Advancing Spatiotemporal Representations in Spiking Neural Networks via Parametric Invertible Transformation
Authors: Yinsong Yan
Yujie Wu
Jibin Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Spiking Neural Networks
Spatiotemporal Representations
Neuromorphic Computing
Primary area: applications to neuroscience & cognitive science
TL;DR: We propose a parametric invertible transformation to enhance the spatiotemporal representations of spiking neural networks.

Abstract: Spiking Neural Networks (SNNs) are regarded as energy-efficient neural architectures due to their event-driven, spike-based computation paradigm. However, existing SNNs suffer from two fundamental limitations: (1) the constrained representational space imposed by binary spike firing mechanisms, which restricts the network's capacity to encode complex spatiotemporal patterns, and (2) the ineffective design of surrogate gradient functions that leads to gradient mismatch issues and suboptimal learning dynamics. To address these challenges, we propose the Parametric Invertible Transformation (PIT), which operates in a conjugate manner with neuronal dynamics to achieve adaptive modulation and augmented spike representations simultaneously. Second, we design an auxiliary gradient correction term to mitigate the gradient mismatch issue and oscillation phenomena during training. Moreover, we introduce a theoretical framework for analyzing the spatiotemporal representation space of SNNs. Extensive experiments on both static and neuromorphic datasets demonstrate state-of-the-art performance with our proposed method. This approach lays the theoretical foundation for expanding the spatiotemporal representations of SNNs, offering a viable pathway for developing low-latency and high-performance neuromorphic processing systems in resource-constrained environments. The code is available at https://github.com/YinsongYan/ICLR26.

Published as a conference paper at ICLR 2026
ADVANCING SPATIOTEMPORAL REPRESENTATIONS IN SPIKING NEURAL NETWORKS VIA PARAMETRIC IN- VERTIBLE TRANSFORMATION
Yinsong Yan', Yujie Jibin
Jibin Wu2.2*
1 Department of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University
2 Department of Computing, The Hong Kong Polytechnic University
j jibin. wulpol yu. edu. hk
ABSTRACT
Spiking Neural Networks (SNNs) are regarded as energy-efficient neural archi- tectures due to their event-driven, spike-based computation paradigm. However, existing SNNs suffer from two fundamental limitations: (1) the constrained repre- sentational space imposed by binary spike firing mechanisms, which restricts the network's 's capacity to encode complex spatiotemporal patterns, and (2) the ineffec- tive design of surrogate gradient functions that leads to gradient mismatch issues and suboptimal learning dynamics. To address these challenges, we propose the Parametric Invertible Transformation (PIT), which operates in conjugate manner with neuronal dynamics to achieve adaptive modulation and augmented spike repre- sentations simultaneously. Second, we design an auxiliary gradient correction term to mitigate the gradient mismatch issue and oscillation phenomena during training. Moreover, we introduce a theoretical framework for analyzing the spatiotemporal representation space of SNNs. Extensive experiments both static and neuromor- phic datasets demonstrate state-of-the-arn performance with our proposed method. This approach lays the theoretical foundation for expanding the spatiotemporal representations of SNNs, offering a viable pathway for developing low-latency and high persooerrrrmecee neuromorphic processing systems in resource-constraingg environments. The code is available at lhttp:://githbbbco com/ how.rtuuub.oooo//ioggggg.. Yan/ICLR26
INTRODUCTION
Spiking Neural Networks (SNNs) are designed to emulate the neural dynamics of biological systems by utilizing asynchronous spikes for communication (Maass) 1997 Roy 2019). In contrast to Artificial Neural Networks (ANNs), which rely on continuous activations, SNNs operate with binary spikes, providing huge potential for more efficient computations by transforming dense Multiply- Acc-Aucumul (MAC) operations into sparse Accumulate (AC) operations. Combined with the event-driven computation paradigm, where computations are triggered only upon the receipt of spikes, these characteristics offer inherent sparsity and low energy consumption, making them promising and appealing for scenarios necessitating real-time and energy-constrained processing (Merolla et al..
2014 Davies al. 2018 Pei et al. 2019 Chen et al. 2023 2025 Roy et al. 2019).
Despite these advantages, SNNs face two fundamental challenges that limit their practical effective- ness. First, SNNs suffer from constrained representational capacity due to their low-precision binary spike representation. This inherently leads to varying degrees of information loss and performance degradation in complex classification tasks (Deng & Gul 2021 Guo et al.) 2024). To address this issue, prior works have explored advanced spiking neuron models with complex dynamics (Fang et al. 2021b; Yin et al.) 2021 2023 Chen al.] 2024 Hao al.) (024), normalization techniques (Kim & Panda 2021 1/ Zheng et al.] 2021 1 Duan et al. 2022). 2). multi-bit spike representations (Guo et al.
2024] Xing al.| 2024) Guo et 2022c), attention mechanisms YYo et al. 202 2023b), and
modern architectures inc including ResNer (Fang et al. 202 Hu et al. 2024) and spike-driven trans-
formers (Zhou al. 2023 Yao et
2023a 2024 Zhou al. 2024]]
Yao et 2025). Nevertheless,
"Corresponding author

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Published as a conference paper at ICLR 2026
Tempural Dimension
Potental distributon
Potential Distribution Shifting (Hard to quantize)
Hidden Dimension
Input Currents
1(t) (+(
Becogazioed potastial disiiaation via PTT
Reurganized Potential Distribution (Easy to quantize)
Potential Updating
Xuit- 1)- (())
bit fidaa the buctoon
Reorganizing Potential via Parametric invertible Transformation
Clipping and Firing (Minimizing information loss)
((0) (m(()
Outped qu holagurr
Multi-bit bit Firing
Multi bit Outputs (Spike- driven inference)
Augmenting Spikes via Parametric Invertible Transformation
=(t) l/cksp 600))
Augmered ouggut outpott splies PIT
A Aummentod O Output Spikes (Real values across spatiolemporal dimensions)
Post layer
Original potential distitution
Reorganbeed potential distr bution
Clipped quantization range
Round function
Augmented real-velues outputs
The overall workflow of our proposed method. By incorporating parametric invertible transformation (PIT) into neuronal dynamies in conjugate manner, spiking neurons could adaptively reorganize the potential distribution before firing and emit real-valued outputs across spatial and temporal dimensions. is worth noting that the SNN integrated with PIT could preserve the spike- based inference and event-driven computation paradigm through the reparameterization technique.
Figure 1:
these approaches have not adequately addressed the inherent limitations of the representational space of SNNs, particularly when dealing with data streams characterized by high dynamic variations and complex distributions. Second, existing surrogate gradient methods tailored for binary spikes (Wul Rathi & Roy [021 Li et al. 2021) and integer spike neurons (Luo et al.) 2024 Yao
let al.
2018
al. phenomena during backpropagation (Yin et al. 2019; Spallanzani et al. 2022 Wu et al. 2023b Nagel
2025) exhibit limited adaptability and often introduce gradient mismatch issues and oscillation
(et al. 2022). These two limitations collectively underscore the critical need for developing novel approaches that can simultaneously enhance the representational capacity of SNNs while enabling
stable and efficient training.
In this study, we focus on enhancing the representational capabilities of SNNs by integrating the parametric invertible transformation (PIT) into neuronal dynamics. As shown in Figure [ our proposed PIT operates in conjugate manner before and after the firing operation of spiking neurons, improving spatiotemporal representations while preserving the event-driven and spike-based inference computation paradigm through the reparameterization technique (see Appendix A).. To further address the non-differentiable spike firing operation, we propose an auxiliary gradient correction term designed to mitigate the gradient mismatch issue and oscillation phenomena, facilitating rapid convergence and improved generalization. Additionally, we establish theoretical framework to analyze and measure the spatiotemporal representational capacity of SNNs. Comprehensive experiments on static and neuromorphic visual datasets, including CIFAR1O (Krizheskky et al.) 2009). CIFAR100 (Krizhevsky et al.) 2009), DVS-Gesture (Amir et al. 2017). CIFARIO-DVS (Li al. 2017), and ImageNet-IK Ik (Deng et al. 2009), validate the superiority and effectiveness of our method. Our contributions can be summarized as follows:
We introduce PIT, a parametric invertible transformation, to enhance the spatiotemporal representations of SNNs in a conjugate manner. PIT employs an input-distribttion-awe aware parameterization strategy to dynamically expand representations while preserving spike- driven inference via the reparameterization technique. Besides, we design rectified surrogate gradient term for improved gradient flow, enabling fast and stable convergence.
We provide a theoretical framework to analyze and measure the spatiotemporal represen- tation capacity of SNNs. Based on theoretical framework, demonstrate that the representation capacity of SNNs exhibits a logarithmic relationship with both the firing bit numbers and total time steps.
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Extensive experiments on both static and neuromorphic datasets demonstrate the state-of- the-art performance achieved by our method across various architectures. Notably, when incorporating our method into SEW ResNet34, it surpasses the baseline model with the same architecture after training for only one epoch and ultimately improves performance by 5.62% on the ImageNet dataset.
2 RELATED WORK
Training Methods for SNNs. Current SNN training methods can be broadly categorized into two strategies: (1) ANN-1O-SNN conversion, which establishes mathematical mappings between ReLU activation layers and spiking neuron layers, and (2) direct training. which employs the Backpropagation Through Time (BPTT) algorithm. Direct training allows SNNs to operate with an extremely small time window by employing BPTT alongside surrogate gradient (SG) functions to approximate the non-differentiable firing operation (Wu et al.] 2018 Neftei et al. 2019 Zenke &
Vogels 2021 Lee et al. 2020 Instabilities during deep SNN training have been mitigated using
normalization techniques (Kim & Panda [021] Zheng et al. [2021] Duan et al. 2022 Guo et al.
2023b). regularization terms to maximize output entropy (Guo al. (002aa), membrane potential
rearrangement (Guo et al. 2023a 2022b). and temporal efficient gradient re- weightit (Deng et al.
