score of Weak Reject.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Rebuttal Appendix:https://anonymous.4open.science/r/TMC-0262**\n\n**1.Visualization or mathematical proof of increased heterogeneity.In-depth analysis of how tmc affects neuron activations.**\n\n**1)Temporal Heterogeneity Visualization**\n\nWe compare VGGSNNs trained by TMC and TET on DVSCIFAR10, visualizing the cosine similarity of layer features across time steps in Figure R1. TMC exhibits higher temporal heterogeneity at all layer levels.\n\n**2)TMC Effect**\n\nThe neuron model:\n \\begin{equation} {u_{t+1}^i}=\\lambda(u_t^i-V_{th}s_t^i)+\\sum_j\\mathbf{W_{ij}}s_{t+1}^j.\\tag{R4}\\end{equation}\n\nWith TMC $\\mathbf{W}$'s separability for varied inputs is enhanced, boosting neuron temporal heterogeneity in response to diverse temporal inputs.\n\n**2.Discussion on computational efficiency.**\n\nWe analyze TMC's time and space complex of the logit gradient calculation relative to SDT and TET. (T:Time Step, B:Batch Size, C:Class Number)\n\n**1)SDT**:Time complexity $O(T\\*B\\*C)$, space complexity $O(B\\*C)$. \n\n**2)TET**:Time complexity $O(T\\*B\\*C)$, space complexity $O(T\\*B\\*C)$. \n\n**3)TMC**:Based on TET, it introduces additional calculation of the regularization term $\\theta_t^{\\lambda_t}$. At each timestep, generating $\\theta_t$ needs $O(B\\*C)$ time complexity and $O(1)$ space complexity. Calculating $\\theta_t^{\\lambda_t}$ needs $O(1)$ for both time and space complexity. Overall, TMC's logit gradient calculation time complexity is $O(T\\*B\\*C)+O(T\\*B\\*C)+O(T\\*B\\*C)+O(T)$, and space complexity is $O(T\\*B\\*C)+O(T\\*B\\*C)+O(T)+O(T)$. \n\nTMC's total gradient computation, including hidden layer and logit gradients, is **close to SDT and TET in computation complexity and training time**. \n\n**3.No formal proof that tmc leads to a more stable or optimal training.Theoretically analyze the convergence properties of tmc.**\n\n**1)Firing Rate Evaluation:**\n\nFiring rates impact SNN gradients. As shown in Figure R2, compared with SDT and TET, TMC-trained VGGSNN on DVS-CIFAR10 exhibits the lowest firing rates in shallow layers, rising with depth, and peaking in deeper layers. This indicates TMC's sensitive response to global features in deeper layers. Higher firing rates also **mitigate the gradient vanishing**.\n\n**2)Gradient Norm Evaluation:**\n\nFigure R3 further compares model parameters update gradient norms of TMC, SDT and TET. TMC shows the highest norm, suggesting **faster convergence**.\n\n**3)Training Stability Evaluation:**\n\nWe theoretically analyze TMC's temporal rescaling factor $g_t$(lines 224-274 left). $g_t$ effectively rescale logit gradients, enhancing optimization over SDT and TET. Figure R4 demonstrates TMC's **stable training**, driving loss to minimal values, unlike SDT and TET, which suffer from overconfidence-induced oscillations.\n\n**4.The paper does not include an ablation study to determine which components of tmc contribute most to the performance gains.Analyze hyperparameter sensitivity of tmc.**\n\n1)In lines 358-384 left and 330-336 right, **we conducted ablation studies** on the regularization term's components—base $\\theta_t$ and exponent $\\lambda_t$. Both enhance performance, but their combined effect is optimal.\n\n2)TMC **does not introduce additional hyperparameters**, ensuring flexibility.\n\n**5.It does not compare tmc to alternative approaches.It does not discuss some recent works on improving training stability through adaptive gradient methods.**\n\n**1)Compare to Alternative Approaches:** In Related Work section and lines 149-155 left, we discussed entropy-based loss function methods and highlighted their homogeneous training effect. Notably, TMC has a different training effect by introducing temporal heterogeneous training. As the code for these methods is not available, we can only compare TMC's classification accuracy with them in Section 4.4.\n\n**2)Compare to Adaptive Gradient Methods:** Recent studies[R10-R12] have introduced adaptive gradient methods, primarily focusing on adjusting hidden layer gradients and using SDT loss to calculate the logit gradient. In contrast, TMC modifies logit gradient calculations and offers plug-and-play compatibility with various hidden layer gradient mechanisms.\n\n**6.Explore tmc to much larger models,real-world applications,deeper networks or more complex architectures.**\n\n**1)Larger Model Evaluation:** Hierarchical Spiking Transformer used in our paper with 64.96M parameters is the large model. \n\n**2)Deeper Model Evaluation:** We verify TMC on ResNet101 on ImageNet dataset with T=4 and compare it with SDT and TET. TMC achieves a higer accuracy of **70.52%** than SDT(68.74%) and TET(67.98%).