PASCAL: Precise and Efficient ANN- SNN Conversion using Spike Accumulation and Adaptive Layerwise Activation
Authors: Pranav Ramesh
Gopalakrishnan Srinivasan
Venue: Accepted by TMLR ()

Abstract: Spiking Neural Networks (SNNs) have been put forward as an energy-efficient alternative to Artificial Neural Networks (ANNs) since they perform sparse Accumulate operations instead of the power-hungry Multiply-and-Accumulate operations. ANN-SNN conversion is a widely used method to realize deep SNNs with accuracy comparable to that of ANNs.~\citeauthor{bu2023optimal} recently proposed the Quantization-Clip-Floor-Shift (QCFS) activation as an alternative to ReLU to minimize the accuracy loss during ANN-SNN conversion. Nevertheless, SNN inferencing requires a large number of timesteps to match the accuracy of the source ANN for real-world datasets. In this work, we propose PASCAL, which performs ANN-SNN conversion in such a way that the resulting SNN is mathematically equivalent to an ANN with QCFS-activation, thereby yielding similar accuracy as the source ANN with minimal inference timesteps. In addition, we propose a systematic method to configure the quantization step of QCFS activation in a layerwise manner, which effectively determines the optimal number of timesteps per layer for the converted SNN. Our results show that the ResNet-34 SNN obtained using PASCAL achieves an accuracy of $\approx$74\% on ImageNet with a 56$\times$ reduction in the number of inference timesteps compared to existing approaches.

Published in Transactions on Machine Learning Research (12/2025)
PASCAL: Precise and Efficient ANN-SNN Conversion using Spike Accumulation and Adaptive Layerwise Activation
Pranav Ramesh
cs22h015 htt.::/smamml..im.m..ooom
Deportment of Computer Science and Engineering Indian Institute of Technology Madrus
Gopalalrishnan Srinivasan
Department of Computer Science and Engineering Indian Institute of Technology Madros
Reviewed on OpenReview: https: // openreveew net/ pet/ffrum? id id=kIdB7Xpiiv
Abstract
Spiking Neural Networks (SNNs) have been put forward as an energy-effiiient altornative to Artificial Neural Networks (ANNs) since they perform sparse Accumnlate operations instead of the power-hungry Multply-and-Accummlat operations. ANN-SNN conversion is a widely used method realize deep SNNs with accuracy comparable to that of ANNs. [u et al. recently proposed the Quantization- CeaatiizttionCliipPlloooo Clip--loor-Shiitt (QCFS) activation to ReLU to minimize the accuracy loss during ANN-SNN conversion. Nevertheless, SNN inferencing requires a large number of timesteps to match the accuracy of the source ANN for real-world datasets. In this work, we propose PASCAL, which performs ANN-SNN conversion in such a way that the resulting SNN is mathematically equivalent to an ANN with QCFS- activation, thereby yielding similar accuracy as the source ANN with minimal inference timesteps. In addition, we propose systematic method to configure the quantization step of QCFS activation in a layerwise manner, which effectively determines the optimal number of timesteps per layer for the converted SNN. Our results show that the ResNet-34 SNN obtained using PASCAL achieves an accuracy of =74% on ImageNet with 56x reduction in the number of inference timesteps compared to existing approaches. The code is available at bttts://gittubb httpp///gittbb.oom/Baaaiikleei. Srainneel--Lbb/PAA. PASCAL
alternative
1 Introduction
Deep Artificial Neural Networks (ANNs) have revolutionized the field of machine learning by achieving unprecedented success for various computer vision and natural language processing tasks (LeCun et al. 2015). Despite their superhuman capabilities, there exists significant. computational efficiency divide between ANNs and the human brain. In recent years, Spiking Neural Networks (SNNs) have emerged as a plausible energy-efficient alternative to ANNs (Roy et al.) 2019 Tavanaei et al. 2019 Guo et al. 2023 The fundamental building block of an SNN, namely, the spiking neuron, encodes input information using binary spiking events over time. The sparse spike-based computation and communication capability of SNNs have been exploited to obtain improved energy efficiency in neuromorphic hardware inplementations Davies
et al. 2021
Training methodologies for SNNs can be classified primarily into two approaches, namely, direct training using spike-based algorithms and ANN-SNN conversion. The efficiency of these methods is compared in A.11
Direct SNN training using spike- based error backpropagation (Lee et al. 2016 Wu et al. 2018 Shrestha & Orchard 2018 Neftci al. 2019 Lee et al.] (2020) has been shown to yield competitive accuracy on complex vision datasets such as ImageNet (Rathi et al.) 2020 Deng et al. 2022). is important to note that SNNs require multiple forward passes, inputs fed sequentially a certain number of timesteps.

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As result, performing backpropagation SNNs is both compute- and memory- because of the need to accumulate error gradients over time. Despite being computationalyy prohibitive, backpropagation methods have been shown to yield competitive accuracy, with inference latencies typically less than a few tens of timestops (Deng et al. 2022).
ANN-SNN conversion, on the other hand, eliminates the need for direct SNN training by using the pre- trained ANN weights and replacing the nonlinear activation (for instance, ReLU) with the equivalent spiking non-linearity (integrate-and-fire). The ANN activation is effectively mapped to the average firing rate of the corresponding spiking neuron. The conversion methods have been shown to provide accuracy comparable to that of the original ANN, albeit with longer inference latencios typically spanning hundreds to thousands of timesteps (Sengupta et al. 2019). This is because the correlation between the ANN activation and spiking neuronal firing rate improves with increasing mumber of timesteps during SNN inference (Cao et al.
2015
Several prior works have proposed techniques to optimize the number of inference timesteps (Dichl et al.
2015
Ruockauer al. 2017 Sengupta et al. 2018 2019 Han Roy 2020 Dong & Gu, 2021). Most notably, Bu et al. proposed novel activation function, namely, the Ceantiiaation-ii-Ploloo
(QCFS) activation, replacement for the ReLU activation during ANN training. The QCFS function, with pre-determined quantization step L, was formally shown to minimize the expected ANN-SNN conversion crror. We note that the QCFS activation works well in practice for CIFAR-10, but for CIFAR-100 and ImageNet, the ANN accuracy can only be matched when the corresponding SNN is inferred for ~ 1024 timesteps despite being trained using only 4 or quantization steps (Bu et al. 2023). In effect, the number SNN inference timesteps needed to match the ANN accuracy depends on the complexity of the dataset.
Our work addresses this issue using precise and mathematically grounded ANN-SNN conversion method, in which the number of inference timesteps depends only the structure of the source ANN, and not on the dataset. We utilize the integrate-ande and-fire spiking neuron with soft reset (Rueckaner et al. 2017 Han et al., 2020), a technique that is critical for achieving near-zero conversion loss. Next, we address key limitation of SNNs, which is their inability to handle negative inputs effectively, since spikes are commonly treated as binary values representing either or 1. We demonstrate that inhibitory (or negative) spikes are essent ial for ensuring the mathematical correctness of our method. Finally, we propose an algorithm determine the optimal quantization step for QCFS activation per layer by computing a statistical layer-sensitivity metric. This enables an optimal trade-off between computational efficiency and accuracy. In summary, we propose PASCAL, an accurate and efficient ANN-SNN conversion methodology, which has the following twofold contributions.
Spike-count and Spike-inhibition based SNN, We develop novel spike-count based approach, which is mathematically grounded and incorporates both excitatory as well as inhibitory spikes, to transform ANN with QCFS activation into sparse SNN. We realize ANN accuracy using fewer timesteps than slate-of-the-arl conversion approaches (refer to Section 4 thus lowering both inference time and energy consumption. We achieve near-lossless ANN-SNN conversion with a small additional overhead caused by introducing a spike counter per ncuron.
Adaptive Layerwise QCFS Activation. We propose a statistical metric to systematically analyze and categorize layers of DNN to selectively use higher precision (or QCFS steps) for certain layers, and lower precision for others. The DNN layers with lower precision need to be unrolled for fewer timesteps, thereby, improving the computational efficiency during SNN inference. This is in contrast. to early-exit SNNs, in which all the layers are executed for the same number of timesteps, which in turn depends on the input complexity (Srinivasan & Roy 2021
Li et al.
2023b
We validate the efficacy of PASCAL on CIFAR-10, CIFAR-100 and ImageNet datasets (described further Appendix A.3), with both VGG-16 and ResNet architectures. In addition, we demonstrate the hardware efficiency of SNN, obtained with both uniform and Adaptive Layerwise (AL) activation.