2022). For integer spike forms of SNNs, prior works (Luo et al. [2024] Yao et al. (2025] directly utilize the clipped rectangular surrogate gradient function to achieve direct training. However, this approach may result in oscillation phenomena that hinder the network from converging to a solution with
good generalizability (Yin et al. 2019 Spallanzani et al. 2022 Wu et al. 2023b Nagel et al. (022). Nevertheless, most existing methods are tailored to binary spike forms of SNNs, while effective and stable training methods for SNNs with integer spike forms remain underexplored. This gap motivates us to design corresponding parameterization techniques and surrogate gradients to enable effective and stable training, particularly in deep networks.
Enhancing Spike Representations for SNNs. At the neuron level, previous works have introduced learnable parameters into spiking neuron models, including learnable membrane decay factors (Bellec
et al. 2018: Fang et al., 2021b; Yao et al. (002)) and learnable threshold leak factors (Yin et al.
2021 2023 Rathi Roy 2021 Additionally, bio-inspired approaches have incorporated multi-
compartment structures to enhance neuronal dynamics for sequential modeling (Zhang et al.) 2024
Chen et al. 2024, Hao al. 2024) and time-series forecasting (Shibo et al. 2025). From the perspective of spike coding strategies, previous studies based on temporal coding (Yu et al., 2021] 2022) utilize augmented spikes to carry complementary information with spike coefficients in addition to spike latencies. Recent works based on rate coding expand binary spike trains into ternary spikes (Guo et al. 2024| Xing et al. (024) or real-valued spikes (Guo et al. (022c),, while ensuring multiplication-free inference through the reparamcterization technique. At the network structure level, recent works propose to enhance SNN architectures by redesigning the standard ResNet backbone (Fang et al. 2021a Hu et al. 2024). and developing spike-driven transformers (Zhoul 2024/Zhou
eet al. [ Yao et al.] 2023a 2024 Zhou et al. 2024 Yao et al., 2025). However, previous studies
have shown limited improvements in spatiotemporal representational space and capacity of SNNs,
as evidenced by our theoretical analysis in Section 3.5 These limitations inspire us to design a differentiable spatiotemporal transformation applied to the neuronal dynamics of SNNs to enhance their representations. Our proposed method leverages an input-distribution aware and spatiotemporal- decoupled parametric strategy to overcome the limitations of binary representations in SNNs, while preserving the advantages of their event-driven and ppsee inference paradigm.
METHODOLOGGY
3.1 PRELIMINARIES
Spiking Neuron. The Leaky Integrate-andFFin (LIF) spiking neuron serves as the most popular fundamental building block of SNNs due to its low computational complexity (Maass) 1997). For implementation, the LIF neuron with soft reset is typically described in a discrete iterative format as follows:
u = Xut_1 + Wi w 1-1
8 = H (u!. 0et)
(2)
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(3) where un and sy denote the membrane potential and output spike of LIF neurons in layer I at time step I, respectively. À is the decay factor of the membrane potential, and W represent the linear synaptic weights corresponding to layer l. (-) is the Heaviside step function, defined as (x) = for æ 2 O and H (x) = 0 for æ < O. When the membrane potential un exceeds its firing threshold Den the neuron will fire a spike its post neurons and reset its membrane potential by subtracting S.
o/. ui Uus
3.2 ANALYZING INFORMATION PROPAGATION FROM A TRANSFORMATION PERSPECTIVE
While the binary spike- based processing paradigm SNNs provides computational efficiency, induces performance degradation compared to full-precision counterparts. A critical bottleneck lies the firing operation, which converts the full-precision membrane potential into binary spike trains that serve information carriers propagating to the post-neurons. For notational simplicity, we omit layer superscripts in the following analysis. The information loss during the transformation of the full-po ooll-prccision membrane potential u into the binary spike form through the firing operation is expressed as:
d(s -=
/" dg (u (1)) r (1)) dl,
L
1=1
where g (-) denotes the firing operation of spiking neurons, and T signifies the total time steps. d(.,-) represents a distance function used to measure the distance between two elements. Based on information theory, .) can be interpreted the mutual information function the Kullback- Leibler (KL) divergence. In Euclidean space, d(.,-) -) can alternatively be represented by the Frobenius norm.
Previous studies have focused on introducing learnable parameters (Fang et al. 2021b Yao et al.
2022. Yin et al. 2021 2023 and multi-compartment structures (Zhang et al., [2024] Chen et al. [2224]
Hao et al.) 2024] Shibo et al. 2025) to enhance the neuronal dynamics before the firing operation.
These approaches can be regarded adding functions with varying complexities before the firing
operation. However, the reduction in information loss achieved by these methods is inherently limited,
as they do not fundamentally expand the output space and capacity of SNNs (see our theoretical
analysis in Section 3.5).
In this paper, we introduce a time-varying invertible transformation applied to neuronal dynamics in conjugate manner, both before and after the firing operation. Mathematically, the corresponding output spikes of SNNs are augmented as:
(5) where fi represents our introduced invertible transformation at time step t, and o represents the composition operator for functions. The corresponding information loss caused by the firing operation formulated as:
s (t) = frogo f. (u (t)).
[' of f. (u (t)) (t)) dt. 1-
(6)
Generally, the henefits of this design twofold: (1) It provides additional flexibility reduce firing errors by improving the neuronal dynamics before firing and expanding the spike representations after firing simultaneously. (2) This conjugate manner ensures variance consistency between the inputs and outputs, which is critical for stable information propagation in deep networks (He et al. 2015). Next, we will elaborate on how to parameterize the invertible transformation to expand the representational space of SNNs in a computationally efficient and input-distribution aware manner.
3.3 INCORPORATING THE PARAMETRIC INVERTIBLE TRANSFORMATION INTO SNNS
In this section, we will elaborate on how to integrate our parametric invertible transformation (PIT) into SNNs. For the 1t layer in the SNN, our introduced parametric invertible transformation f can be expressed in a matrix form as:
f (ul) A!u!.
(7)
where A E R N represents the transformation matrix in layer I at time step t, and ut, R BXN denotes the membrane potential of spiking neurons. B and N signify the batch size and hidden dimensions, respectively.
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Membrane potential distribution
Firing threshold spikes) or truncation line mittt-bitspikes maxx (oc)
Quantiration interval inc (u,)
dis
-5
Membrane potantial
Membrane potentia
Membrane potantial
(a)
(b)
(c)
Figure 2: The comparison of different strategies for the membrane potential quantization. (a) The firing operation for the binary spiking neuron model. (b) The min-max quantization method for the multi-bit spiking neuron model. (c) Our quantization strategy is based on the 3-sigma rule of the normal distribution.
To reduce the information loss incurred by the inherent binary spikes of the LIF neuron model, we adopt the Integer Leaky Integrate-and-Firg (I-LIF) neuron (Luo et al., 2024), which allows emitting integer values during the training stage and maintains its spike-driven propagation during the inference stage through the reparameterization technique. Specifically, the firing dynamics of the I-LIF neuron can be rendered as:
(8) where 1.1 denotes the rounding operator. clip (x, 0, D) confines the input i within range [0, D]. And D is hyperparameter indicating the maximum integer value emitted by the I-LIF neuron.
= clip (,
As illustrated in Eq. (5). our proposed PIT is applied in a conjugate manner both before and after the firing operation. Collectively, incorporating PIT into the I-LIF neuron yields the following neuronal dynamics in discrete form:
u -w/ + W W'st-1
8f Afclip (L(A)) un O..
(10)
o = ut st Eqs. (2) (10). and (11 correspond to the charging, firing, and resetting processes of the spiking neuron, respectively.
(11)
The next crucial question how parameterize introduced PIT, i.e., the transformation matrix A in Eq. (TO), to expand the spatiotemporal representations of SNNs. For the parameterization strategy of the transformation matrix A/, several aspects need to be considered: (1) The parameterization should be efficient in terms of both memory and computation. (2) It could dynamically adapt to the complex spatiotemporal distribution of the input data streams.
To address the first point, we design A in a diagonal form, enabling efficient computation while introducing negligible amount of additional parameters. Specifically, A diag (a!), where af E ER and N denotes the hidden dimension. this way, the computation involved the firing operation in Eq. (TO) is simplified to element- wise production and division, which also facilitates preserving the event-driven and spike- computational paradigm during the inference stage, as detailed in Appendix A Further discussion about the structure of A is also provided in Appendix F
To tackle the second point, we propose an souuttistribution-aae aware initialization strategy for A based on the statistical distribution of the input data, instead of directly applying the min-max method (Xiao) ett al. 2023 Shao et al., 2024), which is heavily influenced by large outliers of inputs. As shown Figure the min-max strategy suffers from two major drawbacks: (1) It leads to an unstable quantization range, thereby slowing down the convergence of the model. (2) It wastes a significant portion of invaluable quantization levels, resulting in larger quantization errors, especially in ultra- low-bit scenarios. i.e., D is small. Our approach mitigates these issues by designing a channel-wiss initialization method based on the statistical distribution of the input tensor. Specifically, we initialize ag based on the 3-sigma rule of the normal distribution as:
a max (Iи (u') 30 (ut) 1, |u (ut) + 30 (u.) DD/DD,
(12)
where y and calculated based the mean and standard deviation of the membrane potential u/ under the first batch of training data. After initialization, a, are updated according to BPTT (refer to Appendix for the detailed learning rules).
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3.4 RECTIFYING SURROGATE GRADIENTS DURING BACKPROPAGATION
The non-differentiable nature of the round function in Eq. (10) poses challenges for training SNNs using the BPTT algorithm. Previous studies have explored various surrogate gradient functions (Wu et et 20188] Rathi & Roy] [022]] Li et al. 2021 to approximate derivatives during backpropagation for binary spikes. For integer spike forms, Luo al. (Luo et al. 2024) employed a rectangular surrogate gradient function to train SNNs by retaining gradients for neurons activated within the [0. D) range, masking all others. This approach resembles the straight- trright-through estimator (STE) (Rosenblati T957, Bengio et al. 2013). where the identity function is used as proxy for the derivative of the rounding function. However, previous studies (Yin et al. 2019; Spallanzani et al., 2022 Wu et al. 2023b Nagel et al., 2022) have demonstrated that directly using STE can lead to gradient mismatches, resulting in oscillation phenomena that hinder the network from converging to a local minimum with strong generalization ability..