\n \n**3)Real-World Applications:** We evaluate TMC on text sequential classification and dynamic action sequential recognition tasks and TMC achieves **SOTA performance**. More detailed results can be seen in Response 3 to Review ksZk. \n\n[R10]Learnable Surrogate Gradient for Direct Training Spiking Neural Networks\n[R11]Sparse Spiking Gradient Descent\n[R12]Gradient Descent for Spiking Neural Network", "claims_and_evidence": "the main claim of the paper is that current direct training methods for snns do not make full use of temporal heterogeneity because their loss gradients remain relatively uniform across time steps. the authors support this claim by analyzing the gradients in standard training methods and showing that they lack diversity. they further claim that their proposed tmc method improves both temporal heterogeneity and accuracy by rescaling these gradients. the experimental results provide reasonable support for this, as the tmc-trained models outperform existing methods on several benchmark datasets. however, the claim that the method enhances temporal heterogeneity is mostly based on indirect evidence (such as improved accuracy) rather than direct visualization or mathematical proof of increased heterogeneity. a more in-depth analysis of how tmc affects neuron activations over time would strengthen this claim.", "methods_and_evaluation_criteria": "the proposed method is well-designed for the problem it addresses, as it directly targets the temporal structure of snns. the authors evaluate their approach using standard benchmark datasets, including imagenet and neuromorphic datasets like dvscifar10 and n-caltech101. these datasets are appropriate for testing the effectiveness of an snn training method. the evaluation primarily focuses on accuracy, expected calibration error (ece), and adaptive ece, which are relevant metrics for both classification performance and model calibration. however, the paper does not provide much discussion on computational efficiency—how much additional training time or memory tmc requires compared to existing methods. since gradient rescaling could introduce extra computation, it would be useful to see an analysis of the trade-offs between performance and efficiency.", "theoretical_claims": "the paper provides a detailed mathematical formulation of the proposed method, including how the gradient rescaling factor is computed. however, there is no formal proof that tmc leads to a more stable or optimal training process. the method is motivated by intuition and empirical results rather than rigorous theoretical guarantees. for example, while the authors argue that their gradient rescaling improves learning dynamics, they do not analyze whether it guarantees faster convergence or prevents issues like vanishing or exploding gradients. a theoretical analysis of the convergence properties of tmc would make the claims more robust.", "experimental_designs_or_analyses": "the experimental design is solid in terms of dataset selection and performance metrics. the authors compare their method against several baselines, including standard direct training (sdt) and temporal efficient training (tet), which are well-known methods in the field. they show consistent improvements in accuracy and calibration errors across multiple datasets. one strength of the experiments is that they include both static and neuromorphic datasets, showing that tmc is broadly applicable. however, there are some missing analyses. for example, the paper does not include an ablation study to determine which components of tmc contribute most to the performance gains. it would be useful to see experiments testing different variations of the gradient rescaling factor to understand its specific effects. also, the paper does not analyze how sensitive tmc is to hyperparameter choices, which is important for practical use.", "relation_to_broader_scientific_literature": "this paper builds on prior work in spiking neural networks, particularly methods that use backpropagation through time for training. it connects to research on temporal heterogeneity in snns, which has been studied in both computational neuroscience and machine learning. the work is also related to research on model calibration in deep learning, as tmc is inspired by techniques like label smoothing and confidence regularization. while the paper does a good job of citing relevant works, it does not compare tmc to alternative approaches that also modify loss gradients for better learning dynamics, such as entropy-based loss functions. discussing these related methods could provide a clearer context for how tmc fits within existing strategies.", "essential_references_not_discussed": "the paper includes most of the key references in snn training and model calibration but does not discuss some recent works on improving training stability through adaptive gradient methods. for example, methods that dynamically adjust learning rates based on gradient variance might have some similarities to tmc. comparing