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(QCFS) activation, Published in Transactions on Machine Learning Research (12/2025)
Table 1: Summary of notations used in this paper.
Symbol
Definition
Symbol
Definition
Layer index
X
Generic imput matrix
QCFS function used in ANN training
Output spike train of layer / Weight matrix
Integrate-and-Fire (IF) layer used in
W
PASC formulation
A single element in X
Heaviside step function
x
Weighted output after layer
4 unrolled timestep of the weiglated after layer I
8.
Threshold curreut layer Spike count
Quantization Step for current layer Quantization Step for previous IF layer term QCFS activation Membrane Potential after & timesteps Batch
lm-1
X
Trainable threshold ANN layer
L
Generic quantization step
mem(()
Bias term
Batch variance
Constant numerical stability of batch normalization operation
Scaling parameler for batch normolization
Shifting parameter for batch normalioa- tion
Input classification model
C
Set of output classes
MatMul operation convolation or fully-comnected layer
fM
Classification map for classification model M with input z
T
Inference timesteps
Ltas
Total number of layers
2 Preliminaries and Related Work
2.1
ANN-SNN Conversion
ANN-SNN conversion has been shown to be a promising approach for building deep SNNs yielding high
enough accuracy for complex visual image recognition (Cao ct al., 2015 Dichl et al., 2015 Pérez- (arzz-Carrasoo
et al.
2013
Rueckauer al., 2016 2017 Sengupla el. al. 2019) and natural language processing tasks (Dieh)
et al.
2016
The conversion algorithms incur the following steps.
The trained weights of the ANN are transferred to architecturally equivalent SNN.
The ANN activation is replaced with an appropriate spiking activation. In discriminative models, ReLU is replaced with Integrate-and-Fire (IF) spiking activation. An IF neuron integrates the weighted sum of inputs into an internal state known as the membrane potential, as described in Section 3 It emits an excitatory (or positive) spike the membrane potential exceeds predetermined firing threshold. The membrane polential is subsequently reset to a suitable value.
Efficient ANN-SNN conversion requires careful threshold initialization at every layer of the network so that the spiking rales are proportional to the ReLU activations. Diehl et al. proposed a model-based threshold balancing scheme, wherein the firing thresholds are determined solely based on ANN activations. [Sengupta] ct al. proposed improved data-based conversion scheme that additionally uses SNN spiking statistics for threshold initialization. However, the aforementioned methods were susceptible to information loss during SNN inference due to the use of hard reset spiking neurons, wherein the membrane potential is reset to zero in the event of spike. Нan et al.] demonstrated near lossless low-latency ANN-SNN conversion with soft reset neurons, wherein the potential is decreased by an amount equal to the neuronal spiking threshold. Subsequently, [Bu et al. showed that the threshold voltage could be learned, instead of configuring heuristically post training, by replacing ReLU with the Quantization-Clip-Fae which further minimizes the conversion error.
2.2 Temporally Efficient SNN Inference
In parallel with improving the ANN-SNN conversion process, there have been efforts to speed up SNN inference by optimizing the number of timesteps needed to achieve the best accuracy. Farly-exit inference
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methods, which incorporate auxiliary classifiers at various stages in the model, have been shown to improve the inference efficiency of both ANNs and SNNs (Panda et al. 2016 Huang et al. 2018 Srinivasan & Roy 2021p. Li et al. proposed unique early-exit approach for SNNs, wherein the number of inference timesteps is modulated based on the imput complexity. In this work, we propose a novel methodology to achieve both precise and lemporally efficient ANN-SNN conversion.
3 Methodology
3.1
Cloo----o---ooiiif (QCFS) Activation
The QCFS function, h, as proposed by Bu et al. can be formulated as
T(z²)) = X clip (1, z'L + 4 / .0.1)
(1)
where 2 is the input to the activation layer, L is the quantization step, and X is the trainable parameter for the activation threshold. At the end of training, = An, where 0n is the final threshold that is subsequently used for SNN inference. [B et al.] showed that the expectation of conversion error reaches 0 when the shift amount, 4, is 2'
3.2 Precise ANN-SNN Conversion (PASC)
A After obtaining a trained ANN, we replace the QCFS function with Integrate-and-Fire (IF) activation and transform the Matrix Multiplication (MatMul) layers for SNN inference so that model computation is unrolled over timesteps. The operations of the transformed and MatMul layers are described below.
Integrate-and-Fire (IF) layer. We consider the general case where the quantization step Ln-1 of the previous layer is different from that of the current layer (I,,). We initialize the layerwise threshold 0" 0. where 6n is obtained from ANN training (refer to Section 3.1 The initial membrane potential is set to 5f similar to prior works (Bu et al. 2023). We outline the procedure in Algorithm I for a generic layer, and Algorithm
L
2 for the imput layer. We formally prove in Section 3.3 that the proposed algorithm enables precise ANN-SNN conversion. The algorithm for a generic layer described below.
1. For the first Ln-1 timesteps, we perform the IF operation (detailed in Section 2.1 train obtained from the preceding layer. We use a spike counter to record the number of spikes produced at this stage.
using the spike
2. In the next stage, we re-initialize the membrane potential to the value obtained at the end of the previous stage. Similarly, we set the initial spike count for this stage to that obtained from stage We then perform the IF operation for the next moz(Ln-1. Ln) timesteps. It is only at this stage that introduce inhibitory spikes. If the membrane potential is less than zero, emit inhibitory spike and update both the membrane potential and the spike count accordingly.
3. For the final stage, we reset the membrane potential to the spike count obtained after stages 1 and 2. We then perform the IF operation for I,, timesteps to generate the final spike train. The spike output of each timestep is stacked to create the final outpul tensor.
The output spike train generated by the PASC algorithm is effectively unrolled across Ln timesteps by the subsequent layer. Therefore, preserve the fact that after each layer, the number of inference timesteps is equal to the value of the quantization step, i.c. T L,, We note that Algorithm 2 is similar to, albeit simpler than, Algorithm spikes) over time. Feeding the input image to the SNN, without encoding the pixel intensities as spike trains, has been adopted in prior works to minimize accuracy loss during inference (Li et al. 2023b Bu et al. 2023
D except that it receives the input image containing real-valued pixels (rather than binary
MatMul layers. The MatMul layer performs either convolution or fully-connected matrix multiplication, followed by batch normalization. These operations generally performed using Multiply-and- antiilly-nd-Accumuaaaaa
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Algorithm PASC Algorithm for Generie IF Layer
Input: An Ln-1 N H x W matrix X, where Ln-1 and are the quantization step and output of the preceding MatMul layer, respectively.
Output: An Ln N x H x W matrix representing the spike output (scaled by threshold) of the current. layer.
3: Initialization: Threshold, 0 = l.. In
Initial membrane potential, mem(() 4* T. Initial spike count,
$ s=0. O.
for E 2.. 11 do
mem(t) mem(t - [
if mem(l) 2 0" then D- = L. // Increment Excitatory spike
8:
mem(t) = mem(t)= H* // Soft reset upon spike
9: Set mem(0) as the final membrane potential after the previous loop.
10: for t E 21. maarraaanaa I,)) 1) do
11: if mem(t) 2 8" then
0m L. // Increment Excitatory spike mem(t)= mem(t)= 0*
12:
13:
14: else if mem(t)) < O then
15:
8 L. mem = mem(t) +
0., Decrement on Inhibitory spike
16:
17: Initialize a new SNN with threshold 0 = and initial membrane potential mem(0) s, where the
spike count (scaled by 8*) computed previously.
18: Initialize empll lensor s', with dimension Ln x N H W, to generate the final spike train.
19: for t 21. }n} do
20: s'(t) = H (t - 1) --) 0", where st()) is the spike output scaled by 0", and H is the Heaviside
step function.
21: mem(l) mem(l 1) - (1)
22: return s', which will be the input for the subsequent MatMul layer.
Algorithm PASC Algorithm for the Input Layer
1: Input: An N x H x W input matrix X.
2: Output: An Ln x N x H x W matrix representing the spike output (scaled by threshold) of the input layer.
Perform the computation as in an ANN, yielding the result h(X) 9n clip 1, 0. + sp) .0.1).