Motivated by this, we rectify the gradient based on the distance between the input and the decision boundary of the rounding function. Specifically, the derivative of the rounding function with respect to the input is calculated in the following form:
0 2
+ X sign (d))(z)) dis(z))
(13)
Ox
where dis(x) = x x [x] 0.5 denotes the distance between the input æ and the decision boundary of the round function. [-] and sign(-) represent the ceiling function and sign function, respectively. Noting that the value of the distance variable falls within the range [-0.5, 0.5). Specifically, the design of Eq. ) can be deemed as adding an l2 penalty on ||0.5 - sign(dis(x)) di)(2)) This regularization encourages the input to move away from the decision boundary, thereby mitigating oscillations of the output values between adjacent quantized states. As a result, the statistics of the rounding function's 's outputs stabilized, reducing oscillation during training and facilitating convergence to a solution with stronger generalizability.
Collectively, by employing the BPTT algorithm along with our rectified surrogate gradient function, all learnable parameters of the model, including the introduced transformation matrix A in Eq. 11) can be effectively optimized (detailed learning rules for learnable parameters provided in the Appendix B
3.5 ANALYSIS OF SPATIOTEMPORAL REPRESENTATION SPACE OF SNNS
This section aims to analyze the spatiotemporal representation space of SNNs from a theoretical perspective. We first provide the mathematical definition of the representation space and capacity based on linear algebra, followed by a thorough comparison of the SNN embedded with the PIT and prior works.
Definition Given s = {51, s2.... corresponding representation space (denoted as Span) generated by it can be expressed follows:
... ,,,,,,, ,, which refers to linearly independent elements, the
N
Span (s} = =Spa (81, 161,82,-11 ... ",,," =
Σ
ky E
(14)
where ky represents the linear combination coefficients (in SNNs, these are typically the weights in the linear layer or the convolution kernels in the convolutional layer).
Definition Given Span {s} = (Ei ky,; k; E <.R.. which denotes the representation space generated by and k. the corresponding representation capacity (dubbed Cap) is evaluated as follows:
Cap(Span (8}) log
Σ kyss ky E RR E {0,
D3
(15)
where |-| denotes the cardinality of the set, i.e., the total number of distinct elements generated by all possible combinations of s given k. In SNNs with multi-bi bit spiking neurons, D refers to the quantization level of 8, and N is the number of linearly independent elements in

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Proposition 1. For SNN with multi-bit spiking neurons emitting the output spike trains {st) {sij Sty E . E (1....,
(1....... , where T, N, and D denote the total time step. hidden dimension, and quantization level, respectively, its corresponding spatiotemporal representation space and representation capacity have the following form:
T
N
Span {8t)
Kysu kg E R. Siy E (0. D.. DI
(16)
Cap(Span (st) = log(T ( (D 1)
(17)
Corollary For a SNN with multi-bit spiking neurons, incorporating PIT generates the output spike trains {A(st) / = {aszSis Osj E (R, 8ij E (0,...,))) The corresponding spatio-tempore
representation space and representation capacity of the SNN embedded with PIT can be described as:
IT Span (A,8:
aiykysij aij E R, E R. (0,...,D) ,D,
(18)
i=
Cap(Span (A,st = log(T (D+ ID)((T(D+))))
(19)
Proposition elucidates the logarithmic relationship between the representation capacity and the quantization bit D as well as the total time step T. Specifically, as the quantization bit D increases, the representation capacity grows at sublinear rate, while the capacity scales linearly with the hidden dimension N. Corollary demonstrates that integrating additional parameters, i.e., {A+] in our
proposed PIT, into the combination coefficients {ki]} N j=1 could increases the representation space of the SNN by offering more degrees of freedom for temporal and spatial variations in output signals. Based Proposition 1 and Corollary 1, Table [ summarizes the comparative results of the spatiotemporal representation space and capacity between previous works and our proposed method.
Detailed proofs of Proposition and Corollary 1 provided in Appendix C
Table 1: Comparison of the representation space and capacity of the currently developed SNNs
Method
Representation Space
Representation Capacity
Vanilla Binary Spike
Σ N Kynnk kydy (y ER, E (0.1))
log(T"
Temary Spike Guo et al. 2024
D Σ Kyau ER (-1.0.1)
(o((T-3))
Trainable Temary Spike Guo et al. 2024
Er Em Iae e R. (-1,0.13)
log(T"
Real Spike
Guo et al. 2022o
D a,lyo l EB, E ( 11]
log(T' 2
Ours
ΣT EN agkesy my E RR,,, R, (0. .))
log(T" (O + 17*)
4 EXPERIMENTS
We conduct extensive experiments on CIFARIO (Krizhevsky al., 2009). CIFARIO0 Krizhevsky et al.) 2009). ImageNet-Ik (Deng et al. 2009). CIFARTO-DVS (Li et al. 2017 and DVS- Gesture (Amir et af.) 2017) datasets to evaluate the performance of our proposed method across various architectures. The Pytorch (Paszke et al. 2019) and SpikingJelly (Fang et al. (023) frame- works are utilized to implement SNN training in this paper. Detailed experimental setups are provided
in Appendix D.2
4.1 COMPARISON WITH THE STATE-OF-THE-ART
Results on static image datasets. We assess the effectiveness of our method on static image datasets, including CIFARIO and CIFARION 100, and ImageNet- lk. The results are summarized in Table On the CIFAR 10 dataset, integrating our proposed PIT into the baseline model achieves remarkable accuracy improvements compared to prior works, including maximizing the output information (IM-Loss (Guo ct al. 2022at) ) and rectifying the membrane potential distribution (RecDis-SNN (Guo al. 2022b))
Furthermore, on the CIFARION dataset, with the introduction of PIT. the classification accuracy of the 19 model surpasses prior methods designed enhance spike representations, including
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Table 2: Comparison with SOTA methods on CIFARIO, CIFARIO0 and ImageNet-Ik. Our results are reported as averages over three experimental runs with different random seeds. We reformulate the time steps for all direct training methods as T x D, where T represents the number time steps and D denotes the upper integer activation value of the firing function. In previous directing training work of SNNs, D is set to by default.
Dataset Method
Type
Architecture
Params
F'x D
Accuracy
TL.
Wa
2033a
Tandem learning
CIFARNet
44,48M
8
89.04%
PT1:
Warer mi
P022
Tandem leaming
VGGIT
9.23M
16
91 24%
training
PLIFNet
93.50%%
DSR
(Meang s 20223
SNN training
ResNe118
11.1M
20 x 1
95,40%
RedDi
2022h
SNN training
ResNe:19
12,20M
2
93.64%
training
ResNe119
Dict-SNN (Rathi & Roy
2021
SNN training
ReiNe120
11,25M
10 1
5x1
92.54%
V1.78%
Depike Ll
training
ResNet18
95.13%
93.6678 xx1
CIFARIO
STBP-1dB Zheng al.
2021
SNN training
RexNe119
12.70M
2x1
92.92%
92. 34%
TET DDng et al.
2022
SNN training
ResNe119
12,70M
2x1
4
19.16%
94.44%
Trainable Temary Spike Gio
2024
training
ResNe119
12,2MM
1x2
95.58%
Real Spike Goa et al
20226
SNN training
ResNe119
12.70M
2x
4x1
95 25.60%
PIT (Ours)
SNN training
ReaNe118
11.8M
1x4
95.86%
0:17
0.
ResNe119
12,20M
1x2
96.31%
U.OS
QCFS Bu all 20223
ANN2SNN
ReaNc120
11.35
61
70.19%
LTL (Ying A012
Tandem learning
ResNer20
11,25M
31
76.08%
Diet-SNNN
training
ResNer20
11,25M
RecDis-SNN (Gi 8
A1Z70
SNN training
ReaNe119
12.70M
4x1
74.10%
IM-Loss Guo
SNN training
VGG16
14.72M
5
70.18%
Dspike Li et 2021
SNN training
ResNe118
11.8M
4
2x
73.354
CIFAR100
TET [Dnng
2023
SNN training
ResNe:19
12,70M
2X
77.87%
Trainablc Tomary Spike (Gao et al. 2024
SNN training
ReaNe119
12.70M
2x
1x2
80 20%
78.45%
Spike Gno 2022c
training
ReaNe120
VGG16
14.72M
12.70M
5x
70.6255
ResNe118
1118M
1x2
76.88 % 0.12
(Ours)
training
ResNe119
12,70M
1x2
1 4
78.83%
81.59% % 0.09
80.12%
0.10
STBP-MdNN Zheng et
training
RexNe134
TET Deng al. p1Az
SNN mainnng
ResNet34
21,79M
6.x
64.79%
GLIF You ol
SNN training
ResNe134
21.79M
4 1
67.52%
DSR
SNN
ResNe118
11,69M
67.74%
TEBN
Doan d
2022
SNN training
ReaNe134
21.79M
4
68.28'5
ResNe118
11.69M
6
63.10%
MS-ResNat (Hu
2024
SNN training
ResNe134
21.79M
6x
69.42%
Trainable Temary Spike
Gan et al. 2024
SNN training
ResNe134
ReaNe188
21.79M
11.69M
4
4
70.74%
08'68%
Spike Goo 2022c
training
ReaNeTT
11,67M
4x
63.68%
ImagoNer-IK Ik
ResNe134
21.79M
4x1
67.69%
ReNNc18
1L88M
4x1
63.18%
ResNet34
21.79M
4x
67 67.04%
SEW Resierr
Pang 21218
training
ResNet50
4x 1
67.78%
RcsNe1101
44,55M
4
68.76%
ResNec152
60.19M
4
69.26 26266
Spike-Driven Transformer
You
training
Spike-driven Transfomer
1681M
4x
E.Spikefiommer Yao
2025
SNN mainingg
E E-SpikeFormer-M
E-SpikcFormmrr3
10.02M
S.IMM
1x4
78.50% 1x4
75.30%
ResNei1X
69.3%
ResNe154
21.79M
1x4
72.66%
0.27
PIT (Ours)
SNN training
Spise-driven Transfomer
168IM
1x1
73.45%
0.26
E.SpiielPormerSS
5.11
0.22
E. E-SpikeFormer.k
10.02M
1x
79.41 %
0.24
Real Spike (Guo al., 2022ch and Trainable Terarry Spike (Guo al. 2024). by 14.99% and 1.39%, respectively.