tmc to these approaches could provide additional insights into its uniqueness and potential limitations.", "other_strengths_and_weaknesses": "one of the main strengths of this paper is that it introduces a relatively simple yet effective modification to the loss function that improves performance across multiple datasets. the empirical results are strong, showing state-of-the-art accuracy in some cases. another strength is that the paper addresses an important issue in snn training—the lack of sufficient temporal heterogeneity—by introducing a novel way to enhance it. however, there are some weaknesses. the paper does not provide a theoretical justification for why tmc should always improve training, and there is little discussion of potential downsides, such as increased training time or instability. also, while the results are impressive, the authors do not explore the scalability of tmc to much larger models or real-world applications.", "other_comments_or_suggestions": "it would be helpful to include a computational cost analysis comparing the training time of tmc with other methods. a more detailed discussion of hyperparameter sensitivity would also improve the paper. the writing is mostly clear, but some sections could be better structured to make the key ideas easier to follow.", "questions_for_authors": "1. how much additional computational overhead does tmc introduce compared to standard direct training methods? does it significantly increase training time?\n2. have you tested whether tmc works well with much deeper networks or more complex architectures?\n3. do you have any direct visualizations of how tmc changes the temporal activity of snn neurons over time? this could provide stronger evidence that it increases temporal heterogeneity.\n4. is there any risk that rescaling the gradients could introduce instability in training, such as oscillations or divergence? if so, how is this mitigated?", "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": "QZmN5nYlHi", "reviewer": "Reviewer_KZ4r", "summary": "This paper proposes a temporal confidence calibration method for SNNs, improving both the model's performance and heterogeneity, as demonstrated across several static classification tasks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Rebuttal Appendix:https://anonymous.4open.science/r/TMC-0262**\n\n**1.Provide more results on dynamic datasets to make the method more convincable, like action recognition datasets including DVS-Gesture, SL-Animals, etc.**\n\n**1)DVS-Gesture**: We train VGGSNN with T=10 on DVS-Gesture and compare the performance of TMC with current works in Table R1 in the appendix. TMC achieves **SOTA performance with an accuracy of 99.12%**.\n\n**2)SL-Animals-DVS**: VGGSNN (instead of a larger model due to time constraints) is trained using TMC with T=16. Comparison with SDT and TET shows TMC achieving a **higher accuracy of 70.05%** against SDT (66.75%) and TET (68.34%), demonstrating the superiority of TMC.\n\n**2.Is $\\hat{P}_t$ the maximum value of softmax on the time-averaged logits? Then how does a temporally perfectly calibrated model trained with gt increase its temporal confidence monotonically to match Definition 3.1?I didn't see much relationship between the proposed method and Definition 3.1, nor a relationship between Definition 3.1 and 3.2.**\n\n**Clarification of Definition 3.1 and Our Mechanism:**\n\n**1)Element Definition(lines 165-169 left)** \n\n+ Temporal Predicted Confidence: $\\hat{P}_t=max(Softmax(\\overline{z_t}))$.\n+ Accumulated Logit Outputs at t-th time step: $\\overline{z_t}=\\frac{1}{t}\\sum_i^tz_i$.\n+ $t \\in {[1,2,...,T]}$. \n+ **$\\hat{P}_t$ is different at different time steps**, because $\\overline{z_t}$ varies across time steps.\n\n**2)Definition 3.1:Temporally Perfectly Calibrated SNN(lines 169-176 left)**\n\n+ Introduce model calibration into SNN's time dimension, where a temporally perfectly calibrated SNN satisfies **Test Accuracy is Equal to Confidence at each timestep**: ${\\hat{\\textit{P}_t}} = \\mathbb{P}(\\hat{\\textit{y}}=\\textit{y}|\\hat{\\textit{P}_t})$. \n\n**3)Definition 3.2:Temporal Gradient Scaling Factor(lines 177-176 left)**\n\nModel calibration is realized using a gradient rescaling factor to rescale the gradient of cross-entropy(CE) loss in the training of ANNs. To realize the temporal model calibration in SNN, we propose the temporal gradient scaling factor $g_t$ (lines 177-180 left) to rescale the gradient of CE loss at each time step, i.e., TET loss, in Equation 6.