Temporal Misalignment in ANN-SNN Conversion and its Mitigation via Probabilistic Spiking Neurons
Authors: Velibor Bojkovic
Xiaofeng Wu
Bin Gu
Venue: ICML 2025 poster (2025)
Keywords: Spiking Neural Networks
SNN
ANN-SNN conversion
probabilistic spiking
Primary area: applications->computer_vision
TL;DR: Identification of a phenomenon in ANN-SNN conversion and its mitigation through novel approach in ANN-SNN conversion.

Abstract: Spiking Neural Networks (SNNs) offer a more energy-efficient alternative to Artificial Neural Networks (ANNs) by mimicking biological neural principles, establishing them as a promising approach to mitigate the increasing energy demands of large-scale neural models. However, fully harnessing the capabilities of SNNs remains challenging due to their discrete signal processing and temporal dynamics. ANN-SNN conversion has emerged as a practical approach, enabling SNNs to achieve competitive performance on complex machine learning tasks. In this work, we identify a phenomenon in the ANN-SNN conversion framework, termed *temporal misalignment*, in which random spike rearrangement across SNN layers leads to performance improvements. Based on this observation, we introduce biologically plausible two-phase probabilistic (TPP) spiking neurons, further enhancing the conversion process. We demonstrate the advantages of our proposed method both theoretically and empirically through comprehensive experiments on CIFAR-10/100, CIFAR10-DVS, and ImageNet across a variety of architectures, achieving state-of-the-art results.

Temporal Misalignment in ANN-SNN Conversion and its Mitigation via Probabilistic Spiking Neurons
Velibor Bojković" ! Xiaofeng Wu
Bin Gu
Abstract
ANN acc
70
Spiking Neural Networks (SNNs) offer energy-efficient alternative to Artificial Neural Networks (ANNs) by mimicking biological neu- ral principles, establishing them as promising ap- proach to mitigate the increasing energy demands of large-scale neural models. However, fully har- nessing the capabilities of SNNs remains chal- lenging due their discrete signal processing and temporal dynamics. ANN-SNN conversion has emerged as a practical approach, enabling SNNs to achieve competitive performance on complex machine learning tasks. this work, we identify a phenomenon in the ANN-SNN conversion frame- work, termed temporal misalignment, in which random spike rearrangement across SNN layers leads to performance improvements. Based on this observation, we introduce biologically plau- sible two-phase probabilistic (TPP) spiking neu- rons, further enhancing the conversion process. We demonstrate the advantages of our proposed method both theoretically and empirically through comprehensive experiments CIFAR- 10/100, CIFARIO-DVS and ImageNet across variety of architectures, achieving state-of-the-art results. Codes are available at https://githi: com/ shi
60
Accuracy (%)
50
40
Raseline (OCFS)
Permute
30
TPP (ours) QCFS
T=4
T-16
T-32
T-64
Latency
Figure I. The initial experiment: After ANN-SNN conversion, we compared the accuracy of the baseline model QCFS (Bu al., 2022c) with its "permuted" version and our proposed TPP neurons (setting is VGG-16 ImageNct, ANN 74.29%).
ley, 1952; Izhikevich, 2003). While biological energy ef- ficiency has historically influenced neural network design, SNNs uniquely achieve this through event-driven process- ing: weighted inputs integrate into membrane potentials, emitting binary spikes only upon crossing activation thresh- olds. This differs significantly from artificial neural net- works (ANNs) (Braspenning et al., 1995), which are based continuous floating point operations that require energy- intensive and computationally costly multiplication opera- tions (Roy et al., 2019). The spike-driven paradigm inher- ently circumvents these costly computations, suggesting that SNNs may offer promising approach for energy-efficient AI. Recent developments in neuromorphic hardware (Pei et al., 2019; DeBole et al., 2019; loi; Ma et al., 2023) have enabled efficient SNN deployment. These specialized chips are inherently designed for spike-based computation, driv- ing breakthroughs across domains: object detection (Kim et al., 2020b; Cheng et al., 2020), tracking (Yang al., 2019), event-based vision (Zhu al., 2022; Ren al., 2024). speech recognition (Wang al., 2023a), and generative Al through models like SpikingBERT (Bal & Sengupta, 2024) and SpikeGPT (Zhu et al., 2023; Wang et al., 2023b). These advancements underscore the potential of SNNs as a viable alternative to conventional ANNs.
1. Introduction
Spiking neural networks (SNNs), often referred to as the third generation of neural networks (Maass, 1997), closely mimic biological neuronal communication through dis- crete spikes (McCulloch & Pitts, 1943; Hodgkin Hux-
"Equal contribution 'Department of Machine Learning, Mo- hamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates FFaculty of Data Science, City University of Macau, Macau, China "Scoool of Artificial Intelligence, Jilin University, China. Correspondence Velibor Bojkovic <veli- bor.bojkovic ibbuai..ac.cc.
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Training SNNs is inherently challenging due the same

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Temporal Misalignment and Probabilistic Neurons
p00 - pi- stt-
Accumulate
Fire
W
Shuffe
|4, b 1s 4)
U/s Berzonill (ctte
(7F-1)R7 0.1))
14 12 1s 4)
B5 4 t)]
Figure 3. Two-Phase Probabilistic (TPP) spiking neuron mecha- nism operates in two phases: accumulation of inputs and proba- bilistic spiking based on membrane potential. The model uses a Bernoulli process for spike generation, with membrane potential updates over time steps.
Figure 2. Spike train permutation: spikes at different time steps are shuffled to alter their temporal order.
characteristics that confer their advantages: their discrete processing of information. Unsupervised direct training, inspired by biological learning mechanisms, leverages local learning rules and spike timing to update weights (Diehl & Cook, 2015). While these methods are computationally effi cient and can be executed on specialized hardware, SNNs trained in this manner often underperform compared to mod- els trained with alternative approaches. Further research is needed to better understand and improve this method. In contrast, supervised training can be categorized into direct training via peapooltempora backpropagation (e.g., surro- gate gradient methods (Neftci et al., 2019)) and ANN-SNN conversion methods (Diehl et al., 2015; Cao et al., 2015). The present work focuses on the latter approach.
the spike outputs of spiking layers in the corresponding SNN models. To evaluate the importance of spike timing relative to firing rates, we randomly permuted the spike trains after each spiking layer. Specifically, when process- ing samples through the baseline model, we rearranged the temporal order of spikes within each spike train after each spiking layer (see Figure 2). The permuted spike trains were then passed to the subsequent layer, and this process was repeated for each layer until the output layer. The results of one of these initial experiments, comparing the performance of the "permuted" model with the baseline, are presented in Figure 1. For each latency, we conducted multiple trials with different permutations, and the "permuted" model con- sistently outperformed the baseline, achieving the original ANN accuracy at lower latencies. The impact of permuta- tions on performance was particularly pronounced at lower latencies.
The core idea of ANN-SNN conversion is to leverage pre- trained ANN models to train SNNs. This process begins by transferring the weights from the ANN to the SNN, which shares the same architecture, and initializing the spiking neuron parameters (e-g., thresholds, time constants) so that the spike rates approximate the activation values of the corresponding ANN layers. This method is advantageous because it typically requires no additional computation for training the SNN, thereby eliminating the need for gradient computation or limiting it to fine-tuning the SNN.
We refer to this phenomenon as "temporal misalignaentt in ANN-SNN conversion and further investigate it by pro- viding a conceptual interpretation in the form of bursting probabilistic spiking neurons, which are designed to mimic the effects of permutations in SNNs. The proposed neurons operate in two phases, as illustrated in Figure 3. In the first phase, they accumulate input, often surpassing the threshold, while in the second phase, they emit spikes probabilistically with varying temporal probabilities. Our proposed spiking neurons characterized by two key properties: (1) the accumulation of membrane potential beyond the threshold, resulting in firing phase, commonly referred to as bursting, and (2) probabilistic firing. Both properties exhibit biolog- ical plausibility and have been extensively studied in the neuroscience literature (see Section 3.3).
The initial experiment: Temporal information in SNNs after ANN-SNN conversion. We begin by identifying a counter-intuitive phenomenon that, to the best of our knowl- edge, has not been previously documented. Specifically, we investigate the extent to which individual spike timing affects the overall performance of the SNN model during ANN-SNN conversion. We explore this question in the con- text of various types of conversion-related issues described in the literature, such as phuse lug (Li et al., 2022) and un- evenness of the spike input (Bu et al., 2022c), where prior work has indicated that the timing of spike trains in SNNs obtained through ANN-SNN conversion may not be optimal and can lead to performance degradation under low-latency conditions.
The main contributions of this paper are summarized as:
We recognize and study the "temporal misalignment" phe- nomenon in ANN-SNN conversion, and we propose frame- work for its exploitation in ANN-SNN conversion utilizing two-phase probabilistic spiking neurons. We provide the- oretical insights into their functioning and superior perfor- mance, as well as support for their biological grounding.
In our initial experiment, we began by examining several widely-used ANN-SNN conversion methods and analyzing
2