4: Initialize SNN with threshold, 0" 6n., Initial membrane potential, mem(() = 7(x). Initialize an empty tensor s' with dimension L, N H W generate the final spike train. for E 21, } do
L,,
-0) where st) is the spike output scaled by 0", and H(x) is the Heaviside 7: mem(t)= mem(t - 1) - s(())
s'(t)= Hmmmm
step function.
return s
(MAC) operations during ANN inference. It is important to note that the MatMul layers receive spike trains over L, timesteps from the preceding IF layer during SNN inference. The binary spike at cach timestep d=. is represented by {0, 1} 0*, where the threshold 0" = T. We would like to point out that the quantity obtained by scaling a spike unit magnitude with the threshold is referred to as postsynaptic potential in the literature (Bu al. 2023).. Therefore, all computations in MatMul layers are unrolled over time and performed on binary input. The former MAC operations are replaced by repeated additions carried out on the spike train.
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3.3 Demonstration of Correctness of PASC Algorithm
We characterize the correctness of PASC algorithm by defining the notion of mathematical equivalence (Definition 3.20, which captures whether two models produce the same classification maps (Definition 3.1 for every imput matrix.
Definition 3.1. We define the classification map a model M with respect to a given input as the probability distribution function fi C [0, where C is the of output classes. The function f essentially specifies the class-wise prediction probabilities of the model.
Definition 3.2. Two classification models M and N are said to be mathematically equivalent their classification maps (Definition 3.1) are identical for all imputs. In other words, for all inputs 2, fM = fğr iee. [m(()) [x),)) Vc E C.
In order to establish this property at the final layer, we show that we maintain the following invariant at the end of each layer
Ln
(2)
Here, 2f denotes the tth unrolled dimension of the output tensor computed using the PASC formulation, and a' is the output tensor in the corresponding ANN with QCFS activation. We state theorems for two classes of operations. The notations used in each theorem are specified in Table
1. MatMul operation (either convolution fully-connected) + Batch Normalization (Theorem 3.3
Theorem 3.3. Consider layer with quantization step L, performing MatMul operation followed by Batch Normalization, i.e. x++1 - (s'aW)+b-R + 3. Assume that z' = Et * Then, if 2 Važ+e
(V---))- + B ViE 21, L}. where b' = 1.2 μ = and B' = then 2/41 -E'= 24 141 7. Val+c
2. Integrate-and-Fire operation (Theorem 3.4
Theorem 3.4. Let zt be the input to QCFS activation layer of an ANN with threshold 0n Suppose that the quantization step is La-1 for the input. Consider a set of tensors z for i E 21.2.... Ln such that E, =2 2¹. Let X be matrix formed by stacking 2/, f.e. X 2/ vi {1, Ln f1.2. L,), where represents
Then, (a) h(21)=(()))))))))))))))) = clip (t.. s'f=+ 0, E, the IF operation detailed in Algorithm [ and La is the quantization step for the output. Moreover, (b) h(X) contains values only from the set {0, B.
Both of these thcorems inductive in nature, also state Theorem 3.5 to show that Invariant 2 holds immediately after the input layer.
Theorem 3.5. Consider a model with quantization step Ln and threshold 0n for the input layer. For a given input 2, let the output of an ANN after the first QCFS activation layer be z' and the corresponding output in the PASC formulation be st. Then, (a) Eta, (i) {0, }-
z'_ ='. Moreover, (b) s contains values only from the sel
We now state the overall equivalence between an SNN with PASC formulation and the corresponding ANN with QCFS activation in Theorem 3.6
Theorem 3.6. Let M be the PASC formulation of any ANN model N with QCFS activation. Then, M is mathematicalty equivalent to N' (Definition 3.2).
The proofs for the aforementioned theorems are presented in Appendix A.1

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Algorithm 3 Computing the AL Metric
The output of the QCFS activation layer of an ANN with quantization step L, for a batch of training images. We show results for I 3000.
Input:
Convert the imput to a histogram H, storing the number of elements belonging to each level of QCFS activation with quantization step I.
3: Compute the Skewness (Equation 4), the Van Der Eijk's Agreement A (refer Section 3.4.1 and the kurtosis K (Equation 5 using the histogram H. Calculate the AL metric M A (g2 +
Algorithm Comprehensive Training Methodology using Adaptive Layerwisc QCFS Activation
1: Train a model with L, = La, on the input dataset until reaches convergence. This a standard training iteration with uniform La across all the QCES layers. A pre-trained model can be used, available off-tho-shelf, for the subsequent steps.
Obtain the AL metric using Algorithm 3 for this training iteration, and partition the layers into clusters (refer Section 3.4.5). At this stage, the user is free choose the number of clusters X and the value of L for each cluster, depending on the desired accuracy-efficiency trade-off.
3: Re-train the model, with uniform L La. for a fraction = of the total number of epochs used in the initial training step.
4: Update the value of Ln for each QCFS layer and train incrementally until convergence.
3.4 AL: Adaptive Layerwise QCFS Activation
In this section, we describe the proposed Adaptive Layerwise QCFS activation methodology (referred to as AL) to determine the optimal quantization step for every layer. The AL algorithm consists of the following steps.
Computing the AL metric, presented in Algorithm
Using the metric to train a new model, with adaptive layerwise quantization step, as detailed in Algorithm
The AL metric statistical indicator of the frequency distribution of various quantization levels in the QCFS layers. We use this metric to ascertain the optimal L, per layer required to approximate the original distribution. The proposed metric is combination of three individual statistical measures, each of which is explained in the following sections.
3.4.1 Van Der Eijk's Agreement
We use Van Der Eijk's agreement sufficient frequency. It is specified by
Eijk 2001 to determine how many bins in the histogram H have
A S-1
K-1
where A indicates the Van Der Eijk's agreement, S is the number of non-empty categories, and K is the total number of categorics. We employ a threshold a E (0. to ascertain if a bin is non-empty, as detailed below. If the frequency of the tt bin, Wo, satisfies Wo, 2 a- Σ, Wb, we call the bin non-empty; otherwise, we call it empty. We show the effect of a on the final metric Appendix A.2 The insights we draw from are twofold:
A then there are few bins which are full. This indicative of a unimodal distribution, which can be approximated well with lower precision. Such a layer is amenable to reduction in quantization step.
If A 2 0, then there are many bins which are full. This means that the activation requires higher precision, and hence would not support reduction in the quantization step T..
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3.4.2 Skewness
Skewness is defined as the deviation of the given distribution from a normal distribution (Joanes & Gill 1998 It is given by
E -i=1(a (2; x)3 n1 Ei=1(6 (I, 2)2)$²
mg 9 =
where n is the number of samples, my is the third moment, 8 is the sample standard deviation, and 2; are the samples whose mean T. We interpret the value of skewness, as explained below.
larger positive skewness indicates that the peak of the given distribution is close zero. Hence, the quantization step can be reduced without impacting the peak of the distribution.
smaller positive skewness indicates that the given distributtin more closely approximates normal distribution. This renders reduction in the quantization step infeasible.
We use the metric g² + 1, where is as defined in Equation adopted in prior works (Tabä et al. 2022
3.4.3 Kurtosis
Kurtosis K is formulated
h4 K
(n + 1)m (n 1)(n 2)
> 1 (x, - σ)
(5)
k2
where ha is the fourth moment, b is the second moment, is the number of samples and i; the samples whose mean is T (Joanes & Gill. 1998). We interpret kurtosis in the following manner.
A distribution with a large (positive) kurtosis indicates sharp and narrow peak, which makes it possible to reduce the quantization step.
Conversely, a low kurtosis indicates distribution with a broader peak, which warrants more quantization levels (or larger quantization step).
3.4.4 Final Metric
The final AL motric, M, computed by taking the product of all the three aforementioned statistical measures, as specified below
M = A (1)
(6)
A higher value of M indicates that greater reduction in the quantization step is possible, and vice versa.
3.4.5 Adaptive Layerwise QCFS Activation based Training Methodology
We estimate the AL metric for all the QCFS layers of pretrained ANN. We 1-dimensional cluster- ing (Grønlund et al. 2018) to divide the layers into x clusters based on the AL metric values, where x is user-defined parameter. The goal is to group QCFS layers with similar AL metric values and assign suitable quantization step. We subscquently retrain the source ANN from the beginning. This approach is inspired by the Lottery Ticket Hypothesis (Frankle & Carbin 2019); the difference being that our subnetwork is not obtained by pruning, but rather by activation quantization. We employ a higher quantization step for the initial epochs to maximize the effect of the learning rate scheduler [Loshchilov & Hutter 2016). We then configure the quantization step per layer and incrementally fine-tum the model until training converges, outlined in Algorithm
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Table Accuracy comparison between SNNs, inferred using PASCAL, and their source ANNs across various models and datasets.