\n\n**4)Effect of $g_t$**\n\n+ Note that Definition 3.1 constructs the concept of temporally perfectly model calibration in the field of SNN. Our work focuses on the rate-coding SNNs and we provide a further instantiation specific to rate-coding SNNs (lines 196-202 left). Specifically, current work suggests that the accuracy of rate-coding SNNs increases monotonically with time steps. Thus, for the temporally perfectly rate-coding SNNs, confidence should increase monotonically with time steps. \n+ To match definition 3.1, $g_t$ should meet: at earlier time steps, it should shrink the effect of CE loss logit gradient ($\\Delta Z_t^{TET}$) to reduce the confidence. Conversely, $g_t$ should enhance the effect of $\\Delta Z_t^{TET}$ to increase the confidence.\n\n**5)Method Design (Section 3.3)**\n\nIn Section 3.3, we propose TMC loss function with a new regularization term to realize the effect of $g_t$. With temporal gradient theoretical analysis (lines 244-274 left and 220-235 right), we derive the temporal gradient rescaling factor $g_t^{TMC}$ in our method, which can optimize the rate-coding SNN to increase its confidence monotonically with time steps.\n\n**3.Some sentences in section 3.1 are false.**\n\n+ Thank you for your correction. We follow the loss function definition of SDT in [R13] and the reference [1] in line 139 left should be changed to [R13]. [R13]  defines the loss function of SDT by calculating the cross-entropy loss between the average pre-synaptic input of the output layer and the true label. \n+ TET calculates the cross-entropy loss between the pre-synaptic inputs of the output layer with the true labels at each time step. To be more accurate, we should change \"the membrane potential of the last layer\" in our paper to \"pre-synaptic input of the output layer\".\n\n**4.The definition of the confidence score is quite weird, not bounded in [0, 1].**\n\nWe would like to provide clarification regarding the \"confidence score\" in our paper.\n\n+ Before Equation 9, our analysis focuses on absolute confidence values, that is, the maximum predicted probability, which is inherently bounded within the interval [0, 1].\n+ For Equation 9, to enhance the loss function's sensitivity to overconfidence, we adopted the ratio of confidence/(1 - confidence) as $\\theta_t$. This essentially serves as a more effective regularization of confidence.\n\n**5.Result on N-Caltech101 is not SOTA.**\n\nOn N-Caltech101, our method achieves 86.03% accuracy at T=10, while [3] achieves 87.86% accuracy at T=16. **Reevaluating our method at T=16, we achieve 88.24% accuracy, surpassing [3]**.\n\n[R13] Deng, S.,Li,Y.,Zhang, S., andGu, S. Temporal efficient training of spiking neural network via gradient re-weighting. arXivpreprintarXiv:2202.11946,2022.", "claims_and_evidence": "The proposed method is simple yet efficient, with a clear motivation at increasing the model's temporal heterogeneity. Comprehensive Experiments on static datasets clearly demonstrate its superiority over SDT and TET. The SOTA results on CIFAR10DVS and Imagenet are compelling.", "methods_and_evaluation_criteria": "1. The presented comparative and ablation experiments make sense for evaluating the method's effectiveness.\n\n2. Evaluating only static datasets is fair yet a little weak regarding a method focusing on temporal heterogeneity. I hope the authors can provide more results on dynamic datasets to make the method more convincable, like action recognition datasets including DVS-Gesture, SL-Animals, etc.", "theoretical_claims": "In definition 3.1, is $\\hat{P}_t$ the maximum value of softmax on the time-averaged logits? If so, the accuracies of the SNN at all time steps are equal. Then how does a temporally perfectly calibrated model trained with gt increase its temporal confidence monotonically to match Definition 3.1? (lines 199-202)", "experimental_designs_or_analyses": "The experiments are conducted fairly and reasonably.", "relation_to_broader_scientific_literature": "It could benefit the brain-inspired learning area a lot, especially the training of classification tasks that leverage cross-entropy loss on spiking neural networks.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "1. Some sentences in section 3.1 are false. [1] Calculate Loss on the output spikes. TET calculates the loss on the output values of an MLP followed by a spike layer.\n\n2. The definition of the confidence score is quite weird, not bounded in [0, 1]. Leveraging it to punish overconfidence seems far deviated from the confidence calibration motivation.\n\n3. Following the above, I didn't see much relationship between the proposed method and Definition 3.1, nor a relationship between Definition 3.1 and 3.2. The paper needs careful revising regarding the writing.\n\n4. The result on N-Caltech101 is not SOTA. See [3].\n\n[1] Spatio-Temporal Backpropagation for Training High-Performance Spiking Neural Networks\n\n[2] Temporal efficient training of spiking neural network via gradient re-weighting\n\n[3] Rethinking the membrane dynamics and optimization objectives of spiking neural networks", "other_comments_or_suggestions": "No", "questions_for_authors": "See above", "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": "De0wVyMrsh", "reviewer": "Reviewer_g8s8", "summary": "This paper systematically summarizes previous logit gradient calculation schemes, including SDT and TET, then proposes a new temporal gradient rescaling method to enhance the learning capability of SNNs.