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Temporal Misalignment and Probabilistic Neurons
We present a comprehensive experimental validation demonstrating that our proposed method outperforms state- of-the-art conversion as well as the other training methods in terms of accuracy on CIFAR-10/104 CIFARIO-DVS, and ImageNet datasets.
adaptive threshold configurations (Stöckl & Maass, 2021;
Ho & Chang, 2021; Wu et al., 2023), and (iv) spike cod- ing optimization (Kim et al., 2020a; Sengupta et al., 2018). Recent innovations focus on ANN activation function adap- tations, including thresholded ReLU (Ding et al., 2021) and quantization strategies (Bu et al., 2022c; Liu et al., 2022; Hu et al., 2023; Shen et al., 2024). However, these approaches introduce inherent accuracy compersionooo tradeoffs. Parallel efforts modify integrate-and-fire neuron dynamics (Li & Zeng, 2022; Wang et al., 2022a; Liu et al., 2022), with (Liu et al., 2022) proposing a dual-phase mechanism resembling our approach.
2. Background and Related Work
The base model employed in this work is Integrate-and- Fire (IF) spiking neuron, whose internal dynamics, after discretization, are given by the equations
v0 [1] (t - 1] w()((-1) . (t) - 0 -000 s(t - 1].
Crucial for achieving high conversion efficacy, threshold initialization methodologies employ layer-wise activation maxima or percentile statisties (Rueckauer et al., 2017a; Deng & Gu, 2021; Li et al., 2021a; Wu et al., 2024), aug- mented with post-conversion weight calibration (Li et al., 2021a; Bojković et al., 2024). Contemporary strategies can be categorized into two paradigms: (1) ANN activation quantization for temporal efficiency at the cost of accu- racy, and (2) SNN neuron modification preserving ANN expressiveness with extended temporal requirements. Our methodology adheres the second paradigm.
s [t H (v [t] )())
(1) Here, 000) is the threshold, (-) is the Heaviside function, while the superscript/ denotes the layer in the SNN. Later, we will modify these equations and use more advanced neu- ron models. By expanding the equations over t 1,....?! and rearranging the terms, we obtain
T ((0) -1=18 (((
(1
-w th
(2)
T
T
[ [0
(3) Direct Training. This approach leverages spatio- pempoal spike patterns through backpropagation -troonggttttttiio with differentiable gradient approximations (O'Connor et al., 2018; Zenke & Ganguli, 2018; Wu et al., 2018; Bellee et al., 2018; Fang et al., 202;a;b; Zenke & Vogels, 2021: Mukhoty et al., 2024). Advancements encompass joint optimization of synaptic weights and neuronal parameters (threshold dy- namics (Wei et al., 2023), leakage factors (Rathi & Roy, 2023)), novel loss formulations for spike distribution regu- larization (Zhu et al., 2024; Guo et al., 2022), and hybrid anversion-training pipelines (Wang et al., 2022b). State-of- the-art developments introduce ternary spike representations for enhanced information density (Guo et al., 2024) and re- versible architectures for memory-efficient training (Zhang & Zhang, 2024).
T
On the ANN side, the transformation between layers is given by
a A (w(aaa
(4)
where 11) is the activation function. The ANN-SNN con- version process begins by transferring the weights (and bi- ases) of a pre-trained ANN to an SNN with the same archi- tecture. By comparing the equations for the ANN outputs (4) and the average output of the SNN (2) (Rueckauer et al., 2017a), the goal is to achieve a relation of the form
d
(5)
T
The most commonly used activation function A is ReLU, due to its simplicity and non-negative output, which aligns well with the properties of neurons. It is important to note the importance of the three components in the conversion: 1) the threshold value 0, 2) the initialization v(0). 3) the ANN activation function A.
3. Methodology
3.1. Motivation
In ANN-SNN conversion methodologies, constant and rate coding are commonly utilized in the resulting spiking neural network models, based on the principle that the expected input at each time step matches the original input the ANN model. Notably, the encoding lacks temporal information, as spike timing does not convey additional information. In constant encoding, this is evident, while in rate encoding, for a fixed input channel, the spike probability remains constant across all time steps, with the channel value assumed to lie between O and 1.
2.1. Related work
ANN-SNN Conversion. This methodology aligns ANNs and SNNs through activation-firie rate correspondence, as initially demonstrated in (Rueckauer et al., 2017a; Cao et al., 2015). Subsequent research has systematically improved conversion fidelity through four principal approaches: (i) weight normalization (Dichl et al., 2015), (ii) soft-reset mechanisms (Rueckauer et al., 2017b; Han et al., 2020), (iii)
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Temporal Misalignment and Probabilistic Neurons
The resulting SNN model is initialized to approximate the outputs of the original ANN model based on the principle that, for each spiking neuron, the expected number of spikes it generates should closely match the output of the corre- sponding ANN neuron. In particular, it is assumed that no temporal information is present throughout the SNN model; that is, the spike train outputs of each SNN layer should convey no additional temporal information beyond spike firing rates.
layer, rather than directly processing spikes from spiking layer.
TPP neurons:
To address the aforementioned questions, we propose two-phase probabilistic spiking neurons (TPP) (see Figure 3). Specifically, in the first phase, the neurons will only accumulate the (weighted) input coming from the previous layer, while in the second phase, the neurons will spike. More precisely, suppose that at a particular layer l the spiking neurons accumulate the whole output of the previous layer, without emitting spikes. Denote the accumulated membrane potential by ((] The subsequent spiking phase is governed by the following equations:
Previous studies examining conversion emrrrs and their clas- sifications (Li et al., 2022; Bu et al., 2022c; Ding et al., 2021; Bojković et al., 2024) suggest that SNNs obtained through ANN-SNN conversion may generate spikes that are suboptimally positioned in the temporal domain, lead- ing to degradation in model performance, particularly at low latencies. Our initial experiments (see Introduction and Figure 1) further validate this observation while also revealing a novel insight: random temporal displacements of spike trains after spiking layers significantly enhance model performance. This phenomenon, which we refer to as temporal misalignment - wherein the original spike trains exhibit temporal misalignment, thereby impairing model performance serves as the foundation and motiva- tion for our proposed method, which is claborated upon in the next section. Additional experiments on permutations in the ANN-SNN context, along with an explanation of their impact on model performance, are provided in Appendix G.
s [t] B (1) (T - 1 11 [0][t] = (t - 1] - (().
1
1 -1). (6)
where = 1, 1...., T. Here, (xx) is a Bernoulli random variable with bias x. extended for x E R in natural way (B(x) = ((max(min(z, 1), Oj)). If the weights of the SNN network are not normalized, the produced spikes will be scaled with the thresholds ((0) [t), before being sent to the next layer.
We observe that the presence of - +++ in the denom- inator of the bias in B demonstrates that the probability of spiking depends not only on the current membrane po- tential, but also on the temporal step: in the absence of spiking, for the same membrane potential, the probability of spiking increases through time. Figure 3 provides a visual representation of the functioning of TPP neurons.
3.2. From permutations to Bursting Probabilistic Spiking Neurons
This work aims to address the following question: How to incorporate the action of permutation of the output spike trains into the dynamics of the spiking neurons? We ap- proach this problem in two steps.
The following theorem provides a comprehensive character- ization of the functioning of TPP neurons and their applica- tions in ANN-SNN conversion when approximating ANN outputs (here ReLUUg()) = min(ReL.U()), 0)).
Consider the scenario where spike trains from layer é are to be permuted. A general approach involves introducing "permutator" -a a subsequent layer tasked with collecting all incoming spikes and re-emitting them in a permuted manner, as illustrated in Figure 2. This inherently implies the two- phase nature of the "permutator": specifically. during the first phase, incoming spikes accumulated, and firing is deferred until the onset of the second phase, where spikes are emitted.
Theorem 1. Let x be the input of the ANN layer with ReLU activation and suppose that, during the accumulation phase, the corresponding SNN layer of TPP neurons accu- mulated T (x))) quantity of voltage.
(a) Suppose that for some = ........ T, the TPP layer pro- duced s00) [1]),.., 500 - 1] vector spike trains for the first - 1 steps, and the residue voltage for neuron is higher than zero. Then,
The second step focuses on the output mechanism of the "permutator". Specifically, it is desirable to design a spiking neuron mechanism that retains the stochastic component of the permutations. This consideration motivates the adoption of probabilistic firing in spiking neurons.
(T (1-++1)
000
E
S
[t]
+
0
t-1
S
(4)
[
T
T
-ReLUgOa OCO (x!1).
(b) If
( are the output vectors of spike trains of the TPP neurons during T time steps, then
The final question we explore is whether more compact ap- proach can be achieved by employing probabilistic spiking neurons that aggregate weighted inputs from the previous
1
[ = RRLU pc) (x()).
(8)
T
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Temporal Misalignment and Probabilistic Neurons
ReRLU pas )x is a multiple of g T. or
Furthermore, suppose that these spike trains are modulated with weights wi:-.-, ,NN, respectively. For given permu- tation π = (TI....,,N N,), let TTs denote the permutation of the spike train St. Then, for every t1, 12 (1,2....,TT
((0
T
g0)
T ge (X
(9)
T
T
((1)
1=1
(())
T (())
ReLU pc (X ]
000
E wasss E Σ [t]s,
or T
(10)
Remarks: Theorem is proved in the Appendix B while Theorem 2 is proved in the Appendix G.
if FRRLUggy gas (x ) is not multiple of g0) T
(c) Suppose that max X S 0 and that the same weights Wl++) act the outputs of layer (1) of ANN and SNN as above, and let x(+1) (resp. T- -x((+1) ) be the inputs to the (I + 1)th ANN layer (resp. the accumulated voltage for the (1 (1++7 1)th SNN layer of TPP neurons), Then
3.3. Bio-plausibility and hardware implementation of TPP neurons
Our proposed neurons have two distinct properties: The two-phase regime and probabilistic spike firing. Both prop- erties are biologically plausible and extensively studied in the neuroscience literature. For example, the two phase regime is related to firing after a delay of biological spik- ing neurons, where neuron collects the input beyond the threshold value and fires after delay after some condition is met. It could also be related to the bursting, when a bio- logical neuron starts emitting bursts of spikes, after certain condition is met, effectively dumping their accumulated po- tential (Izhikevich, 2007; Connors & Gutnick, 1990; Llinás & Jahnsen, 1982; Krahe & Gabbiani, 2004).
O T (11)
[|x(+1]] 8(42) 1bx S [|
We contrast the statement (a) with Theorem 2 of (Bu et al., 2022c). Specifically, the authors demonstrate that if the membrane potential is initialized at half of the thresh- old, the expectation of the conversion error (layerwise) is O. However, this result in (Bu et al., 2022c) relies on the un- derlying assumption that the layerwise distribution of ANN activation values is uniform, which does not generally hold in practice (see, for example, (Bojković et al., 2024)). Our result (a) above shows that after every t S T time steps, our expected spiking rate aligns well with the clipping of the ReLU activation by the threshold, as it should, without imposing any prior assumptions on the distribution of the ANN activation values.
On the other side, stochastic firing of biological neurons has been well studied as well, and different aspects of noise introduction into firing have been proposed. We refer to (Shadlen & Newsome, 1994; Faisal et al., 2008; Softky & Koch, 1993; Maass & Natschläger, 1997; Pagliarini et al., 2019; Stein et al., 2005) for some examples.
Regarding the implementation of TPP neurons on neuro- morphic hardware, two phase regime can be easily achieved various modern neuromorphic chips that support pro- grammable spiking neurons. Stochastic firing can be achieved through random sampling which, for example, is supported by IBM TrueNorth (Merolla et al., 2014), In- tel Loihi (Davies et al., 2018), BrainScaleS-2 (Pehle et al., 2022), SpiNNaker (Furber et al., 2014) neuromorphic chips. For example, TrueNorth incorporates stochastic neuron mod- els using on-chip pseudo-randon number generators, en- abling probabilistic firing patterns that mirror our approach. Similarly, Loihi (Gonzalez et al., 2024) supports stochastic operations by adding uniformly distributed pseudorandom noise to neuronal variables, facilitating the implementation of probabilistic spiking neurons.
Moreoveor, our result demonstrates that the activity of TPP neuron adapts to the output it already produced. In particular, as long as the neuron is still active and contains residual membrane potential, the expectation of its output at the next time step takes into account the previously produced spikes and will yield the ANN counterpart.
The results (b) and (c) show that during the accumulation phase, the TPP neuron closely approximate ANN neurons with ReLU activation. In particular, the only remaining source of errors in layerwise approximation is the clipping error due to the threshold 6, and the quantization error due to the discrete outputs of the spiking neurons. We also note in equations (9) and (10) two possibilities for the output, which come from the probabilistic nature of spiking.
To reduce the overall latency for processing inputs with our models, which yields linear dependence the number of layers (implied by the two phase regime), we note that as soon as particular layer has finished the firing phase, it can start receiving the input from the previous one: The process of classifying a dataset can be serialized. This has already been observed, for example in (Liu et al., 2022).
When it comes to the initial permutation effect on perfor- mance of the model, we connect with the uniformization of the output, as the next result suggests.
Theorem 2. Suppose we have N spiking neurons that produced spike trains si[1), (2())...., (T), 1, 1,...... N.
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Temporal Misalignment and Probabilistic Neurons
Neuromophic hardware implementation of this serialization has been proposed as well, see for example (Das, 2023; Song et al., 2021; Varshika et al., 2022).
2022a), SRP (Hao al., 2023a), DDI (Bojković et al., 2024) and FTBC (Wu et al., 2024).
Table compares the performance of our proposed methods with state-of-the-ar ANN-SNN con- version methods on ImageNet. Our method outperforms the baselines across all simulation time steps for VGG-16, and RegNetXX44GF. For instance. on VGG- 16 at T 32, our method achieves 74.72% accuracy, surpassing other base- lines even at T = 128. Moreover, at T = 128, our method nearly matches the original ANN performance with only a 0. 12% drop in VGG-16 and a 0.14% drop in ResNet-34.
ImageNet dataset:
4. Experiments
In this section, we verify the effectiveness and efficiency of our proposed methods. We compare it with tfate-of-the-ee methods for image classification via converting ResNet-20, ResNet-34 (He et al., 2016), VGG-16 (Simonyan & Zisser- man, 2015), RegNet (Radosavovic et al., 2020) on CIFAR- 10 (LeCun et al., 1998; Krizhevsky et al., 2010), CIFAR- 100 (Krizhevsky& & Hinton, 2009), CIFARIO-DVS (Li et al., 2017) and ImageNet (Deng et al., 2009). Our experiments use Py'Torch (Paszke et al., 2019), PyTTorch vision mod- els (maintainers & contributors, 2016), and the PyTorch Image Models (Timm) library (Wightman, 2019).
We see similar patterns in combining our methods with RTS and QCFS baselines, which use modified ReLU activations to reduce conversion errors. Table I shows these results. For instance, applying TPP with QCFS on ResNet-34 at T' 16 improves performance from 59.35% to 72.03%, a 12.68% increase. Similarly, for VGG-16 at T = 16, combining TPP with QCFS boosts performance from 50.97% to 73.98%, a 23.01 % increase. Using TPP with RTS also shows signifi- cant improvements, such as a 12.82% increase for VGG-16 at T = 16. These results demonstrate the benefits of inte- grating TPP with other optimization approaches, solidifying its role as comprehensive solution for ANN-SNN conver- sion challenges.
To demonstrate the wide applicability of the TPP neurons and the framework we propose, we combine them with three representative methods of ANN-SNN conversion from re- cent literature, each of which has their own particularities. These methods are: QCFS (Bu al., 2022b), RTS (Deng & Gu, 2021), and SNNC (Li et al., 2021a). The particularity of QCFS method is that it uses step function instead of ReLU in ANN models during their training, in order to obtain higher accuracy in lower latency after the conversion. RTS method uses thresholded ReIUU activation in ANN models during their training. so that the outliers are eliminated among the activation values, which helps to reduce the conversion error. Finally, SNNC uses standard ANN models with ReI.U acti- vation, and performs grid search on the activation values to find optimal initialization of the thresholds in the converted SNNs.
We further evaluate the performance of our methods on CIFAR- 100 dataset and present the results in Table 1. We observe similar patterns as with the ImageNet. When comparing our method with ANN-SNN conversion methods which use non-ReLUU activations, e.g. QCFS and RTS, our method constantly outperforms RTS on ResNet-20 and VGG16. QCFS baseline suffers from necessity to train ANN models from scratch with custom activations, while our method is applicable to any ANN model with ReLU -like activation. Furthermore, custom activation functions sometimes sacrifice the ANN performance as can be seen from the corresponding ANN accuracies.
CIFAR dataset:
We initialize our SNNs following the standard ANN-SNN conversion process described in Section 3 (and detailed in A), starting with a pre-trained model given by the baseline, or with training an ANN model using default settings in QCFS (Bu et al., 2022b). RTS (Deng & Gu, 2021), and SNNC (Li et al., 2021a). ANN ReI.U activations were replaced with layers of TPP neurons initialized properly. All experiments were conducted using NVIDIA RTX 4090 and Tesla A100 GPUs For comprehensive details on all setups and configurations, see Appendix C.2.
CIFARIO-DVS dataset: We evaluate our method on the event- based CIFAR10-DVS (L. et al., 2017) dataset, com- paring it with state-of-the-art direct training and ANN-SNN conversion methods (Table 2). Our approach demonstrates superior performance, achieving 82.40% accuracy at just timesteps and further improving to 83.20% 64 timesteps. Notably, our method outperforms the direct training method Spikformer (Zhou et al., 2022) (80.90%) and the ANN-SNN conversion method AdaFire (Wang et al., 2025) (81.25 (.....).
4.1. Comparison with the State-of-the-art ANN-SNN Conversion methods
We evaluate our approach against previous state-of-the- art ANN-SNN conversion methods, including ReLU- Threshold-Shift (RTS) (Deng & Gu, 2021), SNN Calibration (SNNC-AP) (Li et al., 2021a), Quantization Clip-Floor-Shift activation function (QCFS) (Bu et al., 2022b), SNM (Wang et al., 2022a), Burst (Li & Zeng, 2022), OPI (Bu et al.,
4.2. Comparison with other types of SNN training methods and models
We compare our approach with several state-of-the-art di- rect training and hybrid training methods as presented in Table 2. The comparison is founded on performance metrics
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Temporal Misalignment and Probabilistic Neurons
Table 1. Comparison between method and the other ANN- SNN conversion methods on ImageNet and CIFAR- 100. We provide the average accuracy and the associated standard deviation across experiments.
Datasct
Arch.
Mothod
ANN
T=4
T-8
T=16
T=32
T-64
T=128
(Deng & Gu, 202
75.66
59.52
67.54
SNNC-AP'LL et 2021a//eM
75.66
64.54
71.12
73.45
QCFS (Bu et al, 20226)xsx
74.32
59.35
69.37
72.35
73.15
SRP (Hao et al., 20233/AMu
74.32
66.71
67.62
68.02
68.40
68.61
ResNet-34
FTBC(+QCFS) (Wo al., 2024)Rccv
74.32
65.28
71.66
73.57
74.07
74.23
Ours (TPP) QCFS
74.32
37.23 (0.07)
67.32 (0.06)
72.03 (0.02)
72.97 (0.03)
73.24 (0.02)
73.30 (0.02)
Ours (TPP) SNNC w/o Cali.
75.65
2.69 (0.03)
49.24 (0.23)
69.97 (0.10)
74.07 (0.06)
75.23 (0.03)
75,51 (0.05)
SNNC-AP'(I et al., 202 202
75.36
63.64
70.69
73.32
(WangWang al., PCAI
73.18
71.50
72.86
RTS (Deng & Gn, 2021)
72.16
55.80
67.73
70.97
71.89
ImageNet
QCFS (Bu et al. 20226/XLR
74.29
30,97
68.47
72.85
73.97
Burst (Li & Zeng. PCA4
74.27
70.61
73.32
73.00
OBI ol., 2022aAAM
74.85
6.25
36.02
64.70
72.47
74.24
VGG-16
SRP (Hao et al., 2023a)*MAr
74,29
66.47
68.37
69.13
69.35
69.43
FTBCI-QCTS) (Wu et al., 2024)80CV
73.91
58.83
69.31
72,98
74.05
74.16
74.21
Ours (TPP) RTS
72.16
30.50 (1. 19)
56.69(0.67)
67.34 (0.25)
70.63 (0.11)
71.75 (0.05)
72.05 (0.03)
Ours (TPP) QCFS
74.22
68. 39 (0.08)
72.99 (0.05)
73.98 (0.07)
74.23 (0.03)
74.29 (0.00)
74.33 (0.01)
Ours (TPP) SNNC w/o Call.
75.37
54.14 (0.59)
69.75 (0.27)
73.44 (0.02)
74.72 (0.06)
75.14 (0.02)
75.25 (0.03)
RTS (Deng & Gu, 202
80.02
0.218
3.542
48.60
RegNetX-IGF
SNNC-AP al. 202
80.02
55.70
70.96
75.78
Ours (TPP) SNNC w/o Call.
78.45
22.71
66.51 (0:44)
75.54 (0.07)
77.83 (0.04)
TSC & Roy, 20C0)a0cV
68.72
$8.42
RMP (1an et 2020
68.72
27.64
46.91
$7.69
SNNC-AP (Li et al. 202
77.16
76.32
77.29
77.73
77.63
RTS (Deng & Gu, 20T
67.08
63.73
68.40
69.27
69.49
OPI et ol., 2022a4AAi
70.43
23.09
52,34
67.18
69.96
70,51
ResNet-20
QCFS YBu al, 20226)CLR
67.09
27.87
49.53
63.61
67.04
67.87
67.86
Burst(Lii & Zeng. 2022)UCAS
79.83
Ours (TPP) QCFS
67.10
46.88 (0.40)
64.77 (0.20)
67.25 (0.12)
67.74 (0.06)
67.77 (0.05)
67.79 (0.04)
Ours (TPP)'
SNNC SNNC w/o Cali.
81.89
39.67 (0.99)
71. (0.68)
78.97 (0.24)
81 11.06 (0:05)
81.61 (0.08)
81.62 (0.05)
CIFAR-100
TSC'(Han & Roy, 20ey
71.22
69.86
SNM[ al., 212
74.13
71.80
73.69
73.95
SNNC.AP'(Li et 2021a
77.89
73.55
77.10
77.86
KISiDeng & Gu, 202R
76.13
23.76
43.81
56.23
67.61
73.45
75.23
OPI'(Bu el al., 2022ayMau
76.31
60.49
70.72
74.82
75.97
76.25
QCFSTBu al.. 2012
76.21
73.89
75.98
76.53
76.54
76.60
VGG-16
DDI (Bojković et al., 2024)ASSTIIS
70.44
51,21
53.65
57.12
61.61
70.44
73.82
FTBCI-QCTS) (Wu el al., 2024)80CV
76.21
71.47
75.12
76.22
76.48
76.48
76.48
Ours (TPP) RTS
76.13
37.88 (0.35)
81 (0.27)
73.05 (0.12)
75. (0.17)
(0.12)
75.9 (0.08)
Ours (TPP) + QCFS
73.93 (0.22)
76.03 (0.23)
(0.07)
76.55 (0.03)
76.55 (0.07)
76.52 (0.04)
Ours (IPP)'+ SNNC w/o Call.
77.87
59.23 (0.63)
73.16 (0.17)
76.05 (0.26)
77.16 (0.09)
77.56 (0.13)
77.64 (0.04)
Without modification ReI.U ANNs. +: Using authors' provided models and
Self implemented
like accuracy and the number of timesteps utilized during inference on the CIFAR- 100 and ImageNet datasets. We benchmark our method against prominent approaches such as LM-H (Hao et al., 2023b), SEENN (Li et al., 2023), Dual-Phase (Wang et al., 2022b), TTS (Guo et al., 2024), RMP-Loss (Guo et al., 2023), RecDis-SNN (Guo et al., 2022), SpikeConv (Liu et al., 2022), and GAC-SNN (Qiu et al., 2024). We showcase the best accuracy comparable to state-of-the-art methods achieved by our approach with minimal timesteps. We prioritize accuracy, but direct train- ing and hybrid training opt for lower number of timesteps and sacrifice accuracy. We outperform LM-H (Hao al., 2023b) and Dual- (Wang et al., 2022b) for VGG-16 100. For ResNet-20 on CIFAR-100, we have higher accuracy but longer timesteps. Additionally, for ResNet- 34 on the ImageNet dataset, the accuracy of our method
with QCFS with 16 timesteps is higher than that of Spike-
Conv (Liu et al., 2022) with the same number of timesteps. We also achieve higher accuracy with longer timesteps as expected. Overall, our approach demonstrates promising performance and competitiveness in comparison with the existing SNN training methods.
4.3. Spiking activity
The event driven nature of various neuromorphic chips im- plies that the energy consumption is directly proportional to the spiking activity, i.e., the number of spikes produced throughout the network: the energy is consumed in the pres- ence of spikes. To this end, we tested our proposed method (TPP) for the spike activity and compared with the base- lines. For a given model, we counted the average number of spikes produced after each layer, per sample, for both
7