On the more challenging ImageNet- dataset, we incorporate our proposed PIT into the standard SEW ResNet (Fang et al., 2221a) architecture and compare it with state-of-the--n methods, including BN-based approaches (tdBN (Zheng et al.) 2021). TEBN (Duan et al. (022)) and spike representation enhancement methods (Real Spike (Guo et al. 2022c) and Trainable Ternary Spike (Guo al.)
2024 ). As reported in Table by applying PIT to the SEW ResNet18 and SEW ResNet34 models, achieve accuracy improvements of 6.21 % and 5.62% over the baselines, respectively. Notably. the SEW ResNet34 integrated with PIT surpasses the classification accuracy the 152-layer SEW
ResNet model, demonstrating the effectiveness and superiority of method.
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100
1000
Training Accuracy (%)
70
Test Accuracy (%)
80
Training Loss
60
40
40
SEW Reshet- 34
SEW ResNet-34
SEW Reshet
20
ResNet 34+PIT
RosNNet-4455P
34-FP
(73.314)
50
200
100 Epochs
200
Epochs
Epochs
(b)
(c)
Figure Comparison of (a) training loss, (b) training accuracy, and (c) test accuracy on ImageNet-Ik,
Results on neuromorphic datasets, Table I1 and Table 12 Appendix E present the experimental results on the CIFARIO-DVS and DVS-Gesture datasets, respectively. On the CIFARIO-DVS dataset, when utilizing VGG11 under equivalent inference latency (T previous approaches aimed at improving spike representation. including Real Spike (Guo et al.) 2022cb and Trainable Ternary Spike (Guo et al.) 2024), terms of accuracy by 5.92% and 3.90%, respectively. Moreover, the quantization level D increases, the model's classification performance can be further improved. Similar conclusions can be drawn from the comparison in Table 12 on the DVS-Gesture dataset.
10), our method surpasses
4.2 ANALYSIS OF OPTIMIZATION DYNAMICS
Visualization of learning curves. To investigate the optimization procedure during the training stage. we plot the learning curves of our methods and their vanilla counterparts ImageNet- 1k in Figure 3 is evident that the introduction PIT facilitates rapid model convergence and ultimately achieves higher performance. Specifically, the SEW ResNet34 integrated with PIT (red line) converges rapidly and eventually achieves a superior accuracy level (72.66%), closely matching ANN counterpart (ResNet34 (He et al. 2016)). which achieves 73.31%.
Visualization of membrane potential distribution. We visualize the membrane potential distribution after the last convolutional layer of VGG11 for vanilla LIF, I-LIF (Luo et al. 2024), and our PIT on CIFARIO-DVSS as shown in Figure 4 The surface plot is smoothed using Gaussian filtering for better visualization. We observe that vanilla LIF exhibits minimal discriminative ability for inputs different timesteps, showing uniform distributions along the temporal dimension. contrast, both I-LIF and our PIT demonstrate better temporal resolution. Notably, PIT dynamically adjusts the potential distribution for inputs at different timesteps, particularly at = 3 and T = 4, highlighting the ability of our method better capture the temporal structure the data stream.
Membrere Poteriell
Membraae Poteriial
Membrane Potental
cis
cs
5.
cta
Channal
20
Time Stop
Channel
23
Time Stop
Chanmel
Time Stop
420
420
(b)
Figure 4: Comparison of the membrane potential distribution after the last convolutional layer of VGG11 for (a) vanilla LIF, (b) I-LIF, and (c) PIT CIFARIO-DVS.
Table 3: Energy cost comparison on CIFARIO and CIFARI00.
Dataset
Architecture
Method
S-ACE (G)
NS-ACE (G)
Params (M)
Accuracy
ResNet18
LIF
4x1
35,84
6.45
11.18
94,22%
CIFARIO
ResNet18
QAT
N/A
35.84
11.18
95.145i
ResNet18
PIT
1x4
35.84
4.31
11.18
95.86%
ResNet18
LIF
4x L
35.84
7.88
11.18
73.25%
CIFAR100
ResNet18
QAT
N/A
35.84
11.18
77.91%
ResNet18
PIT
1x4
35.84
11.18
78.83%

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Published as a conference paper at ICLR 2026
5 ABLATION STUDIES
In this section, we conduct comprehensive ablation studies to investigate the individual contributions of the core components within proposed framework and evaluate its computational overhead.
Sensitivity to hyperparameter X in the rec- tified surrogate gradient function. Table 4
Table 4: Impact of hyperparameter À in the rectified surrogate gradient function.
presents the sensitivity analysis of À in our rec- tified surrogate gradient function (Eq. (3)). We vary À from 0.1 to 0.001 and observe con- sistent performance across this range, with ac- curacy variations less than 1% on CIFAR10 and CIFAR 100. The stable performance within the broad range of [0.001, 0.1] indicates that our gradient correction mechanism robust to hyperparameter selection, eliminating the need for extensive tuning in practical applications.
Dataset
Architecture
Params
T
Accuracy
ResNet18
1118M
1x
95.73%
CIFARIO
ResNet18
1L18M
1x4 4
0.01
95.86%
ResNet18
1118M
1
0.001
95.70%
ResNet18
1L18M
1x4
0.1
78. 10%
CIFARIOO
ResNet18
11.18M
1x4 4
78.83%
ResNet18
1L18M
1 4 0.001
79.01%
Memory overhead analysis. Ta- ble S compares the training mem- ory footprint of our method against baseline SNNs. On CIFARIO, our approach incurs modest memory overhead (0.22 GB and 0.25 GB for ResNet18 and ResNet19, respectively) while achieving substantial accuracy gains of 1.64% and 1 1.23%. On ImageNet-Ik, despite requiring additional memory, our method delivers significant performance improvements of 6.21% and 5.62% compared to SEW R and SEW ResNet34 (Fang et al. (021a). respectively. This favorable trade-off between performance and memory demonstrates the practical viability of our approach.
Table 5: Memory overhead comparison during training.
Datnset
Architecture
Model
Params
x 7>
Memory
Accuracy
ResNet18
LIF
11.18M
1x
94.22%
CIFARIO
ResNe119
LIF
12.70M
4
4,45 GB
95.49%
ResNou18
PIT
11.18M
95.86%
ResNe119
PIT
12.70M
1 1x4
4,70 GB
96.72%
SEW ResNet18
11.6MM
13.65 GB
63.18%
ImageNet-1k 1k
SEW ResNet34
IF
21.79M
xx1
18.86 GB
67. .4.04%
SEW ResNetl8
PIT
GB
69.39%
SEW ResNet34
PIT
21.79M
1 1x4 24.66 GB
72.66%
Energy efficiency analysis. Following Shen et al. (2024), evaluate energy efficiency using Synaptic Arithmetic Computation Effort (S-ACE) and Neuromorphic Synaptic Arithmetic Compu- tation Effort (NS-ACE) metrics. Table B reports comparisons with vanilla SNNs and Quantization Aware Training (QAT) baseline that uses the same firing count and per-channel scale learning as in our PIT. The S-ACE and NS-ACE are computed under 16-bit weight precision. At the same S-ACE budgets, our method consistently outperforms both LIF and QAT baselines, with accuracy improvements of 1.64% and 0.72% on CIFAR1O, and 5.58% and 0.92% on CIFARIO0 Notably, our approach achieves lower NS-ACE than SNNs with vanilla LIF, which can be attributed to the adaptive modulation mechanism of PIT that reduces firing activity. These results demonstrate that our method simultaneously improves both accuracy and energy efficiency.
DISCUSSION AND COMPARISON TO EXISTING WORKS
This paper analyzes information propagation in SNNs from a continuous-valued transformation perspective, offering a fundamental understanding beyond conventional ANN-to-SNN conversion approaches. We provide three key insights for the community: (1) Mitigating information loss requires adaptively adjusting the firing mechanism based on the time-evolving membrane potential distribution and is insufficient when merely applying batch normalization to input currents (Kim) & Panda, [021] Duan et al., (022). (2) We propose an auxiliary gradient correction term that adaptively addresses gradient mismatch and oscillation, surpassing pre-defined surrogate functions in facilitating convergence and generalization. (3) The inherent temporal dynamics of SNNs necessitate vire-ng and heterogeneous methods that consider spatiotemporal dimensions, rather than the homogeneous strategies commonly adopted in the quantization literature (Esser et al. 2020).
7 CONCLUSION
In this work, we introduced a parametric invertible transformation into neuronal dynamics to enhance the spatiotemporal representations of SNNs. By further incorporating an input-distribution-avaa parametric strategy and a rectified surrogate gradient function into SNNs, we demonstrated state-of- the-art performance across broad range of tasks. Our theoretical analysis further provides insights into the enhanced spatiotemporal representations of our approach, paving the way for low- latenay and high-accuracy neuromorphic computing systems.
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ACKNOWLEDGEMENT
This work was partially supported by the National Natural Science Foundation China (Grant No. 62306259), the Research Grants Council of the Hong Kong SAR (Grant No. C5052-23G, PolyU25216423, and PolyU15217424), and the Hong Kong Polytechnic University (P0058445).