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Rebuttal Appendix:https://anonymous.4open.science/r/TMC-0262**\n\n**1. I tend to think the proposed method can provide a new perspective for the SNN community to a certain extent. The research perspective of Proposition 3.3 seems interesting. The authors conduct research on temporal heterogeneity and gradient allocation problems for different time steps.**\n\nWe sincerely appreciate your feedback and are greatly encouraged by it.\n\n**2. As shown in Tab.2 and Tab.4, the proposed method seems to be more effective for neuromorphic datasets (e.g. DVS-CIFAR10), but the performance improvement on ImageNet-1k (large-scale dataset) and SNN Transformer architecture appears to be relatively limited (<1%). For CIFAR100 dataset, the reported accuracy of this work lags behind TCL and ETC.**\n\nHere, we highlight the performance advantages of our TMC method on neuromorphic datasets, ImageNet and CIFAR100. Furthermore, we evaluate TMC on real-world applications and the results demonstrate that TMC **achieves SOTA performance**. \n\n**1) Neuromorphic Datasets** \n\n+ **Dataset Feature Description:** These datasets capture the dynamic changes in pixel intensity and record the resulting spike events using dynamic vision sensors (DVS). Compared to static or frame-based datasets, they contain **rich spatio-temporal components** by interacting the spatial and temporal information and follow the event-driven processing fashion triggered by binary spikes. \n+ **TMC Advantages:** Our method employs temporal heterogeneous learning using a temporal logit gradient rescaling strategy. This is beneficial for **capturing more task-relevant spatio-temporal dynamics features** compared to existing temporal homogeneous methods, such as SDT and TET. Therefore, TMC exhibits significant advantages on neuromorphic datasets.\n\n**2) ImageNet Dataset** \n\n+ **Dataset Feature Description:** ImageNet is known for its high-resolution and complex scenes. Current efforts to enhance the performance of ImageNet mainly concentrate on refining the model architecture, often resulting in the **introduction of a large number of additional parameters** to achieve performance gains. \n+ **TMC Advantages:** It is notable that our proposed method TMC is **plug-and-play**. When TMC is integrated into the Hierarchical Spiking Transformer (current SOTA model) in our paper, it enhances both classification and calibration performance on ImageNet, further demonstrating the effectiveness of our method.\n\n**3)CIFAR100 Dataset**\n\n+ **Dataset Feature Description:** CIFAR-100 is characterized by its low-resolution images, which are often blurry and contain relatively less information. \n+ **TMC Advantages:** With a small number of time steps, i.e., **T=2**, our method, which employs a temporal heterogeneous learning approach, is capable of capturing a richer set of features. Specifically, when T=2, TMC achieves a classification accuracy of 76.35%, outperforming other methods such as TEBN (75.86%), RMP-Loss (74.66%), and ETC (75.96%). \n+ **Limitation of the Dataset:** However, when **the number of time steps is increased**, our method may capture noisy features due to the blurriness of the images, which is not beneficial to performance improvement. This suggests that our method has more advantages when handling dynamic and complex tasks.\n\n**4) Real-Word Application**\n\nMoreover, we evaluate our method on real-world applications, including text sequential classification and dynamic action sequential recognition tasks, where **TMC achieves state-of-the-art (SOTA) performance**.\n+ **Text Sequential Classification Task.** We apply our method in the direct training phase of SpikingBert model proposed in [R1] on the widely used dataset Quora Question Pair (QQP). We compare the classification performance of TMC with SDT (as used in the original paper) and TET. TMC achieves a higher accuracy of **87.86\\%** than SDT(86.82\\%) and TET(87.03\\%).\n+ **Dynamic Action Sequential Recognition Task.** We train VGGSNN with T=10 on DVS-Gesture and compare the performance of our method with current works in Table R1 of Rebuttal Appendix. Results show that TMC achieves SOTA performance with an accuracy of 99.12%. \n\n**3.We futher evaluate the training process of TMC on VGGSNN on DVSCIFAR10 with T=10.** \n\n**1)Gradient Norm Evaluation:**\n\nIn Figure R3 in the appendix, we evaluate the trend of gradient norms for model parameter updates during the training of TMC, SDT, and TET. TMC exhibits the highest norm, indicating **faster convergence**.