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Temporal Misalignment and Probabilistic Neurons
Timestop
Timestop
2
2
1
1
2
1
1
1
2
2
2
(a) Layer 4
(b) Layer 8
(c) Layer 12
three modes (baseline, shuffle, and prohabilistic) in VGG-16 on CIFAR-100. For comparative analysis, the first two timesteps (t = 1, = 2) from a total of eight timesteps (T' = 8) are selected for each mode. The baseline mode (blue) attains an accuracy of 24.22%, while the shuffle mode (light green) enhances accuracy to 70.54%, and the probabilistic mode (dark orange) further improves accuracy to 73,42%. The distributions are presented prior neuronal firing, with the red dashed line indicating the threshold voltage (Vth) for the respective layer (see Appendix F),
Figure 4. The membrane potential distributions of the first channel (randomly selected)
Table Comparison of direct and hybrid training methods SNNs on CIFAR-100, ImageNet and CIFARIO-DVS Baseline results include the highest reported accuracy and corresponding latency.
the energy consumed is approximately the same as that for the baseline in time T. The complete tables are present in Appendix E.4, where we provide more detailed picture of spike activities.
Method
Category
Timestops
Accuracy
VGG-16 [CIFAR-100]
Spike count after time T
15
(logarithmic scale)
T=8
T=64
Baseline
LM-H (Hao al., 20236/KCLR
Hybrid Training
73.11
13
T=16
T=120
TPP
SEENN-I al., 2023yMoris
Training
1.15"
72.76
T=32
Dual-Phase (Wang et 2022h)
Hybrid Tmining
70.08 75.06
11
Ours (TPP) OCFS
ANN-SNN SNN
73.93
Re [CIFAR-100]
9
LM-H (Hao et al, 20233jACL
Hybrid Triining
57.12
TTS (Guo et al., 2024y4aal
Direct Training
74.02
Ours (TPP) + SNNC w/o Cali.
ANN-SNN
16
78.97
5
13 15
Layer
ResNet-34 34 [ImageNet]
Figure 5. Spike counts of VGG-16 on CIFAR- 100 of RTS baseline compared with RTS+TPP, Note: The bar height from bottom indicates the spike counts after each timestep T, and the color of longer Ts overlaid by shorter Ts (see Appendix E.4)
SEENN-I (Li et al., 2023/Noetts
Direct Training
3.38
64.66
RMP-Lass (Gio et al., 2023)
Direst Training
65.17
ReeDisSNNN (Gno et al., 2022)//
Direct Training
67.33
SpikeConn (Liu et al., 2042
Hybrid Training
16
70.57
GAC-SNN (Qia al., 2024y A044
Direct Training
6
70:42
TTS (Gun et al., 2024yAMM
Direct Training
70.74
SPENN-I (1.i et al,, 2023)NootrS
ANN-SNN
29.53
71.84
4.4. Membrane potential distribution in early time steps
Ours (TPP) + QCFS
ANN-SNN
16
72.03
In Figure 4 we compare the membrane potential distribu- tions for baseline models and two methods that we studied in the paper, the permutations applied on spike trains and TPP method. Once again, it can be seen another reason for performance degradation of baseline models low latency, as the membrane potential is not variable enough produce spike informative spike trains, which is particularly visible in deeper layers. On the other side "permuted" and TPP models show sufficient variability throughout the layers.
Ours (TPP)+ SNNC w/o Cali.
ANN-SNN
32
74.07
ReaNet- 18 [CIFARI-DYSS]
TA-SNN (Yso et al., 20ey
Direct Training
10
72.00
PLIF (Fang et al., tCey
Direct Training
20
74,80
Dipkie (1 et al,, 2extrs
Direct Training
10
75.40
DSR (Meng et al,, FVR
Direct Training
10
77.30
Spikformer (Zhou et al, 201R
Direct Training
10
80,90
AdaFire (Wang al., 20255AAM AAM
ANN-SNN
81.25
77.90
8
82.40
Ours (TPP) SNNC wio Cali.
ANN-SNN
16
82.80
32
83.00
64
83.20
By increasing the latency, the baseline models can recover some of the variability and spike production, as can be seen in Figure 5. But, due the misplacement of spike trains through temporal dimension, they are still not able to pick up the ANN performance.
"The average number of timestcps during inference on the test datasct.
the baseline and our method. Figure shows the example of RTS and RTS + TPP. Both the baseline and our method exhibit similar spike counts. In particular, our method con- stantly outperforms the baselines, and possibly in doing it needs longer average latency per sample (I"). However,
8

---

Temporal Misalignment and Probabilistic Neurons
Accuracy
Accurecy (%)
Baseline (0,12,33)
70
(0. 1.2,2,3
1,1,3,21
3)
00,
00 0.3.2.1 1)
0.0...3)
0,0.3,2
0,2.0.
1.3
e,3.0.1
6.1.0,3
2)
1,1.0 3)
O.
0,0,3,
6, 0)
31.3.2.0
03,21 3,3.1 01
0)
3.3.00,2 2)
21, 01
0. 0... O.0
0,0.1,2)
00,2.
(0, 103.31
(0,
Permutation Order
Figure 6. Accuracy comparison for all !! permutations of input order over '' 4 time steps using QCFS with VGG-16 CIFAR-100
Results of permuted orders outperform the original, non-permuted order (0, 1, 3). Baseline accuracy is 69.31%, The ANN accuracy 76.21%.
4.5. Energy Efficiency
accuracy.
Following prior work such (Rathi & Roy, 2023; Yao et al., 2023), we computed the energy cost for both ANNs and SNNs using 45nm CMOS technology. The energy consumption of 32-bit MAC operation in ANNs (4.6 pJ) is higher than that of an addition operation SNNs (0.9 pJ) (Horowitz, 2014). The results are reported in Table 3. We refer to Appendix E.5 for details.
We also refer to the Appendix G for further experiments concerning the effect of permutations in ANN-SNN conver- sion.
approximately
5. Conclusions and future work
This work identified the phenomenon of "temporal mis- alignment" in ANN-SNN conversion, where random spike rearrangement enhances performance. We introduced two- phase probabilistic (TPP) spiking neurons, designed to in- trinsically perform the effect of spike permutations. We show biological plausibility of such neurons well the hardware friendlines of the underlying mechanisms. We demonstrate their effectiveness through exhaustive exper- iments on large scale datasets, showing their competing performance compared to SOTA ANN-SNN conversion and direct training methods.
Table 3. Energy consumption and Top-1 accuracy evaluated on ImageNet with default input resolution of 3x224x224, The baseline model is used for comparison, and the results for the proposed method (TPP) are highlighted in bold.
Arch
Method
T
(%)
Energy(mJ)
QCFS
32
68.47
61.67
TPP(ours)
4
68.39
7.98
VGG-16
QCFS
64
72.85
123.02
TPP(ours)
8
72.99
15.66
In the future work, we aim to study the effect of permuta- tions and probabilistic spiking combination with directly trained SNN models.
SNNC
32
63.64
13.27
TPP(ours)
8
69.75
5.69
ResNet-34
SNNC
64
70.69
23.51
TPP(ours)
16
73.44
12.16
Acknowledgements
We extend our sincere gratitude to the anonymous review- ers for their valuable comments on this paper. The author X. Wu was supported by the Science and Technology De- velopment Fund (FDCT) of Macau Special Administrative Region (0055/2023/ITP2, 0071/2024/TPP2,
4.6. Effect of Temporal Reordering
To validate the effectiveness of temporal permutations on SNN performance, exhaustively evaluate all T!! 24 possible orderings of the spike trains over T = 4 time steps using the QCFS with VGG-16 16 on CIFAR-100. As shown in Figure 6, all 23 permuted spike orders outperform the de- fault chronological order (0, 2,2,3) 3) without any reordering. The accuracy of the default-order SNN is 70.36% %. while the ANN reference achieves 76.21 %.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
The best-performing permutation 1,2, achieves 75.35%-onl 0.86% below the ANN accuracy, demonstrat- ing that reordering consistently and significantly improves
9