ETHICS STATEMENT
We acknowledge and adhere to the ICLR Code of Ethics in all aspects of this research. This paper focuses on analyzing and expanding spatiotemporal representations of SNNs. In general, there are no direct negative social impacts associated with this work. The SNNs studied in this paper are dedicated to operating in event-driven and spike-based computational paradigm, offering significant potential for energy efficiency and contributing to the reduction of carbon dioxide emissions. The method proposed in this work is applied to neuronal dynamics in conjugate manner, enhancing the SNNs' spatiotemporal representations while preserving the advantages of the spike-driven inference computational paradigm. This work provides theoretical foundation for analyzing and improving the spatiotemporal representation space and capacity of SNNs.
REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our work, we provide comprehensive documentation of all experi- mental and theoretical components. The complete descriptions of datasets used in our experiments, including detailed data processing steps and training configurations, are provided Appendix [ The learning rules for our PIT method during backpropagation are thoroughly explained in Appendix B Complete proofs for all theoretical claims and theorems presented in the main text can be found in Appendix
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Appendix
A IMPLEMENTATION OF SPIKE-DRIVEN INFERENCE
In this section, we illustrate the equivalence between real-valued training and binary spike-driven inference using the reparameterization technique. This enables the SNN integrated with our PIT to retain the advantages of spike-driven inference.
Reparameterization technique. Due to the introduction of our designed PIT, the output of spiking neurons (as illustrated in Eq. (10) takes real-valued form. This results in the multiplication of weights and activations in the SNN being converted into MAC operations instead of AC operations, thereby losing the computational efficiency advantage of SNNs. To address this issue, we follow training-inference decoupled technique (Guo et al., 2022c 2024), which converts the real-valued outputs into binary spikes during the inference stage while maintaining computational efficiency through weight folding.
The key insight is that the PIT matrix A!- can be folded into the weight matrix of the next layer through reparameterization technique similar to that employed in Batch Normalization (loffe & Szegedy 2015). This folding operation enables the preservation of spike-driven computation while accommodating the real-valued intermediate representations during training.
Here, we take a fully connected layer as an example to illustrate the reparameterization technique. For fully connected layer, the inputs from the preceding layer are multiplied by weight matrix to produce the output features, which can be expressed as follows:
O W st
(20)
where Of denotes the outputs of layer time step E, and W represents the weight matrix associated with layer 1. For standard SNNs, the inputs s consists of binary values. While in our model, the inputs are augmented by our introduced PIT, thus possessing the real-valued form. Recall the firing
operation in Eq. (10), we have:
8 + Aclip (1(4!-1)") u, 7,, A!! Σ
(21)
where ل denotes the output spikes generated by the vanilla Integrate- and---nd---ii Fire neuron model with Soft Reset (IF-SR) at time step Specifically, this involves feeding the IF-SR with the input
(A) (A!-1) / at the first time step, and subsequently subtracting at each following time step until
D. This temporal dimension expanding strategy has also been validated by prior works (Luo et al.
2024 Yao et al. 2025).
In this manner, the calculation process during inference of Eq.
20 can be illustrated as follows:
0 = w's 8 W
1-1
= (w' A 4-11
Σ 8
(22)
d=1
d=1
By folding our PIT matrix, i.e., A! into the original weight matrix (W') of the SNN, the whole model could maintain the spike-driven computational paradigm during the inference stage.
Layer-specific implementation details. The folding mechanism varies depending on the layer architecture, and we provide specific implementations for the most commonly used layer types in SNNs.
Fully connected layers: For a fully connected layer, the weight matrix of the (-th layer is W IR MNN, where N and denote the input and output dimensions, respectively. Since A E N the product W A 1-1 e R X can be treated as a new weight matrix Wi e M ensuring the spike-basee inference paradigm. Importantly, A!-1 can be folded into the original weight matrix regardless of whether it is diagonalized, providing flexibility in the PIT design.
2. Convolutional layers: For convolutional layers, assume that the convolution kernel of the 1-th layer is Wi E RCn X X XCuee where Cmn and Cout denote the input and output channels,
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respectively, and K and K₁, represent the kernel width and height. In practice, A is typically set to diagonal form for computational efffciency, i.c,, A!- diag(a;")) where at- R To fold the PIT matrix into the convolution kemnll, the diagonal elements a!-1 are absorbed into the channel dimension of W via the Hadamard product:
1-1
W Wha Lany
Vi (1, ..n,,.,,innn.1
(23)
where W represents the folded weight matrix and " denotes the i-th element of a
This reparameterization ensures that during inference, the network operates with binary spikes while preserving the representational space enhanced by the PIT during training. The folded weights W encapsulate both the original synaptic weights and the learned PIT parameters, enabling efficient spike-driven computation without sacrificing the benefits of real-valued training.
B DETAILED DERIVATIONS OF LEARNING RULES DURING BACKPROPAGATION
By utilizzig the BPTT algorithm (Wu et al. 2018 Neftci et al., 2019: Zenke Vogels 2021 et al. 2020), the gradients of weight parameters in the SNN can be calculated follows:
Lee
Oc
T OL Üs ouk
OC os ds Ou!
(24)
ow
ds Ou, ow
Recall the neuronal dynamics of the spiking neuron model integrated with PIT, which can be described in discrete form as follows:
u Xwt_1 W V's?-1
(25)
of A/clip (L(A)) w/ (.1.0.
(26)
u/¹ ui s/c
(27)
According to the firing operation illustrated in Eq. 26 have:
a AAcllpp (I(A!)
u1,, 0, (,))
os!
0 ((4!)
0 (A4))
m!
(28)
01 (A -1 u4] A
sign (0s (A)) ut s D).
a ((4!)
-1 ut
where sign(-) denotes the sign function. To overcome the non-differentiable nature the round function in Eq. (28), we calculate its derivative based the distance between the input and the decision boundary of the round function (as illustrated in Eq. (3)):
a| ([(A!) u/1
=1 A (0.55 sign dis (AA) u dis ((4))=14:) =e: w.)).
(29)
0 ((A:) u/)
where dis ((A)) ut) (A!) -1 nn (A:))
uiJ 0.5 that refers to the distance between the input and the decision boundary of the round function. [.] represents the ceiling function. À denotes the hyperparameter that controls the strength of the auxiliary rectifying term, which is set to 0.01 in our experiments. Noting that the value of the distance variable dis falls within the range [-0.5, 0.5].
((A))
Thus, combining Eqs.
(28 and 29
we could deduce:
0s! Ou:
as
a
((AL)
ui)
0 ((A!)-1 mt)
Ouk
= =(1+ A (0.5 -sign (dis ((A)) ui!) dis ((A)) -1 wi))) sign (os (A!) uf <
D (30)
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Regarding the gradient of learnable parameters (i.e., A/) involved in our introduced PIT, could
calculate according to the chain rule as follows:
ac dL Üs!
(31)
0A
ds; dA
(clip (I(A)) uf1,0,
0s 0A clip (I(A!) u/1,0,D) - ((41) uu!) sign (o (A))
clip (I(A!) 141,0,1) + A
A -1 ut S =
(32)
Collectively, all learnable parameters in model could be trained in end-to-end manner by utilizing the BPTT algorithm along with our designed surrogate gradient function.
C PROOF OF THEOREMS
Proposition For SNN with multi-bit spiking neurons emitting the oulput spike trains {8:) {si Bij {0.... D} Wij (1.....Ny. i {1.... ,T), where T, N, and D denote the total time step, hidden dimension, and quantization level, respectively, its corresponding spatiotemporal representation space and representation capacity have the following form:
N kysi
Span {st)
ky E R. Siy --,
(33)
==
Cap(Span { log (D 1)
(34)
Proef. For SNN with multi-bit spiking neurons, at each time step t, its output spike s, consists of N elements, each denoted as Stj, where j E {I,.... _N.. According to Definition I, the representation space generated by N linearly independent elements of s; has the following form:
[ppa((s)=
Σ kjstj ky E IR, Stg E (0,...,,) D}
(35)
Over T time steps, the spike trains {st ) aggregated. The spatiotemporal representation space therefore constructed by summing across all T time steps:
Span (
(8 sT
kysis hy E R. siy (0,... D)
(36)
Here, the inner summation j== kisii represents the contribution of the N dimensions at time step i. The outer summation Σ aggregates the contributions across time steps. This establishes the form of the spatiotemporal representation space in Eq. (33
Next, we measure its representation capacity based on Definition 2. Specifically, we first calculate the cardinality (dubbed Card) of its representation space, i.e., {pan(s:) which yields:
Card(Span 81 TT))
ΣΣ k38g kg E ER 8ij E {0, ...... ,D)
(37)
Given coefficients {k,) J. in linear layers convolutional kernels of the SNN, the cardinality of the representation space is determined by the distinct values that {5. can take, given their quantization level D. Each Siy is quantized to D + discrete levels: {0, 1, D). Across dimensions, the total number of possible combinations of {st, \ at a single time step i (D + 1 Over time steps, each time step introduces a new combination of {sig Thus, the total number of distinct configurations across 7 time steps is: T (D + N.
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Finally, taking the logarithm of the cardinality, the corresponding representation capacity is: Cap(Spare M2=1) = log (T (D + 1)
(38)
This establishes the expression for the representation capacity illustrated in Eq. 34
Corollary 1. For SNN with multi-bit spiking neurons, incorporating PIT generates the output spike trains {A( {aıySı Oij IR, Sij E {0.... ;. The corresponding spatio-temporal representation space and representation capacity of the SNN embedded with PIT can be described as:
N
Span {A(s))E=1 1T
aiykySis ais E R, E R, E 10, (0,.,,D))
(39)
i=1
Cap(Span (A,8t) = log(T (D+ 1l0g(T((D+1))))
(40)
Proaf. For a SNN with multi-bit spiking neurons, our introduced PIT extends the spike train ) by introducing parameters diy IR, which are applied to each element Sij. The resulting transformed spike trains are:
(A+*:)?= = {a.581g Ozy RR E (O.... D}, (1...., (1..,.... (41)
PIT scales each spike value Siy by real-valued parameter aij. effectively expanding the representation space by introducing additional degrees of freedom across spatial and temporal dimensions.