Dataset
Model
Values of Quantization Step across Lay-
ANN Accuracy
SNN Accuracy
CIFAR-10
VGG-16
1,4444444444444444444444444
95.66 %
95.61 %
CIFAR-10
VGG-16
A.B.AA......................
95.79 %
95.82%
CIFAR-10
VGG-16
4,4444411111444444444444
93.57 %
93.62 %
CIFAR- 100
VGG-16
144444444444444444444444444
76.28 %
76.06 %
CIFAR- 100
VGG-16
A.S.8..5...................
77.01 %
76.77
ImageNet
VGG-16
166.6616..0000111111116666666660
74.16 %
74.22 %
ImageNet
ResNet-34
5.B.A...8...3388555.....55
74.182%
74.31%
CIFAR-10
ResNet-18
A8..A.......33...........
96.51 %
96.51
CIFAR- 100
ResNet-18
78.43 %
78.45%
CIFAR-100
ResNet-18
424414422211222211111111
76.38%
76.19 %
4
Results
4.1
PASCAL: Accuracy Results
Table 2 shows the classification accuracy of SNNs, obtained using the PASCAL methodology, compared to ANNs with QCFS Activation. Our method mirrors the accuracy the source ANN for both uniform and non-umiform values of L, notwithstanding minor differences arising due to floating-point imprecision. We report competitive accuracy for SNNs inferred using the PASCAL formulation across model architectures (VGG-16, ResNet-18, and ResNet-34) and datasets (CIFAR-10, CIFAR-100, and ImageNet)..
4.2 PASCAL: Energy Efficiency Results
4.2.1 Energy of PASCAL-SNN with respect to vanilla SNN
SNNs gain their energy efficiency as result of replacing compute-intenes intensive Multply-and-Accumiite (MAC) operations by more energy-efleient Accumulate (AC) operations. The energy consumed by the entire net work, ESNN, can be split into two parts as follows.
ESNN EAC + EIF
Unlike previous works (Rathi & Roy 2021 we also take into account the energy consumed by the Integrate-
and-Fire layer (Err). Similar to Equation 7 we express the total energy consumed by the corresponding SNN trained using the PASCAL approach as shown below.
EPASC_SNN = ÉPASC_AC + ÉPASC_IF
(8)
We note that the energy consumed by AC operations does not change in PASCAL compared to the vanilla SNN. This is because for given quantization step L, both networks have MatMul layer that is unrolled across T L timesteps. Therefore,
EAC- The only difference lies in the IF layer, which is more complex in PASCAL. Consider network with uniform quantization step L. According to A Algorithm [ the number of soft reset operations is 3L - 1, while the nunber of spike accunulation operations (i.e., addition and subtraction of the spike counter) 2L 1. In total, each IF layer in PASC framework performs (5L- 2) operations, yielding EPASC_IF = (5L - 2) EIF. We quantify the excess energy consumed by the PASC SNN by comparing with the energy consumed by the vanilla SNN. Specifically, we define the per-layer energy ratio r'g
have EPASC
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EPASC_SNO SNN
EAC + (5L (5L-2) EEF
E ESNN
E'Ac + L E/F
We now analytically compute the ratio from Equation 9 for both convolutional and fully connected layers. To facilitate this, we introduce an auxiliary ratio r' that represents the number of operations in an IF layer relative to those in the preceding MatMul layer. This allows us to express T' in terms of as
1 ++ (5L- 2) -)
(10)
1 L.
Finally, we proceed to compute r' layerwise separately for both the convolution and fully connected layers, using notation outlined in Table
Convolution Layer
We assume constant quantization step for further calculations in this section. Each IF layer performs Lx Co Hg Wo accumulation operations. Therefore, we compute the ratio between the number of operations in the IF layer to the number of operations in the previous convoluttonn layer as
F #OPIF
(11)
EAC
#OP
C.KK, K SpikeRate;
Conr
where SpikeRate, is the total spikes layer across all timesteps averaged over the number of neurons in the layer.
Fully Connected (FC) Layer
Following similar reasoning to the previous section, can deduce r' for the fully connected layer
F/F #OP1P EAC #OPEC
P'e
(12)
C. SpikeRater
We use the above equations defining r' and r'g to find the normalized number of inference timesteps Torm in the PASC framework as
)
T Tnorm
Σ
T,
(13)
LLOt
where Ltot is the total number of layers and is the mumber of time-steps of the corresponding vanilla SNN. For this computation, we assume that the SpikeRater = 0.75 for all layers. In addition, we also define the overall energy ratio
Etot
TE
PPACC_SNN SNN
PASC
(14)
SNN
which characterizes the overall energy cost of our method with respect to the traditional SNN. We show Appendix A.10 that the ratio rg is as low 1.001 in practice, indicating that increasing the number of IF operations does not adversely affect the overall energy consumption. This shows the ability of the proposed PASC framework to provide lossless ANN-SNN conversion for negligible increase in energy consumption.
4.2.2 Energy comparison of vanilla SNN with ANN
Having established that PASCAL preserves the energy benefits of vanilla SNNs, we now analytically compute the energy efficiency of SNN relative to its source ANN. We estimate this energy efficiency by comparing the number of operations between the source ANN (#OPANN) and the converted SNN (#OPsNN), following the approach proposed by Rathi & Roy and formulated as
#OPsNN Ratea x #OPANN-
(15)
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Table 3: Energy comparison between ANN and SNN based on the method proposed by Rathi & Roy The FP32 MAC (AC) energy is estimated as 4.6 p. (0.9 p.)) and the INT8 MAC(AC) energy is estimated as 0.23 pJ (0.03 pJ) (Horowitz
2014 The spike rate per layer for various models is shown in Appendix A.9
Architecture
Dataset
Normalized Normalized
#OPSNN
(c)
EnergyANN (FP32)
EnergyANY (LNT8)
#OProtal
Ener N
EnergusM NNSN
(quantization
#OPANN (a) #OPsNN(b)
c.1.6 cc)b00..)
cx0.23+ (1-c)--000)
slep)
VGG-16 (L=4)
CIFAR-10
1.0
0.66
0.005
(Appendix
A.5
7.49
11.03
VGG-16 (I=4)
CTFAR- 100
1.0
0.62
0.005
(Appendix
A.5
7.96
11.70
VGG-16 (I=166)
ImageNet.
1.0
0.73
0.006
(Appendix A.6
6.76
9.94
ResNet-18 (L=8)
CIFAR-IN 10
1.0
0,86
0,008
(Appendix
5.72
8.38
ResNet-18 (L=8)
CIFAR 100
1.0
0.91
0.008
(Appendix
A.7
5.42
7.95
sotel Spike Rater
Average Spike Rate; across layers, is computed as
where [Leot] is the total number of layers.
Los
80
Metric
Metric
Metric
30
40
10
12
14
16
0
2
10
22
26
10
12
14
16
Layer
Layer
(b)
(c)
Figure AL metric for different models and datasets, with the threshold a L for Agreement, where
Leot is the number of layers. AL metric for (a) VGG-16 (L = 4) on CIFAR-10, (b) ResNet-18 (L 8) on 10, and (c) VGG-16 (L = 8) on CIFAR-100.
Table SNN accuracy and effective timesteps obtained using the Adaptive Layerwise (AL) activation method for different models.
AL Metric
Model
Dataset
L
Layerwise L
Teff
Accuracy (%)
Figure
a.)
VGG-16
CIFAR-IB 10
1,2,4
2,1,2,1,2,1,1, 1,1,444,4444
2.13
93.73
Figure
b)
ResNet-18
CIFAR-10 10
1,4
4.4.4,1,444411444 1111111111
3.14
95.34
Figure
c)
VGG-16
CIFAR-
3
1,4,8
88,4,,,,4,,4 1,1,1,4,888888
4.54
74.07
100
VGG-16
ImageNet
16,8,4
16,66,6,8888.884444444444
9.92
71.188
VGG-16
ImageNet
16,8
16, 16, 16, 8, 16, 8, 8, 8, 8, 8, 12.40
73.98
8, 8, 8, 8, 8)
ResNet-34
ImageNet
16,8
[16 16, 16, 8, 16, 8, 16, 8, 16, 12.94
75.78
8, 16, 16, 8, 16, 8, 16, 8, 16.