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Temporal Misalignment and Probabilistic Neurons
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Temporal Misalignment and Probabilistic Neurons
A. Conversion steps
Copying ANN architecture and weights. ANN-SNN conversion process starts with a pre-trained ANN model, whose weights (and biases) will be copied to an SNN model following the same architecture. In this process, one considers ANN models whose non-activation layers become linear during the inference. In particular, these include fully connected, convolutional, batch normalization and average pooling layers.
Approximating ANN activation functions. The second step of the process considers the activation layers and their activation functions in ANN. Here, the idea is to initialize the spiking neurons in the corresponding SNN layer in such a way that their average spiking rate approximates the values of the corresponding activation functions. For the ReLU (or ReLU -like such as quantized or thresholded ReI.U) ) activations, this process rather well understood. The spiking neuron threshold is usually set to correspond the maximum activation ANN channel or layerwise, to be some percentile of it. If we denote by f the ANN actiavtion, then ideally, after setting the thresholds, would like to have
0
T
S(v(T)) ~ T
Σ (0)
(12)
If we recall the equations for the IF neuron (equations (1) the article)
(1)
[t] [1
(0 1| (0000))) 50-1111- -0000
s|t 1].
(13)
(0)) [1] H (1
(14)
see that the value with which we comparing the membrane potential (threshold) is the same the value with which we are scaling the output spikes. In particular, as soon our membrane potential has reached 0, it will produce the value 0. This can be loosely described as, whatever the input is, the output will be approximately that value (or zero, if the input is negative), which is exactly what ReLU does.
Absorbing thresholds. Finally, we notice that, once we produce spike s(0) [1] the value 00) 500 [c] will be sent to the next layer, and will further be weighted with weights W(++) and the bias g(++1) will be applied. As we want SNNs to operate only using ones and zeros (to avoid multiplication due energy efficiency), the values g(() will be absorbed into W(++1), i.e W(++)) +- g(++1)
B. Proof of Theorem 1
We prove the main theorem from the article, which we restate here.
Theorem 1. Let X!) be the input of the ANN layer with ReLU activation and suppose that, during the accumulation phase, the corresponding SNN layer of TPP neurons accumulated T (!) quantity of voltage.
(a) Suppose that for some = 1...., T, the TPP layer produced (1) [1],...,./
s00
1) vector spike trains for the first t 1
steps, and the residue voltage for neuron i is higher than zero. Then,
T t 1) g(1)
0
1-1
E
8
T
ReLU o0o (X.")..
(b) 1f ( [1] 1!,....,,............. s [T] are the output vectors of spike trains of the TPP neurons during T time steps, then
0())
[ ReI.U.0 (x,)), (x
(8)
ReLLU (x(2) is multiple of T
(()
g())
T
(C)
T
ReLU (x(1))
(9)
T
T
-
i=1
(()
T
ReLU pco (x(")) 0
(10)
(())
T
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Temporal Misalignment and Probabilistic Neurons
if ReRLUge gas pas((x) (x,)) is not a multiple of g T
(c) Suppose that max X <. O and that the same weights W(++) act on the ouppus of layer (1) of ANN and SNN above, and let X ((++) (resp. T. x be the inputs to the (1 (1+ 1)th ANN layer (resp. the accumulated voltage for the (L + 1)
SNN layer of TPP neurons), Then
0
||x0++ - { bo 1 1| 000
(1)
T
Proof. We start by rewriting equation (6) as
50 [t] = B
-t -1
(15)
( (T - +)
Σ (00 i1)
[-1
T. T-X() -009
= B
(16)
00) (T 11+1)
i=1
B 1
T-t+1 (T.
x()
Σ [1 1
(17)
00
obtained by unrolling through time the expression for the membrane potential.
We next consider various settings of the theorem. For the first three statements, we can assume that the vector X(0) is one dimensional, i.c. (1) R. We start by first considering the situation where X > g0) so that the output of the ANN is ReLU (X ()) g(0), In that case, we notice that the spiking neuron will fire at every time step t = 1,..., T, because the bias for the Bernoulli random variable will always be bigger than or equal to 1, as follows from the previous equations. Similarly, if x() < O, the bias will always be non-positive, and both the output of ANN and of SNN will be O. Consequently, the first three statements follow directly for both of these cases.
< 1) and let be minimal non-negative integer such that p(0) < T x() < (a + 1). 0, i.e. a [ gのक In particular, 0 I < T. We proceed to prove statement (b) above. We note that in general, Ei=1 s(1 S a + 1, because after at most a + 1 spikes, the residue membrane potential would become negative. Also, due to the condition of the statement that the residue membrane potential is still non-negative, we may assume that Σ s(0 [i] S a. Then, it becomes clear that the bias in the last equation (17) is non-negative number smaller than 1, and it follows that
Suppose now that 0 I (1) < (ll) (or, equivalently, 0 <
X giT
(T
X
t-1
E
11
T
Σ
(0)
(18)
T
-
++1
(()
so that we finally have
((0)
(T - + 1)
(
1- 1
E
88
+
(19)
T
T
1=1
g(1) (T - t + 1)
x(I T
0 T
s
1 [i]
+
(20)
T
T-t+1
(C)
( ReLU gct) (x))).
(21)
Statement (a) follows from (6), while for the statement (c) for the first case, we notice that after every spike, the residue membrane potential is multiple of g(0), hence we will have exactly spikes (notation above), while the second case follows
from statement (b). Finally, statement (d) is direct consequence of the above discussion.
C. Experiments Details
C.1. Datasets
CIFAR-10: The CIFAR-10 dataset (Krizhevsky et al., 2010) contains 60,000 color images of 32x32 pixels each, divided into 10 distinct classes (e.g., airplanes, cars, birds), with each class containing 6,000 images. The dataset is split into 50,000
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Temporal Misalignment and Probabilistic Neurons
training images and 10,000 test images.
CIFAR-100: The CIFAR-IO0 dataset (Krizhevsky et al., 2010) consists of 60,000 color images of 32x32 pixels, distributed across 100 classes, with each class having 600 images. Similar to CIFAR- 10, it is divided into 50,000 training images and 10,000 test images.
CIFARIO-DVS:
The CIFARIO-DVS (Li et al., 2017) dataset is neuromorphic vision benchmark derived from the CIFAR- 10 dataset, converting 10,000 static images (1,000 per class across 10 categories like airplanes and cars) into event streams using Dynamic Vision Sensor (DVS). Generated via Repeated Closed- Loop Smooth (RCLS) movement to simulate realistic motion, the dataset captures spatio-temporal intensity changes asynchronous ON/OFF events (128x128 resolution) with inherent noise (e.g.. 60Hz LCD artifacts, later corrected). Designed for moderate complexity between MNIST-DVS and N-Caltech101, it challenges event-driven algorithms, achieving initial low accuracies 22-22229952 with methods like spiking neural networks and SVM, highlighting its utility for advancing neuromorphic research. Publicly available on Figshare' CIFARIO-DVE bridges traditional and event-based vision, fostering innovation brain-inspired computing and real-world dynamic scene analysis.
The ImageNet dataset (Deng al., 2009) comprises 1,281,167 images spanning 1,000 classes in the training set, with validation set and test set containing 50,000 and 100,000 images, respectively. Unlike the CIFAR datasets, ImageNet images vary in size and resolution. The validation set is frequently used as the test set in various applications.
ImageNet:
C.2. Configuration and Setups
C.2.1. OURS QCFS
CIFAR: We followed the original paper's training configurations train ResNet-20 and VGG-16 CIFAR-100. The Stochastic Gradient Descent (SGD) optimizer with momentum of 0.9 was used. The initial learning rate was set to 0.02, with a weight decay of 5 x 10-4 A cosine decay scheduler adjusted the learning rate over 300 training epochs. The quantization steps were set to 8 for ResNet-20 and 4 for VGG-16. All models were trained for 300 epochs.
ImageNet: We utilized checkpoints for ResNet-34 and VGG-16 from the original paper's GitHub repository. For ImageNet, was set to and 16 for ResNet-34 and VGG-16, respectively.
C.2.2. OURS RTS
CIFAR: We trained models using the recommended settings from the original paper.
ImageNet: We used pre-trained checkpoints for ResNet-34 and VGG-16 from the original paper's GitHub repository. Subsequently, all ReLU layers were replaced with spiking neuron layers.
For all datasets, we initialize TPP membrane potential to zero, while in the baselines we do as they propose.
C.2.3. OURS SNNC w/o CALIBRATION
We adhered the original paper's configurations train ResNet-20 and VGG-16 on CIFAR-100. The SGD optimizer with momentum of 0.9 used. The initial learning rate set 0.01, with weight decay of x 10-4 for models with batch normalization. A cosine decay scheduler adjusted the learning rate over 300 training epochs. All models were trained for 300 epochs with a batch size of 128.
CIFAR:
ImageNet: We used pre-trained checkpoints for ResNet-34 and VGG-16 from the original paper's GitHub repository. Subsequently, all ReLU layers were replaced with our proposed spiking neuron layers.
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Temporal Misalignment and Probabilistic Neurons
D. Algorithms
The baseline SNN neuron forward function (Algorithm 1) initializes the membrane potential zero and iteratively updates by adding the layer output each timestep. Spikes are generated when the membrane potential exceeds defined threshold, and the potential is reset accordingly. This function captures the core dynamics of spiking neurons. The Shuffle Mode (Algorithm 2) is extension of the baseline forward function. After generating the spikes across the simulation length, this mode shuffles the spike train.
The TPP Mode (Algorithm 3) introduces a probabilisticc component to the spike generation process. Instead of deterministic threshold- based spike generation, uses a Bernoulii process where the probability of spiking is determined by the current membrane potential relative the threshold adjusted for the remaining timesteps.
Algorithm SNN Neuron Forward Function and Additional Modes
Require: SNN Layer & Input tensor x; Threshold 0; Simulation length T. function BASELINESNN(C, x, 0,
2:
0 (Initialize membrane potential)
3: for to do
4:
+- v + (((()) s (v x
5:
6:
v - v
7: Store s(l)
8: end for
9: return
10: end function
Algorithm SNN Neuron Forward Function of Shuffle Mode
Require: SNN Layer E; Input tensor x; Threshold 0; Simulation length T.
1: function SHUFFLEMODE(,, x, 0, T)
2: V f 0 [Initialize membrane potential]
3: for
to 7 do
4:
+ ((x(t) s (v 0
5:
6:
+ -
7: Store st))
8:
end for
9: Shuffle the stored spikes
return shuffled
11: end function
Algorithm 3 SNN Neuron Forward Function of TPP Mode
Require: SNN Layer E; Input tensor x; Threshold 0; Simulation length T. 1: function TPPMODE(E, x, 0,
2: V +- Σ/= x(1) (Initialize membrane potential with the sum of inputs)
for to do
4:
p Clamp(v (0 (T - 1)),
5:
< (ernoull(pp) 0
6:
v-
7:
Store s(t)
8: end for
9: return
end function
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Temporal Misalignment and Probabilistic Neurons
E. Additional Experiments
E.1. SNNC
We show extra experiment results about the comparison among permutation method and two-phase probabilistic method. We validated ResNet-20 and VGG-16 on the CIFAR-10/100 dataset, and ResNet-34, VGG-16 and RegNetX -4GF on ImageNet with batch and channel-wise normalization enabled. Using a batch size of 128, the experiment was run five times with different random seeds to ensure reliable and reproducible results.
Table Comparison between our proposed methods and ANN-SNN conversion SNNC method on CIFAR-IO. The average accuracy and standard deviation of the TPP method are reported over 5 experiments.
Architecture
Method
ANN
T=1
T=2
T=4
T=8
T=16
T=32
T=64
SNNC-AP (Li et al., 2021a)
96.95
$1.20
66.07
83.60
92.79
95.62
96.58
96.85
ResNet-20
Ours (Permute)
96.95
34.05
61,46
90.54
95.05
96.12
96.62
96.77
Ours (TPP)
96.95
10.05 (0.02)
17.30 (0.52)
79.19 (0.67)
93.72 (0.05)
95.87 (0.09)
96.67 (0.04)
96.80 (0.01)
SNNC-AP AP (Li al., la)
95.69
60.72
75.82
82.18
91.93
93.27
94.97
95.40
Ours (Permute)
95.69
38.01
64.40
84.65
92.24
92.80
93.33
94.10
VGG-16
Ours (TPP)
95.69
11.46 (0.35)
32.24 (1.40)
86.85 (0.42)
94.34 (0.12)
94.86 (0.06)
95.48 (0.03)
95.60 (0.04)
Table Comparison between our proposed methods ANN-SNN conversion SNNC method on CIFAR-100 and standard deviation of the TPP method are reported over 5 experiments.
average accuracy
Architecture
Method
ANN
T=2
T=8
T=16
T=32
SNNC-AP (Li et al., 2021a)
81.89
17.91
34.08
54.78
72.28
78.57
81.20
81.95
ResNet-20
Ours (Permute)
81.89
5.64
19.54
52.46
75.21
79.76
81.12
81.52
Ours (TPP)
81.89
1.94 (0.11)
5.15 (0.44)
39.67 (0.99)
71.05 (0.68)
78.97 (0.24)
81.06 (0.05)
81.61 (0.08)
SNNC-AP et 202
77.87
28.64
34.87
50.95
64.30
71.93
75.39
77.05
Ours (Permute)
77.87
12,50
34.98
60.81
69.42
72.78
73.50
75.14
VGG-16
Ours (TPP)
77.87
2.05 (0.27)
15.90 (0.71)
59.23 (0.65)
73.16 (0.17)
76.05 (0.26)
77.16 (0.09)
77.56 (0.13)
Table Comparison between our proposed methods and ANN-SNN conversion SNNC method on ImageNet. The average accuracy and standard deviation of the TPP method are reported over 5 experiments.
Architecture
Method
ANN
T=8
T=16
T=32
T=64
T=128
SNNC-AP (Li al., 202
75.65
64.54
71.12
73.45
ResNet-34
Ours (Permute)
75.65
10.51
57.57
70.94
74.00
75.06
75.47
Ours (TPP)
75.65
2.69 (0.03)
49.24 (0.23)
69.97 (0.10)
74.07 (0.06)
75.23 (0.03)
75.51 (0.05)
SNNC-AP
2021a)
75.37
63.64
70.69
73.32
Ours (Permute)
75.37
38.61
67.29
73.35
74.34
74.82
75.11
VGG-16
Ours (TPP)
75.37
54.14 (0.59)
69.75 (0.27)
73.44 (0.02)
74.72 (0.06)
75.14 (0.02)
75.25 (0.03)
SNNC-AP (Li al., 202
80.02
$5.70
70.96
75.78
Ours (Permute)
78.45
43.45
68.12
75.63
77.63
RegNetX-4GF
Ours (TPP)
78.45
22.71 (2.98)
66.51 (0.44)
75.54 (0.07)
77.83 (0.04)
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Temporal Misalignment and Probabilistic Neurons
E.2. RTS
Table 7. Comparison between our proposed methods and ANN-SNN conversion RTS method CIFAR-10/100 and ImageNet. The
average accuracy and standard deviation of the TPP method are reported over experiments.
Dataset
Architecture
Method
ANN
T=4
T=8
T=16
T=32
T=64
T=128
RTS (Deng & Gu, 2021)
94,99
88.64
91.67
93.64
94.50
94.76
94.91
VGG-16
Ours (Permute)
94,99
91.22
93.70
94.50
94,86
94,88
94.97
Ours (TPP)
94,99
91.49 (0.21)
94.11 (0.09)
94.72 (0.08)
94.84 (0.06)
94.91 (0.02)
94.98 (0.02)
CIFAR-10
RIS (Deng Gu, 2021)
91.07
27.08
40.88
65.13
84.75
90.12
90.76
ResNet-20
(Permute)
91.07
68.18
86,57
90.81
91.04
Ours (TPP)
(0.22)
(0.14)
(0.08)
90.86 (0.14)
90.94 (0.04)
(0.03)
RTS (Deng Gu, 2021)
76.13
23.76
43.81
56.23
67.61
73.45
75.23
CIFAR-100
VGG-16
Ours (Permute)
76.13
35.31
62.84
71.20
74.34
75.53
75.92
Ours (TPP) RTS
76.13
37,88 (0.35)
65.81 (0.27)
73.05 (0.12)
75.17 (0.17)
75 .4 (0.12)
75,90 (0,08)
RTS (Deng & Gu, 2021)
72.16
55.80
67.73
70.97
71.89
ImageNet
VGG-16
Ours (Permute)
72.16
33.77
58.31
67.80
70.89
71.65
71.95
Ours (TPP)
72.16
30.50 (1.19)
56.69(0.67)
67.34 (0.25)
70.63 (0.11)
71.75 (0.05)
72.05 (0.03)
E.3. QCES
Table Comparison between our proposed methods and ANN-SNN conversion QCFS method CIFAR-10/100 and ImageNet. The average accuracy and standard deviation of the TPP method are reported over experiments.
Dataset
Architecture
Method
ANN
T=4
T=8
T=16
T=32
T=64
QCFS' et al., 2022c)
95.76
94.33
95.21
95.65
95.87
95.99
VGG-16
Ours (Permute)
95.76
95.15
95.58
95.83
95.95
95.97
Ours (TPP)
95.76
95. 95.28(0.09)
95.84(0.1)
95.95(0.05)
95.98(0.06)
95.97 (0.03)
CIFAR-10
QCFS (Bu et al., 2022c)
92.43
79.45
88.56
91.94
92.79
92.82
ResNet-20
Ours (Permute)
92.43
84.85
91.24
92.67
92.82
92.85
Ours (TPP)
92.43
86.24(0.18)
92. 02.08(0.11)
92.70X0.1)
92.78(0.04
92.68(0.06)
QCFS et al., 2022c)
76.3
69.29
73.89
75.98
76.52
76.54
VGG-16
Ours (Permute)
76.3
74.28
75.97
76.54
76.60
76.64
Ours (TPP)
76.3
74.0(0.15)
76.06(0.08)
76.37(0.1)
76.55(0.09)
76.51(0.07)
CIFAR-100
QCFS et al., 2022c)
67.0
27.44
49.35
63.12
66.84
67.77
ResNet-20
Ours (Permute)
67.0
45.33
62.81
66.93
67.85
67.96
Ours (TPP)
47.0(0.2)
64.66(0.25)
67.28(0.12)
67.61(0.1)
67.77(0.06)
QCFS (Bu et al., 2022c)
74.29
50.97
68.47
72.85
ImageNet
VGG-16
Ours (Permute)
73.89
55.54
71.12
73.65
74.28
74.28
Ours (TPP)
74.22
68.39 (0.08)
72.99 (0.05)
73.98 (0.07)
74.23 (0.03)
74.29 (0.00)
20