Based Definition 1, aggregating over ' time steps, its corresponding spatio-temporal representation space of the model becomes:
N ΣΣ
Span (A,s:)?=1
aiykysis aij E R, E R, (o,...,.............. D}
(42)
This establishes the form of the spatio-temporal representation space in Eq. 33
According to Definition its representation capacity determined by the logarithmic seale of number distinct elements in the set Span {A
T
N
I ag E R, E R, E .., D} }
Cap(Span (A, (A,s:)T= log
Σ
aiykysi
(43)
Given introduced parameters a.j PIT and ky in linear layers or convolutional kernels of the SNN, the cardinality of the representation space is determined by the distinct values that Sij can take, given their quantization level D. Each 8új is quantized D + 1 discrete levels: 10,1, ... D}. Across N dimensions, the total number of possible combinations of { at a single time step is (D+1) 1) Over T' time steps, each time step introduces new combination of { Thus, the total number of distinct configurations across T time steps is T (D N
Finally, taking the logarithm of the cardinality of the representation space, its representation capacity yields:
Cap(Span {A.st) log (T-(D+))) (D++1)) ()
(44)
This establishes the expression for the representation capacity in Eq. (40
D EXPERIMENTAL SETTINGS
D.1 DATASETS
CIFAR-10 and CIFAR-100. The CIFAR-10 and CIFAR-100 datasets Krizhevsky et al., 2009) consist of 32x32 color images categorized into multiple classes. Specifically, CIFAR- 10 comprises 60,000 images distributed across 10 classes, with 50,000 images allocated for training and 10,000 for testing, while CIFAR- 100 contains images spanning 100 distinet classes. Both datasets have
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been preprocessed to achieve zero mean and unit variance. Image data augmentation techniques, including AutoAugment (Cubuk et al. 2019) and Cutout (De Vries & Taylor 2017). are employed. following the methodologies outlined in prior studies (Guo al., 2022a; Bu et al. 2022 Wang et al., 2023). The pixel values are directly input into the model's input layer at each timestep using a direct
encoding method (Rathi & Roy 2.2T). Following previous works (Huang et al. 2024 Zheng et al.) et al. 2022), the Spiking ResNet18 and ResNet19 are utilized as backbone models for CIFAR-IO and CIFAR-100.
ImageNet- 1k. The ImageNet-IK Ik dataset (Deng et al. 2009) consists of 1,281,167 training images and 50,000 validation images, distributed across 1,000 distinct classes. The images in ImageNet- IK are normalized to have zero mean and unit variance. During training, the images are randomly resized and cropped to dimensions of 224x224 pixels, followed by horizontal flipping. For validation, the images are first resized to 256x256 pixels and subsequently center-cropped to 224x224 pixels. Similar the methodology applied to the CIFAR datasets, the images are converted into temporal sequences using direct encoding (Rathi & Roy 2021 Fang al. 202 For performance comparison, the SEW ResNet (Fang et 200I architecture is used as the backbone model.
The CIFARIO-DV'S dataset (Li et al. 2017) is neuromorphic dataset derived from CIFAR-10 through conversion using a Dynamic Vision Sensor (DVS) camera. It comprises 10,000 wvent-based images with an expanded resolution of 128x pixels. The integration of events into frames is performed using the SpikingJelly framework (Fang et al. 2023 No data augmentation or
CIFARIO-DVS.
TEBN techniques (Duan et al. 2022 applied to the CIFARTO-DE dataset. For performance comparison, the Spiking-VGii VGGII (Huang et al., 2024) architecture (referred to as VGG1I) is adopted the backbone model.
DVS-Gesture. The DVS-Gesture dataset (Amir et al. 2017) is a neuromorphic dataset that captures 11 distinct gestures performed by 29 participants under three different lighting conditions. It contains a total of 1,342 samples, with an average duration of 6.5 seconds per sample. The dataset is divided into a training set with 1,208 samples and test set with 134 samples. Following prior work (Fang et al. 2021b), the events are integrated into frames using the SpikingJelly framework (Fang et al. 2023). ). No data augmentation techniques are applied to the DVS- dataset, and the Spiking- VGG11 I (Huang et al., 2024) (VGGI1) architecture is employed the backbone for performance evaluation.
D.2 TRAINING SETUP
Training Details. Table lists the key hyperparameters and configurations required for training on the static datasets (CIFAR-10, CIFAR-100, ImageNet- 1k), and neuromorphic datasets including Our experiments on CIFAR-10, CIFAR- 100, CIFARIO-DVS, and DVS-Gesture datasets are conducted using NVIDIA GeForce RTX 3090 GPUs, each equipped with 24 GB of memory. The training process on ImageNet- Ik is executed on eight NVIDIA RTX A6000 GPUs, each equipped with 48 GB of memory.
CIFARIO-DVS and
Table 6: Training hyperparameters and configurations.
CIFAR-10
CIFAR-100
ImageNet-
CIFARIO-DVS
DVS-Gesture
Optimizer
SGD
SGD
AdamW
SGD
SGD
Epoch
200
200
200
300
300
Learning rate
le-1
le-1
Se-2
5e-2
5e-2
Batch size
128
128
256
128
16
Weight decay
5e-5
5e-4
5e-4
5e-4
Se-4
Momentum
0.9
Le schedule
Cosine
Cosine
Cosine Cross-entropy
Cosine TET
Cosine
Loss function
Cross-cntropy
Cross-entropy
TET
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E MORE RESULTS
E. ABLATION STUDY
To investigate the impact of the proposed components on the model's performance, we conducted several ablation experiments the CIFAR-10 10 dataset using ResNet18 as the backbone architecture. To exclude the effects introduced by multi-bit spikes of the I-LIF neuron, we set vanilla ResNet18 and D 1 for our model. As reported in Table integrating PIT into the model significantly improves its performance, with accuracy increasing from 86.73% to 91.34%, marking a notable enhancement of 4.61% 1.1.. Furthermore, by leveraging our proposed input distribution-awann initialization strategy and gradient correction term, the model's performance is further enhanced. These results demonstrate that, under ultra-low latency conditions (7 1, D 1), the introduction of PIT significantly enhances the performance SNNs. Furthermore, these performance gains are not due to the use of multi-bit spikes but arise directly from the PIT itself.
for the
Table 7: Comparison of different configurations of our proposed PIT using ResNet18 as backbone on CIFAR-10.
Configurations
Learnable PIT Suput-Distribution-Awred Init
Gradient Rectifying
Accuracy
ResNet18 (i)
X
X
86.73%
X
X
91.34%
(ii)
X
92.53%
Ours
93.23%
Table 8: Comparison of different configurations of LIF, I-LIF, and proposed PIT CIFAR-10.
Model
Neuron Model
Learnable PIT
Multi-bit Spikes
Params
T x D
Accuracy
ResNet18
LIF
X X
11.18M
4 1
94.22%
ResNet18
I-LIF
11.18M
1 1x4
94.83%
ResNet18
I-LIF
X
11.18M
4 1
95.34%
ResNet18
I-LIF
11.18M
1
95.86%
To compare the performance of our proposed PIT with LIF and I-LIF (Luo et al. 2024 under the same inference latency, conducted additional ablation study using ResNet18 as the backbone model on CIFAR10 10. The comparison results, as shown in Table X demonstrate that our proposed PIT achieves 1.12% improvement in accuracy compared to vanilla ResNet18 (LIF). Furthermore, combining PIT with multi-bit spikes provides performance gains of 1.64% and 1.03% compared to vanilla R8 (LIF) and ResNet18 (I-LIF), respectively, further validating the effectiveness of our proposed method.
To investigate the trade-off between temporal length (I) and firing levels (D), we evaluate different T't x D configurations on CIFAR 10 and CIFAR100 using ResNet18 as the backbone model. Table 9 demonstrates that configurations with fewer timesteps and more firing levels consistently outperform their counterparts. Specifically, the 1 x 4 configuration achieves the highest accuracy on both datasets (95.86% and 78.83%), surpassing the 4 configuration by 1.03% and 1.92%, respectively. This trend indicates that for static image datasets without inherent temporal structure, the multi-level firing mechanism contributes more significantly to model performance than temporal length (T).
Table 9: Impact of T x configurations on SNNs' performance.
Dataset
Architecture
Method
Params
T x D
Accuracy
ResNet18
PIT
11.18M
4
94.83%
CIFARIO
ResNet18
PIT
11.18M
2 2
95.47%
ResNet18
PIT
11.18M
1 1x
95.86%
ResNet18
PIT 11.18M
4
76.91%
C0
R8
PIT
11.18M
2
77.65%
ResNet18
PIT
11.18M
1 1x4
78.83%
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E.2 COMPARISON OF ENERGY CONSUMPTION
In this section, we provide a detailed comparison of the energy cost between LIF, I-LIF (Luo et al. 2024 Yao al. 2025), and our spiking neuron models based Synaptic Operations (SynOps) and
Neuron Operations (NeuOps) incurred during neuron updates. SynOps consist of Accumulate (AC) and Aaatipll-An--cccumiee
(MAC) operations. Following previous studies (Han et al. 2015) Horowitz 2014), we assume that the operations are performed using 32-bit floating-pon -oinnn arithmetic under 45 nm CMOS technology. where an AC operation consumes 0.9 pJ and a MAC operation consumes 4.6 pJ. We conduct an extended analysis of energy cost using ResNet18 88 as the backbone model for the CIFARIO dataset.
Table 10: Comparison of the energy cost for LIF, I-LIF, and our PIT using ResNet18 18 the backbone model on CIFARIO.