8, 16, 8, 16, 8, 16, 8, 16, 8, 8,
8, 8, 8, 8
Table 3 shows that converted SNNs are up to 7x more energy efficient than the corresponding source ANNs. The energy results account for the fact that the first SNN layer alone performs MAC operations since it receives input image with real-valued pixel intensities. The formulac used to compute the ANN and SNN operations detailed in Appendix A.4
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Table 5: SNN accuracy and inference timesteps comparison with QCFS and SEENN-L.
PASC formulation with uniform Quantization Step
PASC with Adaplive Layerwise (AL) activation
Model
Dataset
Method
T
Acc.
Model
Dataset
Method
Acc.
QCES
16
93.95%
QCFS
8
94.82%
CIFAR-10
CIFAR-10
VGG-16
PASC (Tnorm)
4.21
95.61%
SEENN-1
2.01
95.08%
ResNet-18
QCFS
16
72.80%
PASCAL (Ters)
3.14
95.34%
CIFAR-100
PASC (Tnorm)
4.21
76.06%
SEENN-1
6.19
65.48%
CIFAR-100
QCFS
64
72.35%
PASCAL (Ters)
2.63
76.19 %
ResNet-31
ImageNet
QCFS
256
74.22%
QCFS
16
72.80%
VGG-16
CIFAR-100
SEENN-1
29.53
71.84%
PASCAL (Ter))
4.54
74.07%
PASC (T norm)
8.17
74.31%
VGG-16
ImageNet.
PASCAL (Ters)
12.40
73.98 %
QCFS
1024
74.32%
ResNet-34
ImageNet
PASCAL (Ter))
12.94
75.78 %
VGG-16
ImageNet
PASC (T (Tnomm)
18.37
74.22%
Table 6: SNN accuracy and inference timesteps comparison with other state-of-the-ate works.
Prior Work
Dataset
Model
Accuracy of Prior Work(%)
Accuracy of PASC(%)
Hao et al.
2023
CIFAR-100
VGG-16
76.77 (T-16)
76.77 (Tnorm 8.79)
Hao et al.
2023
ImageNet
ResNet-31
73.93 (T-32)
74.31 (Tnorm -8.17)
Li et al.
(023a
CIFAR-10
ResNet-18
94.27 (T=11.51)
96.51 (Tnarm = -8.22)
Li
(2023a
ImageNet
VGG-16
73.30 (T=49.54)
74.22 (Tnorm -18.37)
Wu et al.
(2024)
ImageNet
ResNet34
71.66 (T=16)
74.31 (T (Tnorm- = -16.67)
4.3 Adaptive Layerwise (AL) Activation Results
Figure shows the layerwise AL metric (Equation 6 for different models and datascts. We perform clustering with various values of X, and compute the quantization step per layer. We also evaluate the effective inference timesteps, Tefs, which is specified by
Teff
Σrt-TT'
(16)
Lrot
where Ltot is the total number of layers and T is the number of time-steps of the corresponding vanilla SNN for the same layer. Table 4 shows that adapting the quantization step in a layerwise manner achieves competitive accuracy using fewer effective timesteps, thereby improving temporal efficiency. For instance, ResNet-18 provides 95.34% accuracy on CIFAR-10 using Tess ss 3.14, which is comparable to the 96.51% accuracy achieved with a uniform Thorm = 8.22 (see Table
4.4 Comparison with the State- Art
We compare results, presented in Table 5 with those of the QCFS framework (Bu et al. 2023 and the SEENN-1 method (Li et al. 2023b). In addition to these basclines, we also benchmark our approach against various other state-of-the-art works listed in Table 6 Our results indicate that PASCAL consistently yields SNNs that provide competitive accuracy using fewer timesteps across models and datasets, outperforming the QCFS framework (Bu et al. (023). We report comparable performance the SEENN-1 method (Li al. 2023b) on CIFAR-10. However, we demonstrate significant performance improvements, both in terms of accuracy and effective timesteps, for CIFAR-100 and ImageNet. This trend is also observed when comparing
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PASCAL remains competitive smaller datasets and achieves superior results larger ones. The efficacy of PASCAL improves with dataset complexity since it relies on mathematically equivalent ANN-SNN conversion.
method with more recent works in Table 6
5 Conclusion
In this paper, we proposed PASCAL, which a precise and temporally efficient ANN-SNN conversion method. We showed that the proposed methodology enables ANNS be converted into SNNs with guarantee of mathematical equivalence, leading to near-zero conversion loss. We formally proved that the converted SNN performs the same computations, unrolled over timesteps, the source ANN with QCFS activation. We demonstrated the energy efficiency of the proposed approach using analytical model. In addition, we also proposed an Adaptíve Layerwise (AL) activation methodology to further improve temporal efficiency during SNN inference. We presented an algorithmic approach to choose an optimal precision for each layer, which reduces the effective number of inference timesteps while providing competitive accuracy.
6 Acknowledgements
This rescarch was supported in part by the RISC-V Knowledge Centre of Excellence (RKCoE), sponsored by the Ministry of Flectronics and Information Technology (MeitY). We would like to acknowledge the Robert Bosch Centre for Data Science and Al (RBCDSAI) and the Centre for Development of Advanced Computing (C-DAC) for providing us GPU computing resources. Finally, we would like to acknowledge the work of Varun Manjunath (Electrical Engineering, IIT Madras, Class of 2025) for benchmarking our work on neuromorphic architectural simulators.
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A Appendix
A.1 Proof for equivalence between the PASC formulation and an ANN with QCFS activation
Proof for Theorem 3.5
For the input layer, the MatMul operation performed in the PASC formulation is the same as the corresponding ANN with QCFS activation. Let the output after the MatMul Operation be X. Consider a single element x X. In the case of ANN, the output after QCFS
XLn On
On Inn
Mxx) - On clip
+ 4 0,,1 = c.
(17)
Tim
Upon further simplification, we obtain
k
if l Inn, and
(k
2) Ln
1) 8..
< x < (k + 2n.
(18)
Ln
if x 2 Ln 2) An
In
0
if x 2.
8a
Algorithm 2 specifies the Integrate-and-Firr algorithm for the input layer. It also first generates π(X) z' (refer to Line 3). Subsequently, it generates the final spike train st. We denote the spike train for given input X by s'-- Note that zf only consists of integral multiples of 0 = Zz. Therefore, after repeated application of the Heaviside step function on I, and stacking the output over time, we get
On In
if c otherwise
(1)
(19)
Now, Eln (1) = c. t. M))), Vo X. Therefore, Ete 81(1) = proving part (a). Using Equation 19
that sf contains values from the set {0, 6n theorem.
r.,, } for all values of X. This proves part (b) of the
We begin with the remark that 0, = X' after the training is complete. Consider a single element x E zl and the corresponding elements I; E 2/, Vi E {1,2, ... Ln-1). In an ANN with QCFS activation (see Figure
Proof for Theorem 3.4
if < k < I., and
T.n
(k 1) 8= < (k + 1::
T((x))
(20)
On
if x 2 (Im }) 0n
Im
if x 2
8p
Table 7: Mapping between the value of at the end of stage 2, and the final number of spikes at the end of
stage 3.
Value of
Number of Spikes
2h, 1221 22824 Ln
Ln
Ln >8>0
022 s2 (Ln 1)
0
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0.25
0.20
QCFS(X)
0.15
0.10
0.00
0.1
0.0
0.1
0.2
0.3
0.4
X
Figure 2: A sample QCFS activation layer with L = 4 and 0, 0.25.
After the first stage Algorithm let the value be k then obtain inequality of the form
f.,
If there was spike in the last timestep,
I.
0n 2.h.
Σ; Jq kn
On
L. Upon simplificatin,, we get
In-1
0n Ln
Σ
xq (ki + 2)
(21)
On the other hand, if there was no spike in the last timestep, we instead obtain
Ln-1
Σ x. < (h1 2) On Ln
(22)
Although have onc-directional incquality in either case, we nevertheless need more processing to obtain the second direction of the inequality to choose the correct level. We consider the aforementioned two cases. Case 1. Let Inequality 21 be the final inequality after stage 1. Inhibitory spikes are not possible in this case. The QCFS function clips the activation if the accumulated input E. x; is greater than (Ln - } 8a. Therefore, the IF step needs to be done without input at least T1 = Ln, times to ensure that the correct level is obtained for all inputs. The bound arises because the minimum value of k- = At the end of stage 2, let the value of be kg T...