---

Temporal Misalignment and Probabilistic Neurons
E.4. Spiking activity
The percentage difference between the baseline and our method in TPP mode is calculated follows:
Percentage Difference Ours Raseline Baseline
100.
15
Spike count after time (logathhinc
TwB Roselne
Tocbi Tw112
Spika time (logaritinic scaiel
Spike time (logrithicc
13
13
12
11
11-
Layer
(a) RTS
(b) QCFS
(c) SNNC
Figure 7. Spike counts of VGG-16 on CIFAR-100 after different timesteps (T). Note: The bar height from bottom indicates the spike counts after each timestep and the color of longer Ts is overlaid by shorter Ts.
Table 9. Comparison of firing counts percentage difference between the baseline and our proposed TPP method for VGG-16 on CIFAR-100 using QCFS.
Layer
T=4
T=8
T=16
T=32
T=64
T=128
1.073
0.528
0,261
0.136
0.065
0.033
2
2.629
1.022
0.438
0.206
0.102
0.050
3
0.049
0.230
0.185
0.109
0.056
0.028
-0.867
-0.664
-0.419
-0.228
-0.118
-0.060
5
0.073
0.515
0.350
0.182
0.090
0.044
0.701
0.010
-0.098
-0.074
-0.041
-0.021
-1.071
-0.865
-0.470
-0.246
-0.122
-0.063
8
1.009
1.193
0.731
0.385
0.196
0.096
9
0.504
0.417
0.205
0.108
0.051
0.024
-0.112
0.842
0.647
0.375
0.198
0.100
11
2.071
2.438
1.614
0.898
0.465
0.235
12
0.797
0.943
0.756
0.461
0.247
0.127
4.503
2.156
1.209
0.655
0.343
0.171
14
25.898
13.883
7.770
3.852
1.887
0.936
15
33.585
16,864
8.945
4.474
2.227
1.108
21

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Temporal Misalignment and Probabilistic Neurons
Table 10. Comparison firing counts percentage difference between baseline and our proposed TPP method ResNet-34 on ImageNet using QCFS.
Layer
T=4
T-8
T=16
T=32
T=64
T=128
0.587
0.306
0.149
0.079
0.036
0.018
2
00,911
-0.435
-0.212
-0,108
-0.053
-0.055
0.353
0.189
0.082
0.036
0.019
0.010
-2.786
-1,583
-0.920
40.506
-0.270
-0.141
5
0.469
0.277
-0.107
-0.020
-0.019
-0.011
-3,955
-1.865
-0.705
-0.344
-0. 16166
-0.886
-0.381
0.321
-0.090
-0.031
-0.020
-0.013
8
6.615
3.261
1.494
0.628
0.290
0.131
15.116
-3.006
-1.555
-0.794
-0.391
-0.195
10
-2.938
3.431
3.096
1.794
0.975
0.498
11
1.184
0.466
0.359
0.102
0.053
0.022
-17.739
-7.302
-1.788
-0.609
-0.270
-0.132
13
0.105
-0.138
-0.287
-0.292
-0.166
-0.087
14
-8.597
-2.626
0.006
0.327
0.289
0.140
-0.522
-0,214
-0.273
-0,299
-0.173
-0.094
16
-11.99
-5.194
-1.990
-0.813
-0.405
-02177
17
-3.828
-1.192
-0,320
-0.192
-0. 1011
-0.058
-6.869
-2,392
-0.644
0.007
-0.002
0.001
19
0.092
-0.299
-0.181
-0.138
-0.074
-0.035
20
-5.639
-0,308
0.923
0.796
0.448
0.234
0.399
-0,968
-0.796
-0.509
-0.275
-0.145
22
-4.474
3.712
4.440
3.033
1.700
0.880
0.456
-0,901
-0.703
-0.533
-0.281
-0.145
-5,863
4.241
5.617
3.797
2.090
1.074
25
1.433
-0.464
-0.774
-0.632
-0.347
-0.182
26
-5,034
4.908
6.328
4.362
2.459
1.271
0.661
-0.914
-1.156
-0.931
-0.530
-0.284
28
-15.667
4.763
9.616
6.975
4.062
2.096
-9.747
1.663
3.836
2.455
1.384
0.673
-0.151
16.639
15.387
9,638
5.334
2.769
31
-5.403
0.917
1.957
1.555
1.009
0.574
32
17.796
6.777
3.728
3.231
2.507
1.583
-4.935
-2.141
2.055
2.931
2.395
1.561
22

---

Temporal Misalignment and Probabilistic Neurons
Table 11. Comparison of firing counts percentage difference between the bascline and our proposed TPP method for VGG-16 ImageNet using QCFS.
Layer
T=4
T=8
T=16
T=32
T=64
T=128
5.487
2.776
1.444
0.712
0.363
0.179
2
0,418
0.173
-0.055
0.007
0.007
0.006
3
-2.375
-0.883
-0.351
-0.128
-0.062
-0.031
4
6.170
2.181
0.627
0.121
0.024
-0.002
5
-3,338
-0,318
0.327
0.306
0.173
0.097
7.036
2.769
0.993
0.385
0.173
0.078
7
-5.722
-3,482
-1.661
-0.800
-0.400
-0.200
8
-6.155
0.310
1.411
0.955
0.507
0.269
9
-0.718
1.172
0.725
0.337
0.162
0.081
10
-12.833
-9.060
-4882
-2.359
-1.145
-0.564
11
12,966
11.241
7.718
4.443
2.344
1.188
12
-11.194
-14.874
-12.032
-7.889
-4.437
-2.395
13
-37.388
-30.782
-20.701
-12.296
-6.527
-3,377
14
23.619
-12.312
-3.929
-0.233
0.585
0.382
15
-10.988
-18.476
-13.953
-7.904
-4.091
-2.015
23

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Temporal Misalignment and Probabilistic Neurons
E.5. Energy consumption estimation
We follow what most did such as (Rathi & Roy, 2023; Yao al., 2023) We computed the energy cost for ANNs and SNNs in 45nm CMOS technology. The energy cost for 32-bit ANN MAC operation (4.6pJ) is 5. xmmoe than SNN addition operation (0.9pJ) (Horowitz, 2014).
Following prior work such as (Rathi & Roy, 2023; Yao et al., 2023), we computed the energy cost for both ANNs and SNNs using 45nm CMOS technology. The energy consumption of a 32-bit MAC operation in ANNs (4.6 pJ) is approximately 5.1x higher than that an addition operation in SNNs (0.9 pJ) (Horowitz, 2014).
In ANNs, the computational cost per layer, denoted as CNNN. is defined as:
{
Wk h1 Cin bout
Woutt Couds
Convolution layer
ANN
fin Souts
Fully-connected layer
(22)
where Wg and hy denote the kernel width and height; can and Cout are the numbers of input and output channels; hout and Wont denote the height and width of the output feature map; and fins fout represent the input and output feature dimensions fully-connected layers.
For an SNN, the number of operations per layer is given by:
(())
(23)
SNN
CNN
where the spike rate w for layer / is defined as the average spike rate per neuron over all inference timesteps:
(1.e) spikes
1
(24)
T
(U) neurons
Here, T' denotes the total number of inference timesteps, N (1.) the number of neurons in layer l.
is the number of spikes timestep in layer and N (
spikes
neurons
24