Model
Input Resolution
T D
ACs (M)
MACs (M)
Params (M)
Energy (nJ)
R8 (LIF)
32 x 32 3
4 X
68.04
2.33
11.18
71.94
ResNet18 (I-LIF)
32
32 x
1 4
56.67
2.33
11.18
61.71
ResNet18 (PIT)
x 32 x
1x4 49.30
2.88
11.18
57.64
The results reported in the Table TOble[OOeene demonstrate that our model (the last row) achieves lower energy consumption. The energy efficiency of our model can be attributed to the PIT introduced, which dynamically adjusts the membrane potential distribution before firing and significantly reduces the firing rate, verified in Figure Collectively, although method introduces small number of additional MAC operations, our design minimizes overall energy consumption by reducing the firing rates of spiking neurons.
E.3 EXPERIMENTAL RESULTS ON CIFARIO-DVS
Table I1 reports the experimental results the CIFARIO-DVS dataset. Compared to previous approaches that focus on adjusting the membrane potential distribution before firing, such as MPBN (Guo et al., 2023b and IM-Loss (Guo et al. 2022a), method achieves superior per- formance with a lower-parameter network architecture (VGGTT I) under the same equivalent timesteps. Furthermore, compared to prior methods like Real Spike (Guo al. 2022c) and Trainable Ternary
Spike (Guo et al. 2024), integrating our method into VGGIT 10, 1) achieves improve- ments of 5.92% and 3.92%, respectively.
Table 11: Comparison with SOTA methods on CIFARIO-DVS. Our results are reported as averages over three experimental runs with different random seeds.
Method
Type
Architecture
Params
T' D)
Accuracy
STBP-idBN (Zheng et al.
2021
SNN training
ResNet19
12.70M
10
67.80%
PLIF (Fang al. 2021b
SNN training
PLIF Net
37.31M
74.80%
MPBN
21246
SNN training
1esNet19
12.70M
74.40%
IM-Loss Guo
SNN training
ResNet19
12.70M
10
72.60%
Dspike (Li al.
202T
SNN training
ResNet18
11.18M
10
75.40%
TET (Dngg et al.
2027
SNN training
VGG11
9.33M
10
77.33%
Trainable Temary Spike Guo et al. 2024
SNN training
ResNet19
12.70M
10
79.80%
VGG11
9.33M
10 x
76.60%
ResNeul
12.70M
10
72.85%
Real Spike (uo al. 2022c
SNN training
VGG
14.72M
10
74.58%
VGGIT
9.33M
10x1
80.50% 1 0.08
PIT (Ours)
SNN training
VGG11
9.33M
10
81.50% 0.13
E.4 EXPERIMENTAL RESULTS ON DVS- SYS-GESTURE
To further evaluate the effectiveness of our method on neuromorphic datasets, we conducted experi- ments on the DVS- Gesture dataset. The preprocessing steps followed the procedure outlined in prior work (Fang al. 2021b). with the data interpolated to T 10 using the SpikingJelly framework (Fang et al. 2023). VGGII I was adopted as the backbone, consistent with previous work (Huang et al., 2024). As shown in Table T2 it is evident that our method achieves oate-of-the-ar classification
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3.0
2.0
1.75
1.50
1:25
1.5
1.5
1.00
1.0
0.75
1.0
0.5
6.5
0.5
-1.99.0..9
-1.98.2b.99
-0 1:25
0.25
1.75
11.00
-1.00
(a)
(b)
1.00
2.9
1.00
25
0.75
2.5
6.75
2.9
0.9
025
0.25
0.00
-0.35
0.5
-0.00
-5.50
-1.00
-0.50
-0.22
0.00
0.23
-1.00
-0.75
-0.30
-0.25
0.00
0.25
0.50
0.75
(c)
(c)
Figure 5: Comparison of the loss landscape of (a, c) vanilla ResNet18 and (b, d) ResNet18 integrated with PIT models in terms of 3D surface and 2D contour plots on CIFARIO.
accuracy under the same network architecture. Specifically, our method, i.c., VGGI1 integrated with PIT (T 20, D = 1), surpasses Real Spike (Guo et al. 2022c) by 2.02%. Moreover, VGG11 integrated with PIT (T = 20, = 2) outperforms Trainable Ternary Spike (Guo et al. 2024 by 1.84%, demonstrating the effectiveness of our approach under equivalent inference timesteps.
Table 12: Comparison with SOTA methods on DVS-Gesture. Our results are reported as averages over three experimental runs with different random seeds.
Method
Type
Architecture
Params
D
Accuracy
PLIF (Pang al.
2021b
SNN training
PLIF Net
37.31M
20
97.57%
CLIF HHunn a. 2024
SNN truining
VGG11
9.33M
20 1
97.92%
Trainable Ternary Spike (Guo et al. 2024
SNN training
VGGII
9.33M
20 x 2
96.42%
Real Spike (Guo et al. 202ze
SNN training
VGGII
9.33M
20 x
96.21%
VGGIT
9.33M
20
98.23% 0.05
PIT (Ours)
SNN training
VGG11
9.33M
20 x 2
98.26 %
E.5 EXTENDED VISUALIZATION OF Loss LANDSCAPE
To evaluate the impact of the proposed PIT on the optimization procedure, we conducted comparative analysis of the loss landscape our model against its vanilla counterpart. ResNet18 was employed as the backbone architecture, with 7 1 and D set to exclude any effects introduced by the multi-bit bit spikes of the I-LIF neuron. By utilizing the technique introduced by Li et al. (018). we plot the 3D surface and 2D contour of the loss landscape of models in Figure From Figure 5 could observe that the model integrated with PIT exhibits a markedly smoother loss landscape around the local minima, which facilitates more stable and faster convergence during training.
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Published as a conference paper at ICLR 2026
E.6 EXTENDED VISUALIZATION OF FIRING RATES STATISTICS
To analyze the impact of our method on firing rates, we track the average firing rates of the ResNet18 integrated with PIT and its vanilla counterpart across each layer over time, as shown in Figure 6 Specifically, the ResNet18 integrated with PIT exhibits lower average firing rates (purple line) and demonstrates descending trend with respect to the time step. In contrast, the vanilla R tends to fire more frequently in later time steps and shows higher firing rates, particularly in the earlier layers. This phenomenon can be attributed to the ability of our input-distriutiio--vaaa PIT to dynamically adjust the potential distribution before the firing operation, which is absent in vanilla spiking models.
Firing Rates
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Figure 6: Comparison of the firing rates statistics for (a) vanilla ResNet18 and (b) the ResNet18 integrated with PIT models on CIFARIO.
E.7 EXTENDED VISUALIZATION OF PIT DISTRIBUTION ACROSS LAYERS
To inspect the evolving dynamics of our proposed PIT during the training stage.
ResNet18
= 1, D = 4) as the backbone and visualize the parameter distribution involved in PIT across layers the beginning of training and after convergence CIFAR-IO and CIFAR- 100 datasets, as illustrated in Figure]) 7 It can be observed that, at the beginming of training, the parameter distribution of PIT is relatively concentrated. After convergence, the parameter distribution of PIT exhibits significant inter-layer differences, demonstrating that PIT can dynamically learn and adjust according the features different levels.
F DISCUSSION ON THE STRUCTURE OF PIT
The conceptualization of PIT draws inspiration from the adaptive learning mechanisms inherent in biological systems, particularly the mechanisms underlying neural plasticity and the modulation of membrane potentials (Salaj et al. (021). As evidenced by previous neuroscience findings, neurons exhibit remarkable ability modify their neurophysiological characteristics in response dynamically changing environmental stimuli. This plasticity plays a critical role in enabling efficient information encoding and processing, as it allows neurons to flexibly adjust their neuronal activities and optimize their responses to external inputs.
In this work, to conserve parameters and memory, the transformation matrix of PIT designed in a diagonal form. However, in principle, it can adopt more general structure. Specifically, the off-diagonal elements of the transformation matrix are non-zero, indicates that neuron receives modulation signals (the sub-threshold voltages) of neighboring neurons before firing. This design mimics the excitatory-inhibitory (E-I) balance observed biological neural groups (Zhou) & Yu 2018). Such sub-threshold modulation mechanism could be effectively simulated by proposed PIT method. Additionally, neurons in biological systems may exhibit bursting behavior when responding to complex stimuli, which also be represented by applying our proposed PIT after the firing operation. The design of integrating PIT into neuronal dynamics in a conjugate manner encapsulates both adaptive regulation and efficient neural coding.
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Published as a conference paper at ICLR 2026
Hidden Dimension
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Figure Comparison of the parameter distribution in PIT across the layers of the ResNet18 integrated with the PIT model at the first epoch (a) and the last epoch (b) on CIFARIO. Comparison of the parameter distribution in PIT across the layers of the ResNetl integrated with the PIT model the first epoch (c) and the last epoch (d) on CIFAR100.
G THE USE OF LARGE LANGUAGE MODELS
In this paper, Large Language Models (LLMs) are only used for polishing and correcting grammatical errors. Their usage does not affect the core methodology, scientific rigor, originality of this research.