T.
Pn Since an inhibitory spike is not possible, we get
On
On
Σ
34
k2
2 0
(23)
Ln
Ln
Also, if 22 < Ln, then we also have the following inequality.
On
On
On Lu
+
Σ
k2
(24)
2 Ln
In
Otherwise, we only have the incquality specified in Equation
23 In effect,
k
0n La
Ln.
if (k 4) On
L. < Σ < (k:
1 On
(25)
2
Ln
8n
k Ln if Σ x 2 Ln L,
1 1)
On
If Inequality 22 is the final inequality, we need inhibitory spikes to determine the correct level. This is different from Case 1, because we have an upper bound on the value of E/n at. In effect, any level below ki should be possible. Since kı < Ln-1- -1, 1, this stage requires T2 = Ln-1 timesteps. Here, normal spikes are not possible. Using a similar analysis as presented in Case 1, we obtain
Case 2.
8a k 0 k
if (k 1) t.
I Σ I3 < (k: + 1 On 2 Ln
(26)
<
if Σ x4
Tea
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Considering Cases 1 and 2, rum stage 2 for maz(Ti, max(Ln-11, L.) timesteps.
Stage 3 is used to generate a final spike train from the value of s obtained after stage 2. This layer is used to generate & stacked output ((X). As a result of this computation, we get that 2, h(X) = min(maz(s, 0), Ln).
The final number of spikes, parametrized by the value at the end of stage 2, given in Table 7 Using 25 and 26
we can conclude that
k
On In
if < h < Lmn and
(h: 2) 8n
I.
Σ
I x (k + 2
8n
=π(x)
(27)
Lp
On
if x 2 [In
2 2
1 Dn
0
if r <
2-Ln
Vz E X. This proves part (a). st) is the output of the Heaviside function, i.c. h(x)[1] E 10,0- = 2. Vj. thus proving part (b).
]
Proof for Theorem 3.3 We begin by simplifying the right- side.
(2) * W +
4)
(28)
V +E
L
1=1
Σ 7
2 W
+ V0² +
L
p
1.
(29)
0² +E
i=1
i=1
= S1 + S2 + S3
(30)
Here,
S
* W
(31)
00 +e
7
W
(32)
V0² +E
vo +E
7
(+/11 3)
b p
(33)
V0 +c
7
=1411 -8 3-b+
(34)
Also,
L b
S2
L 4
(35)
S3 Σ 3
3
(36)
Therefore,
S + S2 +
S3
2 1-1
3 8- b+ + - 3
(37)
141
(38)
Proof for Theorem 3.6
Consider an input z to the models M and N. We invoke Theorems 3.5
3.4 and 3.3 to show that Invariant 2
holds after the final layer. In other words, after the final layer, 2' Σi= 2/ where is the quantization step of the final layer I. In SNN, the output probabilities are derived by taking the mean timesteps after the
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final layer. Therefore, the tensor used for classification in the PASC formulation = -EEE 2 = .2. Therefore,
2" 2=|c]
[m(()
(39)
E. 2-
z'
(40)
>. =1cc
L
2/[c]
(41)
Σ. 2/
(c)
(42)
Vc E C. Thus, M and N are mathematically equivalent.
A.2 Effect of threshold of Van Der Eijk's A the Final Metric
The effect of A is pronounced in datasets such CIFAR-100. It primarily affects the value of the metric in the input layer. The effect can be seen in Figure g
60
80
80
Metric
Metric
Metric
40
40
8
20
12
14
16
2
6
10
12
14
16
0
8
20
12
14
26
Layer
Layer
Layer
(a)
(b)
(c)
Layerwise AL-metric with the threshold a = [ooel for Agreement, where Leot the number of layers, computed using VGG-16 for CIFAR-100. (a) k = 1 (b) & = 0.75 (c) k = 0.5.
Figure
A.3 Introduction of Datasets
CIFAR-10. The CIFAR-10 dataset (Krizhevsky al. 2009) consists of 60000 32 32 images in 10 classes. There are 50000 training images and 10000 test images.
CIFAR-100. The CIFAR-100 dataset (Krizhevsky et al.
2009 consists of 60000 32 x 32 images in 100
classes. There are 50000 training images and 10000 Lest images.
ImageNet. We use the ILSVRC 2012 dataset (Russakovsky et al. images and 50000 testing images.
2015 which consists of 1,281,167 training
A.4 Comparison of the number of MACs and ACs between ANN and SNN
The table below shows the number of operations for both convolution and fully connected layers. Table S
shows the notation used for this analysis. The energy cost of addition and multiplication in 45mm CMOS is
shown in Table I0
Type of Layer Fully Connected
Nature of Activation
# *MACs
# A Cs
ANN
C. X C.
0
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Published in Transactions on Machine Learning Research (12/2025)
Type of Layer Fully Connected
Nature of Activation
#MACs
#ACs Cj x Co x SaikeRater, along with Co X SpikeRater multiplics for threshold scaling
QCFS Activation with I. 0 levels, inferred using PAS-
CAL
Convolution
ANN
C, Kn Kw x W. x
0
H.
Convolution
QCFS Activation with L levels, inferred using PAS- CAL
C2xCo KA x Ki x W. Ho x SpikeRate, (along with Co Ho W. Spike Roles multiplies for threshold scaling)
A.5
Layerwise count of operations in the VGG-16 architecture for CIFAR-10 and CIFAR-100
Layer
Input Size
Output Size
#MACs
Conv1 (Convolution)
32 x 32 x 3
32 x 32 x 64
32 x 32 x x 64
1, 769, 472
Convl 2 (Convolution)
32 32 64
32 x x
32 x x x4x3333xt0
348 736
MaxPooll
32 32 64
x 64
Conv2
(Convolution)
16 16 64
16 128
16 x 16 64x 18, 874, 368
Conv2 2 (Convolution)
16 16 128
16 x 16 X 128
16 16 x2X 37, 7,8,74,,77 736
MaxPool2
16 16 128
X 8 X 128
Conv3 I (Convolution)
8 X 8 x 128
8 X 8 x 256
8 8 x 128 x x3x3xxx 256 18, 874, 368
Conv3 2 (Convolution)
8 X 8 x 256
8 x 8 x 256
8 x 8 x 256 x 3 x 3 256
37, 748, 736
Conv3 3 (Convolution)
8 x 8 x 256
x 8 x 256
8 x 8 x 256 x 3 x 256
37, 748, 736
MaxPool8
8 X X 256
X 4 X 256
Conv4 | (Convolution)
4 x x 256
x 4 x 512
4 4 x 256 x x x 512 18, 874, 368
Conv4 (Convolution)
4 X 512
4 x 512
4 4 512 xxx
37, 748, 736
Conv4 3 (Convolution)
4 4 512
4 512
4 4x4x 512 x 3 512
37, 748, 736
MaxPool4
4 x x 512
x 2 x 512
Conv5_1 (Convolution)
2 x 512
x 2 x 512
2 2 512 X 3 x 512 =
9,437, 184
Conv5 2 (Convolution)
2 x x 512
X 2 X 512
2 2 x 512 X x 5
9,437, 184
Conv5_3 3 (Convolution)
2 x 2 x 512
x 2 X 512
2 2 x 512 x 3 x 3 x 512
9, 1437,118
MaxPool5
2 x 2 x 512
1 x 1 x 512
FC1 (Fully Connected)
1 1 x 512 = 512
4096
512 x 4096 = 2, 097, 152
FC2 (Fully Connected)
4096
4096
4096 x 4096 = 16, 777, 216
FC3 (Fully Connected)
4096
100 (CIFAR-100),
10
40,960
(CIFAR-10)
(CIFAR-10) 10)
409,600 (CIFAR-100)
20

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Published in Transactions on Machine Learning Research (12/2025)
Layer
Input Size
Output Size
#MACs
Total Operations
332,480,512 (CIFAR-
100)
332,111,872 (CIFARIO)
A.6 Layerwise count of operations in the VGG-16 architecture for ImageNet
Layer
Input Size
Output Size
#MACs
Convl 1 (Convolution)
224 x 224 X 3
224 x 224 x 64
224 x x
64 86, 704, 128
Convl 2 (Convolution)
224 X 224 X 64
224 X 224 X 64