\n\n**2)Training Stability Evaluation:**\n\nWe visualize the training loss variation during the training of TMC, SDT and TET in Figure R4 in the appendix. The results demonstrate TMC's **stable training, driving loss to minimal values**, while SDT and TET suffer from overconfidence-induced oscillations.\n\n\n[R1] Spikingbert: Distilling bert to train spiking language models using implicit differentiation.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "I tend to think the proposed method can provide a new perspective for the SNN community to a certain extent.", "theoretical_claims": "I have read the theoretical claims in Sec. 3.2-3.3 and Appendix.", "experimental_designs_or_analyses": "I have checked experimental designs mentioned in Section 4.", "relation_to_broader_scientific_literature": "The research field of this work is related to the BPTT training methods based on surrogate gradients in the SNN community.", "essential_references_not_discussed": "Not found yet", "other_strengths_and_weaknesses": "1. The research perspective of Proposition 3.3 seems interesting. The authors conduct research on temporal heterogeneity and gradient allocation problems for different time-steps.\n\n2. As shown in Tab.2 and Tab.4, the proposed method seems to be more effective for neuromorphic datasets (e.g. DVS-CIFAR10), but the performance improvement on ImageNet-1k (large-scale dataset) and SNN Transformer architecture appears to be relatively limited (<1%). For CIFAR-100 dataset, the reported accuracy of this work lags behind TCL and ETC.", "other_comments_or_suggestions": "This work can be seen as a further exploration on TET [1], which proposes a new loss function calculation scheme around the gradient allocation problem at different time-steps.\n\n[1] Deng, S., Li, Y., Zhang, S., and Gu, S. Temporal efficient training of spiking neural network via gradient reweighting. ICLR 2022.", "questions_for_authors": "See Strengths And Weaknesses Section.", "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:

Meta-review: This paper introduces *Temporal Model Calibration (TMC)*, a gradient rescaling strategy aimed at improving the temporal heterogeneity and overall performance of Spiking Neural Networks (SNNs). It proposes to rescale the cross-entropy loss gradients at each time step, thereby enhancing the diversity of logit gradients. The method is validated on a broad range of static and neuromorphic datasets, achieving competitive or state-of-the-art results.

**Strengths noted by reviewers include:**

- Simple yet effective design with no new hyperparameters and compatibility with a wide range of SNN architectures.
- Consistent performance improvement over SDT and TET on both static and neuromorphic datasets, including larger models like ResNet101 and SpikingBERT.
- Broad applicability validated through experiments on vision and sequential tasks, including DVS-Gesture and SL-Animals-DVS.
- Extensive rebuttal with detailed theoretical clarifications, new visualizations, calibration error evaluations (ECE/AdaECE), and computational complexity analysis.

**Concerns raised by reviewers and how they were addressed:**

- **Novelty and conceptual distinction from TET**: Reviewers noted that TMC appears to be an incremental improvement over TET. The authors argued that while TMC builds on TET, it introduces a principled calibration-based gradient rescaling mechanism that better handles confidence evolution over time.
  
- **Theoretical justification**: While no formal convergence proofs were provided, the authors offered analysis of the temporal rescaling factor, firing rate dynamics, and gradient norms, along with empirical evidence supporting faster and more stable convergence.

- **Calibration definition and clarity**: Reviewer KZ4r raised issues with Definitions 3.1 and 3.2. The rebuttal clarified the formulation and provided both theoretical objectives and empirical confirmation that temporal confidence converges to accuracy.

- **Limited gains on certain datasets**: Reviewer g8s8 noted smaller gains on ImageNet and CIFAR100. The authors responded with deeper analysis, showing TMC's strengths on dynamic or complex temporal tasks, and explained limitations due to low resolution or static nature of some benchmarks.

- **Visualization and interpretability**: Reviewer GDhn requested more direct evidence of increased heterogeneity. The authors provided cosine similarity plots across time steps, showing increased diversity in intermediate representations.

Overall, this paper offers a thoughtful and empirically solid contribution to SNN training, particularly in calibrating temporal dynamics via gradient rescaling. While the theoretical novelty remains somewhat incremental, the work is well-executed, addresses a relevant problem, and shows strong generalization across tasks and architectures. The rebuttal was detailed and resolved several key reviewer concerns.
Decision: Accept (poster)