---

Temporal Misalignment and Probabilistic Neurons
F. Membrane potential Distribution
Timestep
Timesiep
Timesieg
Membrane Potential Layer
CI
Membrane Potential Layer
Membrane Potential Layer
Timestep
Timestep
Timestep
Membrane Potential of Layer 4
Membrane Potential of Layer 5
Membrane Potential of Layer
Timestap
Membrane Potential Layer
Membrane Potential Layer
Membrane Potential Layer
Timestep
Timestep
Timesteg
Membrane Potential of Layer
Membrane Potential of Layer
Membrane Potential of Layer
Timestop
Membrane Potential of Layer 13
Figure 8. The membrane potential distributions of the first channel (randomly selected) across three modes (baseline, shuffle, and probabilistic) in VGG-16 CIFAR- 100. For comparison, the first timesteps (t=1, t=2) from total of eight timesteps (T=8) selected for each mode. The baseline mode (blue) achieves an accuracy of 24.22%, while the shuffle mode (light green) improves accuracy 70.54%, and the probabilistic mode (dark orange) further increases accuracy to 73.42%. The distributions are shown before firing, and the red dashed line indicates the threshold voltage (Vth) for the layer.
25

---

Temporal Misalignment and Probabilistic Neurons
Nembrate Fotentia Layer
Hembrate Poterbal of Layer?
Membrran Potential of Layer?
Verrzeane Potential of Layer
Trwetpp
Tiee
Tiradea
Nembrane Poterbia Layer5
Hombranc Poterbal Layer
Mambrane PoionBal Layer
Memorann Poionbal Layar
Tewerep
Teradea
Nembrane Potentia Layer
Nembrane Potential Layer
Mumbrane Petertial Layer
Hombrane Potantial of Layer
Tmeespp
Nembrane Potential Layer
Figure 9. The membrane potential of the first channel (randomly selected) from layer in SNNC baseline mode using VGG-16 on CIFAR-100 achieves accuracy of 24.22% before firing.
The first two timesteps exhibit an abnormal distribution compared to those at 1=4 to 1=8. This discrepancy arises from the initially incorrect membrane potential before firing, which affects the firing rate and propagates errors layer by layer. A detailed quantifiable error analysis is provided in Appendix B. Furthermore, shown in Figure 10, shuffling the membrane potential effectively alleviates this effect.
26

---

Temporal Misalignment and Probabilistic Neurons
Mambrane
Polential
Potantial
Layer
Theat2
3
Memtrran Layer
Layer
Potertial Layer
Potontial Layer
Membiase Potentia of Layer
Figure 10. Membrane potential of the first channel (randomly selected) before firing in SNNC shuffle mode using VGG-16 on CIFAR-100. The achieved accuracy 70.54%, indicating impact of random spike rearrangement.
27

---

Temporal Misalignment and Probabilistic Neurons
Polential
Layer
Theat2
3
Memtrane Rotential Layer
Layer
Layer
Layer
Potential Layer
Figure 11. Membrane potential of the first channel (randomly selected) before firing in SNNC probabilistic mode using VGG-16 16 on CIFAR-100. The accuracy increases 73.42%.
28

---

Temporal Misalignment and Probabilistic Neurons
G. Permutations and ANN-SNN conversion
Heuristics behind permutations We come back to the original motivation, and the mysterious effect of temporal misalign- ment. To this end, we notice that permutations may act "uniformizer" of the inputs to the spiking neuron, which highly related to notions of phase lag or unevenness the inputs (see (Li al., 2022) and (Bu al., 2022c), respectively).
Theorem 2.
I,...,,, N. Fur- thermore, suppose that these spike trains are modulated with weights w.....; wN, respectively. For a given permutation (TI,...;TN) let T8; denote the permutation of the spike train 8;. Then, for every t,, .1,
Suppose we have N spiking neurons that produced spike trains 8; s;(1), ..) si
Ex Σ wesss[tall Ex Σ w:Ts;(12]]
Proof. It is enough to prove that for each i
N, E [T[S E. [Tx)
(25)
Let A(4)) be the cardinality of the set of all the permutations that end up with a spike in step 4, and note that the probability of having spike at & is then A(t:), But, for each permutation that ends up with spike at t, one can find a permutation that ends up with a spike at l2 (by simply applying a cyclic permutation) and moreover this correspondence is bijective. In The equation (25) and the statement follow.
particular A(t,) is independent
The previous result deals with the expected outputs with respect to the permutations. When comes to the action of single permutation, we make the following observation. The effect of a single permutation mostly visible on spike trains that have low number of spikes. This, in turn, related the situation where the input to the neuron is low throughout time, and takes longer for neuron to accumulate enough potential in order spike, hence the neuron spikes later time during latency. In this case, single permutation of the output spike(s) actually move the spikes forward in time (in general) and such contributes to the elimination of the unevenness error, which appears when the input neuron in the beginning is higher than the average input through time (hence. the neuron produces superfluous spikes in the beginning, which shouldn't 't the case).
Table 12, Recorded accuracy after < T time steps, when the baseline model is "permuted" in latency T. Seuting is VGG-16, CIFAR-100.
Method
ANN
l=2
t=8
t=16
t=32
QCFS (Bu et al., 2022c)
49,09
63.22
69.29
73.89
75.98
76.52
Ours (Permute)
T=4
68.11
71.91
74.2
Ours (Permute)
T=8
71.76
74.11
75.53
75.86
Ours (Permute)
T=16
72.75
74.27
75.63
76.0
76.39
Ours ((Permute)
T=32
73.15
75.23
75.74
76.27
76.59
76.52
RTS (Deng & Gu, 2021)
1.0
1.03
23.76
43.81
56.23
67.61
Ours (Permute)
T=4
22.9
30.78
34.54
Ours ((Permute)
T=8
45.11
52.7
59.2
62.58
Ours ((Permute)
T=16
54.58
64.37
68.6
70.8
71.79
Ours (Permute)
T=32
62.76
69.12
71.76
73.31
74.09
74.6
Remarks:
1. In Table 12 we combine permutations with baseline models in fixed latency T., Afterwards, we record the accuracies of such "permuted" model for lower latencies t. We can notice a sharp increase in the accuracies compared to the baselines, and in particular, the variance accuracies across reduced.
2. Baseline analysis:
(a) SNN models converted from a pretrained ANN aim to approximate the ANN activation values with firing rates. In particular, in lower time steps, the approximation is too coarse as the firing rate has only few possibilities to use to approximate the ANN (continuous) values. For example, in T 1, the baselines are attempting to approximate ANN activations with binary values 0 and 0.
29

---

Temporal Misalignment and Probabilistic Neurons
(b) Moreover, at each spiking layer, the spiking neurons at early time steps, use only the outputs of the previous spiking layer from the same, early, time steps. As this information is already coarse, the approximation error accumulates throughout the network, finally yielding in models that are underperfoming in low latencies. (c) With longer latencics, the model is using more spikes and is able to approximate the ANN values more accurately, and to correct the results from the first time steps.
3. Effect of permutations:
(a) When performing permutations on spike trains after spiking layers in the baseline models, the input to the next spiking layer lower time steps, no longer depends only on the outputs of the previous layer in the same lower time steps, but depends on the outputs in all time steps T..
(b) In particular, when spiking layer is producing spikes at time step t = 1, it does so "taking into account" (via permutation) outputs at all the time steps from the previous spiking layer.
(c) As way of example, consider two spiking neurons N1 and N₂, where N2 receives the weighted input from Ni. If spiking neuron N1 in one layer has produced spike train then spiking neuron N2 at the first time step will use I as the approximation and will receive the input W 1 from neuron N1. However, after a generic permutation of the probability of having zero at the first time step of output of neuron N₁ is 4 (as oppose to having 1 with probability 1). and at the first time step neuron N2 will most likely receive the input W 0 = 0 from neuron N,, which a rather better approximation for W .225 than W itself.
0, in approximating ANN value of .25,
3
(d) This property of receiving input at lower t but taking into account the previous layer spike outputs at all the time steps is not only exclusive to lower Indeed, at every time step / < T, the input at a spiking layer is formed by taking into account spiking train outputs from the previous layer at all the time steps, but having already accounted for for the observed input at the first < I steps.
(e) In general, the permutations overall increase the performance of the baselines because the spike trains are "uniformized" in accordance to their rate, and the accumulation error is reduced. If a layer has produced spike outputs that well approximate the / layer in ANN,, then, after a generic permutation, at each time step starting with the first, the next layer is receiving the most likely binary approximation those rates.
(f) This is nothing but Theorem 2 in visible action.
gg) Besides Table 12, we provide further evidence on how permutation affect the baselines through the observed membrane potential in the following sections.
30

---

Temporal Misalignment and Probabilistic Neurons
G.I. The effect of permutations performance: Further experiments
76.25
76.00
75.75
Accuracy
75.50
75.25
75.00
74.75
74.50
6
31
41
51
6 (4. 71
6
10
(1.
Fixed positions
Figure Accuracy comparison for permutations time steps, fixing given pairs time steps. Setting is VGG-16, CIFAR-100. The
baseline (QCFS) accuracy is 73.89%, ANN accuracy is 76.21%. %.
72
71
Accuracy
70
69
68
4
6
8
10
12
14
Layer position
Figure Accuracy the model when permutation applied on single layer using QCFS bascline. Setting VGG-16 16, T CIFAR-100 1000 Baseline accuracy is 69.31%, 5, ANN accuracy is 76.31%
31

---

Temporal Misalignment and Probabilistic Neurons
60
Accuracy
40
30
20
4
6
10
12
Layer position
Figure 14. Accuracy of the model when permutation is applied on a single layer using SNNC baseline. Setting VGG-16, 7 = 8,
CIFAR-100. Baseline accuracy without calibration is 24. 24.22%, ANN accuracy is 77.87%
32