224 x 224 X 64 3 x 3 64 = 849, 688, 064
MaxPooll
224 x 224 X 64
112 x 112 x 64
Conv2 1 (Convolution)
112 x 112 x 64
112 X 112 X 128
112 x 112 x 64 x 3 x 3 x 128 924, 844, 032
Conv2 2 (Convolution)
112 x 112 x 128
112 X 112 X 128
112 x 112 X 128 x 3 x 3 x 128 = 1, 849, 688, 064
MaxPool2
112 x 112 x 128
56 56 x 128
Conv3 1 (Convolution)
56 x 56 x 128
56 x 56 x 256
56 x 56 x 128 X 3 x 3 x 256 = 924, 844, 032
Conv3 2 (Convolution)
56 x 56 x 256
56 x 56 x 256
56 x 56 x 256 x 3 x 3 x 256 = 1, 849, 688, 064
Conv3 3 (Convolution)
56 x 56 x 256
56 x 56 x 256
56 x 56 x 256 X 3 x 3 x 256 = 1, 849, 688, 064
MaxPool3
56 x 56 x 256
28 x 28 x 256
Conv4 1 (Convolution)
28 x 28 x 256
28 x 28 x 512
28 x 28 x 256 x 3 x 3 512 924, 844, 032
Conv4 2 (Convolution)
28 x 28 x 512
28 X 28 x 512
28 x 28 x 512 x 3 x 3 x 512 849, 688, 064
Conv4 3 (Convolution)
28 x 28 x 512
28 x 28 x 512
28 x 28 X 512 x 3 x 3 x 512 1, 849, 688, 064
MaxPool4
28 x 28 x 512
14 X 14 X 512
Conv5 1 (Convolution)
14 x 14 x 512
14 x 14 x 512
14 x 14 x 512 x 3 x 3 512
016
Conv5 2 (Convolution)
14 X 14 x 512
14 X 14 X 512
14 X 14 x 512 x 3 x 3 x 512 462, 016
Conv5 3 (Convolution)
14 x 14 x 512
14 X 14 x 512
14 x 14 x 512 x 3 x 3 x 512 462, 422, 016
MaxPoo15
14 x 14 x 512
7 x 7 x 512
FCI (Fully Connected)
7 x 7 x 512 = 25088
4096
25088 x 4096 102, 760, 448
FC2 (Fully Connected)
4096
4096
4096 x 4096 16, 777,210
FC3 (Fully Connected)
4096
1000
4096 x 1000 = 4,096, 000
Total Operations
15,470,264,320
A.7
Layerwise count of operations in the ResNet-18 architecture for CIFAR-10 and CIFAR-100
Layer
Input Size
Output Size
#MACs
Initial Conv
3 x 32 x 32
64 x 32 32
1,769,472
Residual Block 1.1 Convl
64 x 32 x 32
64 x 32 32
37,748,736
Residual Block 1.1 Conv2
64 x 32 x 32
64 x 32 32
37,748,736
Residual Block 1.2 Convl
64 X 32 x 32
64 x 32 32
37,748,736
Residual Block 1.2 Conv2
64 x 32 x 322
64 x 32 32
37,748,736
Residual Block 2.1 Convl
64 x 32 x 322
128 x 16 16
18,874,368
Residual Block 2.1 Conv2
128 x 16 x 16
128 x 16 16
9,437,184
Residual Block 2.1 Shortcut
64 x 32 x 32
128 x 16 16
2,097,153
Residual Block 2.2 Convl
128 x 16 x 16
128 x 16 16
9,437,184
Residual Block 2.2 Conv2
128 x 16 x 16
128 x 1.6 16
9,437,18
Residual Block 3.1 Convl
128 x 16 x 16
256 x 8 x 8
4,718,592
Residual Block 3.1 Conv2
256 8 x 8
256 x 8 x 8
2,359,290 ,96
Residual Block 3.1 Shortcut
128 x 16 x 16
256 x 8 x 8
524,288
Residual Block 3.2 Convl
256 8 x 8
256 x 8 x 8
2,359,296
Residual Block 3.2 Conv2
256 8 x 8
256 8 x 8
2,359,290
Residual Block 4.1 Conv1
256 8 x 8
512 x 4 x
1,179,648
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Published in Transactions on Machine Learning Research (12/2025)
Layer
Input Size
Output Sizc
#MACs
Residual Block 4.1 Conv2
512 x 4 x
512 x x 4
589, 189,,24
Residual Block 4.1 Shortcut
256 x 8 x
512 x x 4
262, 144
Residual Block 4.2 Conv1
512 x 4 X
512 x X 4
589,824
Residual Block 4.2 Conv2
512 x 4 x
512 x x 4
589,824
Final FC Layer
512
100 (CIFAR-100)
51,200 (CIFAR-100)
10 (CIFAR-10)
5,120 (CIFAR-10)
Total Operations
217,630,720
(CIFAR-T00)
217,584,640 (CIFAR-10)
A.8 Training Configurations
In order to implement PASCAL, we adopt the code framework of QCFS Bu ct al. (2023) We use the
Stochastic Gradient Descent optimizer (Boltou 2012) with a momentum parameter of 0.9. The initial learning
rate is set to 0.1 for CIFAR-10 and ImageNet, and 0.02 for CIFAR-100. A cosine decay scheduler [Lohhchllov & Hutter 2016) used adjust the learning rate. We apply a 10-4 weight decay for CIFAR datascts while applying a 10 weight decay for ImageNet.
A.9 Layerwise spike rate over the test set
We plot the spike rate over the entire test set for various model architectures and datasets in Figure 4
1.5
CIFAR-10 (L-4)
CIFAR-10 (L-8)
CIFAR-100 (L+4)
CIFAR-100 (L-8)
1.5
Spike Rate
Spike Rate
1.0
1.0
0.5
0.5
0.0
0.0
12
14
8
12
16
Layer
Layer
(a)
(b)
ImageeNt (L=16)
2.0
Spike Rate
1.5
0.5
0.0
10 12 14
Layer
(c)
Figure 4: Layerwise Spike Rate over the test-set, (a) for L = using VGG-16 for both CIFAR-10 and CIFAR-100, (b) for -= 8 using ResNet-18 for both CIFAR-10 and CIFAR-100, and (c) for = 16 using VGG-16 for ImageNet.
22

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Published in Transactions on Machine Learning Research (12/2025)
Table 8: Notation used in computing MACs and ACs.
Notation
Meaning
C2
Number of Input Channels/Teer Features
Co
Number of Output Channels Features
Ka.
Convolution Kernel Height
Kw
Convolution Kernel Width
Wo Output matrix width (after convolution)
H. Output matrix height (after convolution)
Table 10: Energy cost of addition and multiplication in 45mm CMOS (Horowitz
2014
FP ADD (32 bit)
0.9pJ
FP MULT (32 bit)
3.7pJ
FP MAC (32 bit)
(0.9 + 3.7)
4.6pJ
INT8 ADD
0.03pJ
INT8 MUL
0.2p.
INT8 MAC
(0.03 + 0.2)
0.23pJ
A.10
Hardware Quantification
A.10.1
Energy
We implemented a custom SNN accelerator in-house to quantify the energy consumed per inference with and without the PASC IF layer. The accelerator a modified version of the LoAS accelerator (Yin et al.) 2024), enhanced support the PASC IF layer. We report the energy metries for the VGG-16 SNN, over the CIFAR-10 test set.
Table 14: Energy metrics of VGG-16 SNN with and without PASC IF. Total Inference Energy Energy Contributed by the IF Layer
Type of IF Default IF (T 4) PASC IF (T 4)
445.47 mJJ
0.08 m.J 0.63 mJ
446.02 mJ
Our results show that the VGG-16 SNN with PASC IF neurons consumes only 0.1% more energy compared to the default IF implementation.
In an effort to further validate the energy efficiency results, we re-implemented an ANN accelerator SparTen addition the aforementioned SNN accelerator LoAS Yin et al. (2024). Both the accelerators were designed to support INT8 quantization for the weights. In addition, Spar Ten enhanced to use the QCFS function to implement quantized activation. We synthesized the Register Transfer Level (Verilog RTL) design of the accelerators using the Synopsys Design Compiler on 40 nm CMOS technology to determine the operating frequency (560 MHz) and power consumption. We then developed cycle-accurate simulator of the respective accelerators to estimate inference latency and energy. Table [5 summarizes the inference energy of VGG-16 ANN and SNN, obtained empirically from our cycle-accurate simulators, over the CIFAR-10 test set.