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Reviews:
[{"review_id": "rtnLHrSjnT", "reviewer": "Reviewer_CsiP", "summary": "This paper proposes Parametric Invertible Transformation (PIT), a linear transform integrated conjugately around the firing/quantization step of spiking neurons. The motivation is to enhance the representational capabilities of SNNs. A rectified surrogate gradient is introduced that penalizes proximity to rounding decision boundaries to mitigate oscillations. A theoretical section defines “representation space/capacity” and shows that PIT can expand the space by offering more degrees of freedom while preserving spike-driven inference. Experiments on static and neuromorphic datasets demonstrate that the method improves the performance across ResNet-like SNNs and spike-driven transformers.", "questions": "See Weaknesses.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. Good empirical performance. The reported multi-dataset and multi-architecture improvements demonstrate substantial gains under various scenarios.\n2. Ablation results in Appendix show the importance of each component.\n3. Theoretical energy consumption analysis show that PIT yields lower energy through firing-rate reduction.", "weaknesses": "1. Expressivity claim is overstated. PIT is largely a reparameterization: the linear transformation is a per-channel rescaling of weights which does not change the function class. It is akin to learnable pre-quantization scaling or BN-like affine parameters rather than a fundamentally more expressive operator. Gains plausibly stem from better quantization alignment and training dynamics, not representational power. This aligns with the paper’s own capacity formula that is identical to the non-PIT case. It largely undercuts the title/abstract’s emphasis on “advancing spatiotemporal representations”.\n2. Time-varying linear transformation is not friendly for neuromorphic deployment. The transformation $A_t$ varies over time, meaning that per-t weight variants at inference is required. Standard neuromorphic hardware does not support such kind of operations, and it prevents the model from extending to more time steps.\n3. Positioning vs. prior works. PIT’s diagonal scaling and distribution-aware initialization feel close to learnable pre-quantization scaling (similar to Real Spike’s reparameterization). The difference is primarily the conjugate placement and the specific initialization method. A clearer differentiation for contributions is required.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "J0tL1saRAV", "reviewer": "Reviewer_p6Qe", "summary": "This paper introduces a innovative solution to the long-standing problem of limited representational power and training instability in Spiking Neural Networks (SNNs). Its core contribution is the Parametric Invertible Transformation (PIT), a conjugate transform applied before and after the neuron's firing process. The elegant design allows the model to learn with rich, ANN-like continuous representations during training, while a reparameterization trick recovers the efficient, event-driven spike-based computation at inference.", "questions": "1. On the Fairness of the T×D Comparison: The paper uses T×D as a measure of equivalent inference latency, comparing their method (e.g., T=1, D=4) with prior works (e.g., T=4, D=1 or T=2, D=2).While T×D can be a proxy for total operations in some cases, different combinations of T and D can have different implications for actual hardware latency, memory access patterns, and the model's dynamic properties. For instance, a larger T implies longer sequential dependencies, while a larger D might require more complex computation within a single step. Could the authors discuss this trade-off in more detail? What are the limitations of this comparison metric?\n\n2. On the Sensitivity of the Rectified Surrogate Gradient: The rectified surrogate gradient proposed in Eq. (13) is an interesting design that aims to reduce training oscillations by penalizing inputs near the rounding boundary. The computation introduces a hyperparameter λ, which is set to 0.01 in Appendix B. How sensitive is the model's performance to the choice of λ? A sensitivity analysis on the value of λ would be valuable for demonstrating the robustness and practical utility of this gradient correction term.\n\n3. On the Distinction from Quantized Neural Networks (QNNs): In the low-latency regime of T=1 and D>1, the proposed model strongly resembles a Quantized Neural Network, where the integer spike value acts as a quantized activation and the neuron's dynamics effectively become a stateful quantization function. Could the authors elaborate on the fundamental distinctions between their approach (specifically in the T=1 case) and mainstream QNN research? While the inspiration from neuronal dynamics is clear, what are the key methodological or performance advantages of framing this problem within the SNN paradigm, as opposed to approaching it from a pure network quantization perspective? This clarification would help position the work more precisely.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The novelty of the PIT concept. Instead of incrementally improving existing SNN components, it fundamentally rethinks how information flows and transforms within a neuron.\n2. Empirical support. The experimental results provide evidence for the method's effectiveness and generality with SOTA results on a wide range of standard datasets.", "weaknesses": "1. Redefining the SNN Paradigm: By introducing continuous-valued transformations during training, the PIT method blurs the lines between traditional SNNs and ANNs with specialized quantization. This raises the question of whether we are still strictly in the SNN domain or creating a new class of SNN-inspired, highly efficient hybrid networks. The paper could benefit from a more explicit discussion of its philosophical positioning within the broader neural network landscape.\n2. Unaddressed Trade-off between Temporal Dynamics and Bit-Depth. The paper relies on the T×D metric to claim comparable latency with prior work, assuming that trading temporal steps (T) for higher bit-depth (D) is an equitable exchange. This assumption is could be a weakness as it overlooks a critical trade-off. By decomposing a D-bit spike into D sequential sub-steps within a single logical timestep (as detailed in Appendix A), the model's architecture fundamentally shifts from a truly sequential processor to a static one with enhanced precision. A model with high T and low D (e.g., T=4, D=1) can integrate information across distinct and potentially distant time steps, making it suitable for tasks with genuine temporal dependencies. In contrast, the authors' preferred configuration of low T and high D (e.g., T=1, D=4) largely sacrifices this temporal processing capability in favor of higher representational richness within an isolated moment. The paper fails to acknowledge or discuss this fundamental architectural trade-off. This limits the claimed generality of the method, as its effectiveness on truly temporal tasks (like video analysis or complex time-series forecasting) compared to traditional high-T SNNs is not established and remains questionable. Furthermore, the equivalence of latency under the T×D metric is an idealization that may not hold on real hardware, where micro-looping overhead for high D could introduce different latency characteristics compared to processing distinct high-T steps.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bhKRfhmSzu", "reviewer": "Reviewer_rRf7", "summary": "This paper proposes a Parametric Invertible Transformation (PIT) and provides a theoretical analysis of the spatiotemporal representation space and capacity of Spiking Neural Networks (SNNs). Experimental results demonstrate improved accuracy on several datasets.", "questions": "See in Weakness", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "+ The paper is well written and clearly organized.\n\n+ The proposed PIT effectively enhances information representation by transforming both membrane potentials and spikes in spiking neurons.\n\n+ The introduction of an auxiliary gradient correction term further facilitates SNN optimization.\n\n+ The paper provides comprehensive empirical comparisons with state-of-the-art models.", "weaknesses": "- While energy efficiency is presented as the main motivation of this work, the validation in this aspect appears rather limited. Simply calculating the computational energy is not sufficient; memory access cost should also be considered. A more detailed analysis or discussion on this point is recommended.\n\n- Compared with E-SpikeFormer, the proposed method achieves only a 0.7% improvement under the same model size. It would be helpful to include a more in-depth comparison with E-SpikeFormer to better demonstrate the advantages and strengths of your approach.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NSl7HupifS", "reviewer": "Reviewer_KE4i", "summary": "The paper builds on I-LIF–style neurons and instantiates firing as uniform integer quantization of membrane potential with a learnable per-channel step. Concretely, it applies a conjugate scaling scheme ($A^{-1}$ before rounding and $A$ after) so that $s_t^l=A_t^l,\\mathrm{clip}(\\lfloor (A_t^l)^{-1}u_t^l\\rceil,0,D)$ and $v_t^l=u_t^l-s_t^l$, alongside the standard charging equation. The scale $A_t^l$ is parameterized per channel, initialized by a channel-wise $3\\sigma$ rule, and trained end-to-end with a rectified surrogate-gradient (STE-like) update. For $T=1$ and small $D$ (e.g., $D=4$), the neuron reduces to a one-shot multi-level activation quantizer. Inference remains event-driven by folding $A$ into downstream weights. Experiments train the SNN directly rather than via ANN→SNN conversion, and the paper positions the approach as an engineering refinement that reconciles SNN charge–discharge–reset semantics with practical quantization.", "questions": "1. Explicit equivalence: acknowledge that Eq. (10) is a uniform quantizer and provide the mapping $s_t^l = Q_{A_t^l,D}(u_t^l)$ with $Q_{A,D}(z)=A,\\mathrm{clip}(\\mathrm{round}(z/A),0,D)$. Clarify what is fundamentally new beyond scale learning plus STE.\n\n2. QAT baselines: add ANN+QAT baselines with identical level counts (e.g., 5 levels for $D=4$), per-channel scale learning, and the same rounding and folding; compare accuracy.\n\n3. Temporal isolation: sweep $T$ and report accuracy.\n\n4. Energy evidence and fair accounting. Since this reduces to quantization (e.g., $T\\times D=1\\times 2/4$), report experiments against an ANN+QAT baseline with the same levels/scales and matched energy budgets.\n\n---\n\nShen et al., “Are Conventional SNNs Really Efficient? A Perspective from Network Quantization,” CVPR 2024; https://arxiv.org/pdf/2311.10802  ￼", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper presents a coherent synthesis: framing firing as uniform integer quantization wrapped by conjugate scaling, with per-channel learnable steps and a pragmatic $3\\sigma$ initializer, plus a rectified surrogate gradient near quantization boundaries. In terms of quality, the components are principled and practical—easy to implement in standard toolchains, trainable end-to-end, and compatible with event-driven deployment.", "weaknesses": "1. With temporal dynamics effectively disabled (e.g., $T=1$ so no carryover via $v_t$), Eqs. (9–11) collapse to a static per-layer mapping\n$a^{(l)} = Q_{A^{(l)},D}\\big(W^{(l)} a^{(l-1)}\\big)$,\nwhere $Q_{A,D}(z)=A\\,\\mathrm{clip}(\\mathrm{round}(z/A),0,D)$.\nThis is a per-channel uniform activation quantizer with a learnable step, i.e., a QAT-style layer; the model no longer exhibits SNN temporal dynamics. When $T\\times D = 1\\times 4$, this is exactly the one-shot 5-level quantizer $Q_{A,4}(z)=A,\\mathrm{clip}(\\mathrm{round}(z/A),0,4)$; with $T=1$ the temporal state is negligible, so the layer behaves like a QAT activation rather than a classical LIF gate. The paper does not clearly acknowledge this equivalence and risks overstating novelty relative to standard QAT (per-channel scale learning with STE).\n\n2. Missing baselines and fairness. For $T=1$ and $D=2/4$ in the experiments, the correct baseline is an ANN with uniform QAT using the same level count and per-channel scale learning (same rounding rule, same folding of scales). These head-to-head comparisons are missing, so it is unclear whether the gains come from SNN dynamics or simply from quantization.\n\n3. Temporal dynamics conflation. Although the scheme preserves charge–discharge–soft-reset, firing is no longer a spike-time thresholding mechanism but a multi-level amplitude code. Claims attributed to SNN dynamics should be isolated by sweeping $T$ and $\\lambda$. With $T=1$, Eq. (9) contributes little temporal effect; with $T>1$, residual carryover arises from $v_t = u_t - s_t$. Accuracy results should be reported.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)