Reviews:
[{"review_id": "rhmMVQDGen", "reviewer": "Reviewer_BQ8A", "summary": "This paper presents a new framework for ANN-SNN conversion, which is motivated by an interesting phenomenon called “temporal misalignment”. The authors observe that the performance of converted SNN becomes better if they rearrange the temporal order of output spike trains of each layer. Based on such observation, the authors propose a new method for ANN-SNN conversion, using a two-phase probabilistic spiking neuron which mimics the effect of “permuting spike trains”. The method achieves SOTA results on various image classification datasets.", "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": "We thank the reviewer for their time and constructive comments.\n\n**Claims And Evidence: Not Clear**\n\nIt is important to ensure not only that the expected number of spikes matches the value of the ANN activation, but also that these spikes are distributed in a uniform manner. In particular, even though all spiking neurons in one layer emitted the expected number of spikes, when we apply the subsequent weights the next layer will not necessarily receive the expected input corresponding to the input of ANN layer. The key issue here is the unevenness error, as thoroughly discussed in AAAI’23, “Reducing ANN-SNN Conversion Error through Residual Membrane Potential”, as it can happen that the expected number of spikes arrives too early or too late. \n\nSo, ideally, we expect the spikes received from the preceding layer to be uniformly distributed. However, as spikes propagate through deeper layers, their timing often becomes increasingly irregular, leading to deviations from the expected spike count and resulting in unevenness error. \n\nFor your convenience, we compared our TPP method with a naive probabilistic spiking neuron, where after the accumulation of the membrane potential $v$, the neuron emits spikes with the constant bias $v/(\\theta\\cdot T)$ (so spiking becomes a stationary process). The following tables show that our design is outperforming the stationary process. Our Theorem 1. (b) offers an explanation, as the dynamic bias takes into account already emitted spikes and offers a more meaningful approximation of the ANN outputs. We will provide full tables in the revised manuscript.\n\n[Table1 and Table2](https://anonymous.4open.science/r/ICML2025_rebuttal-6705/figs_tables/t1_t2.jpg)\n\n**Experimental Designs Or Analyses**\n\nThe shuffling is performed in the following way. Once a baseline method is fixed, we proceed in order, layer by layer. After each spiking layer we would collect spike trains of that layer, then we would permute them in temporal dimension (that is we rearrange the spikes in time). Such permuted spike trains are then passed to the next layer, and we continue this process until the output layer. The permutation we apply after each layer is random, based on the current seed. \n\nGeneralization to TPP: Conceptually, a permutation would require to 1) collect the spike trains and 2) rearrange them in temporal dimension. These two correspond to two phases in our TPP neuron. In the second phase, TPP is producing the spike train, and the total number of spikes is given by Theorem 1. (c). Furthermore, at any given time step, there is a non-zero probability to have a spike (as long as there is non-zero residue voltage). Since in the end, produced spike trains will have \"predetermined\" number of spikes and there is a possibility to have a spike at every given time, it is as if TPP was acting as a permutation on a hypothetical spike train with the same number of spikes.\n\nWe hope this clarification makes the connection between permutation and TPP neurons more explicit, but please also kindly refer to Section 3.2 in our paper.\n\n**Weaknessess**\n\n**Temporal misalignment in baselines** We consider the situation of an ANN neuron with ReLU activation and how its output is approximated with a vanilla spiking neuron and our proposed TPP neuron. \n\n**Proposition**\n    Consider an ANN neuron with ReLu activation and let activation values follow distribution with PDF $p(x)$. Let further $\\theta>0$ be the threshold of the corresponding vanilla SNN neuron. Then:\n\n- The expectation of a spike output at step $t=1$ of a vanilla spiking neuron is $\\int_{\\theta}^\\infty p(x)dx$. \n\n- The expectation of having a spike at step $t=1$ of a TPP neuron is $\\int_{0}^\\infty p(x)dx$ which is the same as the expectation of the output of the ANN neuron with ReLU activation.     \n\nIn particular, this result shows that TPP neurons with their designed probabilistic spiking approximate the output of the ANN neurons as starting with the first time step, while for vanilla spiking neurons this approximation is coarser (see also Figures 8 for an empirical evidence of the above Proposition). Our Theorem 1 (b) then shows how this approximation evolves throughout the rest of the simulation time. \n\n**Different spike counts**\nNote that in Figure 4 and Figure 7 the distribution of membrane potential is presented for time steps 1 and 2. This is to empirically show that baselines do not have enough membrane potential to produce spikes in the first time steps, which eventually causes approximation errors as we discussed above. However, in Figures 5 nad 8, we are comparing spike counts for latencies 8 and higher. In particular, baselines are producing more spikes, however, these spikes trains are not \"evenly\" or \"optimally\" positioned in time, hence the gap in performance compared to our method. \n\nWe hope that this discussion contributes to the clarity of the paper, but we will incorporate your objections in the revised manuscript.", "claims_and_evidence": "Clear:\nThe bio-plausibility and hardware implementation of the proposed TPP neurons is well presented.\n\nThe effectiveness and efficiency of proposed method is verified on various image classification datasets with several network architectures, and the results are sound.\n\nNot Clear:\nThe proposed TPP neuron seems ignore any temporal information of input spike trains, since the first phase will accumulate the input and IF neuron has no decay. Then why is the Bernoulli random firing mechanism needed? I think the output timing of spikes for TPP neuron won’t affect the next layer, so it won’t change the final result as long as TPP neuron just fire the expected number of spikes?", "methods_and_evaluation_criteria": "The comparison with other ANN-SNN conversion methods is thorough and fair. Apart from accuracy, the spiking activity and membrane potential distribution are provided.", "theoretical_claims": "No checks", "experimental_designs_or_analyses": "No issues for experimental designs in Table1 and Table 2.\n\nIn figure1 and figure4, even in appendix G, it’s not very clear to me how the authors do “shuffle” or “permute”. The analysis doesn’t make sense if it’s just a random permute of spike trains, since the authors claim the permute can significantly improve the performance.\n\nIt is not clear how “permutation” inspires the authors to propose the TPP neurons. Section 3.2 is not very informative about this. Why TPP is doing (random) permutation?", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Weakness:\nThe analysis for “temporal misalignment” is more empirical and not theoretical to me. I’m not sure why other methods have “temporal misalignment” and why the proposed TPP neuron can solve the “temporal misalignment”. Based on the figure 4 and figure 7, baseline and probabilistic modes should have completely different spike counts and firing rate, but in the figure 5 and figure 6 baseline and TPP have almost the same spike counts.\n\nAs mentioned in previous sections, it is not clear how “permutation” is related to the TPP neurons.  At least it is not clearly presented in the paper.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": "J92Mn8mdnS", "reviewer": "Reviewer_BthZ", "summary": "The paper investigates the ANN-SNN (Artificial Neural Network to Spiking Neural Network) conversion process, identifying a phenomenon called \"temporal misalignment,\" where random permutations of spike trains across SNN layers improve performance. The authors propose a novel two-phase probabilistic (TPP) spiking neuron model to address this, featuring an accumulation phase followed by probabilistic spiking based on a Bernoulli process. Main findings include improved accuracy and reduced latency in SNNs compared to baseline methods, validated across datasets like CIFAR-10/100, CIFAR10-DVS, and ImageNet with architectures such as VGG-16, ResNet-20/34, and RegNet", "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": "We thank the reviewer for their time in assessing our paper and constructive comments. \n\n**Essential References Not Discussed:** \n\nThank you for pointing these out, we will include them in the revised manuscript. \n\n**Questions For Authors:**\n\n**1.** We reported in Tables 8-10 the number of spikes as, in general, this is a fair \"measure\" to use when comparing various methods for the same architectures and latency, on neuromorphic hardware. However, we provide further tables that compute the energy consumption on specialized hardware. In particular, our approach follows Merolla, et al. \"A million spiking\nneuron integrated circuit with a scalable communication network and interface\" where synaptic operations were used to calculate the energy, as this approach has been adapted in recent relevant literature. To estimate the energy consumptions per one SOP (FLOP), we use the values for the neuromorphic processor Qiao et al. \"A reconfigurable on-line learning spiking neuromorphic processor comprising 256 neurons and 128K synapses\". To be fair, we also calculated potential MAC operations in our and baseline models coming from the first layer, as the sample images use constant encoding. \n\nIn the following table (please refer to Table 17), we compare the energy consumption for some of the baseline methods and our approach, together with the approximated ANN energy consumption. We compare the energy consumption based on the accuracy. As our method reaches the ANN accuracy using much lower number of SOPs, it provides a more energy efficient alternative. However, we acknowledge that the stochastic component of our method can induce further energy consumption, but we were not able to test for that. We will provide full tables and more comparison results in the updated manuscript. \n\nhttps://drive.google.com/drive/folders/1sbhqT9Nabl8BUIDs9dt8OFpsz1sMZK6w\n\n**Generalization** We applied our proposed method on a simple Reinforcement Learning (DQN) example. We compared performance of the baseline QCFS (L=8) and TPP on CartPole task with over 20 evaluation epochs. The following table reports the average episode length (with standard deviation in parentheses) for different time horizons (T). A higher episode length indicates better performance, with a maximum possible score of 500 steps. The baseline struggles at T=4 but improves with higher latency, while TPP achieves significantly better performance at lower T values and reaches optimal performance faster.\n\n[Reinforcement Learning](https://anonymous.4open.science/r/ICML2025_rebuttal-6705/figs_tables/t16.jpg)\n\nIn our current work in progress, we consider generalization of TPP neurons that are specific to approximate various non-linear activation functions, as a first step towards conversion of various ANN architectures. In particular, design that we are currently exploring is the following (please see the following Figure for a visual explanation).   \n\n[activation function](https://anonymous.4open.science/r/ICML2025_rebuttal-6705/figs_tables/act-func.png)\n\nWe consider a general nice enough activation function $f$, and a sequence of thresholds $\\theta_1,\\dots,\\theta_T$, chosen in such a way that the steps with length $\\theta_i$ and height $s=1$ approximate the function $f$ in an ``optimal'' way (we do not discuss how to choose $\\theta_i$ and what optimality would actually mean). If we denote by $\\Theta_t:=\\sum_{i=1}^t\\theta_i$, the modified spiking mechanism of our TPP neuron becomes $s[t] = B(\\frac{v[t-1]}{\\Theta_{T-t+1}})$, $v[t+1] = v[t]-s[t]\\theta_t$. We note that the mechanism generalizes our TPP in that for ReLU activation we had $\\theta_t=\\theta$ for all $t$. This situation requires more sophisticated approach for theoretical insights, and we decided to keep it separate from our submission.  \n\n**3.** We provide the required tests in the following tables (please refer to Tables 3-15). The results confirm that the reported changes in TPPs spiking changes are meaningful.\n\nhttps://drive.google.com/drive/folders/1sbhqT9Nabl8BUIDs9dt8OFpsz1sMZK6w", "claims_and_evidence": "The claims are generally well-supported, but the assertion that TPP neurons are universally superior might overreach, as evidence is limited to specific datasets and architectures. Generalization to other tasks is untested, and the paper lacks discussion on failure cases or limitations.", "methods_and_evaluation_criteria": "The focus on accuracy alone neglects energy efficiency, a key SNN advantage, which could enhance the evaluation’s completeness.", "theoretical_claims": "Yes, theorems seem correct", "experimental_designs_or_analyses": "Spiking activity analysis (Tables 8-10) shows percentage differences, but the interpretation is unclear without statistical significance tests or error bars, reducing confidence in firing rate claims.\n\nCIFAR10-DVS (Table 2): Shows TPP outperforming direct training methods. The event-based dataset choice is apt, but the lack of latency or energy metrics limits depth.\n\nImageNet Results (Table 1): Compares TPP with multiple baselines across timesteps. The use of five runs with averages and deviations (partially shown) supports validity, but incomplete data (e.g., missing T=4 for some methods) hampers full assessment.", "relation_to_broader_scientific_literature": "The paper builds on ANN-SNN conversion literature (e.g., Rueckauer et al., 2017a; Diehl et al., 2015) and spiking neuron dynamics (e.g., Izhikevich, 2007). The temporal misalignment concept extends prior work on phase lag (Li et al., 2022) and input unevenness (Bu et al., 2022c), offering a novel interpretation.", "essential_references_not_discussed": "Hunsberger & Eliasmith (2015), \"Spiking Deep Networks with LIF Neurons\" (Neural Computation)\nNeftci et al. (2017), \"Event-Driven Deep Neural Networks\" (IEEE)\nDiehl et al. (2016), \"Conversion of Continuous-Valued Deep Networks to Efficient Event-Driven Networks\" (arXiv):\nLee, Donghyun, et al. \"TT-SNN: tensor train decomposition for efficient spiking neural network training.\" 2024 Design, Automation & Test in Europe Conference & Exhibition (DATE). IEEE, 2024.", "other_strengths_and_weaknesses": "Strengths:\nOriginality in identifying and leveraging temporal misalignment.\nPractical significance via state-of-the-art results on standard benchmarks.\nClear exposition of TPP mechanics (Figures 2-3, Algorithms 1-3).\n\nWeaknesses:\nLimited discussion of computational cost or energy efficiency, critical for SNNs.\nOveremphasis on accuracy without addressing trade-offs (e.g., latency vs. precision).\nClarity suffers from incomplete tables (e.g., Table 1 lacks full data) and missing proof details).", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1) Energy Efficiency: Why was energy consumption not evaluated alongside accuracy, given SNNs’ energy-efficient premise? Including this could strengthen the paper’s practical impact—e.g., if TPP reduces energy use, it bolsters the contribution; if not, it reveals a limitation.\n2) Generalization: How does TPP perform on non-classification tasks (e.g., reinforcement learning or time-series prediction)? Evidence of broader applicability could elevate the method’s significance; lack thereof might narrow its scope.\n3) Statistical Significance: Can you provide p-values or confidence intervals for the firing count differences (Tables 8-10)? This would clarify if TPP’s spiking changes are meaningful, potentially affecting the perceived robustness of the approach.", "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": "pZXppUSv8j", "reviewer": "Reviewer_UeYr", "summary": "The authors in this paper report a seemingly unusual phenomena in the ANN to SNN conversion framework, wherein by random spike arrangement of the output of SNN layers there was a increase in performance. Following this, the authors introduce a probabilistic neuronal model namely TPP, which improves the accuracy of the resulting SNN closer to the baseline ANN. The authors evaluate their proposed approach on vision-based datasets such as CIFAR-10/100, ImageNet, etc and achieved SOTA performance.", "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": "**Other Strenths and Weaknesses**\n\n(a) Theoretical justification: \n\nWe argue that ``temporal misalignment'' happens primarily due to the fact that in ANN-SNN conversion, SNN models will need a few time steps to accumulate enough potential to start firing. \n\nWe start by noting that in ANN-SNN conversion, at a particular activation layer of ANN, the thresholds for the corresponding spiking neurons SNN layer are chosen based on the distribution of the corresponding ANN activations, standard choice being the maximum activation or its percentile. In any case, there will be substantial activations that fall below this threshold. We argue that TPP neuron will produce more spikes at a first time step, than a vanilla spiking neuron, because of its probabilistic nature. To make this formal, we propose the following:\n\n**Proposition**\nConsider an ANN neuron with ReLu activation and let activation values follow distribution with PDF $p(x)$. Let further $\\theta>0$ be the threshold of the corresponding vanilla SNN neuron. Then:\n\n- The expectation of a spike output at step $t=1$ of a vanilla spiking neuron is $\\int_{\\theta}^\\infty p(x)dx$. \n\n - The expectation of having a spike at step $t=1$ of a TPP neuron is $\\int_{0}^\\infty p(x)dx$ which is the same as the expectation of the output of the ANN neuron with ReLU activation.     \n\nMany of the baselines propose to start with an initial membrane potential, in order to ``encourage'' early spiking, however, there will always be a substantial part of the values that will fall below the threshold and the above proposition still holds. \n\nEmpirical evidence for this is presented in Figures 4 and 7, to which we kindly refer. This absence of spikes in the first time steps further causes unevenness error of the spike outputs, as thoroughly discussed, for example, in AAAI’23, “Reducing ANN-SNN Conversion Error through Residual Membrane Potential.” Ideally, we expect the spikes received from the preceding layer to be uniformly distributed.\n\nIn particular, this result shows that TPP neurons with their designed probabilistic spiking approximate the output of the ANN neurons as starting with the first time step, while for vanilla spiking neurons this approximation is coarser. Our Theorem 1 (b) then shows how this approximation evolves throughout the rest of the simulation time. \n\n**Other Comments Or Suggestions**\n\n**Going beyond convolution architectures** In our current work in progress, we consider generalization of TPP neurons that are specific to approximate various non-linear activation functions, as a first step towards conversion of various ANN architectures. In particular, design that we are currently exploring is the following (please see the following Figure for a visual explanation).   \n\n[activation function](https://anonymous.4open.science/r/ICML2025_rebuttal-6705/figs_tables/act-func.png)\n\n\n\nWe consider a general nice enough activation function $f$, and a sequence of thresholds $\\theta_1,\\dots,\\theta_T$, chosen in such a way that the steps with length $\\theta_i$ and height $s=1$ approximate the function $f$ in an ``optimal'' way (we do not discuss how to choose $\\theta_i$ and what optimality would actually mean). If we denote by $\\Theta_t:=\\sum_{i=1}^t\\theta_i$, the modified spiking mechanism of our TPP neuron becomes $s[t] = B(\\frac{v[t-1]}{\\Theta_{T-t+1}})$, $v[t+1] = v[t]-s[t]\\theta_t$. We note that the mechanism generalizes our TPP in that for ReLU activation we had $\\theta_t=\\theta$ for all $t$. This situation requires more sophisticated approach for theoretical insights, and we decided to keep it separate from our submission.  \n\n**Questions For Authors**\n\nFor the experiments in the main body of the paper, we performed permutations with 5 different seeds and report the mean of the results. However, we kindly refer to Appendix G where we provide further experiments concerning permutations as well as theoretical insights into their functioning (Appendix G, Theorem 1). We would also like to point one curiosity here, as it is still beyond our understanding. Namely, in Figure 11, we report results when for latency $T=4$ and baseline model, a chosen permutation is applied to all the spiking layers of the model (so the permutation does not vary from layer to layer in a random way). We tested for all the 24 possible permutations of 4 elements and for all the 24 permutations, the performance improved.\n\n**Supplementary material**\n\nWe provide anonymized code that was used in some of the experiments. \n\n[Code](https://anonymous.4open.science/r/ICML2025_rebuttal-6705/TPP_QCFS/README.md)", "claims_and_evidence": "The primary claim is substantiated with empirical results. However, there is no strong theoretical proof behind the underlying temporal misalignment phenomena.", "methods_and_evaluation_criteria": "The experimental result bolsters the core contribution of the work.", "theoretical_claims": "The authors provided theoretical proofs regarding the design formulation of the TPP neurons. However, I could not find any substantial proof for the underlying temporal misalignment phenomena.", "experimental_designs_or_analyses": "The authors used standard datasets such as CIFAR-10/100, Imagenet, etc. to evaluate their approach. They also examined the membrane potential distribution across different models as a means of understanding why the base models underperformed.", "relation_to_broader_scientific_literature": "This work pertains to the domain of ANN-SNN conversion, which is relevant in the broader neuromorphic community.", "essential_references_not_discussed": "The authors discussed all major references.", "other_strengths_and_weaknesses": "Strengths: (a) Interesting observation regarding temporal misalignment. (b) proposed model achieves SOTA performance. \n\nWeaknesses: (a) No strong theoretical justification behind the observed event. (b) The work mainly explores convolutional architectures. Exploring transformer based architectures as well can provide a more complete picture.", "other_comments_or_suggestions": "Can explore transformer based architectures as well.", "questions_for_authors": "(1) For SNNs operating on longer timesteps, how many runs were done while doing the permuted spikes experiment. It is very unusual that model performance will be consistently higher for all permutations.", "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": "THQZhS43xl", "reviewer": "Reviewer_CXU9", "summary": "This paper analyzes the spike temporal dynamics in the ANN-SNN conversion framework and investigates the impact of spike firing timing on the stability of the conversion. It identifies a phenomenon termed temporal misalignment, where random spike rearrangements across SNN layers can lead to performance improvements. Furthermore, the paper introduces a bursting probabilistic spiking neuron to simulate and enhance this phenomenon, supported by theoretical justification, which improves performance on several tasks.", "questions": "", "limitations": "", "rating": 4, "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": "We thank the reviewer for their time in assessing our paper and their constructive comments.\n\n**3.** We consider the setting from Theorem 1: we assume that the accumulated membrane potential in the first phase is $v:=TX$, that the threshold $\\theta=1$ and that $0\\leq TX\\leq T$ (other cases being trivial). Let us also put $S[t]=\\sum_{i=1}^ts[i]$. We consider two different situations:\n\nCase 1. $TX$ is an integer. In particular, Theorem 1 (b) tells us that at each time step $t$, as long as $v\\geq S[t-1]$, the expectation of having a spike at $t$ given $S[t-1]$ is $\\frac{v-S[t-1]}{T-t+1}$. We note that the last quantity is never strictly bigger than 1 and never strictly below 0. In particular, the conditional expectation (of mem. potential before spiking) $E[v[t+1]\\mid S[t-1]]=v-S[t-1]-\\frac{v-S[t-1]}{T-t+1}=v[t]+\\frac{v[t]}{T-t+1}$. The total expectation then becomes $E[v[t+1]]=E[v[t]]-\\frac{E[v[t]]}{T-t+1}=E[v[t]]\\frac{T-t}{T-t+1}$. Initially, we have $E[v[1]]=v=TX$. Solving this recurrence gives $E[v[t+1]]=v\\prod_{i=1}^t\\frac{T-i}{T-i+1}=v\\frac{T-t+1}{T}$. Hence, $E[s[t]]=\\frac{E[v[t]]}{T-t+1}=\\frac{v}{T}$. \n\n\nIn particular, since the total number of spikes during $T$ steps is $v$ (Theorem 1. (c)) it follows that when $v$ is an integer in $[0,T]$, $S[t]$ is a random variable that follows **hypergeometric** distribution $H(T,v,t)$. The variance of the firing rate is then $V[\\frac{1}{t}S[t]]=\\frac{v}{tT}(1-\\frac{v}{T})\\cdot\\frac{T-t}{T-1}$. Note that at the final step $T$, the firing rate is precisely $X$ with variance 0. \n\nCase 2. When $TX$ is not an integer, we no longer have a nice expression for the probability of having a spike at $t$. Namely, the modified expression becomes $E[s[t]]=\\frac{\\min(\\max(v[t],0),1)}{T-t+1}$ as $v[t]$ in this case can be strictly negative or be strictly larger than 1. Hence, we cannot solve the resulting recurrence in a closed form. However, we do note that the final number of spikes at step $T$ is either $\\lfloor TX\\rfloor$ or $\\lceil TX\\rceil$ (Theorem 1. (c)).  Both cases occur with non-zero probabilities $p$ and $1-p$n Then, the firing rate at the end is either $\\frac{\\lfloor TX\\rfloor}{T}$ with probability $p$, either $\\frac{\\lceil TX\\rceil}{T}$ with probability $1-p$. The expected firing rate and its variance are $\\frac{\\lfloor TX\\rfloor}{T}+\\frac{1-p}{T}$ and $\\frac{p(1-p)}{T^2}$. In particular, we see that the firing rate stabilizes with $T$. \n\n\nIt seems that our spiking process offers a continuous extension of hypergeometric distribution (with continuous $v$), but we were not able to precisely detect this distribution in the existing literature. Further theoretical study of this process seems like a promising continuation of this work. \n\nTo shed more light on the situation, we provide the following  plots. In the first plot, we fix $T=16$ and vary $v$ continuously. We plot probability (blue) and variance (orange) of the event $S[T]=i$, when $v$ varies through $[i,i+1]$.  \n\n[Prob var](https://anonymous.4open.science/r/ICML2025_rebuttal-6705/figs_tables/prob-var.png)\n\nWe also plot the probabilities for various steps $t$ of having a spike at that time step, in dependence of the inital $v$. We notice that when $v$ is an integer, the probability is uniform for all $t$, while in other cases it varies from step to step. \n\n[Prob spike](https://anonymous.4open.science/r/ICML2025_rebuttal-6705/figs_tables/prob-spike.png)\n\n**2.** We thank for the references provided.\nComparison with [1] and [2]: We focus on Section 5.2 in [2] and in particular Theorem 3. We note the assumption that the assumption of the theorem is that the firing mechanism of spiking neurons are stationary processes with a fixed given number of produced spikes. However, in our case, as we discussed above, our proposed firing mechanism is dynamic, with bias depending on the emitted spikes. Furthermore, the number of spikes TPP neuron produces is ``guaranteed'' ($\\lfloor TX\\rfloor$ or $\\lfloor TX\\rfloor+1$) which offers more control in conversion. Based on our ablation studies (please refer to tables provide to Reviewer BQ8A), TPP neurons constantly outperform probabilistic neurons with fixed bias, and we expect that implementation of TPP with method of [2] would further ameliorate their respective results. We will deal with this more thoroughly in the future.\n\n\n**1.** You are on the point in your description of the motivation behind this work and the shuffling effect (See also Theorem 1 in Appendix G). However, we do emphasize subtle characteristics of our proposed Bernoulli firing process.\n\n**4.**  Our work in progress deals with extension of TPP neurons to approximate non-linear (activation) functions. Please refer to **Going beyond convolution architectures** answer to the Reviewer UeYr.\n\n**5.** We acknowledge that the stochastic component may induce extra energy consumption for TPP neurons. However, we argue that low number of SOPs to achieve high accuracy may compensate for this energy consumption.", "claims_and_evidence": "I appreciate the idea presented in this work; however, the writing is somewhat opaque. Despite reading the detailed appendices, I remain unsure whether I fully grasp the concepts. Below, I outline my understanding, which I hope the authors can confirm or correct.\n\n1. My understanding is that the effectiveness of the temporal shuffle is due to the following reasons:\n  - SNNs require potential accumulation to fire, leading to delayed responses, with most spikes occurring in later time steps, meaning that many neurons are unable to complete firing before time ending.\n- To mitigate this, we desire earlier spike emissions. By shuffling effect, neurons may \"overdraft\" and fire earlier based on their expected firing rate, instead of waiting for potential accumulation.\n- The method proposed in this paper uses a Bernoulli distribution to simulate this behavior, ensuring that the spike emission rate **at each time step** meets the conversion requirements, rather than only ensuring accuracy for the final result after $T$ step, as in standard ANN-SNN rate coding.\n﻿\nBased on this understanding, I raise the following question:\n\n2. The concept of shuffle described in the paper seems not fundamental. It merely appears to achieve an even distribution of spikes. A more precise term of your method might be \"expected firing rate estimation.\" This idea bears resemblance to the approaches in [1, 2], where spike attention outputs are predicted in the time dimension to ensure consistent expectations at each time step. Both papers address mechanisms for stabilizing variable multiplications in ANN-SNN conversion for Transformers, but could offer a simpler explanation for the phenomenon explored in this paper.\n\n3. While Theorem 1 demonstrates that the proposed SNN neuron firing rate is an unbiased estimate of the ANN firing rate at each time step, the stability of this firing rate should not be overlooked. The randomness introduced by the Bernoulli distribution could cause significant variance in the results of the same inference task. With the proposed algorithm, this variance may not converge to zero over time (and may even diverge), leading to perturbations during both training and testing. Does this introduce any stability concerns for the algorithm? Could there be considerable room for improvement? Related theoretical analysis may be found in [1,2], and supplementary empirical analysis would also be beneficial.\n\n[1] Jiang, Yizhou, et al. \"Spatio-Temporal Approximation: A Training-Free SNN Conversion for Transformers.\" The Twelfth International Conference on Learning Representations. 2024.\n\n[2] Huang, Zihan, et al. \"Towards High-performance Spiking Transformers from ANN to SNN Conversion.\" Proceedings of the 32nd ACM International Conference on Multimedia. 2024.", "methods_and_evaluation_criteria": "The evaluation criteria are appropriate, using standard datasets and backbone networks.\n\n4. However, the algorithm presented in this paper appears limited to traditional convolutional networks with ReLU activation and may not extend to Vision Transformers with GELU activation and attention operations. This severely restricts its potential applications. Could the authors discuss possible future directions for overcoming this limitation?", "theoretical_claims": "I reviewed the proof of Theorem 1 in Appendix B and found no apparent issues.", "experimental_designs_or_analyses": "The experimental design and implementation appear sound.\n\n5. However, the probabilistic neuron introduced in this work includes a stochastic component, which incurs computational and time costs. These costs are hardware-dependent (e.g., they could significantly affect performance on NVIDIA GPUs), potentially rendering the power consumption analysis incomplete and impacting the universality of its application.", "relation_to_broader_scientific_literature": "This paper builds on prior work in the ANN-SNN conversion domain, particularly related to phase lag and unevenness, and introduces a novel analysis and solution regarding neuron dynamics. These aspects are clearly discussed in the paper.", "essential_references_not_discussed": "The authors might consider referencing the latest work on Transformer conversion algorithms, as noted above.", "other_strengths_and_weaknesses": "While the paper's overall structure is clear, the writing clarity and organization could be improved. Despite some efforts to reduce the reading difficulty (e.g., Section 3.2, lines 191–215), the explanation could be more direct in clarifying the essence of the algorithm rather than the process.", "other_comments_or_suggestions": "I have no additional comments.", "questions_for_authors": "My questions are aligned with those discussed in points 1–5 above. I hope the authors will address point 1 and provide clarification on point 2. Additionally, I particularly recommend a more explicit theoretical and practical analysis of point 3.\n\nI am generally enthusiastic about this work but may reconsider my rating if some key issues remain unclear.", "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 proposes a novel ANN-SNN conversion framework based on the phenomenon of temporal misalignment by introducing a two-phase probabilistic spiking neuron (TPP) to improve the temporal distribution of spike trains across layers, thereby enhancing the performance of the converted SNN. Reviewers generally acknowledge the method’s advantages in accuracy improvement and performance stability, with strong results on datasets such as CIFAR and ImageNet. Some concerns remain regarding the TPP neuron’s ability to preserve input temporal information, energy efficiency, and comparisons with existing conversion methods, as well as the need for additional theoretical analysis and more comprehensive ablation studies. Despite these issues, the overall idea is innovative and the experimental results are sufficiently compelling, therefore, acceptance is recommended.
Decision: Accept (poster)