A.10.2 Latency
We would like to emphasize that the proposed PASC IF neuron adds only a minor overhead to inference latency and energy cost, since the end-to-end latency and energy consumption of the network are dominated by the MatMul layers, which incur more operations than the IF layers, as discussed in Soction 4.2 We quantify the overhead by implementing the PASC IF neuron on the LoAS accelerator Yin al. (2024). which employs fully temporal-parallel dataflow architecture designed to minimize both data movement across
23

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Published in Transactions on Machine Learning Research (12/2025)
Table 15: Comparison of inference energy consumption across different networks.
Type of Network
Total Inference Energy (mJ)
ANN (L=4)
3724.865
Default IF (T=4) PASC IF (T=4)
445.47
446.02
timesteps and the end-to-end latency of sparse SNNs. Table I6 reports the inference latency of VGG-16 SNN, over the CIFAR-10 test set, for both the default IF and PASC IF configurations.
Table 16: Inference latency comparison between different TF types.
Type of IF
Total Inference Latency (s)
Default IF (T=4)
0.40
PASC IF (T=4)
0.45
A.11
Comparison between PASCAL and direct SNN training
We compare our method against the direct training approach proposed in Deng al. (2022 p. The results
presented in Table 17 We show significant improvements in accuracy across datasets, primarily for complex datasets like ImageNet.
Table 17: Comparison between PASCAL and direct SNN training (TET).
Datasct
Method
Model
Timesteps
Accuracy
TET PASCAL (ours)
ResNet-19
94.44
CIFAR-10
VGG-16
4.21
95.61
TET
ResNet-19
4
74.47
CIFAR-100
PASCAL (ours)
VGG-16
4.21
76.06
TET
Spiking ResNet-34
64.79
ImageNet.
PASCAL
(ours)
RosNet-34
8.17
74.31
24


Reviews:
[{"review_id": "Qd2PUqzv6Q", "reviewer": "Reviewer_xLuW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "\\+ The paper is well written and motivated\n\\+ The paper is well organized and easy to read\n\\+ I appreciate the demonstration that the original NN and the SNN are mathematically equivalent, although I did not check all the proofs.\n\\+ Results are quite impressive with almost no-loss in accuracy in the conversion and an energy gain up-to 10x.\n\n\\- While the SNN are evaluated in terms of performance and energy consumption, I could not find any information about the inference speed of the proposed model, although in the intro the authors talk about an increased speed. Also, if the adaptation is layer-dependent, the number of time-steps can vary between layer. Thus, the final speed is based on the slowest layer, i.e. the layer with more time-steps?\n\n\\- Related work and preliminary knowledge are mixed-up in section 2. I would have appreciated two separated sections, one with related works and another one in the beginning of the methodology with the basic knowledge and formulation needed for the method.\n\n\\- Why in table 5, for the algorithm with Adaptive Layer, there are no results on ImageNet? I think it would make more sense to have the same models and datasets for the PASC and the PASCAL models.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a new approach for converting a neural network (NN) into a spiking neural network (SNN) with almost zero loss in performance and reduced energy consumption.\nThe main two contributions of the paper are:\n\n- a spike-count and inhibition SNN, which has a formal prof of the quality of the proposed conversion. The NN is first converted into its quantized version and then into the SNN, with a fewer time-steps and therefore reduced inference time and energy consumption\n\n- An adatptive layer-wise quantisation, in which some layers are required to be in high precision, while others can keep a lower precision, without affecting the final network performance.\nThe proposed approach is evaluated on standard computer vision datasets such as CIFAR-10 and 100 adn ImageNet, with VGG-16 and ResNet architectures.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "-(crititcal) I would like to see more information about inference time for the converted SNN and how they compare with previous approches.\n-(critical) Is there a reason to not evaluate PASC and PASCAL on the same datasets and models?\n- I would appreciate a better separation between related work and basics of the method.\n- I would appreciate a bit more information about the hardware necessary to fully exploit SNN.", "broader_impact_concerns": "This work will allow to convert NN models into their SNN, without relevant loss in performance, but with a remarkable gain in energy consumption. This will enable tu run NN on very small and energy efficient devices. This, in turn would enable a broader expansion of AI algorithms on very constrained environnements. This could introduce an higher risk in the control and usage of these algorithms.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "77kRNbXMSH", "reviewer": "Reviewer_NtDp", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. This paper shows the equivalence between converted and original models at the network level.\n\n2. Experiments are carried out on various network architectures and datasets, showing high performance with fewer time steps compared with previous ANN-SNN conversion methods.\n\nWeakness:\n\n1. A longstanding problem for the ANN-SNN conversion methods is their inadequate comparison to network quantization. The proposed method is to minimize the difference from an activation quantized neural network, but no significant practical advantage is demonstrated over quantized ANNs. In the energy estimation, this paper simply assumes FP32 MAC operations for ANNs, but ANNs can actually be quantized to 8-bit or even 4-bit precision (for both weight and activation) with minimal accuracy loss, given advances in the network quantization techniques. These operations are much cheaper. Some previous work revisited SNNs from the network quantization perspective [1], offering a potential way to compare under more fair settings.\n\n2. To minimize the difference from the activation quantized neural network, this paper introduces a lot of requirements to neuron models, such as multiple stages, both excitatory and inhibitory spikes, etc. However, there is no discussion of the plausibility and costs of them considering neuromorphic hardware that SNNs target. Actually, the development of neuromorphic hardware, such as Loihi-2, enables the support for several bits in a spike signal, akin to quantized activations. Then, why are these highly demanding techniques required, compared with directly leveraging multiple bits? The requirement of heterogeneous stages actually eliminates the potential advantage of asynchronization and parallelization of neuromorphic computing.\n\n[1] Are Conventional SNNs Really Efficient? A Perspective from Network Quantization. CVPR 2024.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes PASCAL, a new ANN-SNN conversion framework aiming to achieve small conversion error with minimal inference timesteps. It builds on the QCFS method while proposing a spiking neuron model incorporating several stages of excitatory and inhibitory spikes, and an adaptive layerwise activation method to dynamically configure per-layer quantization steps. Experiments on several static datasets show the high performance and low latency of the method, with high estimated energy efficiency.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Properly discuss/compare the results with quantized ANNs.\n\n2. Justify the plausibility and costs of the proposed neuronal operations on potential neuromorphic hardware.", "broader_impact_concerns": "No.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wiKeVsqT1X", "reviewer": "Reviewer_SB3M", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n- Comprehensive evaluation and comparison against existing methods for ANN-SNN conversion on computer vision models and datasets.\n- The use of “inhibiting spike” to adjust the internal spike count in order to match the ANN with QCFS activation is interesting. \n- The provided proofs in the Appendix seem to be sound and valid, though I did not verify them in detail.\n\nWeaknesses\n- The energy efficiency of the method is estimated, instead of measured with traditional vs neuromorphic hardware when running the ANNs and SNNs. Though this limitation is understandable. \n- While the authors have compared their method against other ANN-SNN conversion methods, specifically in Table 6, I think the work can be further strengthened by adding baseline SNNs that were directly trained on the 3 vision datasets, or their spiking/neuromorphic equivalent (CIFAR10-DVS, CIRFAR100-DVS and ES-ImageNet), to demonstrate the purpose of ANN-SNN conversion.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a method for converting ANNs to SNNs for image classification tasks. The authors introduced a spike-count-based approach to derive an activation function that is mathematically equivalent to an ANN with QCFS activation. Moreover, the authors proposed an alternative quantization method to adaptively determine the number of quantization steps for each layer of the network, instead of using a fixed number of steps. Experimental results on standard image classification benchmarks (CIFAR-10/100 and ImageNet) and multiple architectures (VGG-16, ResNet-18/34) demonstrate that PASCAL achieves accuracy comparable to that of the original ANN. The paper also includes an estimated energy efficiency analysis, demonstrating the potential reduction in energy use as compared to ANNs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the _Strengths and Weaknesses_ section above. Overall, I don’t have any major concerns with the paper.", "broader_impact_concerns": "I do not believe this work requires a broader impact statement.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)